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Applications of the STFT

This chapter briefly tours selected applications involving spectral audio signal processing, with associated examples in matlab. Related historical background and techniques appear in Appendix G. Therefore, this chapter can focus more on practical methods known to the author regarding applications the Short-Time Fourier Transform (STFT) (defined and discussed in §7.1). The following applications are considered:

Fundamental Frequency Estimation from Spectral Peaks

Spectral peak measurement was discussed in Chapter 5. Given a set of peak frequencies $ f_i$ , $ i=1,\ldots,N_f$ , it is usually straightforward to form a fundamental frequency estimate ``$ F_0$ ''. This task is also called pitch detection, where the perceived ``pitch'' of the audio signal is assumed to coincide well enough with its fundamental frequency. We assume here that the signal is periodic, so that all of its sinusoidal components are harmonics of a fundamental component having frequency $ F_0$ . (For inharmonic sounds, the perceived pitch, if any, can be complex to predict [54].)

An approximate maximum-likelihood $ F_0$ -detection algorithm11.1 consists of the following steps:

  1. Find the peak of the histogram of the peak-frequency-differences in order to find the most common harmonic spacing. This is the nominal $ F_0$ estimate. The matlab hist function can be used to form a histogram from the measured peak-spacings.

  2. Refine the nominal $ F_0$ estimate using linear regression. Linear regression simply fits a straight line through the data to give a least-squares fit. In matlab, the function polyfit(x,y,1) can be used, e.g., p = polyfit([0,1],[1,1.5],1) returns p = [0.5,1], where p(1) is the slope, and p(2) is the offset.

  3. The slope p(1) of the fitted line gives the $ F_0$ estimate.

A matlab listing for $ F_0$ estimation along these lines appears in §F.6.

Useful Preprocessing

In many cases, results are improved through the use of preprocessing of the spectrum prior to peak finding. Examples include the following:

  • Pre-emphasis: Equalize the spectrum so as to flatten it. For example, low-order linear-prediction is often used for this purpose (the ``flattened'' spectrum is that of the prediction error). In voice coding, first-order linear prediction is typically used [162].

  • Masking: Small peaks close to much larger peaks are often masked in the auditory system. Therefore, it is good practice to reject all peaks below an inaudibility threshold which is the maximum of the threshold of hearing (versus frequency) and the masking pattern generated by the largest peaks [16]. Since it is simple to extract peaks in descending magnitude order, each removed peak can be replaced by its masking pattern, which elevates the assumed inaudibility threshold.

Getting Closer to Maximum Likelihood

In applications for which the fundamental frequency $ F_0$ must be measured very accurately in a periodic signal, the estimate obtained by the above algorithm can be refined using a gradient search which matches a so-called harmonic comb to the magnitude spectrum of an interpolated FFT $ X(\omega)$ :

$\displaystyle {\hat f}_0$ $\displaystyle \isdef$ $\displaystyle \arg\max_{{\hat f}_0} \sum_{k=1}^K
\log\left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]$  
  $\displaystyle =$ $\displaystyle \arg\max_{{\hat f}_0} \prod_{k=1}^K \left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]
\protect$ (11.1)


K &=& \mbox{number of peaks, and}\\
k &=& \mbox{harmonic number of $k$th peak}\\
\epsilon &=& \mbox{some small value on the order of the spectral magnitude}\\
& & \mbox{\lq\lq noise floor'' level}

The purpose of $ \epsilon>0$ is an insurance against multiplying the whole expression by zero due to a missing partial (e.g., due to a comb-filtering null). If $ \epsilon=0$ in (10.1), it is advisable to omit indices $ k$ for which $ k{\hat f}_0$ is too close to a spectral null, since even one spectral null can push the product of peak amplitudes to a very small value. At the same time, the product should be penalized in some way to reflect the fact that it has fewer terms ( $ \epsilon>0$ is one way to accomplish this).

As a practical matter, it is important to inspect the magnitude spectra of the data frame manually to ensure that a robust row of peaks is being matched by the harmonic comb. For example, it is typical to look at a display of the frame magnitude spectrum overlaid with vertical lines at the optimized harmonic-comb frequencies. This provides an effective picture of the $ F_0$ estimate in which typical problems (such as octave errors) are readily seen.

References on $ F_0$ Estimation

An often-cited book on classical methods for pitch detection, particularly for voice, is that by Hess [106]. The harmonic-comb method can be considered an approximate maximum-likelihood estimator for fundamental frequency, and more accurate maximum-likelihood methods have been worked out [65,297,230,231]. Another highly regarded method for $ F_0$ estimation is the YIN algorithm [55]. For automatic transcription of polyphonic music, Klapuri has developed methods for multiple $ F_0$ estimation [189,127,126,124].11.2Finally, a rich source of methods may be found in the conference proceedings for the field of Music Information Retrieval (MIR)11.3 Of course, don't forget to try a Web search for ``F0 estimation'' and the like.


Cross-synthesis is the technique of impressing the spectral envelope of one sound on the flattened spectrum of another. A typical example is to impress speech on various natural sounds, such as ``talking wind.'' Let's call the first signal the ``modulating'' signal, and the other the ``carrier'' signal. Then the modulator may be a voice, and the carrier may be any spectrally rich sound such as wind, rain, creaking noises, flute, or other musical instrument sound. Commercial ``vocoders'' (§G.10G.5) used as musical instruments consist of a keyboard synthesizer (for playing the carrier sounds) and a microphone for picking up the voice of the performer (to extract the modulation envelope).

Cross-synthesis may be summarized as consisting of the following steps:

  1. Perform a Short-Time Fourier Transform (STFT) of both the modulator and carrier signals (§7.1).

  2. Compute the spectral envelope of each time-frame (as described in the next section).

  3. Optionally divide the spectrum of each carrier frame by its own spectral envelope, thereby flattening it.

  4. Multiply the flattened spectral frame by the envelope of the corresponding modulator frame, thereby replacing the carrier's envelope by the modulator's envelope.

For an audio example of cross-synthesis (a ``talking organ''), see

Spectral Envelope Extraction

There are many definitions of spectral envelope. Piecewise-linear (or polynomial spline) spectral envelopes (applied to the spectral magnitude of an STFT frame), have been used successfully in sines+noise modeling of audio signals (introduced in §10.4). Here we will consider spectral envelopes defined by the following two methods for computing them:

  1. cepstral windowing to lowpass-filter the log-magnitude spectrum (a ``nonparametric method'')

  2. using linear prediction (a ``parametric method'') to capture spectral shape in the amplitude-response of an all-pole filter in a source-filter decomposition of the signal (where the source signal is defined to be spectrally flat)

In the following, $ X_m(\omega_k)$ denotes the $ m$ th spectral frame of the STFT (§7.1), and $ Y_m(\omega_k)$ denotes the spectral envelope of $ X_m(\omega_k)$ .

Cepstral Windowing

The spectral envelope obtained by cepstral windowing is defined as

$\displaystyle Y_m \eqsp \hbox{\sc DFT}[w \cdot \underbrace{\hbox{\sc DFT}^{-1}\log(\vert X_m\vert)}_{\hbox{real cepstrum}}]$ (11.2)

where $ w$ is a lowpass-window in the cepstral domain. A simple but commonly used lowpass-window is given by

$\displaystyle w(n) \eqsp \left\{\begin{array}{ll} 1, & \vert n\vert< n_c \\ [5pt] 0.5, & \vert n\vert=n_c \\ [5pt] 0, & \vert n\vert>n_c, \\ \end{array} \right.$ (11.3)

where $ n_c$ denotes the lowpass ``cut-off'' sample.

The log-magnitude spectrum of $ X_m$ is thus lowpass filtered (the real cepstrum of $ x$ is ``liftered'') to obtain a smooth spectral envelope. For periodic signals, $ n_c$ should be set below the period in samples.

Cepstral coefficients are typically used in speech recognition to characterize spectral envelopes, capturing primarily the formants (spectral resonances) of speech [227]. In audio applications, a warped frequency axis, such as the ERB scale (Appendix E), Bark scale, or Mel frequency scale is typically preferred. Mel Frequency Cepstral Coefficients (MFCC) appear to remain quite standard in speech-recognition front ends, and they are often used to characterize steady-state spectral timbre in Music Information Retrieval (MIR) applications.

Linear Prediction Spectral Envelope

Linear Prediction (LP) implicitly computes a spectral envelope that is well adapted for audio work, provided the order of the predictor is appropriately chosen. Due to the error minimized by LP, spectral peaks are emphasized in the envelope, as they are in the auditory system. (The peak-emphasis of LP is quantified in (10.10) below.)

The term ``linear prediction'' refers to the process of predicting a signal sample $ y(n)$ based on $ M$ past samples:

$\displaystyle y(n) \eqsp -a_1 y(n-1) - a_2 y(n-2) - \cdots - a_M y(n-M) + e(n) \protect$ (11.4)

We call $ M$ the order of the linear predictor, and $ \{a_i\}_{i=1}^M$ the prediction coefficients. The prediction error (or ``innovations sequence'' [114]) is denoted $ e(n)$ in (10.4), and it represents all new information entering the signal $ y$ at time $ n$ . Because the information is new, $ e(n)$ is ``unpredictable.'' The predictable component of $ y(n)$ contains no new information.

Taking the z transform of (10.4) yields

$\displaystyle Y(z) \eqsp \frac{E(z)}{A(z)}$ (11.5)

where $ A(z) = 1 + a_1z^{-1}+ \cdots a_M z^{-M}$ . In signal modeling by linear prediction, we are given the signal $ y(n)$ but not the prediction coefficients $ a_i$ . We must therefore estimate them. Let $ {\hat A}(z) = 1 + {\hat a}_1z^{-1}
+ \cdots {\hat a}_M z^{-M}$ denote the polynomial with estimated prediction coefficients $ {\hat a}_i$ . Then we have

$\displaystyle Y(z) \eqsp \frac{{\hat E}(z)}{{\hat A}(z)}$ (11.6)

where $ {\hat E}(z)$ denotes the estimated prediction-error z transform. By minimizing $ \vert\vert\,{\hat E}\,\vert\vert _2$ , we define a minimum-least-squares estimate $ {\hat A}$ . In other words, the linear prediction coefficients $ {\hat a}_i$ are defined as those which minimize the sum of squared prediction errors $ {\hat e}(n)$

$\displaystyle \left\Vert\,{\hat e}\,\right\Vert _2^2 \eqsp \sum_n {\hat e}^2(n)$ (11.7)

over some range of $ n$ , typically an interval over which the signal is stationary (defined in Chapter 6). It turns out that this minimization results in maximally flattening the prediction-error spectrum $ E(z)$ [11,157,162]. That is, the optimal $ {\hat A}(z)$ is a whitening filter (also called an inverse filter). This makes sense in terms of Chapter 6 when one considers that a flat power spectral density corresponds to white noise in the time domain, and only white noise is completely unpredictable from one sample to the next. A non-flat spectrum corresponds to a nonzero correlation between two signal samples separated by some nonzero time interval.

