Spectrum Analysis Windows

In spectrum analysis of naturally occurring audio signals, we nearly always analyze a short segment of a signal, rather than the whole signal. This is the case for a variety of reasons. Perhaps most fundamentally, the ear similarly Fourier analyzes only a short segment of audio signals at a time (on the order of 10-20 ms worth). Therefore, to perform a spectrum analysis having time- and frequency-resolution comparable to human hearing, we must limit the time-window accordingly. We will see that the proper way to extract a ``short time segment'' of length $ N$ from a longer signal is to multiply it by a window function such as the Hann window:

$\displaystyle w(n) = \cos^2\left(\frac{n}{N}\pi\right), \quad n = -\frac{N-1}{2}, \ldots, -1,0,1,\ldots,\frac{N-1}{2}$ (4.1)

We will see that the main benefit of choosing a good Fourier analysis window function is minimization of side lobes, which cause ``cross-talk'' in the estimated spectrum from one frequency to another.

The study of spectrum-analysis windows serves other purposes as well. Most immediately, it provides an array of useful window types which are best for different situations. Second, by studying windows and their Fourier transforms, we build up our knowledge of Fourier dualities in general. Finally, the defining criteria for different window types often involve interesting and useful analytical techniques.

In this chapter, we begin with a summary of the rectangular window, followed by a variety of additional window types, including the generalized Hamming and Blackman-Harris families (sums of cosines), Bartlett (triangular), Poisson (exponential), Kaiser (Bessel), Dolph-Chebyshev, Gaussian, and other window types.

The Rectangular Window

The (zero-centered) rectangular window may be defined by

$\displaystyle w_R(n) \isdef \left\{ \begin{array}{ll} 1, & -\frac{M-1}{2} \leq n \leq \frac{M-1}{2} \\ 0, & \mbox{otherwise} \\ \end{array} \right.$ (4.2)

where $ M$ is the window length in samples (assumed odd for now). A plot of the rectangular window appears in Fig.3.1 for length $ M=21$ . It is sometimes convenient to define windows so that their dc gain is 1, in which case we would multiply the definition above by $ 1/M$ .

Figure 3.1: The rectangular window.
\includegraphics[width=3.5in]{eps/rectWindow}

To see what happens in the frequency domain, we need to look at the DTFT of the window:

\begin{eqnarray*}
W_R(\omega )
& = & \hbox{\sc DTFT}_\omega(w_R) \isdef \sum_{n=-\infty}^\infty
w_R(n)e^{-j\omega n}, \quad \omega\in[-\pi,\pi) \\
& = & \sum_{n=-\frac{M-1}{2}}^{\frac{M-1}{2}} e^{-j \omega n}
= \frac{e^{j \omega \frac{M-1}{2}} - e^{-j \omega \frac{M+1}{2}} }{1 - e^{-j \omega }}
\end{eqnarray*}

where the last line was derived using the closed form of a geometric series:

$\displaystyle \sum_{n=L}^U r^n = \frac{ r^L - r^{U+1}}{1-r}$ (4.3)

We can factor out linear phase terms from the numerator and denominator of the above expression to get
$\displaystyle W_R(\omega)$ $\displaystyle =$ $\displaystyle \frac{e^{-j \omega \frac{1}{2}}}{e^{-j\omega \frac{1}{2}}}
\left[ \frac{ e^{j \omega \frac{M}{2}}-e^{-j\omega\frac{M}{2}}}
{e^{j \omega\frac{1}{2}}-e^{-j\omega\frac{1}{2}}} \right]$  
  $\displaystyle =$ $\displaystyle \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}
\isdef M\cdot \hbox{asinc}_M(\omega)
\protect$ (4.4)

where $ \hbox{asinc}_M(\omega)$ denotes the aliased sinc function:4.1

$\displaystyle \hbox{asinc}_M(\omega) \isdef \frac{\sin(M\omega/2)}{M\cdot \sin(\omega/2)} \protect$ (4.5)

(also called the Dirichlet function [175,72] or periodic sinc function). This (real) result is for the zero-centered rectangular window. For the causal case, a linear phase term appears:

$\displaystyle W^c_R(\omega) = e^{-j\frac{M-1}{2}\omega} \cdot M \cdot \hbox{asinc}_M(\omega)$ (4.6)

The term ``aliased sinc function'' refers to the fact that it may be simply obtained by sampling the length-$ \tau$ continuous-time rectangular window, which has Fourier transform sinc$ (f \tau)\isdeftext \sin(\pi f \tau)/(\pi f\tau)$ (given amplitude $ 1/\tau$ in the time domain). Sampling at intervals of $ T$ seconds in the time domain corresponds to aliasing in the frequency domain over the interval $ [0,1/T]$ Hz, and by direct derivation, we have found the result. It is interesting to consider what happens as the window duration increases continuously in the time domain: the magnitude spectrum can only change in discrete jumps as new samples are included, even though it is continuously parametrized in $ \tau$ .

As the sampling rate goes to infinity, the aliased sinc function therefore approaches the sinc function

sinc$\displaystyle (x) \isdef \frac{\sin(\pi x)}{\pi x}. \protect$ (4.7)

Specifically,

$\displaystyle \lim_{\stackrel{T\to 0}{MT=\tau}} \hbox{asinc}_M(\omega T) =$   sinc$\displaystyle (\tau f). \protect$ (4.8)

where $ \omega =
2\pi f$ .4.2

Figure 3.2: Fourier transform of the rectangular window.
\includegraphics[width=\textwidth ,height=2.25in]{eps/rectWindowRawFT}

Figure 3.2 illustrates $ W_R(\omega) =
M\cdot\hbox{asinc}_M(\omega)$ for $ M=11$ . Note that this is the complete window transform, not just its real part. We obtain real window transforms like this only for zero-centered, symmetric windows. Note that the phase of rectangular-window transform $ W_R(\omega)$ is zero for $ \vert\omega\vert<2\pi/M$ , which is the width of the main lobe. This is why zero-centered windows are often called zero-phase windows; while the phase actually alternates between 0 and $ \pi$ radians, the $ \pi$ values occur only within side-lobes which are routinely neglected (in fact, the window is normally designed to ensure that all side-lobes can be neglected).

More generally, we may plot both the magnitude and phase of the window versus frequency, as shown in Figures 3.4 and 3.5 below. In audio work, we more typically plot the window transform magnitude on a decibel (dB) scale, as shown in Fig.3.3 below. It is common to normalize the peak of the dB magnitude to 0 dB, as we have done here.

Figure 3.3: Magnitude (dB) of the rectangular-window transform.
\includegraphics[width=\twidth]{eps/rectWindowFT}

Figure 3.4: Magnitude of the rectangular-window Fourier transform.
% latex2html id marker 73365
\includegraphics[width=\twidth,height=0.3125\theight]{eps/rectWindowFTzeroX}

Figure 3.5: Phase of the rectangular-window Fourier transform.
% latex2html id marker 73369
\includegraphics[width=\twidth,height=0.3125\theight]{eps/rectWindowPhaseFT}

Rectangular Window Side-Lobes

From Fig.3.3 and Eq.$ \,$ (3.4), we see that the main-lobe width is $ 2\cdot 2\pi/M=4\pi/11 \approx 1.1$ radian, and the side-lobe level is 13 dB down.

Since the DTFT of the rectangular window approximates the sinc function (see (3.4)), which has an amplitude envelope proportional to $ 1/\omega$ (see (3.7)), it should ``roll off'' at approximately 6 dB per octave (since $ -20\log_{10}(2)=6.0205999\ldots$ ). This is verified in the log-log plot of Fig.3.6.

Figure 3.6: Roll-off of the rectangular-window Fourier transform.
\includegraphics[width=\textwidth ]{eps/rectWindowLLFT}

As the sampling rate approaches infinity, the rectangular window transform ( $ \hbox{asinc}$ ) converges exactly to the sinc function. Therefore, the departure of the roll-off from that of the sinc function can be ascribed to aliasing in the frequency domain, due to sampling in the time domain (hence the name `` $ \hbox{asinc}$ '').

Note that each side lobe has width $ \Omega_M \isdeftext 2\pi/M$ , as measured between zero crossings.4.3 The main lobe, on the other hand, is width $ 2\Omega_M$ . Thus, in principle, we should never confuse side-lobe peaks with main-lobe peaks, because a peak must be at least $ 2\Omega_M$ wide in order to be considered ``real''. However, in complicated real-world scenarios, side-lobes can still cause estimation errors (``bias''). Furthermore, two sinusoids at closely spaced frequencies and opposite phase can partially cancel each other's main lobes, making them appear to be narrower than $ 2\Omega_M$ .

In summary, the DTFT of the $ M$ -sample rectangular window is proportional to the `aliased sinc function':

\begin{eqnarray*}
\hbox{asinc}_M(\omega) &\isdef & \frac{\sin(\omega M / 2)}{M\cdot\sin(\omega/2)} \\ [0.2in]
&\approx& \frac{\sin(\pi fM)}{M\pi f} \isdefs \mbox{sinc}(fM)
\end{eqnarray*}

Thus, it has zero crossings at integer multiples of

$\displaystyle \Omega_M \isdefs \frac{2\pi}{M}.$ (4.11)

Its main-lobe width is $ 2\Omega_M$ and its first side-lobe is 13 dB down from the main-lobe peak. As $ M$ gets bigger, the main-lobe narrows, giving better frequency resolution (as discussed in the next section). Note that the window-length $ M$ has no effect on side-lobe level (ignoring aliasing). The side-lobe height is instead a result of the abruptness of the window's transition from 1 to 0 in the time domain. This is the same thing as the so-called Gibbs phenomenon seen in truncated Fourier series expansions of periodic waveforms. The abruptness of the window discontinuity in the time domain is also what determines the side-lobe roll-off rate (approximately 6 dB per octave). The relation of roll-off rate to the smoothness of the window at its endpoints is discussed in §B.18.


