## 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:

*cepstral windowing*to lowpass-filter the log-magnitude spectrum (a ``nonparametric method'')- 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, denotes the th spectral frame of the STFT (§7.1), and denotes the spectral envelope of .

### Cepstral Windowing

The spectral envelope obtained by *cepstral windowing* is defined
as

(11.2) |

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

(11.3) |

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

The log-magnitude spectrum of
is thus *lowpass filtered*
(the real cepstrum of
is ``liftered'') to obtain a smooth spectral
envelope. For periodic signals,
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 based on past samples:

We call the

*order*of the linear predictor, and the

*prediction coefficients*. The

*prediction error*(or ``

*innovations sequence*'' [114]) is denoted in (10.4), and it represents all new information entering the signal at time . Because the information is new, is ``unpredictable.'' The predictable component of contains no new information.

Taking the *z* transform of (10.4) yields

(11.5) |

where . In signal modeling by linear prediction, we are given the signal but not the prediction coefficients . We must therefore

*estimate*them. Let denote the polynomial with estimated prediction coefficients . Then we have

(11.6) |

where denotes the estimated prediction-error

*z*transform. By minimizing , we define a minimum-least-squares estimate . In other words, the linear prediction coefficients are defined as those which minimize the sum of squared prediction errors

(11.7) |

over some range of , 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 [11,157,162]. That is, the optimal 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

(11.8) |

where denotes the power spectral density of (defined in Chapter 6), and denotes the variance of the (white-noise) prediction error . Thus, the

*spectral magnitude envelope*may be defined as

EnvelopeLPC | (11.9) |

#### Linear Prediction is Peak Sensitive

By Rayleigh's energy theorem,
(as
shown in §2.3.8). Therefore,

From this ``ratio error'' expression in the frequency domain, we can see that contributions to the error are smallest when . Therefore, LP tends to

*overestimate peaks*. LP cannot make arbitrarily large because is constrained to be monic and minimum-phase. It can be shown that the log-magnitude frequency response of every minimum-phase monic polynomial 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
is always minimum phase (*i.e.*,
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
are computed from the
Bartlett-window-biased *autocorrelation function*
(Chapter 6):

where denotes the th data frame from the signal . To obtain the th-order linear predictor coefficients , we solve the following system of linear

*normal equations*(also called

*Yule-Walker*or

*Wiener-Hopf*equations):

In matlab syntax, the solution is given by `` '', where , and . Since the covariance matrix is symmetric and

*Toeplitz*by construction,

^{11.4}an solution exists using the

*Durbin recursion*.

^{11.5}

If the rank of the
autocorrelation matrix
is
, then the solution to (10.12)
is *unique*, and
this solution is always *minimum phase* [162] (*i.e.*, all roots of
are inside the unit circle in the
plane [263], so
that
is always a *stable* all-pole filter). In
practice, the rank of
is
(with probability 1) whenever
includes a noise component. In the noiseless case, if
is a sum
of sinusoids, each (real) sinusoid at distinct frequency
adds 2 to the rank. A dc component, or a component at half the
sampling rate, adds 1 to the rank of
.

The choice of time window for forming a short-time sample
autocorrelation and its weighting also affect the rank of
. 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 stays within the frame--a so-called ``unwindowed'' method. The autocorrelation method sums over the whole frame and replaces by zero when 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
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,
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
should be significantly
less than twice the number of sinusoids to inhibit modeling the
sinusoids as spectral-envelope peaks. For filtered-white-noise,
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 th spectral frame, computed by linear prediction, is given by

(11.13) |

where is computed from the solution of the Toeplitz normal equations, and is the estimated rms level of the prediction error in the th frame.

(11.14) |

can be driven by unit-variance white noise to produce a filtered-white-noise signal having spectral envelope . We may regard (no absolute value) as the frequency response of the filter in a

*source-filter decomposition*of the signal , where the source is white noise.

It bears repeating that is zero mean when is monic and minimum phase (all zeros inside the unit circle). This means, for example, that can be simply estimated as the mean of the log spectral magnitude .

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); end; 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 10.2 shows the Hamming-windowed synthesized vowel
`speech`, and its spectrum overlaid with the true formant
envelope.

#### 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.

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.

#### 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)'; end % 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); end % 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;

#### 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
(
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`''.

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Cross-Synthesis