If the prediction-error is successfully whitened, then the signal model can be expressed in the frequency domain as

$\displaystyle S_y(\omega) \eqsp \frac{\sigma^2_e}{\vert A(\omega)\vert^2}$ (11.8)

where $ S_y(\omega)$ denotes the power spectral density of $ y$ (defined in Chapter 6), and $ \sigma_e^2$ denotes the variance of the (white-noise) prediction error $ e(n)$ . Thus, the spectral magnitude envelope may be defined as

EnvelopeLPC$\displaystyle _y(\omega) \eqsp \frac{\sigma_e}{\vert A(\omega)\vert}$ (11.9)

Linear Prediction is Peak Sensitive

By Rayleigh's energy theorem, $ \vert\vert\,{\hat e}\,\vert\vert _2= \vert\vert\,{\hat E}\,\vert\vert _2$ (as shown in §2.3.8). Therefore,

$\displaystyle \sum_{n=-\infty}^{\infty} {\hat e}^2(n)$ $\displaystyle =$ $\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}\left\vert{\hat E}\left(e^{j\omega}\right)\right\vert^2 d\omega$  
  $\displaystyle \isdef$ $\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}\left\vert{\hat A}\left(e^{j\omega}\right)Y\left(e^{j\omega}\right)\right\vert^2 d\omega$  
  $\displaystyle =$ $\displaystyle \frac{{\hat\sigma}^2_e}{2\pi}\int_{-\pi}^{\pi}\left\vert\frac{Y\left(e^{j\omega}\right)}%
{{\hat Y}\left(e^{j\omega}\right)}\right\vert^2 d\omega.
\protect$ (11.10)

From this ``ratio error'' expression in the frequency domain, we can see that contributions to the error are smallest when $ \vert{\hat Y}(e^{j\omega})\vert>\vert Y(e^{j\omega})\vert$ . Therefore, LP tends to overestimate peaks. LP cannot make $ \vert{\hat Y}\vert$ arbitrarily large because $ A(z)$ is constrained to be monic and minimum-phase. It can be shown that the log-magnitude frequency response of every minimum-phase monic polynomial $ A(z)$ is zero-mean [162]. Therefore, for each peak overestimation, there must be an equal-area ``valley underestimation'' (in a log-magnitude plot over the unit circle).

Linear Prediction Methods

The two classic methods for linear prediction are called the autocorrelation method and the covariance method [162,157]. Both methods solve the linear normal equations (defined below) using different autocorrelation estimates.

In the autocorrelation method of linear prediction, the covariance matrix is constructed from the usual Bartlett-window-biased sample autocorrelation function (see Chapter 6), and it has the desirable property that $ {\hat A}(z)$ is always minimum phase (i.e., $ 1/{\hat A}(z)$ is guaranteed to be stable). However, the autocorrelation method tends to overestimate formant bandwidths; in other words, the filter model is typically overdamped. This can be attributed to implicitly ``predicting zero'' outside of the signal frame, resulting in the Bartlett-window bias in the sample autocorrelation.

The covariance method of LP is based on an unbiased autocorrelation estimate (see Eq.$ \,$ (6.4)). As a result, it gives more accurate bandwidths, but it does not guarantee stability.

So-called covariance lattice methods and Burg's method were developed to maintain guaranteed stability while giving accuracy comparable to the covariance method of LP [157].

Computation of Linear Prediction Coefficients

In the autocorrelation method of linear prediction, the linear prediction coefficients $ \{a_i\}_{i=1}^M$ are computed from the Bartlett-window-biased autocorrelation function (Chapter 6):

$\displaystyle r_{y_m}(l) \isdefs \sum_{n=-\infty}^\infty y_m(n)y_m(n+l) \eqsp \hbox{\sc DFT}^{-1}\left\vert Y_m\right\vert^2 \protect$ (11.11)

where $ y_m$ denotes the $ m$ th data frame from the signal $ y$ . To obtain the $ M$ th-order linear predictor coefficients $ \{a_1,\ldots,a_M\}$ , we solve the following $ M\times M$ system of linear normal equations (also called Yule-Walker or Wiener-Hopf equations):

$\displaystyle \sum_{i=1}^M a_i r_{y_m}(\vert i-j\vert) \eqsp -r_{y_m}(j), \qquad j=1,2,\ldots,M \protect$ (11.12)

In matlab syntax, the solution is given by `` $ \verb+a=R\p+$ '', where $ \verb+p(j)+ = r_{y_m}(j)$ , and $ \verb+R(i,j)+=r_{y_m}(\vert i-j\vert)$ . Since the covariance matrix $ R$ is symmetric and Toeplitz by construction,11.4 an $ O(M^2)$ solution exists using the Durbin recursion.11.5

If the rank of the $ M\times M$ autocorrelation matrix $ R[i,j]=r_{y_n}(\vert i-j\vert)$ is $ M$ , then the solution to (10.12) is unique, and this solution is always minimum phase [162] (i.e., all roots of $ A(z)$ are inside the unit circle in the $ z$ plane [263], so that $ 1/A(z)$ is always a stable all-pole filter). In practice, the rank of $ R$ is $ M$ (with probability 1) whenever $ y(n)$ includes a noise component. In the noiseless case, if $ y(n)$ is a sum of sinusoids, each (real) sinusoid at distinct frequency $ 0<\omega_i T
< \pi$ adds 2 to the rank. A dc component, or a component at half the sampling rate, adds 1 to the rank of $ R$ .

The choice of time window for forming a short-time sample autocorrelation and its weighting also affect the rank of $ R$ . Equation (10.11) applied to a finite-duration frame yields what is called the autocorrelation method of linear prediction [162]. Dividing out the Bartlett-window bias in such a sample autocorrelation yields a result closer to the covariance method of LP. A matlab example is given in §10.3.3 below.

The classic covariance method computes an unbiased sample covariance matrix by limiting the summation in (10.11) to a range over which $ y_m(n+l)$ stays within the frame--a so-called ``unwindowed'' method. The autocorrelation method sums over the whole frame and replaces $ y_m(n+l)$ by zero when $ n+l$ points outside the frame--a so-called ``windowed'' method (windowed by the rectangular window).

Linear Prediction Order Selection

For computing spectral envelopes via linear prediction, the order $ M$ of the predictor should be chosen large enough that the envelope can follow the contour of the spectrum, but not so large that it follows the spectral ``fine structure'' on a scale not considered to belong in the envelope. In particular, for voice, $ M$ should be twice the number of spectral formants, and perhaps a little larger to allow more detailed modeling of spectral shape away from the formants. For a sum of quasi sinusoids, the order $ M$ should be significantly less than twice the number of sinusoids to inhibit modeling the sinusoids as spectral-envelope peaks. For filtered-white-noise, $ M$ should be close to the order of the filter applied to the white noise, and so on.

Summary of LP Spectral Envelopes

In summary, the spectral envelope of the $ m$ th spectral frame, computed by linear prediction, is given by

$\displaystyle {\hat Y}_m(\omega_k) \eqsp \frac{{\hat g}_m}{\left\vert{\hat A}_m\left(e^{j\omega_k }\right)\right\vert}$ (11.13)

where $ {\hat A}_m$ is computed from the solution of the Toeplitz normal equations, and $ {\hat g}_m = \vert\vert\,{\hat E}_m\,\vert\vert _2$ is the estimated rms level of the prediction error in the $ m$ th frame.

The stable, all-pole filter

$\displaystyle \frac{{\hat g}_m}{{\hat A}_m(z)}$ (11.14)

can be driven by unit-variance white noise to produce a filtered-white-noise signal having spectral envelope $ {\hat g}_m/\vert{\hat A}_m(e^{j\omega_k })\vert$ . We may regard $ {\hat g}_m/{\hat A}_m(e^{j\omega_k })$ (no absolute value) as the frequency response of the filter in a source-filter decomposition of the signal $ y_m(n)$ , where the source is white noise.

It bears repeating that $ \log A(e^{j\omega_k })$ is zero mean when $ A(z)$ is monic and minimum phase (all zeros inside the unit circle). This means, for example, that $ \log {\hat g}_m$ can be simply estimated as the mean of the log spectral magnitude $ \log \vert Y_m(e^{j\omega_k })\vert$ .

For best results, the frequency axis ``seen'' by linear prediction should be warped to an auditory frequency scale, as discussed in Appendix E [123]. This has the effect of increasing the accuracy of low-frequency peaks in the extracted spectral envelope, in accordance with the nonuniform frequency resolution of the inner ear.

Spectral Envelope Examples

This section presents matlab code for computing spectral envelopes by the cepstral and linear prediction methods discussed above. The signal to be modeled is a synthetic ``ah'' vowel (as in ``father'') synthesized using three formants driven by a bandlimited impulse train [128].

Signal Synthesis

% Specify formant resonances for an "ah" [a] vowel:
F = [700, 1220, 2600]; % Formant frequencies in Hz
B = [130,   70,  160]; % Formant bandwidths in Hz

fs = 8192;  % Sampling rate in Hz
	    % ("telephone quality" for speed)
R = exp(-pi*B/fs);     % Pole radii
theta = 2*pi*F/fs;     % Pole angles
poles = R .* exp(j*theta);
[B,A] = zp2tf(0,[poles,conj(poles)],1);

f0 = 200; % Fundamental frequency in Hz
w0T = 2*pi*f0/fs;

nharm = floor((fs/2)/f0); % number of harmonics
nsamps = fs;  % make a second's worth
sig = zeros(1,nsamps);
n = 0:(nsamps-1);
% Synthesize bandlimited impulse train:
for i=1:nharm,
    sig = sig + cos(i*w0T*n);
sig = sig/max(sig);
soundsc(sig,fs); % Let's hear it

% Now compute the speech vowel:
speech = filter(1,A,sig);
soundsc([sig,speech],fs); % "buzz", "ahh"
% (it would sound much better with a little vibrato)

The Hamming-windowed bandlimited impulse train sig and its spectrum are plotted in Fig.10.1.

Figure 10.1: Bandlimited impulse train.
\includegraphics[width=\textwidth ]{eps/ImpulseTrain}

Figure 10.2 shows the Hamming-windowed synthesized vowel speech, and its spectrum overlaid with the true formant envelope.

Figure 10.2: Synthetic vowel in time and frequency domains, with formant envelope overlaid.
\includegraphics[width=\textwidth ]{eps/Speech}

Spectral Envelope by the Cepstral Windowing Method

We now compute the log-magnitude spectrum, perform an inverse FFT to obtain the real cepstrum, lowpass-window the cepstrum, and perform the FFT to obtain the smoothed log-magnitude spectrum:

Nframe = 2^nextpow2(fs/25); % frame size = 40 ms
w = hamming(Nframe)';
winspeech = w .* speech(1:Nframe);
Nfft = 4*Nframe; % factor of 4 zero-padding
sspec = fft(winspeech,Nfft);
dbsspecfull = 20*log(abs(sspec));
rcep = ifft(dbsspecfull);  % real cepstrum
rcep = real(rcep); % eliminate round-off noise in imag part
period = round(fs/f0) % 41
nspec = Nfft/2+1;
aliasing = norm(rcep(nspec-10:nspec+10))/norm(rcep) % 0.02
nw = 2*period-4; % almost 1 period left and right
if floor(nw/2) == nw/2, nw=nw-1; end; % make it odd
w = boxcar(nw)'; % rectangular window
wzp = [w(((nw+1)/2):nw),zeros(1,Nfft-nw), ...
       w(1:(nw-1)/2)];  % zero-phase version
wrcep = wzp .* rcep;  % window the cepstrum ("lifter")
rcepenv = fft(wrcep); % spectral envelope
rcepenvp = real(rcepenv(1:nspec)); % should be real
rcepenvp = rcepenvp - mean(rcepenvp); % normalize to zero mean

Figure 10.3 shows the real cepstrum of the synthetic ``ah'' vowel (top) and the same cepstrum truncated to just under a period in length. In theory, this leaves only formant envelope information in the cepstrum. Figure 10.4 shows an overlay of the spectrum, true envelope, and cepstral envelope.

Figure 10.3: Real cepstrum (top) and windowed cepstrum (bottom).
\includegraphics[width=\textwidth ]{eps/CepstrumBoxcar}

Figure 10.4: Overlay of spectrum, true envelope, and cepstral envelope.
\includegraphics[width=\textwidth ]{eps/CepstrumEnvBoxcarC}

Instead of simply truncating the cepstrum (a rectangular windowing operation), we can window it more gracefully. Figure 10.5 shows the result of using a Hann window of the same length. The spectral envelope is smoother as a result.