Rectangular Window Summary

The rectangular window was discussed in Chapter 53.1). Here we summarize the results of that discussion.

Definition ($ M$ odd):

$\displaystyle w_R(n) \isdef \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$ (4.12)

Transform:

$\displaystyle W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdef \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$ (4.13)

The DTFT of a rectangular window is shown in Fig.3.7.

Figure 3.7: Rectangular window discrete-time Fourier transform.
\includegraphics[width=\twidth]{eps/Rect}

Properties:

  • Zero crossings at integer multiples of

    $\displaystyle \Omega_M \isdef \frac{2\pi}{M} = \hbox{frequency sampling interval for a length $M$\ DFT.}$ (4.14)

  • Main lobe width is $ 2 \Omega_M = \frac{4\pi}{M} $ .
  • As $ M$ increases, the main lobe narrows (better frequency resolution).
  • $ M$ has no effect on the height of the side lobes (same as the ``Gibbs phenomenon'' for truncated Fourier series expansions).
  • First side lobe only 13 dB down from the main-lobe peak.
  • Side lobes roll off at approximately 6dB per octave.
  • A phase term arises when we shift the window to make it causal, while the window transform is real in the zero-phase case (i.e., centered about time 0).


Generalized Hamming Window Family

The generalized Hamming window family is constructed by multiplying a rectangular window by one period of a cosine. The benefit of the cosine tapering is lower side-lobes. The price for this benefit is that the main-lobe doubles in width. Two well known members of the generalized Hamming family are the Hann and Hamming windows, defined below.

The basic idea of the generalized Hamming family can be seen in the frequency-domain picture of Fig.3.8. The center dotted waveform is the aliased sinc function $ 0.5\cdot W_R(\omega) = 0.5\cdot
M\cdot\hbox{asinc}_M(\omega)$ (scaled rectangular window transform). The other two dotted waveforms are scaled shifts of the same function, $ 0.25\cdot
W_R(\omega\pm\Omega_M)$ . The sum of all three dotted waveforms gives the solid line. We see that

  • there is some cancellation of the side lobes, and
  • the width of the main lobe is doubled.

Figure 3.8: Construction of the generalized Hamming window transform as a superposition of three shifted aliased sinc functions.
\includegraphics[width=3in]{eps/shiftedSincs}

In terms of the rectangular window transform $ W_R(\omega) =
M\cdot\hbox{asinc}_M(\omega)$ (the zero-phase, unit-amplitude case), this can be written as

$\displaystyle W_H( \omega ) \isdefs \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M )$ (4.15)

where $ \alpha=1/2$ , $ \beta=1/4$ in the example of Fig.3.8.

Using the shift theorem2.3.4), we can take the inverse transform of the above equation to obtain

$\displaystyle w_H = \alpha w_R(n) + \beta e^{-j\Omega_M n}w_R(n) + \beta e^{j \Omega_M n} w_R(n),$ (4.16)

or,

$\displaystyle \zbox {w_H(n) = w_R(n) \left[ \alpha + 2 \beta \cos \left( \frac{2 \pi n}{M} \right) \right].} \protect$ (4.17)

Choosing various parameters for $ \alpha $ and $ \beta $ result in different windows in the generalized Hamming family, some of which have names.

Hann or Hanning or Raised Cosine

The Hann window (or hanning or raised-cosine window) is defined based on the settings $ \alpha=1/2$ and $ \beta=1/4$ in (3.17):

$\displaystyle w_H(n)=w_R(n) \left[ \frac{1}{2} + \frac{1}{2} \cos( \Omega_M n) \right] = w_R(n) \cos^2\left(\frac{\Omega_M}{2} n\right) \protect$ (4.18)

where $ \Omega_M \isdeftext 2\pi/M$ .

The Hann window and its transform appear in Fig.3.9. The Hann window can be seen as one period of a cosine ``raised'' so that its negative peaks just touch zero (hence the alternate name ``raised cosine''). Since it reaches zero at its endpoints with zero slope, the discontinuity leaving the window is in the second derivative, or the third term of its Taylor series expansion at an endpoint. As a result, the side lobes roll off approximately 18 dB per octave. In addition to the greatly accelerated roll-off rate, the first side lobe has dropped from $ -13$ dB (rectangular-window case) down to $ -31.5$ dB. The main-lobe width is of course double that of the rectangular window. For Fig.3.9, the window was computed in Matlab as hanning(21). Therefore, it is the variant that places the zero endpoints one-half sample to the left and right of the outermost window samples (see next section).

Figure 3.9: The Hann window (hanning(21)) and its DTFT.
\includegraphics[width=4in]{eps/hanningWindow}


Matlab for the Hann Window

In matlab, a length $ M$ Hann window is designed by the statement

w = hanning(M);
which, in Matlab only is equivalent to
w = .5*(1 - cos(2*pi*(1:M)'/(M+1)));
For $ M=3$ , hanning(3) returns the following:
>> hanning(3)
ans =
     0.5
     1
     0.5
Note the curious use of M+1 in the denominator instead of M as we would expect from the family definition in (3.17). This perturbation serves to avoid using zero samples in the window itself. (Why bother to multiply explicitly by zero?) Thus, the Hann window as returned by Matlab hanning function reaches zero one sample beyond the endpoints to the left and right. The minus sign, which differs from (3.18), serves to make the window causal instead of zero phase.

The Matlab Signal Processing Toolbox also includes a hann function which is defined to include the zeros at the window endpoints. For example,

>> hann(3)
ans =
     0
     1
     0
This case is equivalent to the following matlab expression:
w = .5*(1 - cos(2*pi*(0:M-1)'/(M-1)));
The use of $ M-1$ is necessary to include zeros at both endpoints. The Matlab hann function is a special case of what Matlab calls ``generalized cosine windows'' (type gencoswin).

In Matlab, both hann(3,'periodic') and hanning(3,'periodic') produce the following window:

>> hann(3,'periodic')
ans =
     0
     0.75
     0.75
This case is equivalent to
w = .5*(1 - cos(2*pi*(0:M-1)'/M));
which agrees (finally) with definition (3.18). We see that in this case, the left zero endpoint is included in the window, while the one on the right lies one sample outside to the right. In general, the 'periodic' window option asks for a window that can be overlapped and added to itself at certain time displacements ($ (M-1)/2=1$ samples in this case) to produce a constant function. Use of ``periodic'' windows in this way is introduced in §7.3 and discussed more fully in Chapters 8 and 9.

In Octave, both the hann and hanning functions include the endpoint zeros.

In practical applications, it is safest to write your own window functions in the matlab language in order to ensure portability and consistency. After all, they are typically only one line of code!

In comparing window properties below, we will speak of the Hann window as having a main-lobe width equal to $ 4\Omega_M$ , and a side-lobe width $ \Omega_M$ , even though in practice they may really be $ 4\Omega_{M\pm1}$ and $ \Omega_{M\pm1}$ , respectively, as illustrated above. These remarks apply to all windows in the generalized Hamming family, as well as the Blackman-Harris family introduced in §3.3 below.

Summary of Hann window properties:

  • Main lobe is $ 4\Omega_M$ wide, $ \Omega_M \isdeftext 2\pi/M$
  • First side lobe at -31dB
  • Side lobes roll off approximately $ -18$ dB per octave


Hamming Window

The Hamming window is determined by choosing $ \alpha $ in (3.17) (with $ \beta\isdeftext (1-\alpha)/2$ ) to cancel the largest side lobe [101].4.4 Doing this results in the values

\begin{eqnarray*}
\alpha &=& \frac{25}{46} \approx 0.54\\ [5pt]
\beta &=& \frac{1-\alpha}{2} \approx 0.23.
\end{eqnarray*}

The peak side-lobe level is approximately $ -42.76$ dB for the Hamming window [101].4.5 It happens that this choice is very close to that which minimizes peak side-lobe level (down to $ -43.19$ dB--the lowest possible within the generalized Hamming family) [196]:

$\displaystyle \alpha = 0.53836\ldots$ (4.19)

Since rounding the optimal $ \alpha $ to two significant digits gives $ 0.54$ , the Hamming window can be considered the ``Chebyshev Generalized Hamming Window'' (approximately). Chebyshev-type designs normally exhibit equiripple error behavior, because the worst-case error (side-lobe level in this case) is minimized. However, within the generalized Hamming family, the asymptotic spectral roll-off is constrained to be at least $ -6$ dB per octave due to the form (3.17) of all windows in the family. We'll discuss the true Chebyshev window in §3.10 below; we'll see that it is not monotonic from its midpoint to an endpoint, and that it is in fact impulsive at its endpoints. (To peek ahead at a Chebyshev window and transform, see Fig.3.31.) Generalized Hamming windows can have a step discontinuity at their endpoints, but no impulsive points.