Figure 10.5: Overlay of spectrum, true envelope, and cepstral envelope.
\includegraphics[width=\textwidth ]{eps/CepstrumEnvHanningC}

Spectral Envelope by Linear Prediction

Finally, let's do an LPC window. It had better be good because the LPC model is exact for this example.

M = 6; % Assume three formants and no noise

% compute Mth-order autocorrelation function:
rx = zeros(1,M+1)';
for i=1:M+1,
  rx(i) = rx(i) + speech(1:nsamps-i+1) ...
                * speech(1+i-1:nsamps)';

% prepare the M by M Toeplitz covariance matrix:
covmatrix = zeros(M,M);
for i=1:M,
  covmatrix(i,i:M) = rx(1:M-i+1)';
  covmatrix(i:M,i) = rx(1:M-i+1);

% solve "normal equations" for prediction coeffs:

Acoeffs = - covmatrix \ rx(2:M+1)

Alp = [1,Acoeffs']; % LP polynomial A(z)

dbenvlp = 20*log10(abs(freqz(1,Alp,nspec)'));
dbsspecn = dbsspec + ones(1,nspec)*(max(dbenvlp) ...
                   - max(dbsspec)); % normalize
plot(f,[max(dbsspecn,-100);dbenv;dbenvlp]); grid;

Figure 9.16:
\includegraphics[width=\textwidth ]{eps/LinearPredictionEnvC}

Linear Prediction in Matlab and Octave

In the above example, we implemented essentially the covariance method of LP directly (the autocorrelation estimate was unbiased). The code should run in either Octave or Matlab with the Signal Processing Toolbox.

The Matlab Signal Processing Toolbox has the function lpc available. (LPC stands for ``Linear Predictive Coding.'')

The Octave-Forge lpc function (version 20071212) is a wrapper for the lattice function which implements Burg's method by default. Burg's method has the advantage of guaranteeing stability ($ A(z)$ is minimum phase) while yielding accuracy comparable to the covariance method. By uncommenting lines in lpc.m, one can instead use the ``geometric lattice'' or classic autocorrelation method (called ``Yule-Walker'' in lpc.m). For details, ``type lpc''.

Spectral Modeling Synthesis

This section reviews elementary spectral models for sound synthesis. Spectral models are well matched to audio perception because the ear is a kind of spectrum analyzer [293].

For periodic sounds, the component sinusoids are all harmonics of a fundamental at frequency $ \omega_1$ :

$\displaystyle s(t) = \sum_{k=0}^K A_k \sin(\omega_k t + \phi_k) \protect$ (11.15)

where $ t$ denotes time in seconds, $ \omega_k = k\cdot 2\pi/P$ is the $ k$ th harmonic radian frequency, $ P$ is the period in seconds, $ A_k$ is the amplitude of the $ k$ th sinusoidal component, $ \phi_k$ is its phase, and $ K$ is the number of the highest audible harmonic.

Aperiodic sounds can similarly be expressed as a continuous sum of sinusoids at potentially all frequencies in the range of human hearing:11.6

$\displaystyle s(t) = \int_{0}^\Omega A_\omega \sin(\omega t + \phi_\omega) d\omega, \protect$ (11.16)

where $ \Omega$ denotes the upper bound of human hearing (nominally $ 2\pi\cdot 20$ kHz).

Sinusoidal models are most appropriate for ``tonal'' sounds such as spoken or sung vowels, or the sounds of musical instruments in the string, wind, brass, and ``tonal percussion'' families. Ideally, one sinusoid suffices to represent each harmonic or overtone.11.7 To represent the ``attack'' and ``decay'' of natural tones, sinusoidal components are multiplied by an amplitude envelope that varies over time. That is, the amplitude $ A_k(t)$ in (10.15) is a slowly varying function of time; similarly, to allow pitch variations such as vibrato, the phase $ \phi_k(t)$ may be modulated in various ways.11.8 Sums of amplitude- and/or frequency-enveloped sinusoids are generally called additive synthesis (discussed further in §10.4.1 below).

Sinusoidal models are ``unreasonably effective'' for tonal audio. Perhaps the main reason is that the ear focuses most acutely on peaks in the spectrum of a sound [179,305]. For example, when there is a strong spectral peak at a particular frequency, it tends to mask lower level sound energy at nearby frequencies. As a result, the ear-brain system is, to a first approximation, a ``spectral peak analyzer''. In modern audio coders [16,200] exploiting masking results in an order-of-magnitude data compression, on average, with no loss of quality, according to listening tests [25]. Thus, we may say more specifically that, to first order, the ear-brain system acts like a ``top ten percent spectral peak analyzer''.

For noise-like sounds, such as wind, scraping sounds, unvoiced speech, or breath-noise in a flute, sinusoidal models are relatively expensive, requiring many sinusoids across the audio band to model noise. It is therefore helpful to combine a sinusoidal model with some kind of noise model, such as pseudo-random numbers passed through a filter [249]. The ``Sines + Noise'' (S+N) model was developed to use filtered noise as a replacement for many sinusoids when modeling noise (to be discussed in §10.4.3 below).

Another situation in which sinusoidal models are inefficient is at sudden time-domain transients in a sound, such as percussive note onsets, ``glitchy'' sounds, or ``attacks'' of instrument tones more generally. From Fourier theory, we know that transients, too, can be modeled exactly, but only with large numbers of sinusoids at exactly the right phases and amplitudes. To obtain a more compact signal model, it is better to introduce an explicit transient model which works together with sinusoids and filtered noise to represent the sound more parsimoniously. Sines + Noise + Transients (S+N+T) models were developed to separately handle transients (§10.4.4).

A advantage of the explicit transient model in S+N+T models is that transients can be preserved during time-compression or expansion. That is, when a sound is stretched (without altering its pitch), it is usually desirable to preserve the transients (i.e., to keep their local time scales unchanged) and simply translate them to new times. This topic, known as Time-Scale Modification (TSM) will be considered further in §10.5 below.

In addition to S+N+T components, it is useful to superimpose spectral weightings to implement linear filtering directly in the frequency domain; for example, the formants of the human voice are conveniently impressed on the spectrum in this way (as illustrated §10.3 above) [174].11.9 We refer to the general class of such frequency-domain signal models as spectral models, and sound synthesis in terms of spectral models is often called spectral modeling synthesis (SMS).

The subsections below provide a summary review of selected aspects of spectral modeling, with emphasis on applications in musical sound synthesis and effects.

Additive Synthesis (Early Sinusoidal Modeling)

Additive synthesis is evidently the first technique widely used for analysis and synthesis of audio in computer music [232,233,184,186,187]. It was inspired directly by Fourier theory [264,23,36,150] (which followed Daniel Bernoulli's insights (§G.1)) which states that any sound $ s(t)$ can be expressed mathematically as a sum of sinusoids. The `term ``additive synthesis'' refers to sound being formed by adding together many sinusoidal components modulated by relatively slowly varying amplitude and frequency envelopes:

$\displaystyle y(t)= \sum\limits_{i=1}^{N} A_i(t)\sin[\theta_i(t)]$ (11.17)

$\displaystyle A_i(t)$ $\displaystyle =$ $\displaystyle \hbox{amplitude of $i$th partial over time $t$}$  
$\displaystyle \theta_i(t)$ $\displaystyle =$ $\displaystyle \int_0^t \omega_i(t)dt + \phi_i(0)$  
$\displaystyle \omega_i(t)$ $\displaystyle =$ $\displaystyle d\theta_i(t)/dt = \hbox{radian frequency of $i$th partial vs.\ time}$  
$\displaystyle \phi_i(0)$ $\displaystyle =$ $\displaystyle \hbox{phase offset of $i$th partial at time $0$}
\protect$ (11.18)

and all quantities are real. Thus, each sinusoid may have an independently time-varying amplitude and/or phase, in general. The amplitude and frequency envelopes are determined from some kind of short-time Fourier analysis as discussed in Chapters 8 and 9) [62,187,184,186].

An additive-synthesis oscillator-bank is shown in Fig.10.7, as it is often drawn in computer music [235,234]. Each sinusoidal oscillator [166] accepts an amplitude envelope $ A_i(t)$ (e.g., piecewise linear, or piecewise exponential) and a frequency envelope $ f_i(t)$ , also typically piecewise linear or exponential. Also shown in Fig.10.7 is a filtered noise input, as used in S+N modeling (§10.4.3).

% latex2html id marker 27324\psfrag{A1} []{ \Large$ A_1(t)$\ }\psfrag{A2} []{ \Large$ A_2(t)$\ }\psfrag{A3} []{ \Large$ A_3(t)$\ }\psfrag{A4} []{ \Large$ A_4(t)$\ }\psfrag{f1} []{ \Large$ f_1(t)$\ }\psfrag{f2} []{ \Large$ f_2(t)$\ }\psfrag{f3} []{ \Large$ f_3(t)$\ }\psfrag{f4} []{ \Large$ f_4(t)$\ }\psfrag{s} [b]{ {\Huge$\sum$} }\psfrag{noise} [][b]{{\Large$\stackrel{\hbox{[new] noise}}{u(t)}$}}\psfrag{Filter} [][c]{{\Large$\stackrel{\hbox{filter}}{h_t(\tau)}$}}\psfrag{out} []{
{\large$ y(t)= \sum\limits_{i=1}^{4}
A_i(t)\sin\left[\int_0^t\omega_i(t)dt +\phi_i(0)\right]
+ (h_t \ast u)(t) $\ = sine-sum + filtered-noise}}\begin{figure}[htbp]
\caption{Sinusoidal oscillator bank and filtered
noise for sines+noise spectral modeling synthesis.}

Additive Synthesis Analysis

In order to reproduce a given signal, we must first analyze it to determine the amplitude and frequency trajectories for each sinusoidal component. We do not need phase information ($ \phi_i$ in (10.18)) during steady-state segments, since phase is normally not perceived in steady state tones [293,211]. However, we do need phase information for analysis frames containing an attack transient, or any other abrupt change in the signal. The phase of the sinusoidal peaks controls the position of time-domain features of the waveform within the analysis frame.

Following Spectral Peaks

In the analysis phase, sinusoidal peaks are measured over time in a sequence of FFTs, and these peaks are grouped into ``tracks'' across time. A detailed discussion of various options for this can be found in [246,174,271,84,248,223,10,146], and a particular case is detailed in Appendix H.

The end result of the analysis pass is a collection of amplitude and frequency envelopes for each spectral peak versus time. If the time advance from one FFT to the next is fixed (5ms is a typical choice for speech analysis), then we obtain uniformly sampled amplitude and frequency trajectories as the result of the analysis. The sampling rate of these amplitude and frequency envelopes is equal to the frame rate of the analysis. (If the time advance between FFTs is $ \Delta t=5$ ms, then the frame rate is defined as $ 1/\Delta t
= 200$ Hz.) For resynthesis using inverse FFTs, these data may be used unmodified. For resynthesis using a bank of sinusoidal oscillators, on the other hand, we must somehow interpolate the envelopes to create envelopes at the signal sampling rate (typically $ 44$ kHz or higher).

It is typical in computer music to linearly interpolate the amplitude and frequency trajectories from one frame to the next [271].11.10 Let's call the piecewise-linear upsampled envelopes $ {\hat A}_k(n)$ and $ \hat{F}_k(n)$ , defined now for all $ n$ at the normal signal sampling rate. For steady-state tonal sounds, the phase may be discarded at this stage and redefined as the integral of the instantaneous frequency when needed:

$\displaystyle \hat{\Theta }_k(n) \isdefs \hat{\Theta }_k(n-1) + 2\pi T \hat{F}_k(n). \protect$ (11.19)

When phase must be matched in a given frame, such as when it is known to contain a transient event, the frequency can instead move quadratically across the frame to provide cubic phase interpolation [174], or a second linear breakpoint can be introduced somewhere in the frame for the frequency trajectory (in which case the area under the triangle formed by the second breakpoint equals the added phase at the end of the segment).

Sinusoidal Peak Finding

For each sinusoidal component of a signal, we need to determine its frequency, amplitude, and phase (when needed). As a starting point, consider the windowed complex sinusoid with complex amplitude $ {\cal A}_x$ and frequency $ \omega _x$ :

$\displaystyle x_w(n) = w(n){\cal A}_xe^{j\omega_x nT}$ (11.20)

As discussed in Chapter 5, the transform (DTFT) of this windowed signal is the convolution of a frequency domain delta function at $ \omega _x$ [ $ \delta(\omega - \omega_x) $ ], and the transform of the window function, $ W(\omega)$ , resulting in a shifted version of the window transform $ {\cal A}_xW(\omega-\omega_x)$ . Assuming $ M$ is odd, we can show this as follows:

X_w(\omega) &=& \sum_{n=-\infty}^{\infty}[w(n)x(n)]e^{ -j\omega nT}
\qquad\hbox{(DTFT($x_w$))} \\
&=& \sum_{n=-(M-1)/2}^{(M-1)/2} \left[w(n){\cal A}_xe^{j\omega_xnT}\right]e^{ -j\omega nT}\\
&=& {\cal A}_x\sum_n w(n) e^{-j(\omega-\omega_x)nT} \\
&=& \zbox {{\cal A}_xW(\omega-\omega_x)}


\vert X_w(\omega) \vert &=& \vert{\cal A}_x\vert \cdot \vert W(\omega-\omega_x)\vert
\qquad \hbox{(see \fref {peak} below)}\\
\angle X_w(\omega) &=& \angle {\cal A}_x+ \angle W(\omega-\omega_x).

At $ \omega _x$ , we have

\vert X_w(\omega_x)\vert &=& \vert{\cal A}_x\vert\cdot \vert W(0)\vert \\
\angle X_w(\omega_x)\vert &=& \angle {\cal A}_x+ \angle W(0)

If we scale the window to have a dc gain of 1, then the peak magnitude equals the amplitude of the sinusoid, i.e., $ \vert X_w(\omega_x)\vert=\vert{\cal A}_x\vert\isdef a$ , as shown in Fig.10.8.

Figure: Schematic diagram of a window transform amplitude-scaled by $ a$ and frequency-shifted by $ \omega _x$ .

If we use a zero-phase (even) window, the phase at the peak equals the phase of the sinusoid, i.e., $ \angle X_w(\omega_x) =
\angle {\cal A}_x$ .

Tracking Sinusoidal Peaks in a Sequence of FFTs

The preceding discussion focused on estimating sinusoidal peaks in a single frame of data. For estimating sinusoidal parameter trajectories through time, it is necessary to associate peaks from one frame to the next. For example, Fig.10.9 illustrates a set of frequency trajectories, including one with a missing segment due to its peak not being detected in the third frame.

Figure 10.9: Sinusoidal frequency trajectories.

Figure 10.10 depicts a basic analysis system for tracking spectral peaks in the STFT [271]. The system tracks peak amplitude, center-frequency, and sometimes phase. Quadratic interpolation is used to accurately find spectral magnitude peaks (§5.7). For further analysis details, see Appendix H. Synthesis is performed using a bank of amplitude- and phase-modulated oscillators, as shown in Fig.10.7. Alternatively, the sinusoids are synthesized using an inverse FFT [239,94,139].

% latex2html id marker 27619\psfrag{s} []{\Large$s(t)$}\psfrag{tan} []{\Large$\tan^{-1}$}\begin{figure}[htbp]
\caption{Block diagram
of a sinusoidal-modeling \emph{analysis} system
(from \cite{SerraT}).}

Sines + Noise Modeling

As mentioned in the introduction to this chapter, it takes many sinusoidal components to synthesize noise well (as many as 25 per critical band of hearing under certain conditions [85]). When spectral peaks are that dense, they are no longer perceived individually, and it suffices to match only their statistics to a perceptually equivalent degree.

Sines+Noise (S+N) synthesis [249] generalizes the sinusoidal signal models to include a filtered noise component, as depicted in Fig.10.7. In that figure, white noise is denoted by $ u(t)$ , and the slowly changing linear filter applied to the noise at time $ t$ is denoted $ h_t(\cdot)$ .

The time-varying spectrum of the signal is said to be made up of a deterministic component (the sinusoids) and a stochastic component (time-varying filtered noise) [246,249]:

$\displaystyle s(t) \eqsp \sum_{i=1}^{N} A_i(t) \cos[ \theta_i(t)] + e(t),$ (11.21)

where $ A_i(t)$ and $ \theta_i(t)$ are the instantaneous amplitude and phase of the $ i$ th sinusoidal component, and $ e(t)$ is the residual, or noise signal, assumed to be well modeled by filtered white noise:

$\displaystyle e(t) \eqsp (h_t \ast u)(t) \isdefs \int_0^t h_{t}(t-\tau)u(\tau)d\tau,$ (11.22)

where $ u(t)$ is the white noise, and $ h_t(\tau)$ is the impulse response of a time varying linear filter at time $ t$ . Specifically, $ h_t(\tau)$ is the response at time $ t$ to an impulse at time $ t-\tau$ .

Filtering white-noise to produce a desired timbre is an example of subtractive synthesis [186]. Thus, additive synthesis is nicely supplemented by subtractive synthesis as well.

Sines+Noise Analysis

The original sines+noise analysis method is shown in Fig.10.11 [246,249]. The processing path along the top from left to right measures the amplitude and frequency trajectories from magnitude peaks in the STFT, as in Fig.10.10. The peak amplitude and frequency trajectories are converted back to the time domain by additive-synthesis (an oscillator bank or inverse FFT), and this signal is windowed by the same analysis window and forward-transformed back into the frequency domain. The magnitude-spectrum of this sines-only data is then subtracted from the originally computed magnitude-spectrum containing both peaks and ``noise''. The result of this subtraction is termed the residual signal. The upper spectral envelope of the residual magnitude spectrum is measured using, e.g., linear prediction, cepstral smoothing, as discussed in §10.3 above, or by simply connecting peaks of the residual spectrum with linear segments to form a more traditional (in computer music) piecewise linear spectral envelope.

% latex2html id marker 27835\psfrag{s}{\Large $x(n)$}\begin{figure}[htbp]
\caption{Sines+noise analysis diagram
(from \cite{SerraT}).}

S+N Synthesis

A sines+noise synthesis diagram is shown in Fig.10.12. The spectral-peak amplitude and frequency trajectories are possibly modified (time-scaling, frequency scaling, virtual formants, etc.) and then rendered into the time domain by additive synthesis. This is termed the deterministic part of the synthesized signal.

The stochastic part is synthesized by applying the residual-spectrum-envelope (a time-varying FIR filter) to white noise, again after possible modifications to the envelope.

To synthesize a frame of filtered white noise, one can simply impart a random phase to the spectral envelope, i.e., multiply it by $ \exp[j\phi(\omega_k)]$ , where $ \phi(\omega_k)$ is random and uniformly distributed between $ -\pi$ and $ \pi$ . In the time domain, the synthesized white noise will be approximately Gaussian due to the central limit theoremD.9.1). Because the filter (spectral envelope) is changing from frame to frame through time, it is important to use at least 50% overlap and non-rectangular windowing in the time domain. The window can be implemented directly in the frequency domain by convolving its transform with the complex white-noise spectrum3.3.5), leaving only overlap-add to be carried out in the time domain. If the window side-lobes can be fully neglected, it suffices to use only main lobe in such a convolution [239].

In Fig.10.12, the deterministic and stochastic components are summed after transforming to the time domain, and this is the typical choice when an explicit oscillator bank is used for the additive synthesis. When the IFFT method is used for sinusoid synthesis [239,94,139], the sum can occur in the frequency domain, so that only one inverse FFT is required.

% latex2html id marker 27855\psfrag{IFFT} [][c]{{\Large IFFT $\cdot w$}}\begin{figure}[htbp]
\caption{Sines+noise synthesis diagram
(from \cite{SerraT}).}

Sines+Noise Summary

To summarize, sines+noise modeling is carried out by a procedure such as the following:

  • Compute a sinusoidal model by tracking peaks across STFT frames, producing a set of amplitude envelopes $ A_i(t_m)$ and frequency envelopes $ F_i(t_m)$ , where $ m$ is the frame number and $ i$ is the spectral-peak number.

  • Also record phase $ \Theta_m(\omega_k)$ for frames containing a transient.

  • Subtract modeled peaks from each STFT spectrum to form a residual spectrum.

  • Fit a smooth spectral envelope $ H_m(\omega_k)$ to each residual spectrum.

  • Convert envelopes to reduced form, e.g., piecewise linear segments with nonuniformly distributed breakpoints (optimized to be maximally sparse without introducing audible distortion).

  • Resynthesize audio (along with any desired transformations) from the amplitude, frequency, and noise-floor-filter envelopes.

  • Alter frequency trajectories slightly to hit the desired phase for transient frames (as described below equation Eq.$ \,$ (10.19)).

Because the signal model consists entirely of envelopes (neglecting the phase data for transient frames), the signal model is easily time scaled, as discussed further in §10.5 below.

For more information on sines+noise signal modeling, see, e.g., [146,10,223,248,246,149,271,248,271]. A discussion from an historical perspective appears in §G.11.4.

Sines + Noise + Transients Models

As we have seen, sinusoids efficiently model spectral peaks over time, and filtered noise efficiently models the spectral residual left over after pulling out everything we want to call a ``tonal component'' characterized by a spectral peak the evolves over time. However, neither is good for abrupt transients in a waveform. At transients, one may retain the original waveform or some compressed version of it (e.g., MPEG-2 AAC with short window [149]). Alternatively, one may switch to a transient model during transients. Transient models have included wavelet expansion [6] and frequency-domain LPC (time-domain amplitude envelope) [290].

In either case, a reliable transient detector is needed. This can raise deep questions regarding what a transient really is; for example, not everyone will notice every transient as a transient, and so perceptual modeling gets involved. Missing a transient, e.g., in a ride-cymbal analysis, can create highly audible artifacts when processing heavily based on transient decisions. For greater robustness, hybrid schemes can be devised in which a continuous measure of ``transientness'' $ {\cal T}$ can be defined between 0 and 1, say.

Also in either case, the sinusoidal model needs phase matching when switching to or from a transient frame over time (or cross-fading can be used, or both). Given sufficiently many sinusoids, phase-matching at the switching time should be sufficient without cross-fading.

Sines+Noise+Transients Time-Frequency Maps

Figure 10.13 shows the multiresolution time-frequency map used in the original S+N+T system [149]. (Recall the fixed-resolution STFT time-frequency map in Fig.7.1.) Vertical line spacing in the time-frequency map indicates the time resolution of the underlying multiresolution STFT,11.11 and the horizontal line spacing indicates its frequency resolution. The time waveform appears below the time-frequency map. For transients, an interval of data including the transient is simply encoded using MPEG-2 AAC. The transient-time in Fig.10.13 extends from approximately 47 to 115 ms. (This interval can be tighter, as discussed further below.) Between transients, the signal model consists of sines+noise below 5 kHz and amplitude-modulated noise above. The spectrum from 0 to 5 kHz is divided into three octaves (``multiresolution sinusoidal modeling''). The time step-size varies from 25 ms in the low-frequency band (where the frequency resolution is highest), down to 6 ms in the third octave (where frequency resolution is four times lower). In the 0-5 kHz band, sines+noise modeling is carried out. Above 5 kHz, noise substitution is performed, as discussed further below.