Figure 3.10: A Hamming window and its transform.
\includegraphics[width=\twidth]{eps/hammingWindow}

The Hamming window and its DTFT magnitude are shown in Fig.3.10. Like the Hann window, the Hamming window is also one period of a raised cosine. However, the cosine is raised so high that its negative peaks are above zero, and the window has a discontinuity in amplitude at its endpoints (stepping discontinuously from 0.08 to 0). This makes the side-lobe roll-off rate very slow (asymptotically $ -6$ dB/octave). On the other hand, the worst-case side lobe plummets to $ -41$ dB,4.6which is the purpose of the Hamming window. This is 10 dB better than the Hann case of Fig.3.9 and 28 dB better than the rectangular window. The main lobe is approximately $ 4\Omega_M$ wide, as is the case for all members of the generalized Hamming family ( $ \beta\neq 0$ ).

Due to the step discontinuity at the window boundaries, we expect a spectral envelope which is an aliased version of a $ -6$ dB per octave (i.e., a $ 1/\omega$ roll-off is converted to a ``cosecant roll-off'' by aliasing, as derived in §3.1 and illustrated in Fig.3.6). However, for the Hamming window, the side-lobes nearest the main lobe have been strongly shaped by the optimization. As a result, the nearly $ -6$ dB per octave roll-off occurs only over an interior interval of the spectrum, well between the main lobe and half the sampling rate. This is easier to see for a larger $ M$ , as shown in Fig.3.11, since then the optimized side-lobes nearest the main lobe occupy a smaller frequency interval about the main lobe.

Figure 3.11: A longer Hamming window and its transform.
\includegraphics[width=\twidth]{eps/hammingWindowLong}

Since the Hamming window side-lobe level is more than 40 dB down, it is often a good choice for ``1% accurate systems,'' such as 8-bit audio signal processing systems. This is because there is rarely any reason to require the window side lobes to lie far below the signal quantization noise floor. The Hamming window has been extensively used in telephone communications signal processing wherein 8-bit CODECs were standard for many decades (albeit $ \mu$ -law encoded). For higher quality audio signal processing, higher quality windows may be required, particularly when those windows act as lowpass filters (as developed in Chapter 9).


Matlab for the Hamming Window

In matlab, a length $ M$ Hamming window is designed by the statement

w = hamming(M);
which is equivalent to
w = .54 - .46*cos(2*pi*(0:M-1)'/(M-1));
Note that M-1 is used in the denominator rather than M+1 as in the Hann window case. Since the Hamming window cannot reach zero for any choice of samples of the defining raised cosine, it makes sense not to have M+1 here. Using M-1 (instead of M) provides that the returned window is symmetric, which is usually desired. However, we will learn later that there are times when M is really needed in the denominator (such as when the window is being used successively over time in an overlap-add scheme, in which case the sum of overlapping windows must be constant).

The hamming function in the Matlab Signal Processing Tool Box has an optional argument 'periodic' which effectively uses $ M$ instead of $ M-1$ . The default case is 'symmetric'. The following examples should help clarify the difference:

>> hamming(3) % same in Matlab and Octave
ans =
    0.0800
    1.0000
    0.0800
>> hamming(3,'symmetric') % Matlab only
ans =
    0.0800
    1.0000
    0.0800
>> hamming(3,'periodic') % Matlab only
ans =
    0.0800
    0.7700
    0.7700
>> hamming(4) % same in Matlab and Octave
ans =
    0.0800
    0.7700
    0.7700
    0.0800


Summary of Generalized Hamming Windows

Definition:

$\displaystyle w_H(n) = w_R(n) \left[ \alpha + 2 \beta \cos \left( \Omega_M n \right) \right], \quad n \in {\bf Z}, \; \Omega_M \isdef \frac{2 \pi}{M}$ (4.20)

where

$\displaystyle w_R(n) \isdefs \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$ (4.21)

Transform:

$\displaystyle W_H( \omega ) \isdefs \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M ), \quad \omega\in[-\pi,\pi)$ (4.22)

where

$\displaystyle W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdefs \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$ (4.23)

Common Properties

  • Rectangular + scaled-cosine window
  • Cosine has one period across the window
  • Symmetric ( $ \Rightarrow$ zero or linear phase)
  • Positive (by convention on $ \alpha $ and $ \beta $ )
  • Main lobe is $ 4\Omega_M$ radians per sample wide, where $ \Omega_M\isdef 2\pi/M$
  • Zero-crossings (``notches'') in window transform at intervals of $ \Omega_M$ outside of main lobe

Figure 3.12 compares the window transforms for the rectangular, Hann, and Hamming windows. Note how the Hann window has the fastest roll-off while the Hamming window is closest to being equal-ripple. The rectangular window has the narrowest main lobe.

Figure 3.12: Comparison of window transforms for the rectangular, Hann, and Hamming windows.
\includegraphics[width=\twidth]{eps/RectHannHamm}

Rectangular window properties:

  • Abrupt transition from 1 to 0 at the window endpoints
  • Roll-off is asymptotically $ -6$ dB per octave (as $ T\rightarrow 0$ )
  • First side lobe is $ -13$ dB relative to main-lobe peak

Hann window properties:

  • Smooth transition to zero at window endpoints
  • Roll-off is asymptotically -18 dB per octave
  • First side lobe is $ -31$ dB relative to main-lobe peak

Hamming window properties:

  • Discontinuous ``slam to zero'' at endpoints
  • Roll-off is asymptotically -6 dB per octave
  • Side lobes are closer to ``equal ripple''
  • First side lobe is $ 41$ dB down = $ 10$ dB better than Hann4.7


The MLT Sine Window

The modulated lapped transform (MLT) [160] uses the sine window, defined by

$\displaystyle w(n) = \sin\left[\left(n+\frac{1}{2}\right)\frac{\pi}{2M}\right], \quad n=0,1,2,\ldots,2M-1\,.$ (4.24)

The sine window is used in MPEG-1, Layer 3 (MP3 format), MPEG-2 AAC, and MPEG-4 [200].

Properties:

Note that in perceptual audio coding systems, there is both an analysis window and a synthesis window. That is, the sine window is deployed twice, first when encoding the signal, and second when decoding. As a result, the sine window is squared in practical usage, rendering it equivalent to a Hann window ($ \cos^2$ ) in the final output signal (when there are no spectral modifications).

It is of great practical value that the second window application occurs after spectral modifications (such as spectral quantization); any distortions due to spectral modifications are tapered gracefully to zero by the synthesis window. Synthesis windows were introduced at least as early as 1980 [213,49], and they became practical for audio coding with the advent of time-domain aliasing cancellation (TDAC) [214]. The TDAC technique made it possible to use windows with 50% overlap without suffering a doubling of the number of samples in the short-time Fourier transform. TDAC was generalized to ``lapped orthogonal transforms'' (LOT) by Malvar [160]. The modulated lapped transform (MLT) is a variant of LOT used in conjunction with the modulated discrete cosine transform (MDCT) [160]. See also [287] and [291].


Blackman-Harris Window Family

The Blackman-Harris (BH) window family is a straightforward generalization of the Hamming family introduced in §3.2. Recall from that discussion that the generalized Hamming family was constructed using a summation of three shifted and scaled aliased-sinc-functions (shown in Fig.3.8). The Blackman-Harris family is obtained by adding still more shifted sinc functions:

$\displaystyle w_{BH}(n)= w_R(n)\sum_{l=0}^{L-1} \alpha_l \cos( l \Omega_M n), \protect$ (4.26)

where $ \Omega_M\isdef 2\pi/M$ , and $ w_R(n)$ is the length $ M$ zero-phase rectangular window (nonzero for $ n\in[-(M-1)/2,(M-1)/2]$ ). The corresponding window transform is given by

$\displaystyle W_{BH}(\omega) = \sum_{k=-(L-1)}^{L-1}\alpha_k W_R(\omega + k\Omega_M),$ (4.27)

where $ W_R(\omega) =
M\cdot\hbox{asinc}_M(\omega)$ denotes the rectangular-window transform, and $ \Omega_M = 2\pi/M$ as usual.

Note that for $ L=1$ , we obtain the rectangular window, and for $ L=2$ , the BH family specializes to the generalized Hamming family.

Blackman Window Family

When $ L=3$ in (3.26), we obtain the Blackman family:

$\displaystyle w_{B}(n)= w_R(n)\left[\alpha_0 + \alpha_1 \cos(\Omega_M n) + \alpha_2 \cos(2\Omega_M n)\right]$ (4.28)

Relative to the generalized Hamming family (§3.2), we have added one more cosine weighted by $ \alpha_2$ . We now therefore have three degrees of freedom to work with instead of two. In the Hamming family, we used one degree of freedom to normalize the window amplitude and the second was used either to maximize roll-off rate (Hann) or side-lobe rejection (Hamming). Now we can use two remaining degrees of freedom (after normalization) to optimize these objectives, or we can use one for each, resulting in three subtypes within the Blackman window family.


Classic Blackman

The so-called ``Blackman Window'' is the specific case for which $ \alpha_0 = 0.42$ $ \alpha_1 = 0.5$ , and $ \alpha_2 = 0.08$ . It has the following properties:

  • Side lobes roll off at about $ 18\dB $ per octave (like Hann)
  • Side-lobe level is about $ 58$ dB (worst case)
  • One degree of freedom used to increase the roll-off rate from 6dB/octave (like rectangular) to 18 dB per octave by matching amplitude and slope to 0 at the window endpoints
  • One degree of freedom is used to minimize side lobes (like Hamming)
  • One degree of freedom is used to scale the window


Matlab for the Classic Blackman Window

N = 101; L = 3; No2 = (N-1)/2; n=-No2:No2;
ws = zeros(L,3*N); z = zeros(1,N);
for l=0:L-1
  ws(l+1,:) = [z,cos(l*2*pi*n/N),z];
end
alpha = [0.42,0.5,0.08]; % Classic Blackman
w = alpha * ws;

Figure 3.13 plots the classic Blackman Window and its transform.