Figure 10.13: S+N+T Time-Frequency Map (from [149]).

Figure 10.14 shows a similar map in which the transient interval depends on frequency. This enables a tighter interval enclosing the transient, and follows audio perception more closely (see Appendix E).

Figure 10.14: Quasi-Constant-Q (Wavelet) Time-Frequency Map [149].

Sines+Noise+Transients Noise Model

Figure 10.15 illustrates the nature of the noise modeling used in [149]. The energy in each Bark band11.12 is summed, and this is used as the gain for the noise in that band at that frame time.

Figure 10.15: Bark-band noise modeling (from [149]).

Figure 10.16 shows the frame gain versus time for a particular Bark band (top) and the piecewise linear envelope made from it (bottom). As illustrated in Figures 10.13 and 10.14, the step size for all of the Bark bands above 5 kHz is approximately 3 ms.

Figure 10.16: Amplitude envelope for one noise band (from [149]).

S+N+T Sound Examples

A collection of sound examples illustrating sines+noise+transients modeling (and various subsets thereof) as well as some audio effects made possible by such representations, can be found online at

Discussion regarding S+N+T models in historical context is given in §G.11.5.

Time Scale Modification

Time Scale Modification (TSM) means speeding up or slowing down a sound without affecting the frequency content, such as the perceived pitch of any tonal components. For example, TSM of speech should sound like the speaker is talking at a slower or faster pace, without distortion of the spoken vowels. Similarly, TSM of music should change timing but not tuning.

When a recorded speech signal is simply played faster, such as by lowering its sampling-rate and playing it at the original sampling-rate, the pace of the speech increases as desired, but so does the fundamental frequency (pitch contour). Moreover, the apparent ``head size'' of the speaker shrinks (the so-called ``munchkinization'' effect). This happens because, as illustrated in §10.3, speech spectra have formants (resonant peaks) which should not be moved when the speech rate is varied. The average formant spacing in frequency is a measure of the length of the vocal tract; hence, when speech is simply played faster, the average formant spacing decreases, corresponding to a smaller head size. This illusion of size modulation can be a useful effect in itself, such as for scaling the apparent size of virtual musical instruments using commuted synthesis [47,266]. However, we also need to be able to adjust time scales without this overall scaling effect.

The Fourier dual of time-scale modification is frequency scaling. In this case, we wish to scale the spectral content of a signal up or down without altering the timing of sonic events in the time domain. This effect is used, for example, to retune ``bad notes'' in a recording studio. Frequency scaling can be implemented as TSM preceded or followed by sampling-rate conversion, or it can be implemented directly in a sequence of STFT frames like TSM.

TSM and S+N+T

Time Scale Modification (TSM), and/or frequency scaling, are relatively easy to implement in a sines+noise+transient (S+N+T) model (§10.4.4). Figure 10.17 illustrates schematically how it works. For TSM, the envelopes of the sinusoidal and noise models are simply stretched or squeezed versus time as desired, while the time-intervals containing transients are only translated forward or backward in time--not time-scaled. As a result, transients are not ``smeared out'' when time-expanding, or otherwise distorted by TSM. If a ``transientness'' measure $ {\cal T}$ is defined, it can be used to control how ``rubbery'' a given time-segment is; that is, for $ {\cal T}=1$ , the interval is rigid and can only translate in time, while for $ {\cal T}=0$ it is allowed stretch and squeeze along with the adjacent S+N model. In between 0 and 1, the time-interval scales less than the S+N model. See [149] for more details regarding TSM in an S+N+T framework.

Figure 10.17: Time Scaling for S+N+T Models (from [149]).

TSM by Resampling STFTs Across Time

In view of Chapter 8, a natural implementation of TSM based on the STFT is as follows:

  1. Perform a short-time Fourier transform (STFT) using hop size $ R$ . Denote the STFT at frame $ m$ and bin $ k$ by $ X_m(\omega_k)$ , and denote the result of TSM processing by $ Y_m(\omega_k)$ .

  2. To perform TSM by the factor $ \alpha>1$ , advance the ``frame pointer'' $ m$ by $ R/\alpha$ during resynthesis instead of the usual $ R$ samples.

For example, if $ \alpha=2$ ($ 2\times $ slow-down), the first STFT frame $ X_0(\omega_k)$ is processed normally, so that $ Y_0=X_0$ . However, the second output frame $ Y_{1/2}$ corresponds to a time $ m=1/2$ , half way between the first two frames. This output frame may be created by interpolating (across time) the STFT magnitude magnitude spectra of the first. For example, using simple linear interpolation gives

$\displaystyle Y_{1/2}(\omega_k) \eqsp \frac{\left\vert X_0(\omega_k)\right\vert + \left\vert X_1(\omega_k)\right\vert}{2} \,e^{j\theta(\omega_k)}$ (11.23)

where the phase $ \theta(\omega_k)$ is chosen to preserve continuity and/or the amplitude envelope from frame to frame under the overlap-add (more on this below). Generalizing to arbitrary TSM factors $ \alpha $ , we obtain

$\displaystyle Y_{m}(\omega_k) \eqsp \left[ (1-\eta)\, \vert X_{\left\lfloor m\right\rfloor }(\omega_k)\vert + \eta\,\vert X_{\left\lceil m\right\rceil }(\omega_k)\vert \right]\, e^{j\theta_m(\omega_k)}$ (11.24)

where $ \eta\isdeftext m-\lfloor m\rfloor $ , and $ m$ is advanced by $ R/\alpha$ each frame-step.

In general, TSM methods based on STFT modification are classified as ``vocoder'' type methods (§G.5). Thus, the TSM implementation outlined above may be termed a weighted overlap-add (WOLA) phase-vocoder method.

Phase Continuation

When interpolating the STFT across time for TSM, it is straightforward to interpolate spectral magnitude, as we saw above. Interpolating spectral phase, on the other hand, is tricky, because there's no exact way to do it [220]. There are two conflicting desiderata when deciding how to continue phase from one frame to the next:

Sinusoids should ``pick up where they left off'' in the previous frame.
The relative phase from bin to bin should be preserved in each FFT.
To satisfy condition (1), it is necessary to replace the original phase of each frame by the phase corresponding to smooth continuation across time from the previous frame (which is generally an interpolated frame) for each FFT bin. Altering the phase of a spectral frame changes its amplitude envelope in the time domain. Thus, it may no longer looks like a windowed signal segment. Using the WOLA framework (§8.6) helps because the post-window guarantees a smooth cross-fade from frame to frame. Random amplitude-modulation distortion is generally heard as reverberation, also called phasiness [140].

When condition (2) is violated, the signal frame suffers dispersion in the time domain. For steady-state signals (filtered noise and/or steady tones), temporal dispersion should not be audible, while frames containing distinct pulses will generally become more ``smeared out'' in time.

It is not possible in general to satisfy both conditions (1) and (2) simultaneously, but either can be satisfied at the expense of the other. Generally speaking, ``transient frames'' should emphasize condition (2), allowing the WOLA overlap-add cross-fade to take care of the phase discontinuity at the frame boundaries. For stationary segments, phase continuation, preserving condition (1), is more valuable.

It is often sufficient to preserve relative phase across FFT bins (i.e., satisfy condition (2)) only along spectral peaks and their immediate vicinity [142,143,141,138,215,238].

TSM Examples

To illustrate some fundamental points, let's look at some TSM waveforms for a test signal consisting of two constant-amplitude sinusoids near 400 Hz having frequencies separated by 10 Hz (to create an amplitude envelope having 10 beats/sec). We will perform a $ 2\times $ time expansion ($ 2\times $ ``slow-down'') using the following three algorithms:

The vocoder STFT frame size is set to 45 ms, the analysis hop size is 1/4 frame, or 11.25 ms, so the analysis frame rate is 89 Hz.

The results are shown in Figures 10.18 through 10.23.

Phase-Continued STFT TSM

Figure 10.18 shows the phase-continued-frames case in which relative phase is not preserved across FFT bins. As a result, the amplitude envelope is not preserved in the time domain within each frame. Figure 10.19 shows the spectrum of the same case, revealing significant distortion products at multiples of the frame rate due to the intra-frame amplitude-envelope distortion, which then ungracefully transitions to the next frame. Note that modulation sidebands corresponding to multiples of the frame rate are common in nonlinearly processed STFTs.

Figure 10.18: Phase-continued vocoder waveforms at 2X expansion.

Figure 10.19: Phase-continued vocoder spectra at 2X expansion.

Relative-Phase-Preserving STFT TSM

Figure 10.20 shows the relative-phase-preserving (sometimes called ``phase-locked'') vocoder case in which relative phase is preserved across FFT bins. As a result, the amplitude envelope is preserved very well in each frame, and segues from one frame to the next look much better on the envelope level, but now the individual FFT bin frequencies are phase-modulated from frame to frame. Both plots show the same number of beats per second while the overall duration is doubled in the second plot, as desired. Figure 10.21 shows the corresponding spectrum; instead of distortion-modulation on the scale of the frame rate, the spectral distortion looks more broadband--consistent with phase-discontinuities across the entire spectrum from one frame to the next.

Figure 10.20: Phase-locked vocoder waveforms at 2X time expansion.

Figure 10.21: Phase-Locked Vocoder Spectra at 2X Time Expansion.


Finally, Figures 10.22 and 10.23 show the time and frequency domain plots for the SOLA-FS algorithm (a time-domain method). SOLA-type algorithms perform slow-down by repeating frames locally. (In this case, each frame could be repeated once to accomplish the $ 2\times $ slow-down.) They maximize cross-correlation at the ``loop-back'' points in order to minimize discontinuity distortion, but such distortion is always there, though typically attenuated by a cross-fade on the loop-back. We can see twice as many ``carrier cycles'' under each beat, meaning that the beat frequency (amplitude envelope) was not preserved, but neither was it severely distorted in this case. SOLA algorithms tend to work well on speech, but can ``stutter'' when attack transients happen to be repeated. SOLA algorithms should be adjusted to avoid repeating a transient frame; similarly, they should avoid discarding a transient frame when speeding up.

Figure 10.22: SOLA-FS waveforms at 2X expansion.

Figure 10.23: SOLA-FS spectra at 2X expansion.

Further Reading

For a comprehensive tutorial review of TSM and frequency-scaling techniques, see [138].

Audio demonstrations of TSM and frequency-scaling based on the sines+noise+transients model of Scott Levine [149] may be found online at http://ccrma.stanford.edu/~jos/pdf/SMS.pdf .

See also the Wikipedia page entitled ``Audio time-scale/pitch modification''.

Gaussian Windowed Chirps (Chirplets)

As discussed in §G.8.2, an interesting generalization of sinusoidal modeling is chirplet modeling. A chirplet is defined as a Gaussian-windowed sinusoid, where the sinusoid has a constant amplitude, but its frequency may be linearly ``sweeping.'' This definition arises naturally from the mathematical fact that the Fourier transform of a Gaussian-windowed chirp signal is a complex Gaussian pulse, where a chirp signal is defined as a sinusoid having linearly modulated frequency, i.e., quadratic phase:

$\displaystyle s(t) \eqsp e^{j\beta t^2}$ (11.25)

Applying a Gaussian window to this chirp yields

$\displaystyle x(t) \eqsp e^{-\alpha t^2} s(t) \eqsp e^{-(\alpha-j\beta) t^2} \isdefs e^{-p t^2}$ (11.26)

where $ p\isdef \alpha-j\beta$ . It is thus clear how naturally Gaussian amplitude envelopes and linearly frequency-sweeping sinusoids (chirps) belong together in a unified form called a chirplet.

The basic chirplet $ x(t)=e^{-p t^2}$ can be regarded as an exponential polynomial signal in which the polynomial is of order 2. Exponential polynomials of higher order have also been explored [89,90,91]. (See also §G.8.2.)

Chirplet Fourier Transform

The Fourier transform of a complex Gaussian pulse is derived in §D.8 of Appendix D:

$\displaystyle \zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{\omega^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0} \protect$ (11.27)

This result is valid when $ p$ is complex. Writing $ p$ in terms of real variables $ \alpha $ and $ \beta $ as

$\displaystyle p \eqsp \alpha - j\beta,$ (11.28)

we have

$\displaystyle x(t) \eqsp e^{-p t^2} \eqsp e^{-\alpha t^2} e^{j\beta t^2} \eqsp e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right].$ (11.29)

That is, for $ p$ complex, $ x(t)$ is a chirplet (Gaussian-windowed chirp). We see that the chirp oscillation frequency is zero at time $ t=0$ . Therefore, for signal modeling applications, we typically add in an arbitrary frequency offset at time 0, as described in the next section.

Modulated Gaussian-Windowed Chirp

By the modulation theorem for Fourier transforms,

$\displaystyle \zbox {x(t)e^{-j\omega_0t}\;\longleftrightarrow\;X(\omega+\omega_0).}$ (11.30)

This is proved in §B.6 as the dual of the shift-theorem. It is also evident from inspection of the Fourier transform:

$\displaystyle \int_{-\infty}^\infty \left[x(t)e^{-j\omega_0 t}\right] e^{-j\omega t} dt \eqsp \int_{-\infty}^\infty x(t)e^{-j(\omega+\omega_0) t} dt \isdefs X(\omega+\omega_0)$ (11.31)

Applying the modulation theorem to the Gaussian transform pair above yields

$\displaystyle \zbox {e^{-pt^2} e^{-j\omega_0 t} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{(\omega+\omega_0)^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0.}$ (11.32)

Thus, we frequency-shift a Gaussian chirp in the same way we frequency-shift any signal--by complex modulation (multiplication by a complex sinusoid at the shift-frequency).

Identifying Chirp Rate

Consider again the Fourier transform of a complex Gaussian in (10.27):

$\displaystyle e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} \, e^{-\frac{\omega^2}{4p}}\;\isdef \;F(\omega)$ (11.33)

Setting $ p=\alpha - j\beta$ gives

$\displaystyle e^{-\alpha t^2} e^{j\beta t^2} \;\longleftrightarrow\; \sqrt{\frac{\pi}{\alpha-j\beta}} \, e^{-\frac{\alpha}{4(\alpha^2+\beta^2)}\omega^2} e^{-j\frac{\beta}{4(\alpha^2+\beta^2)}\omega^2}.$ (11.34)

The log magnitude Fourier transform is given by

$\displaystyle \ln\left\vert F(\omega)\right\vert \eqsp \hbox{constant} -\frac{\alpha}{4(\alpha^2+\beta^2)}\omega^2$ (11.35)

and the phase is

$\displaystyle \angle F(\omega) \eqsp \hbox{constant} -\frac{\beta}{4(\alpha^2+\beta^2)}\omega^2.$ (11.36)

Note that both log-magnitude and (unwrapped) phase are parabolas in $ \omega$ .

In practice, it is simple to estimate the curvature at a spectral peak using parabolic interpolation:

c_m &\isdef & \frac{d^2}{d\omega^2} \ln\vert F(\omega)\vert \eqsp - \frac{\alpha}{2(\alpha^2+\beta^2)}\\ [5pt]
c_p &\isdef & \frac{d^2}{d\omega^2} \angle F(\omega) \eqsp - \frac{\beta}{2(\alpha^2+\beta^2)}

We can write

\zbox {\frac{d^2}{d\omega^2} \ln F(\omega) \eqsp c_m + jc_p \eqsp - \frac{\alpha}{2(\alpha^2+\beta^2)} - j\frac{\beta}{2(\alpha^2+\beta^2)}.}

Note that the window ``amplitude-rate'' $ \alpha $ is always positive. The ``chirp rate'' $ \beta $ may be positive (increasing frequency) or negative (downgoing chirps). For purposes of chirp-rate estimation, there is no need to find the true spectral peak because the curvature is the same for all $ \omega$ . However, curvature estimates are generally more reliable near spectral peaks, where the signal-to-noise ratio is typically maximum. In practice, we can form an estimate of $ \alpha $ from the known FFT analysis window (typically ``close to Gaussian'').

Chirplet Frequency-Rate Estimation

The chirp rate $ \beta $ may be estimated from the relation $ \beta =
\alpha c_m/c_p$ as follows:

  • Let $ {\hat \alpha}$ denote the measured (or known) curvature at the midpoint of the analysis window $ w(n)$ .

  • Let $ [c_m]$ and $ [c_p]$ denote weighted averages of the measured curvatures $ c_m(k)$ and $ c_p(k)$ along the log-magnitude and phase of a spectral peak, respectively.

  • Then the chirp-rate $ \beta $ estimate may be estimated from the spectral peak by

    $\displaystyle \zbox {{\hat \beta}\isdefs {\hat \alpha}\frac{[c_p]}{[c_m]}}

Simulation Results

Figure 10.24: Real Gaussian-windowed chirp (time domain).

Figure 10.24 shows the waveform of a Gaussian-windowed chirp (``chirplet'') generated by the following matlab code:

fs = 8000;
x = chirp([0:1/fs:0.1],1000,1,2000);
M = length(x);
w = exp(-n.*n./(2*sigma.*sigma));
xw = w(:) .* x(:);

Figure 10.25 shows the same chirplet in a time-frequency plot. Figure 10.26 shows the spectrum of the example chirplet. Note the parabolic fits to dB magnitude and unwrapped phase. We see that phase modeling is most accurate where magnitude is substantial. If the signal were not truncated in the time domain, the parabolic fits would be perfect. Figure 10.27 shows the spectrum of a Gaussian-windowed chirp in which frequency decreases from 1 kHz to 500 Hz. Note how the curvature of the phase at the peak has changed sign.

Figure 10.25: Real Gaussian-windowed chirp (spectrogram).

Figure 10.26: Gaussian-windowed chirp (frequency domain).

Figure 10.27: Downgoing chirp.

Figure 10.28: Short chirp--time waveform.
Figure 10.29: Short chirp--spectrum.

FFT Filter Banks

This section, based on [265], describes how to make practical audio filter banks using the Short Time Fourier Transform (STFT). This material bridges the filter-bank interpretation of the STFT in Chapter 9 and the discussion of multirate filter banks in Chapter 11. The filter banks of this section are based entirely on the STFT, with consideration of the basic Fourier theorems introduced in Chapter 2. In Chapter 11, filter banks such as used in audio compression are addressed. However, when not doing compression, i.e., when each channel of the filter bank does not have to be critically sampled, the methods of this section may suffice in many applications.

Audio Filter Banks

It is well known that the frequency resolution of human hearing decreases with frequency [71,276]. As a result, any ``auditory filter bank'' must be a nonuniform filter bank in which the channel bandwidths increase with frequency over most of the spectrum. A classic approximate example is the third-octave filter bank.11.14 A simpler (cruder) approximation is the octave filter bank,11.15 also called a dyadic filter bank when implemented using a binary tree structure [287]. Both are examples of constant-Q filter banks [29,30,244], in which the bandwidth of each filter-bank channel is proportional to center frequency [263]. Approximate auditory filter banks, such as constant-Q filter banks, have extensive applications in computer music, audio engineering, and basic hearing research.

If the output signals from all channels of a constant-Q filter bank are all sampled at a particular time, we obtain what may be called a constant-Q transform [29]. A constant-Q transform can be efficiently implemented by smoothing the output of a Fast Fourier Transform (FFT) [30]. More generally, a multiresolution spectrogram can be implemented by combining FFTs of different lengths and advancing the FFTs forward through time (§7.3). Such nonuniform filter banks can also be implemented based on the Goetzel algorithm [33].

While the topic of filter banks is well developed in the literature, including constant-Q, nonuniform FFT-based, and wavelet filter banks, the simple, robust methods presented in this section appear to be new [265]. In particular, classic nonuniform FFT filter banks as described in [226] have not offered the perfect reconstruction property [287] in which the filter-bank sum yields the input signal exactly (to within a delay and/or scale factor) when the filter-band signals are not modified. The voluminous literature on perfect-reconstruction filter banks [287] addresses nonuniform filter banks, such as dyadic filter banks designed based on pseudo quadrature mirror filter designs, but simpler STFT methods do not yet appear to be incorporated. In the cosine-modulated filter-bank domain, subband DCTs have been used in a related way [302], but apparently without consideration for the possibility of a common time domain across multiple channels.11.16

This section can be viewed as an extension of [30] to the FFT filter-bank case. Alternatively, it may be viewed as a novel method for nonuniform FIR filter-bank design and implementation, based on STFT methodology, with arbitrarily accurate reconstruction and controlled aliasing in the downsampled case. While we consider only auditory (approximately constant-Q) filter banks, the method works equally well for arbitrary nonuniform spectral partitions and overlap-add decompositions in the frequency domain.

Basic Idea

The basic idea is to partition FFT bins into the desired nonuniform bands, and perform smaller inverse FFTs on each subband to synthesize downsampled time-domain signals in each band. A simple example for a length 8 FFT octave filter bank is shown in Fig.10.30. We'll next extend this example to a practically useful level via a series of tutorially oriented matlab examples.

Figure: FFT implementation of one frame of the simple octave filter bank shown in Fig.10.31. Successive frames are non-overlapping (rectangular window advances its full length each frame).

Summing STFT Bins

In the Short-Time Fourier Transform, which implements a uniform FIR filter bank (Chapter 9), each FFT bin can be regarded as one sample of the filter-bank output in one channel. It is elementary that summing adjacent filter-bank signals sums the corresponding pass-bands to create a wider pass-band. Summing adjacent FFT bins in the STFT, therefore, synthesizes one sample from a wider pass-band implemented using an FFT. This is essentially how a constant-Q transform is created from an FFT in [30] (using a different frequency-weighting, or ``smoothing kernel''). However, when making a filter bank, as opposed to only a transform used for spectrographic purposes, we must be able to step the FFT through time and compute properly sampled time-domain filter-bank signals.

The wider pass-band created by adjacent-channel summing requires a higher sampling rate in the time domain to avoid aliasing. As a result, the maximum STFT ``hop size'' is limited by the widest pass-band in the filter bank. For audio filter banks, low-frequency channels have narrow bandwidths, while high-frequency channels are wider, thereby forcing a smaller hop size for the STFT. This means that the low-frequency channels are heavily oversampled when the high-frequency channels are merely adequately sampled (in time) [30,88]. In an octave filter-bank, for example, the top octave, occupying the entire upper half of the spectrum, requires a time-domain step-size of no more than two samples, if aliasing of the band is to be avoided. Each octave down is then oversampled (in time) by an additional factor of 2.

Inverse Transforming STFT Bin Groups

The solution proposed in [265] is to compute multiple time samples for each high-frequency channel, so that one hop of the FFT produces all needed samples in each band. That way all channels can use the same hop-size without redundancy. If the narrowest band gets one sample per hop, then a band $ N$ times as wide must produce $ N$ samples per hop.