Figure 3.13: Classic Blackman window and transform.
\includegraphics[width=\twidth]{eps/blackmanWindow}


Three-Term Blackman-Harris Window

The classic Blackman window of the previous section is a three-term window in the Blackman-Harris family ($ L=2$ ), in which one degree of freedom is used to minimize side-lobe level, and the other is used to maximize roll-off rate. Harris [101, p. 64] defines the three-term Blackman-Harris window as the one which uses both degrees of freedom to minimize side-lobe level. An improved design is given in Nuttall [196, p. 89], and its properties are as follows:

  • $ \alpha_0 = 0.4243801$ $ \alpha_1 = 0.4973406$ , and $ \alpha_2 = 0.0782793$ .
  • Side-lobe level $ 71.48$ dB
  • Side lobes roll off $ \approx 6\dB $ per octave in the absence of aliasing (like rectangular and Hamming)
  • All degrees of freedom (scaling aside) are used to minimize side lobes (like Hamming)

Figure 3.14 plots the three-term Blackman-Harris Window and its transform. Figure 3.15 shows the same display for a much longer window of the same type, to illustrate its similarity to the rectangular window (and Hamming window) at high frequencies.

Figure 3.14: Three-term Blackman-Harris window and transform
\includegraphics[width=\twidth]{eps/blackmanHarris3}

Figure 3.15: Longer three-term Blackman-Harris window and transform
\includegraphics[width=\twidth]{eps/blackmanHarris3Long}


Frequency-Domain Implementation of the
Blackman-Harris Family

The Blackman-Harris window family can be very efficiently implemented in the frequency domain as a $ (2L-1)$ -point convolution with the spectrum of the unwindowed data.

For example, to implement a zero-phase Hann window,

  1. Start with a length $ M$ rectangular window
  2. Take an $ M$ -point DFT
  3. Convolve the DFT data with the 3-point smoother $ W=[1/4,1/2,1/4]$
Note that the frequency-domain implementation of the Hann window requires no multiplies in linear fixed-point data formats [188].

Similarly, any Blackman window may be implemented as a 5-point smoother in the frequency domain. More generally, any $ L$ -term Blackman-Harris window requires convolution of the critically sampled spectrum with a smoother of length $ 2L-1$ .


Power-of-Cosine Window Family

Definition:

$\displaystyle w(n)=w_R(n) \cos^P\left( \frac{ \pi n}{M} \right)$ (4.29)

where $ P$ is a nonnegative integer.

Properties:

  • The first $ P$ terms of the window's Taylor expansion, evaluated at the endpoints are identically 0 .
  • Roll-off rate $ \approx 6(P+1)$ dB/octave.

Special Cases:

  • $ P=0 \Rightarrow$ Rectangular window
  • $ P=1 \Rightarrow$ MLT sine window
  • $ P=2 \Rightarrow$ Hann window (``raised cosine'' = ``$ \cos^2$ '')
  • $ P=4 \Rightarrow$ Alternative Blackman (maximized roll-off rate)

Thus, $ \cos^P$ windows parametrize $ L$ -term Blackman-Harris windows (for $ L=P/2+1$ ) which are configured to use all degrees-of-freedom to maximize roll-off rate.


Spectrum Analysis of an Oboe Tone

In this section we compare three FFT windows applied to an oboe recording. The examples demonstrate that more gracefully tapered windows support a larger spectral dynamic range, at the cost of reduced frequency resolution.

Rectangular-Windowed Oboe Recording

Figure 3.16a shows a segment of two quasi periods from an oboe recording at the pitch C4, and Fig.3.16b shows the corresponding FFT magnitude. The window length was set to the next integer greater than twice the period length in samples. The FFT size was set to the next power of 2 which was at least five times the window length (for a minimum zero-padding factor of 5). The complete Matlab script is listed in §F.2.5.

Figure 3.16: (a) Rectangularly windowed segment of two periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboeboxcar}


Hamming-Windowed Oboe Recording

Figure 3.17a shows a segment of four quasi periods from the same oboe recording as in the previous figure multiplied by a Hamming window, and Fig.3.17b shows the corresponding zero-padded FFT magnitude. Note how the lower side-lobes of the Hamming window significantly improve the visibility of spectral components above 6 kHz or so.

Figure 3.17: (a) Hamming-windowed segment of four periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboehamming}


Blackman-Windowed Oboe Recording

Figure 3.18a shows a segment of six quasi periods from the same oboe recording as in Fig.3.16 multiplied by a Blackman window, and Fig.3.18b shows the corresponding zero-padded FFT magnitude data. The lower side lobes of the Blackman window significantly improve over the Hamming-window results at high frequencies.

Figure 3.18: (a) Blackman-windowed segment of six periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboeblackman}


Conclusions

In summary, only the Blackman window clearly revealed all of the oboe harmonics. This is because the spectral dynamic range of signal exceeded that of the window transform in the case of rectangular and Hamming windows. In other words, the side lobes corresponding to the loudest low-frequency harmonics were comparable to or louder than the signal harmonics at high frequencies.

Note that preemphasis (flattening the spectral envelope using a preemphasis filter) would have helped here by reducing the spectral dynamic range of the signal (see §10.3 for a number of methods). In voice signal processing, approximately $ +6$ dB/octave preemphasis is common because voice spectra generally roll off at $ -6$ dB per octave [162]. If $ X(\omega)$ denotes the original voice spectrum and $ X_p(\omega)$ the preemphasized spectrum, then one method is to use a ``leaky first-order difference''

$\displaystyle X_p(\omega) = (1-0.95\,e^{-j\omega T})X(\omega).$ (4.30)

For voice signals, the preemphasized spectrum $ \vert X_p(\omega)\vert$ tends to have a relatively ``flat'' magnitude envelope compared to $ \vert X(\omega)\vert$ . This preemphasis can be taken out (inverted) by the simple one-pole filter $ 1/(1-0.95z^{-1})$ .


Bartlett (``Triangular'') Window

The Bartlett window (or simply triangular window) may be defined by

$\displaystyle w(n) = w_R(n)\left[1 - \frac{\vert n\vert}{\frac{M-1}{2}}\right], \quad n\in\left[-\frac{M-1}{2},\frac{M-1}{2}\right]$ (4.31)

and the corresponding transform is

$\displaystyle W(\omega) = \left(\frac{M-1}{2}\right)^2\hbox{asinc}_{\frac{M-1}{2}}^2(\omega)$ (4.32)

The following properties are immediate:

  • Convolution of two length $ (M-1)/2$ rectangular windows
  • Main lobe twice as wide as that of a rectangular window of length $ M$
  • First side lobe twice as far down as rectangular case (-26 dB)
  • Often applied implicitly to sample correlations of finite data
  • Also called the ``tent function''
  • Can replace $ M-1$ by $ M+1$ to avoid including endpoint zeros

Matlab for the Bartlett Window:

In matlab, a length $ M$ Bartlett window is designed by the statement

w = bartlett(M);
This is equivalent, for odd $ M$ , to
w = 2*(0:(M-1)/2)/(M-1);
w = [w w((M-1)/2:-1:1)]';
Note that, in contrast to the hanning function, but like the hann function, bartlett explicitly includes zeros at its endpoints:
>> bartlett(3)
ans =
     0
     1
     0
The triang function in Matlab implements the triangular window corresponding to the hanning case:
>> triang(3)
ans =
     0.5000
     1.0000
     0.5000


Poisson Window

The Poisson window (or more generically exponential window) can be written

$\displaystyle w_P(n) = w_R(n)e^{- \alpha \frac{\vert n\vert}{ \frac{M-1}{2} }}$ (4.33)

where $ \alpha $ determines the time constant $ \tau$ :

$\displaystyle \frac{\tau}{T} = \frac{M-1}{2\alpha}\quad\hbox{samples}$ (4.34)

where $ T$ denotes the sampling interval in seconds.

Figure 3.19: The Poisson (exponential) window.
\includegraphics[width=3.5in]{eps/poissonwindow}

The Poisson window is plotted in Fig.3.19. In the $ z$ plane, the Poisson window has the effect of radially contracting the unit circle. Consider an infinitely long Poisson window (no truncation by a rectangular window $ w_R$ ) applied to a causal signal $ h(n)$ having $ z$ transform $ H(z)$ :

\begin{eqnarray*}
H_P(z) &=& \sum_{n=0}^\infty [w(n)h(n)] z^{-n} \\
&=& \sum_{n=0}^\infty \left[h(n) e^{- \frac{ \alpha n}{ M/2 }}\right] z^{-n}
\qquad\hbox{(let $r\isdef e^{-\frac{\alpha}{ M/2 }}$)}\\
&=& \sum_{n=0}^\infty h(n) z^{-n} r^{n}
= \sum_{n=0}^\infty h(n) (z/r)^{-n} \\
&=& H\left(\frac{z}{r}\right)
\end{eqnarray*}

Thus, the unit-circle response is moved to $ \vert z\vert=1/r$ . This means, for example, that marginally stable poles in $ H(z)$ now decay as $ r^{-n}=e^{-\alpha n/(M/2)}$ in $ H(zr)$ .