A fast way to compute multiple time samples from the frequency-samples (FFT bins) of a given band is an inverse FFT, as shown in Fig.10.30. In matlab notation, let X(1:N) denote the FFT (length N) of the current frame of data in the STFT, and denote the lower and upper spectral samples for band k by lsn(k) and hsn(k), respectively. Then we may specify the matlab computation of the full-rate time-domain signal corresponding to band k as follows:

BandK = X(lsn(k):hsn(k));
z1 = zeros(1,lsn(k)-1);
z2 = zeros(1,N-hsn(k));
BandKzp = [z1, BandK, z2];     % (1)
x(k,:) = ifft(BandKzp);
where x(k,:) denotes the output signal vector (length N) for the kth filter-bank channel for the current time-domain STFT window.

Let $ \texttt{Nk}= \texttt{hsn(k)} - \texttt{lsn(k)} + 1$ denote the number of FFT bins in band $ k$ . When $ \texttt{Nk}$ is a power of 2, we can apply an inverse FFT only to the nonzero samples in the band:

xd{k} = ifft(BandK);
where xd{k} now denotes a cell array for holding the ``downsampled'' signal vectors (since the downsampling factor is typically different for different bands).

We may relate x(k,:) and xd{k} by noting that, when Lk = N/Nk is an integer, we have that the relation

BandK == alias(BandKzp,Lk)     % (2)
is true, for each element, where alias(x,K) denotes aliasing of K equal partitions of the vector x2.3.12):11.17
function y = alias(x,L)
  Nx = length(x);
  Np = Nx/L; % aliasing-partition length
  x = x(:);  % ensure column vector
  y = zeros(Np,1);
  for i=1:L
    y = y + x(1+(i-1)*Np : i*Np);

By the aliasing theorem (downsampling theorem) for Discrete Fourier Transforms (DFTs) (§2.3.12), the relation (2) in the frequency domain corresponds to

xd{k} == Lk * x(k,1:Lk:end)
in the time domain, i.e., xd{k} is obtained from x(k,:) by means of downsampling by the factor Lk. This produces N/Lk == Nk samples. That is, for a band that is Nk bins wide, we obtain Nk time-domain samples for each STFT frame, when critically sampled. (At the full rate, we obtain N samples from each channel for each frame.)

We see that taking an inverse FFT of only the bins in a given channel computes the critically downsampled signal for that channel. Downsampling factors between 1 and Lk can be implemented by choosing a zero-padding factor for the band somewhere between 1 and Lk samples.

Note that this filter bank has the perfect reconstruction (PR) property [287]. That is, the original input signal can be exactly reconstructed (to within a delay and possible scale factor) from the channel signals. The PR property follows immediately from the exact invertibility of the discrete Fourier transform. Specifically, we can recover the original signal frame by taking an FFT of each channel-signal frame and abutting the spectral bins so computed to form the original frame spectrum. Of course, the underlying STFT (uniform filter bank) must be PR as well, as it routinely is in practice.

Improving the Channel Filters

Recall that each FFT bin can be viewed as a sample from a bandpass filter whose frequency response is a frequency-shift of the FFT-window Fourier transform9.3). Therefore, the frequency response of a channel filter obtained by summing Nk adjacent FFT bins is given by the sum of Nk window transforms, one for each FFT bin in the sum. As a result, the stop-band of the channel-filter frequency response is a sum of Nk window side lobes, and by controlling window side-lobe level, we may control the stop-band gain of the channel filters.

The transition width from pass-band to stop-band, on the other hand, is given by the main-lobe width of the window transform (§5.5.1). In the previous subsection, by zero-padding the band (line (1) above), we implicitly assumed a transition width of one bin. Only the length N rectangular window can be reasonably said to have a one-bin transition from pass-band to stop-band. Since the first side lobe of a rectangular window transform is only about 13 dB below the main lobe, the rectangular window gives poor stop-band performance, as illustrated in Fig.10.33. Moreover, we often need FFT data windows to be shorter than the FFT size N (i.e., we often need zero-padding in the time domain) so that the frame spectrum will be oversampled, enabling various spectral processing such as linear filtering (Chapter 8).

One might wonder how the length N rectangular window can be all that bad when it gives the perfect reconstruction property, as demonstrated in the previous subsection. The answer is that there is a lot of aliasing in the channel signals, when downsampled, but this aliasing is exactly canceled in the reconstruction, provided the channel signals were not modified in any way.

Going back to §10.7.3, we need to replace the zero-padded band (1) by a proper filtering operation in the frequency domain (a ``spectral window''):

BandK2 = Hk .* X;
x(k,:) = ifft(BandK2); % full rate
BandK2a = alias(BandK2,Nk);
xd{k} = ifft(BandK2a); % crit samp
where the channel filter frequency response Hk may be prepared in advance as the appropriate weighted sum of FFT-window transforms:
Hideal = [z1,ones(1,Nk),z2];
Hk = cconvr(W,Hideal); % circ. conv.
where z1 and z2 are the same zero vectors defined in §10.7.3, and cconvr(W,H) denotes the circular convolution of two real vectors having the same length [264]:
function [Y] = cconvr(W,X)
  wc=fft(W);  xc=fft(X);
  yc = wc .* xc;
  Y = real(ifft(yc));
Note that in this more practical case, the perfect reconstruction property no longer holds, since the operation
BandK2a = alias(Hk .* X, Nk);
is not exactly invertible in general.11.18However, we may approach perfect reconstruction arbitrarily closely by only aliasing stop-band intervals onto the pass-band, and by increasing the stop-band attenuation of Hk as desired. In contrast to the PR case, we do not rely on aliasing cancellation, which is valuable when the channel signals are to be modified.

The band filters Hk can be said to have been designed by the window method for FIR filter design [224]. (See functions fir1 and fir2 in Octave and/or the Matlab Signal Processing Toolbox.)

Fast Octave Filter Banks

Let's now apply the above technique to the design of an octave filter bank.11.19 At first sight, this appears to be a natural fit, because it is immediately easy to partition a power of 2 (typical for the FFT size N) into octaves, each having a width in bins that is a power of 2. For example, when N == 8, we have the following stack of frequency-response vectors for the rectangular-window, no-zero-padding, complex-signal case:

H = [ ...
   0   0   0   0   1   1   1   1 ; ...
   0   0   1   1   0   0   0   0 ; ...
   0   1   0   0   0   0   0   0 ; ...
   1   0   0   0   0   0   0   0 ];
Figure 10.31 depicts the resulting filter bank schematically in the frequency domain.11.20 Thus, H(1,:) is the frequency-response for the top (first) octave, H(2,:) is the frequency-response for the next-to-top (second) octave, H(3,:) is the next octave down, and H(4,:) is the ``remainder'' of the spectrum. (In every octave filter bank, there is a final low-frequency band.

Figure 10.31: Simple octave filter bank for complex signals.

A schematic of the filter-bank implementation using FFTs is shown in Fig.10.30. The division by N called for by ifft(N) is often spread out among the ifft butterfly stages as divisions by 2 (one-bit right-shifts in fixed-point arithmetic) after each of them. For conservation of dynamic range, half of the right-shifts can be spread out among every other forward-FFT butterfly stage [28] as well.

The simple spectral partitioning of Fig.10.31 is appropriate for complex signals--those for which the spectrum is regarded as spanning 0 Hz to the sampling rate. For real signals, we need a spectral partition more like that in Fig.10.32.

Figure 10.32: Simple octave filter bank for real signals.

Unfortunately, the number of spectral samples in Fig.10.32 is 14--not a power of 2. (The previous length 8 complex case maps to length 14 real case because the dc and Nyquist-limit samples do not have complex-conjugate counterparts.) Discarding the sample at the Nyquist limit--number 8 in Fig.10.32--does not help, since that gives 13 samples--still not a power of 2. In summary, there is no obvious way to octave-partition the spectral samples of a real signal while maintaining the power-of-2 condition for each band and symmetrically partitioning positive and negative frequencies.

A real signal can of course be converted to its corresponding ``analytic signal'' by filtering out the negative-frequency components. This is normally done by designing a Hilbert transform filter [133,224]. However, such filters are large-order FIR filters, exactly like we are trying to design! If we design our filter bank properly, we can use it to zero the negative-frequency components.

Spectral Rotation of Real Signals

Note that if we rotate the spectrum of a real signal by half a bin, we obtain $ N/2$ positive-frequency samples and $ N/2$ negative-frequency samples, with no sample at dc or at the Nyquist limit. This is typically desirable for audio signals because dc is inaudible and the Nyquist limit is a degenerate point of the spectrum that, for example, cannot have a phase other than 0 or $ \pi$ . If $ N$ is a power of 2, then so is $ N/2$ , and the octave-band partitioning of the previous subsection can be applied separately to each half of the spectrum:

N = 32;  x = randn(1,N); % Specific example
LN = round(log2(N)); % number of octave bands
shifter = exp(-j*pi*[0:N-1]/N); % half-bin
xs = x .* shifter; % apply spectral shift
X = fft(xs,N); % project xs onto rotated basis
XP = X(1:N/2); % positive-frequency components
XN = X(N:-1:N/2+1); % neg.-frequency components
YP = dcells(XP); % partition to octave bands
YN = dcells(XN); % ditto for neg. frequencies
YPe = dcells2spec(YP); % unpack "dyadic cells"
YNe = dcells2spec(YN); % unpack neg. freqs
YNeflr = fliplr(YNe);  % undo former flip
ys = ifft([YPe,YNeflr],N,2);
y = real(ones(LN,1)*conj(shifter) .* ys);
% = octave filter-bank signals (real)
yr = sum(y); % filter-bank sum (should equal x)
yrerr = x - yr;
  'Total filter-bank sum L2 error = %0.2f %%',...
In the above listing, the function dcells11.21simply forms a cell array in matlab containing the spectral partition (``dyadic cells''). The function dcells2spec11.22is the inverse of dcells, taking a spectral partition and unpacking it to produce a usual spectrum vector.

Improving the Octave Band Filters

To see the true filter-bank frequency response corresponding to Fig.10.31, we may feed an impulse to the filter bank and take a long FFT (for zero padding) of the channel signals to produce the interpolated response shown in Fig.10.33.

Figure: Channel-frequency-response overlay for the octave filter bank shown in Fig.10.31.

The horizontal line along 0 dB in Fig.10.33 was obtained by summing the channel responses, indicating that it is a perfect-reconstruction filter-bank, as expected. However, the stop-band performance of the channels is quite poor, being comparable to the side-lobes of a rectangular window transform (an aliased sinc function). In fact, the stop-band is identical to the rectangular-window side-lobes for the two lowest bands. Notice that the original eight samples of Fig.10.31 still lie along the 0-dB line, and there are still zeros in each channel response beneath the unity-gain point of every other channel's response, so Fig.10.31 can be obtained by sampling Fig.10.33 at those eight points. However, the interpolated response shows that the filter bank is quite poor by audio standards.

Figure 10.34: Channel-frequency-response overlay for a complex-signal octave filter bank, designed using a length 127 Dolph-Chebyshev window (80 dB stop-band attenuation) and length 256 FFT.

Figure 10.34 shows an octave filter bank (again for complex signals) that is better designed for audio usage. Instead of basing the channel filter prototype on the rectangular window, a Dolph-Chebyshev window was used (using the matlab function call chebwin(127,80)). The FFT size is about twice the filter length, thereby allowing for a data frame of equal length (to the filter) without incurring time aliasing, in the usual way for STFTs (Chapter 8). The data window is chosen to overlap-add to a constant, as typical in STFTs, so its choice does not affect the quality of the filter-bank output channel signals. We therefore may continue to use a rectangular data window that advances by its full length each frame. Choosing an odd filter length facilitates zero-phase offline STFT processing, in which the middle FIR sample is treated as occurring at time zero, so that there is no delay in any filter-bank channel [264].