The effect of this radial $ z$ -plane contraction is shown in Fig.3.20.

Figure 3.20: Radial contraction of the unit circle in the $ z$ plane by the Poisson window.
\includegraphics[width=3.5in]{eps/zplane2}

The Poisson window can be useful for impulse-response modeling by poles and/or zeros (``system identification''). In such applications, the window length is best chosen to include substantially all of the impulse-response data.


Hann-Poisson Window

Definition:

$\displaystyle w(n) = \frac{1}{2}\left[1 + \cos\left(\pi\frac{n}{\frac{M-1}{2}}\right)\right] e^{-\alpha\frac{\vert n\vert}{\frac{M-1}{2}}}$ (4.35)

Figure 3.21: Hann-Poisson window (upper plot, circles) and Fourier transform (lower plot). The upper plot also shows (using solid lines) the Hann and Poisson windows that are multiplied pointwise to produce the Hann-Poisson window.
\includegraphics[width=\twidth]{eps/hannPoissonWindow}

The Hann-Poisson window is, naturally enough, a Hann window times a Poisson window (exponential times raised cosine). It is plotted along with its DTFT in Fig.3.21.

The Hann-Poisson window has the very unusual feature among windows of having ``no side lobes'' in the sense that, for $ \alpha\geq 2$ , the window-transform magnitude has negative slope for all positive frequencies [58], as shown in Fig.3.22. As a result, this window is valuable for ``hill climbing'' optimization methods such as Newton's method or any convex optimization methods. In other terms, of all windows we have seen so far, only the Hann-Poisson window has a convex transform magnitude to the left or right of the peak (Fig.3.21b).

Figure 3.22: Hann-Poisson Slope and Curvature
\includegraphics[width=\twidth]{eps/hannPoissonSlope}

Figure 3.23 also shows the slope and curvature of the Hann-Poisson window transform, but this time with $ \alpha $ increased to 3. We see that higher $ \alpha $ further smooths the side lobes, and even the curvature becomes uniformly positive over a broad center range.

Figure 3.23: Hann-Poisson magnitude, slope, and curvature, in the frequency domain, for larger $ \alpha $ .
\includegraphics[width=\twidth]{eps/hannPoissonSlope2}

Matlab for the Hann-Poisson Window

function [w,h,p] = hannpoisson(M,alpha)
%HANNPOISSON  - Length M Hann-Poisson window
Mo2 = (M-1)/2; n=(-Mo2:Mo2)';
scl = alpha / Mo2;
p = exp(-scl*abs(n));
scl2 = pi / Mo2;
h = 0.5*(1+cos(scl2*n));
w = p.*h;


Slepian or DPSS Window

A window having maximal energy concentration in the main lobe is given by the digital prolate spheroidal sequence (DPSS) of order 0 [256,136]. It is obtained by using all $ M$ degrees of freedom (sample values) in an $ M$ -point window $ w(n)$ to obtain a window transform $ W(\omega)\approx\delta(\omega)$ which maximizes the energy in the main lobe of the window relative to total energy:

$\displaystyle \max_w \left[ \frac{ \mbox{main lobe energy} } { \mbox{total energy} } \right] \protect$ (4.36)

In the continuous-time case, i.e., when $ W(\omega)$ is a continuous function of $ \omega\in(-\infty,\infty)$ , the function $ W(\omega)$ which maximize this ratio is the first prolate spheroidal wave function for the given main-lobe bandwidth $ 2\omega_c$ [101], [202, p. 205].4.8

A prolate spheroidal wave function is defined as an eigenfunction of the integral equation

$\displaystyle \int_{-\omega_c}^{\omega_c} W(\nu) \frac{\sin[\pi D\cdot(\omega-\nu)]}{\pi(\omega-\nu)}\, d\omega = \lambda W(\omega) \protect$ (4.37)

where $ D$ is the nonzero duration of $ w(t)$ in seconds. This integral equation can be understood as ``cropping'' $ W(\omega)$ to zero outside its main lobe (note that the integral goes from $ -\omega _c$ to $ \omega_c$ , followed by a convolution of $ W(\omega)$ with a sinc function which ``time limits'' the window $ w(t)$ to a duration of $ D$ seconds centered at time 0 in the time domain. In operator notation,

\begin{eqnarray*}
&& [\hbox{\sc Chop}_{2\omega_c}(W)]*[D\,\mbox{sinc}(D\omega)]\\
&=& \hbox{\sc FT}(\hbox{\sc Chop}_D(\hbox{\sc IFT}(\hbox{\sc Chop}_{2\omega_c}(W)))) \eqsp \lambda W
\end{eqnarray*}

where $ \hbox{\sc Chop}_D(w)$ is a rectangular windowing operation which zeros $ w(t)$ outside the interval $ t\in[-D/2,D/2]$ .

Satisfying (3.37) means that window transform $ W(\omega)$ is an eigenfunction of this sequence of operations; that is, it can be zeroed outside the interval $ [-\omega_c,\omega_c]$ , inverse Fourier transformed, zeroed outside the interval $ [-D/2,D/2]$ , and forward Fourier transformed to yield the original Window transform $ W(\omega)$ multiplied by some scale factor $ \lambda$ (the eigenvalue of the overall operation). We may say that $ W$ is the bandlimited extrapolation of its main lobe.

The sinc function in (3.37) can be regarded as a symmetric Toeplitz operator kernel), and the integral of $ W$ multiplied by this kernel can be called a symmetric Toeplitz operator. This is a special case of a Hermitian operator, and by the general theory of Hermitian operators, there exists an infinite set of mutually orthogonal functions $ W_m(\omega)$ , each associated with a real eigenvalues $ \lambda_m$ .4.9 If $ \lambda_0$ denotes the largest such eigenvalue of (3.37), then its corresponding eigenfunction, $ W_0(\omega)\;\leftrightarrow\;
w_0(t)$ , is what we want as our Slepian window, or prolate spheroidal window in the continuous-time case. It is optimal in the sense of having maximum main-lobe energy as a fraction of total energy.

The discrete-time counterpart is Digital Prolate Spheroidal Sequences (DPSS), which may be defined as the eigenvectors of the following symmetric Toeplitz matrix constructed from a sampled sinc function [13]:

$\displaystyle S[k,l] = \frac{\sin[\omega_c T(k-l)]}{k-l}, \quad k,l=0,1,2,\ldots,M-1$ (4.38)

where $ M$ denotes the desired window length in samples, $ \omega_c$ is the desired main-lobe cut-off frequency in radians per second, and $ T$ is the sampling period in seconds. The main-lobe bandwidth is thus $ 2\omega_c$ rad/sec, counting both positive and negative frequencies.) The digital Slepian window (or DPSS window) is then given by the eigenvector corresponding to the largest eigenvalue. A simple matlab program is given in §F.1.2 for computing these windows, and facilities in Matlab and Octave are summarized in the next subsection.

Matlab for the DPSS Window

The function dpss in the Matlab Signal Processing Tool Box4.10 can be used to compute them as follows:

  w = dpss(M,alpha,1); % discrete prolate spheroidal sequence
where $ M$ is the desired window length, and $ \alpha\in[0,M/2)$ can be interpreted as half of the window's time-bandwidth product $ \Delta
t\cdot \Delta f$ in ``cycles''. Alternatively, $ \alpha $ can be interpreted as the highest bin number $ k$ inside the main lobe of the window transform $ W(\omega_k)=\hbox{\sc DFT}_M(w)$ , when the DFT length is equal to the window length $ M$ . (See the next section on the Kaiser window for more on this point.)

Some examples of DPSS windows and window transforms are given in Fig.3.29.


Kaiser Window

Jim Kaiser discovered a simple approximation to the DPSS window based upon Bessel functions [115], generally known as the Kaiser window (or Kaiser-Bessel window).

Definition:

$\displaystyle w_K(n) \isdefs \left\{ \begin{array}{ll} \frac{ I_0 \left( \beta \sqrt{ 1 - \left( \frac{n}{M/2}\right)^2} \right)}{ I_0(\beta) }, & -\frac{M-1}{2} \leq n \leq \frac{M-1}{2} \\ 0, & \mbox{elsewhere} \\ \end{array} \right.$ (4.39)

Window transform:

The Fourier transform of the Kaiser window $ w_K(t)$ (where $ t$ is treated as continuous) is given by4.11

$\displaystyle W(\omega) = \frac{M}{I_0(\beta)} \frac{\sinh\left(\sqrt{\beta^2 - \left(\frac{M \omega}{2}\right)^2}\right)} {\sqrt{ \beta^2 - \left(\frac{M\omega}{2}\right)^2}} = \frac{M}{I_0(\beta)} \frac{\sin\left(\sqrt{\left(\frac{M \omega}{2}\right)^2-\beta^2}\right)} {\sqrt{\left(\frac{M\omega}{2}\right)^2 - \beta^2}}$ (4.40)

where $ I_0$ is the zero-order modified Bessel function of the first kind:4.12

$\displaystyle I_0(x) \isdefs \sum_{k=0}^{\infty} \left[ \frac{\left(\frac{x}{2}\right)^k}{k!} \right]^2
$

Notes:
  • Reduces to rectangular window for $ \beta=0$
  • Asymptotic roll-off is 6 dB/octave
  • First null in window transform is at $ \omega_0=2\beta/M$
  • Time-bandwidth product $ \omega_0 (M/2) = \beta$ radians if bandwidths are measured from 0 to positive band-limit
  • Full time-bandwidth product $ (2\omega_0) M = 4\beta$ radians when frequency bandwidth is defined as main-lobe width out to first null
  • Sometimes the Kaiser window is parametrized by $ \alpha $ , where

    $\displaystyle \beta\isdefs \pi\alpha$ (4.42)

Kaiser Window Beta Parameter

The $ \beta $ parameter of the Kaiser window provides a convenient continuous control over the fundamental window trade-off between side-lobe level and main-lobe width. Larger $ \beta $ values give lower side-lobe levels, but at the price of a wider main lobe. As discussed in §5.4.1, widening the main lobe reduces frequency resolution when the window is used for spectrum analysis. As explored in Chapter 9, reducing the side lobes reduces ``channel cross talk'' in an FFT-based filter-bank implementation.

The Kaiser beta parameter can be interpreted as 1/4 of the ``time-bandwidth product'' $ \Delta t\cdot \Delta \omega$ of the window in radians (seconds times radians-per-second).4.13 Sometimes the Kaiser window is parametrized by $ \alpha\isdef \beta/\pi$ instead of $ \beta $ . The $ \alpha $ parameter is therefore half the window's time-bandwidth product $ \Delta
t\cdot \Delta f$ in cycles (seconds times cycles-per-second).


Kaiser Windows and Transforms

Figure 3.24 plots the Kaiser window and its transform for $ \alpha = \beta/\pi = 1,2,3$ . Note how increasing $ \alpha $ causes the side-lobes to fall away from the main lobe. The curvature at the main lobe peak also decreases somewhat.

Figure 3.24: Kaiser window and transform for $ \alpha =1,2,3$ .
\includegraphics[width=\twidth]{eps/kaiser123}

Figure 3.25 shows a plot of the Kaiser window for various values of $ \beta = [0,2,4,6,8,10]$ . Note that for $ \beta=0$ , the Kaiser window reduces to the rectangular window.

Figure 3.25: The Kaiser window for various values of the time-bandwidth parameter $ \beta $ .
\includegraphics[width=\twidth]{eps/KaiserTBetas}

Figure 3.26 shows a plot of the Kaiser window transforms for $ \beta = [0,2,4,6]$ . For $ \beta=0$ (top plot), we see the dB magnitude of the aliased sinc function. As $ \beta $ increases the main-lobe widens and the side lobes go lower, reaching almost 50 dB down for $ \beta=6$ .

Figure 3.26: Kaiser window transform magnitude for various $ \beta $ .
\includegraphics[width=\twidth]{eps/KaiserFBetas}

Figure 3.27 shows the effect of increasing window length for the Kaiser window. The window lengths are $ M = [20,30,40,50]$ from the top to the bottom plot. As with all windows, increasing the length decreases the main-lobe width, while the side-lobe level remains essentially unchanged.

Figure 3.27: Kaiser window transform magnitudes for various window lengths.
\includegraphics[width=\twidth]{eps/KaiserFLengths}

Figure 3.28 shows a plot of the Kaiser window side-lobe level for various values of $ \alpha = [0,0.5,1,1.5,\ldots,4]$ . For $ \beta=0$ , the Kaiser window reduces to the rectangular window, and we expect the side-lobe level to be about 13 dB below the main lobe (upper-lefthand corner of Fig.3.28). As $ \alpha =\beta /\pi $ increases, the dB side-lobe level reduces approximately linearly with main-lobe width increase (approximately a 25 dB drop in side-lobe level for each main-lobe width increase by one sinc-main-lobe).

Figure 3.28: Kaiser window side-lobe level for various values of $ \alpha =\beta /\pi $ .
\includegraphics[width=\twidth]{eps/kaiserBeta}


Minimum Frequency Separation vs. Window Length

The requirements on window length for resolving closely tuned sinusoids was discussed in §5.5.2. This section considers this issue for the Kaiser window. Table 3.1 lists the $ \alpha $ parameter required for a Kaiser window to resolve equal-amplitude sinusoids with a frequency spacing of $ \Delta\omega$ rad/sample [1, Table 8-9]. Recall from §3.9 that $ \alpha $ can be interpreted as half of the time-bandwidth of the window (in cycles).


Table: Kaiser $ \alpha $ parameter for various frequency resolutions, assuming an FFT zero-padding factor of at least 3.5.
$ \alpha $ $ \Delta\omega$
1.5 $ 2.1$
2.0 $ 2.5$
2.5 $ 2.9$
3.0 $ 3.4$



Kaiser and DPSS Windows Compared

Figure 3.29 shows an overlay of DPSS and Kaiser windows for some different $ \alpha $ values. In all cases, the window length was $ M=51$ . Note how the two windows become more similar as $ \alpha $ increases. The Matlab for computing the windows is as follows:

  w1 = dpss(M,alpha,1);    % discrete prolate spheroidal seq.
  w2 = kaiser(M,alpha*pi); % corresponding kaiser window

Figure: Comparison of length 51 DPSS and Kaiser windows for $ \alpha =1,3,5$ .
\includegraphics[width=\twidth]{eps/dpsstest}

The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of $ \alpha $ as the bin number of the edge of the critically sampled window main lobe, i.e., when the DFT length equals the window length:

format long;
M=17;
alpha=5;
abs(fft([ dpss(M,alpha,1), kaiser(M,pi*alpha)/2]))
ans =
   2.82707022360190   2.50908747431366
   2.00652719015325   1.92930705688346
   0.68469697658600   0.85272343521683
   0.09415916813555   0.19546670371747
   0.00311639169878   0.01773139505899
   0.00000050775691   0.00022611995322
   0.00000003737279   0.00000123787805
   0.00000000262633   0.00000066206722
   0.00000007448708   0.00000034793207
   0.00000007448708   0.00000034793207
   0.00000000262633   0.00000066206722
   0.00000003737279   0.00000123787805
   0.00000050775691   0.00022611995322
   0.00311639169878   0.01773139505899
   0.09415916813555   0.19546670371747
   0.68469697658600   0.85272343521683
   2.00652719015325   1.92930705688346

Finally, Fig.3.30 shows a comparison of DPSS and Kaiser window transforms, where the DPSS window was computed using the simple method listed in §F.1.2. We see that the DPSS window has a slightly narrower main lobe and lower overall side-lobe levels, although its side lobes are higher far from the main lobe. Thus, the DPSS window has slightly better overall specifications, while Kaiser-window side lobes have a steeper roll off.

Figure: DPSS and Kaiser window transforms, for length $ M=61$ , DPSS cut-off $ \omega _c = 2\pi 3.5/M$ , and Kaiser $ \beta =(M-1)\omega _c/2$ .
\includegraphics[width=\twidth]{eps/dpsskaiser-fd}


Dolph-Chebyshev Window

The Dolph-Chebyshev Window (or Chebyshev window, or Dolph window) minimizes the Chebyshev norm of the side lobes for a given main-lobe width $ 2\omega_c$ [61,101], [224, p. 94]:

$\displaystyle \displaystyle \hbox{min}_{w,\sum w=1} \left\Vert\,\hbox{sidelobes($W$)}\,\right\Vert _\infty \isdefs \hbox{min}_{w,\sum w=1} \left\{\hbox{max}_{\omega>\omega_c} \left\vert W(\omega)\right\vert\right\}$ (4.43)

The Chebyshev norm is also called the $ L-infinity$ norm, uniform norm, minimax norm, or simply the maximum absolute value.

An equivalent formulation is to minimize main-lobe width subject to a side-lobe specification:

$\displaystyle \displaystyle \left. \min_{w,W(0)=1}(\omega_c) \right\vert _{\,\left\vert W(\omega)\,\right\vert \leq\, c_\alpha,\; \forall \vert\omega\vert\geq\omega_c}$ (4.44)

The optimal Dolph-Chebyshev window transform can be written in closed form [61,101,105,156]:

\begin{eqnarray*}
W(\omega_k) &=& \frac{\cos\left\{M\cos^{-1}\left[\beta\cos\left(\frac{\pi k}{M}\right)
\right]\right\}}{\cosh\left[M\cosh^{-1} (\beta)\right]},
\qquad k=0,1,2,\ldots,M-1 \\
\beta &=& \cosh \left[\frac{1}{M}\cosh^{-1}(10^\alpha)\right], \qquad (\alpha\approx 2,3,4).
\end{eqnarray*}

The zero-phase Dolph-Chebyshev window, $ w(n)$ , is then computed as the inverse DFT of $ W(\omega_k)$ .4.14 The $ \alpha $ parameter controls the side-lobe level via the formula [156]

Side-Lobe Level in dB$\displaystyle = -20\alpha.$ (4.45)

Thus, $ \alpha=2$ gives side-lobes which are $ 40$ dB below the main-lobe peak. Since the side lobes of the Dolph-Chebyshev window transform are equal height, they are often called ``ripple in the stop-band'' (thinking now of the window transform as a lowpass filter frequency response). The smaller the ripple specification, the larger $ \omega_c$ has to become to satisfy it, for a given window length $ M$ .

The Chebyshev window can be regarded as the impulse response of an optimal Chebyshev lowpass filter having a zero-width pass-band (i.e., the main lobe consists of two ``transition bands''--see Chapter 4 regarding FIR filter design more generally).