The filter bank is PR in the full-rate case because the underlying STFT is PR, in the absence of modifications, and because the channel filter-bank is constant-overlap-add in the frequency domain (again in the absence of modifications) according to STFT theory (Chapter 8).

Aliasing on Downsampling

While the filter bank of Fig.10.34 gives good stop-band rejection, there is still a significant amount of aliasing when the bands are critically sampled. This happens because the transition bands are aliased about their midpoints. This can be seen in Fig.10.34 by noting that aliasing ``folding frequencies'' lie at the crossover point between each pair of bands. An overlay of the spectra of the downsampled filter-bank outputs, for an impulse input, is shown in Fig.10.35.

Figure: Same as Fig.10.34 obtained by critically downsampling each channel signal, zero-padding, and performing an FFT. All the observable stop-band error happens to cancel out in the filter-bank sum because the input signal is an impulse, in which case the reconstruction remains exact.

Figure 10.36 shows the aliased spectral signal bands (prior to inverse STFT) for a step input (same filter bank). (This type of plot looks ideal for an impulse input signal because the spectrum is constant, so the aliased bands are also constant.) Note the large slice of dc energy that has aliased from near the sampling rate to near half the sampling rate in the top octave band. The signal and error spectra are shown overlaid in Fig.10.37. In this case, the aliasing causes significant error in the reconstruction.

Figure: Same filter bank as in Fig.10.35 but driven by a step input.

Figure: Signal spectrum (an impulse, since the time signal is a step) and error spectrum for the case of Fig.10.36. Note the large error near half the sampling rate.

Restricting Aliasing to Stop-Bands

To eliminate the relatively heavy transition-band aliasing (when critically sampling the channel signals), we can define overlapping bands such that each band encompasses the transition bands on either side. However, unless a full $ 2\times\null$ oversampling is provided for each band (which is one easy solution), the bandwidth (in bins) is no longer a power of two, thereby thwarting use of radix-2 inverse-FFTs to compute the time-domain band signals.

To keep the channel bandwidths at powers of two while restricting aliasing to stop-band energy, the IFFT bands can be widened to include transition bands on either side. That is, the desired pass-band plus the two transition bands span a power-of-two bins. This results in overlapping channel IFFTs. Figure 10.38 shows how the example of Fig.10.34 is modified by this strategy.

Figure 10.38: Channel-frequency-response overlay for an octave filter bank designed using a length 127 Dolph-Chebyshev window (80 dB SBA) and length 256 FFT size.

The basic principle of filter-bank band allocation is to enclose each filter band plus its transition bands within a wider band that is a power-of-two bins wide.11.23 The band should roll off to reach its stop-band at the edge of the wider encompassing band. It is fine to have extra space in the wider band, and this may be filled with a continuation of the enclosed band's stop-band response (or some tapering of it--since we assume stop-band energy is negligible, the difference should be inconsequential). The desired bands may overlap each other by any amount, and may have any desired shape. The encompassing bands then overlap further to reach the next power of two (in width) for each overlapping extended band. (See the gammatone and gammachirp filter banks for examples of heavily overlapping bands in an audio filter bank [111].)

In this approach, pass-bands of arbitrary width are embedded in overlapping IFFT bands that are a power-of-2 wide. As a result of this flexibility, the frequency-rotation trick of §10.7.7 is no longer needed for real filter banks. Instead, we simply allocate any desired bands between dc and half the sampling rate, and then conjugate-symmetry dictates the rest. In addition to a left-over ``dc-Nyquist'' band, there is a similar residual ``Nyquist-limit'' band (a typically negligible band about half the sampling rate). In other words, since the pass-bands may be any width and the encompassing IFFT bands may overlap by any amount, they do not have to ``pack'' conveniently as power-of-two blocks.

The minimum channel bandwidth is defined as two transition bands plus one bin (i.e., the minimum pass-band width is zero, corresponding to one bin, or one spectral sample). For the Dolph-Chebyshev window, the transition bandwidth is known in closed form [155]. In our examples, we have a length 127 window with 80 dB stop-band attenuation in the lowpass prototype [chebwin(127,80)], corresponding to a transition width of 6.35 bins in a length 256 FFT, which was rounded up to 7 bins in software for simplicity of band allocation. Therefore, our minimum channel bandwidth is 15 bins (two transition bands plus one sample for the band center). The next highest power of two is 16, so that is our minimum encompassing IFFT length for any band.

The dc and Nyquist channels are combined into a single channel containing the left-over residual filter-bank response consisting of a low transition down from dc and a high-frequency transition up to the sampling rate (in the complex-signal case). When N is sufficiently large so that these bands contain no audible energy, they may be discarded. We include them in all examples here so as to preserve the (near) perfect reconstruction property of the filter bank. Thus, the 7-bin dc channel is combined with the 7-bin Nyquist channel to form a single 16-bin encompassing residual band that may be discarded in many audio applications (when the initial FFT size is sufficiently large for the sampling rate used).

In the example of Fig.10.38, the initial FFT size is 256, and the channel bandwidths (pass-bands only, excluding transitions), from top to bottom, are 121, 64, 32, 16, and 8 bins. The top band is reduced by 7 bins to leave a transition band to the sampling rate. Similarly, the lowest band lies above a transition band consisting of bins 0-6. The encompassing IFFTs (containing transitions) are lengths 256, 128, 64, 32, 32, for the interior bands, and a length 32 IFFT handles the dc and Nyquist bands (which are combined into a single 14-bin band about dc, which occupies 28 bins when the transition bands are appended). Letting [lo,hi] denote a band by its lower and upper bin limit, the non-overlapping adjacent pass-band edges in ``spectral samples''11.24 of the interior bands are [8, 15], [16, 31], [32, 63], [64, 127], and [128, 248]; the overlapping encompassing IFFT band edges are then [1, 32], [9, 40], [25, 88], [57, 184], [1, 256], i.e., they each contain a pass-band and two transition bands, and have a power-of-2 length. The downsampling factor for each channel can be computed as the initial FFT size (256) divided by the IFFT size ($ 32$ , $ 32$ , $ 64$ , $ 128$ , or $ 256$ ) for the channel.

Figure 10.39 shows the counterpart of Fig.10.35 for this example. In this case, the aliased signal energy comes only from channel-filter stop-bands. For narrow bands, the aliasing is suppressed by at least 80 dB (the side-lobe level of the chosen Dolph-Chebyshev window transform). For bands significantly wider than one bin (the minimum bandwidth in this example is the dc-Nyquist band at 14 bins), the stop-band consists of a sum of shifts of the window-transform side lobes, and these are found to be more than 80 dB down due to cancellation (more than 90 dB down in most bands of this example).

Figure: Same example as in Fig.10.38 but looking at the effects of aliasing due to channel-signal downsampling. Compare to Fig.10.35.

Tightening the IFFTs

In this example the top band is not downsampled at all, and the interior bands are oversampled by approximately 2. This is because the desired pass-band widths started out at a power of 2, so that the addition of transition bands forced the next higher power of 2 for the IFFT size. Narrowing the width of the top band from 121 bins to $ 128-2\cdot 7 = 114$ bins would enable use of a length 128 IFFT for the top band, and similarly for the lower bands. In other words, when the desired spectral partition is that of an ideal octave filter bank, as sketched in Fig.10.31, narrowing each octave-band by twice the transition width of the lowpass prototype filter (and ``covering down'' to keep them adjacent) will produce a relatively ``tight'' FFT filterbank design in which the IFFT sizes remain the same length as in the heavily aliased case discussed above (Fig.10.35). When applied to the octave filter bank, the pass-bands become a little narrower than one octave. We may call this a quasi octave filter bank.

Real Filter Bank Example

Finally, Fig.10.40 shows the appearance of the octave filter bank for real signals. In this case, bands are constructed between dc and half the sampling rate, and conjugate symmetry is used to automatically construct the desired bands between half the sampling rate and the sampling rate. The band allocation algorithm therefore needs no modification for the real-signal case.

Figure: Channel-frequency-response overlay for a real-signal octave filter bank, designed using a length 127 Dolph-Chebyshev window and length 256 FFT size (no downsampling). Compare the complex-signal filter bank in Fig.10.38.

Optimal Band Filters

In the filter-bank literature, one class of filter banks is called ``cosine modulated'' filter banks. DFT filter banks are similar. The lowpass-filter prototype in such filter banks can be used in place of the Dolph-Chebyshev window used here. Therefore, any result on optimal design of cosine-modulated filter banks can be adapted to this context. See, for example, [253,302]. Note, however, that in principle a separate optimization is needed for each different channel bandwidth. An optimal lowpass prototype only optimizes channels having a one-bin pass-band, since the prototype frequency-response is merely shifted in frequency (cosine-modulated in time) to create the channel frequency response. Wider channels are made by summing such channel responses, which alters the stop-bands.

In practice, the Dolph-Chebyshev window, used in the examples of this section, typically yields a filter bank magnitude frequency response that is optimal in the Chebyshev sense, when at least one channel is minimum width, because

  1. there can be only one lowpass prototype filter in any modulated filter bank (such as the DFT filter bank),
  2. the prototype itself is the optimal Chebyshev lowpass filter of minimum bandwidth, and
  3. summing shifted copies of the prototype frequency response (to synthesize a wider pass-band) generally improves the stop-band rejection over that of the prototype, thereby meeting the Chebyshev optimality requirement for the filter-bank as a whole (keeping below the worst-case deviation of the prototype).
All channel bands, whatever their width, are constructed by some linear combination of shifted copies of the lowpass prototype frequency response. The Dolph-Chebyshev window is precisely optimal (in the Chebyshev sense) for any pass-band that is one bin wide. Summing shifts of the window transform to synthesize wider bands has been observed to invariably improve the stop-band rejection significantly. The examples shown above illustrate the margin beyond 80 dB stop-band rejection achieved for the octave filter bank.

The Dolph-Chebyshev window has faint impulsive endpoints on the order of the side-lobe level (about 50 dB down in the 80-dB-SBA examples above), and in some applications, this could be considered an undesirable time-domain characteristic. To eliminate them, an optimal Chebyshev window may be designed by means of linear programming with a time-domain monotonicity constraint (§3.13). Alternatively, of course, other windows can be used, such as the Kaiser, or three-term Blackman-Harris window, to name just two (Chapter 3).

FFT Filter-Bank Summary and Fourier Duality with OLA

We have seen in this section how to make general FIR filter banks implemented using the STFT. This approach can be seen as the Fourier dual of time-domain overlap-add (OLA) STFT processing (Chapter 8).

To summarize in terms of OLA duality, we perform any desired overlap-add decomposition of the spectrum of an STFT frame in the frequency domain. Each STFT spectrum is multiplied, in the frequency domain, by a time-limited spectral window (i.e., an FIR-filter frequency-response). To obtain FFT efficiency for arbitrary spectral bands, we extend the width each spectral band (either zero-padding or ``stop-band-padding'') to the next power of 2; this is reminiscent of the OLA step of zero-padding a time-domain frame to the next power of 2. An inverse FFT is performed on the extended band, either in its natural location, or translated to baseband. For the channel signals to have a common sampling rate, each band is embedded in the same spectral domain from dc to half the common sampling rate (or the full sampling rate in the complex case), and the same-length IFFT is used for each spectral band. On the other hand, when the channel signals are downsampled to their natural bandwidths (rounded up to the next power of 2 in FFT bins), the aliasing distortion is fully controlled by the stop-band rejection of the band-filter used.

Pointers to Sound Examples

See the CCRMA Music 421 website for pointers to demonstrations and sound examples related to applications of the STFT. A PDF presentation including pointers to a good number of sound examples can be found at the author's website.11.25

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