Matlab for the Dolph-Chebyshev Window

In Matlab, the function chebwin(M,ripple) computes a length $ M$ Dolph-Chebyshev window having a side-lobe level ripple dB below that of the main-lobe peak. For example,

w = chebwin(31,60);
designs a length $ M=31$ window with side lobes at $ -60$ dB (when the main-lobe peak is normalized to 0 dB).


Example Chebyshev Windows and Transforms

Figure 3.31 shows the Dolph-Chebyshev window and its transform as designed by chebwin(31,40) in Matlab, and Fig.3.32 shows the same thing for chebwin(31,200). As can be seen from these examples, higher side-lobe levels are associated with a narrower main lobe and more discontinuous endpoints.

Figure: The length $ 31$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -40$ dB.
\includegraphics[width=\twidth]{eps/cheb31R40}

Figure: The length $ 31$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -200$ dB.
\includegraphics[width=\twidth]{eps/cheb31R200}

Figure: The length $ 101$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -40$ dB.
\includegraphics[width=\twidth]{eps/cheb101R40}

Figure 3.33 shows the Dolph-Chebyshev window and its transform as designed by chebwin(101,40) in Matlab. Note how the endpoints have actually become impulsive for the longer window length. The Hamming window, in contrast, is constrained to be monotonic away from its center in the time domain.

The ``equal ripple'' property in the frequency domain perfectly satisfies worst-case side-lobe specifications. However, it has the potentially unfortunate consequence of introducing ``impulses'' at the window endpoints. Such impulses can be the source of ``pre-echo'' or ``post-echo'' distortion which are time-domain effects not reflected in a simple side-lobe level specification. This is a good lesson in the importance of choosing the right error criterion to minimize. In this case, to avoid impulse endpoints, we might add a continuity or monotonicity constraint in the time domain (see §3.13.2 for examples).


Chebyshev and Hamming Windows Compared

Figure 3.34 shows an overlay of Hamming and Dolph-Chebyshev window transforms, the ripple parameter for chebwin set to $ 42$ dB to make it comparable to the Hamming side-lobe level. We see that the monotonicity constraint inherent in the Hamming window family only costs a few dB of deviation from optimality in the Chebyshev sense at high frequency.

\begin{psfrags}
% latex2html id marker 9837\psfrag{freq}{$\omega T$\ (radians per sample)}\begin{figure}[htbp]
\includegraphics[width=\twidth]{eps/chebHammingC}
\caption{}
\end{figure}
\end{psfrags}


Dolph-Chebyshev Window Theory

In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.

Chebyshev Polynomials

Figure 3.34:
\includegraphics[width=\twidth]{eps/first-even-chebs-c}

The $ n$ th Chebyshev polynomial may be defined by

$\displaystyle T_n(x) = \left\{\begin{array}{ll} \cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt] \cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\ \end{array} \right..$ (4.46)

The first three even-order cases are plotted in Fig.3.35. (We will only need the even orders for making Chebyshev windows, as only they are symmetric about time 0.) Clearly, $ T_0(x)=1$ and $ T_1(x)=x$ . Using the double-angle trig formula $ \cos(2\theta)=2\cos^2(\theta)-1$ , it can be verified that

$\displaystyle T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)$ (4.47)

for $ n\ge 2$ . The following properties of the Chebyshev polynomials are well known:
  • $ T_n(x)$ is an $ n$ th-order polynomial in $ x$ .
  • $ T_n(x)$ is an even function when $ n$ is an even integer, and odd when $ n$ is odd.
  • $ T_n(x)$ has $ n$ zeros in the open interval $ (-1,1)$ , and $ n+1$ extrema in the closed interval $ [-1,1]$ .
  • $ T_n(x)>1$ for $ x>1$ .


Dolph-Chebyshev Window Definition

Let $ M$ denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by [155]

$\displaystyle W(\omega) = \frac{T_{M-1}[x_0 \cos(\omega/2)]}{T_{M-1}(x_0)}$ (4.48)

where $ x_0>1$ is defined by the desired ripple specification:

$\displaystyle \vert W(\omega)\vert \le r = \frac{1}{T_{M-1}(x_0)}, \quad \forall\vert\omega\vert\ge\omega_c,$ (4.49)

where $ \omega_c$ is the ``main lobe edge frequency'' defined by

$\displaystyle \omega_c \isdefs 2\cos^{-1}\left[\frac{1}{x_0}\right].$ (4.50)

Expanding $ W(\omega)$ in terms of complex exponentials yields

$\displaystyle W(\omega) = \sum_{n=-M_h}^{M_h} w(n) e^{-j \omega n}$ (4.51)

where $ M_h\isdef (M-1)/2$ . Thus, the coefficients $ w(n)$ give the length $ M$ Dolph-Chebyshev window in zero-phase form.


Dolph-Chebyshev Window Main-Lobe Width

Given the window length $ M$ and ripple magnitude $ r$ , the main-lobe width $ 2\omega_c$ may be computed as follows [155]:

\begin{eqnarray*}
x_0 &=& \cosh\left[\frac{\cosh^{-1}\left(\frac{1}{r}\right)}{M-1}\right]\\
\omega_c &=& 2\cos^{-1}\left(\frac{1}{x_0}\right)
\end{eqnarray*}

This is the smallest main-lobe width possible for the given window length and side-lobe spec.


Dolph-Chebyshev Window Length Computation

Given a prescribed side-lobe ripple-magnitude $ r$ and main-lobe width $ 2\omega_c$ , the required window length $ M$ is given by [155]

$\displaystyle M = 1 + \frac{\cosh^{-1}(1/r)}{\cosh^{-1}[\sec(\omega_c/2)]}.$ (4.52)

For $ \omega_c\ll\pi$ (the typical case), the denominator is close to $ \omega_c/2$ , and we have

$\displaystyle M \approx 1 + \frac{2}{\omega_c}\cosh^{-1}\left(\frac{1}{r}\right)$ (4.53)

Thus, half the time-bandwidth product in radians is approximately

$\displaystyle \beta \isdefs (M-1) \omega_c\approx 2\cosh^{-1}\left(\frac{1}{r}\right),$ (4.54)

where $ \beta $ is the parameter often used to design Kaiser windows3.9).


Gaussian Window and Transform

The Gaussian ``bell curve'' is possibly the only smooth, nonzero function, known in closed form, that transforms to itself.4.15

$\displaystyle \frac{1}{\sigma\sqrt{2\pi}}e^{-t^2 \left / 2\sigma^2\right.} \;\longleftrightarrow\; e^{-\omega^2 \left/ 2\left(1/\sigma\right)^2\right.}$ (4.55)

It also achieves the minimum time-bandwidth product

$\displaystyle \sigma_t\sigma_\omega = \sigma\times (1/\sigma) = 1$ (4.56)

when ``width'' is defined as the square root of its second central moment. For even functions $ w(t)$ ,

$\displaystyle \sigma_t \isdefs \sqrt{\int_{-\infty}^\infty t^2 w(t) dt}.$ (4.57)

Since the true Gaussian function has infinite duration, in practice we must window it with some usual finite window, or truncate it.

Depalle [58] suggests using a triangular window raised to some power $ \alpha $ for this purpose, which preserves the absence of side lobes for sufficiently large $ \alpha $ . It also preserves non-negativity of the transform.

Matlab for the Gaussian Window

In matlab, w = gausswin(M,alpha) returns a length $ M$ window with parameter $ \texttt{alpha} \isdeftext 1/\tilde{\sigma}$ where $ \tilde{\sigma}\isdeftext \sigma/((M-1)/2)$ is defined, as in Harris [101], so that the window shape is invariant with respect to window length $ M$ :

function [w] = gausswin(M,alpha)
n = -(M-1)/2 : (M-1)/2;
w = exp((-1/2) * (alpha * n/((M-1)/2)) .^ 2)';

An implementation in terms of unnormalized standard deviation (sigma in samples) is as follows:

function [w] = gaussianwin(M,sigma)
n= -(M-1)/2 : (M-1)/2;
w = exp(-n .* n / (2 * sigma * sigma))';
In this case, sigma would normally be specified as a fraction of the window length $ M$ (sigma = M/8 in the sample below).

Note that, on a dB scale, Gaussians are quadratic. This means that parabolic interpolation of a sampled Gaussian transform is exact. This can be a useful fact to remember when estimating sinusoidal peak frequencies in spectra. For example, one suggested implication is that, for typical windows, quadratic interpolation of spectral peaks may be more accurate on a log-magnitude scale (e.g., dB) than on a linear magnitude scale (this has been observed empirically for a variety of cases).


Gaussian Window and Transform

Figure 3.36 shows an example length $ M=21$ Gaussian window and its transform. The sigma parameter was set to $ M/8$ so that simple truncation of the Gaussian yields a side-lobe level better than $ -80$ dB. Also overlaid on the window transform is a parabola; we see that the main lobe is well fit by the parabola until the side lobes begin. Since the transform of a Gaussian is a Gaussian (exactly), the side lobes are entirely caused by truncating the window.

Figure 3.36: Gaussian window and transform.
\includegraphics[width=\twidth]{eps/gaussianWindow}

More properties and applications of the Gaussian function can be found in Appendix D.


Exact Discrete Gaussian Window

It can be shown [44] that

$\displaystyle e^{-j\frac{\pi}{8}} \, e^{j\frac{2\pi}{N}\frac{1}{2}n^2} \;\longleftrightarrow\; e^{j\frac{\pi}{8}} \, e^{-j\frac{2\pi}{N}\frac{1}{2}k^2}$ (4.58)

where $ n\in[0,N-1]$ is the time index, and $ k\in[0,N-1]$ is the frequency index for a length $ N$ (even) normalized DFT (DFT divided by $ \sqrt{N}$ ). In other words, the Normalized DFT (NDFT) of this particular sampled Gaussian pulse is exactly the complex-conjugate of the same Gaussian pulse. (The proof is nontrivial.)


Optimized Windows

We close this chapter with a general discussion of optimal windows in a wider sense. We generally desire

$\displaystyle W(\omega) \approx \delta(\omega),$ (4.59)

but the nature of this approximation is typically determined by characteristics of audio perception. Best results are usually obtained by formulating this as an FIR filter design problem (see Chapter 4). In general, both time-domain and frequency-domain specifications are needed. (Recall the potentially problematic impulses in the Dolph-Chebyshev window shown in Fig.3.33 when its length was long and ripple level was high). Equivalently, both magnitude and phase specifications are necessary in the frequency domain.

A window transform can generally be regarded as the frequency response of a lowpass filter having a stop band corresponding to the side lobes and a pass band corresponding to the main lobe (or central section of the main lobe). Optimal lowpass filters require a transition region from the pass band to the stop band. For spectrum analysis windows, it is natural to define the entire main lobe as ``transition region.'' That is, the pass-band width is zero. Alternatively, the pass-band could be allowed to have a finite width, allowing some amount of ``ripple'' in the pass band; in this case, the pass-band ripple will normally be maximum at the main-lobe midpoint ( $ W(0)= 1+\delta$ , say), and at the pass-band edges ( $ W(\epsilon) = W(-\epsilon) = 1-\delta$ ). By embedding the window design problem within the more general problem of FIR digital filter design, a plethora of optimal design techniques can be brought to bear [204,258,14,176,218].

Optimal Windows for Audio Coding

Recently, numerically optimized windows have been developed by Dolby which achieve the following objectives:

  • Narrow the window in time
  • Smooth the onset and decay in time
  • Reduce side lobes below the worst-case masking threshold
See [200] for an overview. This is an excellent example of how window design should be driven by what one really wants.

See §4.10 for an overview of optimal methods for FIR digital filter design.


General Rule

There is rarely a closed form expression for the optimal window in practice. The most important task is to formulate an ideal error criterion. Given the right error criterion, it is usually straightforward to minimize it numerically with respect to the window samples $ w$ .


Window Design by Linear Programming

This section, based on a class project by EE graduate student Tatsuki Kashitani, illustrates the use of linprog in Matlab for designing variations on the Chebyshev window3.10). In addition, some comparisons between standard linear programming and the Remez exchange algorithm (firpm) are noted.

Linear Programming (LP)

If we can get our filter or window design problems in the form

\begin{displaymath}\begin{array}[t]{ll} \mathrm{minimize} & f^{T}x\\ \mathrm{subject}\, \mathrm{to} & \begin{array}[t]{l} \mathbf{A}_{eq}x=b_{eq}\\ \mathbf{A}x\le b\end{array}, \end{array}\end{displaymath} (4.60)

where $ x,f\in{\bf R}^N$ , $ b\in{\bf R}^M$ , $ A$ is $ M\times N$ , etc., then we are done.

The linprog function in Matlab Optimization Toolbox solves LP problems. In Octave, one can use glpk instead (from the GNU GLPK library).


LP Formulation of Chebyshev Window Design

What we want:

  1. Symmetric zero-phase window.
  2. Window samples to be positive.

    $\displaystyle w\left(n\right)\geq 0\qquad \mathrm{for}\quad -\frac{M-1}{2}\leq n\leq \frac{M-1}{2}=L$ (4.61)

  3. Transform to be 1 at DC.

    $\displaystyle W\left(0\right)=1$ (4.62)

  4. Transform to be within $ \left[-\delta ,\delta \right]$ in the stop-band.

    $\displaystyle -\delta \leq W\left(\omega \right)\leq \delta \qquad \mathrm{for}\quad \omega _{sb}\leq \omega \leq \pi$ (4.63)

  5. And $ \delta $ to be small.


Symmetric Window Constraint

Because we are designing a zero-phase window, use only the positive-time part $ h\left(n\right)$ :

$\displaystyle h\left(n\right)=w\left(n\right)\qquad n\geq 0$ (4.64)

$\displaystyle w\left(n\right)=\left\{ \begin{array}{ll} h\left(n\right) & n\geq 0\\ h\left(-n\right) & n<0\end{array} \right.$ (4.65)


Positive Window-Sample Constraint

For each window sample, $ h\left(n\right) \geq 0$ , or,

$\displaystyle -h\left(n\right) \leq 0.$ (4.66)

Stacking inequalities for all $ n$ ,

$\displaystyle \left[\begin{array}{ccccc} -1 & 0 & \cdots & 0 & 0\\ 0 & -1 & & & 0\\ \vdots & & \ddots & & \vdots \\ 0 & & & -1 & 0\\ 0 & 0 & \cdots & 0 & -1\end{array} \right]\left[\begin{array}{c} h\left(0\right)\\ h\left(1\right)\\ \vdots \\ h\left(L-1\right)\\ h\left(L\right)\end{array} \right] \le \left[\begin{array}{c} 0\\ 0\\ \vdots \\ 0\\ 0 \end{array} \right]$ (4.67)

or

$\displaystyle \zbox {-\mathbf{I}\, h \le 0.}$ (4.68)


DC Constraint

The DTFT at frequency $ \omega$ is given by

$\displaystyle W\left(\omega \right)=\sum _{n=-L}^{L}w\left(n\right)e^{-j\omega n}.$ (4.69)

By zero-phase symmetry,
$\displaystyle W\left(\omega \right)$ $\displaystyle =$ $\displaystyle h\left(0\right)+2\sum _{n=1}^{L}h\left(n\right)\cos \left(n\omega \right)$  
  $\displaystyle =$ $\displaystyle \left[\begin{array}{cccc}
1 & 2\cos \left(\omega \right) & \cdots & 2\cos \left(L\omega \right)\end{array}\right]\left[\begin{array}{c}
h\left(0\right)\\
h\left(1\right)\\
\vdots \\
h\left(L\right)\end{array}\right]$  
  $\displaystyle =$ $\displaystyle d\left(\omega \right)^{T}h.$  

So $ W\left(0\right)=1$ can be expressed as

$\displaystyle \zbox {d\left(0\right)^{T}h=1.}$ (4.70)


Sidelobe Specification

Likewise, side-lobe specification can be enforced at frequencies $ \omega_{i}$ in the stop-band.

$\displaystyle -\delta \leq d\left(\omega _{i}\right)^{T}h\leq \delta$ (4.71)

or

$\displaystyle \left\{ \begin{array}{ccc} -d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\\ \;d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\end{array} \right.$ (4.72)

where

$\displaystyle \omega _{sb}\leq \omega _1,\omega _{2}\ldots ,\omega _{K}\leq \pi .$ (4.73)

We need $ K\gg L$ to obtain many frequency samples per side lobe. Stacking inequalities for all $ \omega_{i}$ ,
$\displaystyle \left[\begin{array}{c}
-d\left(\omega _1\right)^{T}\\
\vdots \\
-d\left(\omega _{K}\right)^{T}\\
d\left(\omega _1\right)^{T}\\
\vdots \\
d\left(\omega _{K}\right)^{T}\end{array}\right]h+\left[\begin{array}{c}
-\delta \\
\vdots \\
-\delta \\
-\delta \\
\vdots \\
-\delta \end{array}\right]$ $\displaystyle \le$ $\displaystyle \mathbf{0}$  
$\displaystyle \left[\begin{array}{cc}
-d\left(\omega _1\right)^{T} & -1\\
\vdots & \vdots \\
-d\left(\omega _{K}\right)^{T} & -1\\
d\left(\omega _1\right)^{T} & -1\\
\vdots & \vdots \\
d\left(\omega _{K}\right)^{T} & -1
\end{array}\right]\left[\begin{array}{c}
h\\
\delta \end{array}\right]$ $\displaystyle \le$ $\displaystyle \mathbf{0}.$  

I.e.,

$\displaystyle \zbox {\mathbf{A}_{sb}\left[\begin{array}{c} h\\ \delta \end{array} \right] \le \mathbf{0}.}$ (4.74)


LP Standard Form

Now gather all of the constraints to form an LP problem:

\begin{displaymath}\begin{array}[t]{ll} \mathrm{minimize} & \left[\begin{array}{cccc} 0 & \cdots & 0 & 1\end{array} \right] \left[\begin{array}{c} h\\ \delta \end{array} \right]\\ [5pt] \mbox{subject to} & \begin{array}[t]{l} \left[\begin{array}{cc} d\left(0\right)^{T} & 0\end{array} \right]\left[\begin{array}{c} h\\ \delta \end{array} \right]=1\\ \left[\begin{array}{c} \left[\begin{array}{cc} -\mathbf{I} & \mathbf{0}\end{array} \right]\\ [5pt] \mathbf{A}_{sb}\end{array} \right]\left[\begin{array}{c} h\\ \delta \end{array} \right]\le \mathbf{0}\end{array} \end{array}\end{displaymath} (4.75)

where the optimization variables are $ [h, \delta]^T$ .

Solving this linear-programming problem should produce a window that is optimal in the Chebyshev sense over the chosen frequency samples, as shown in Fig.3.37. If the chosen frequency samples happen to include all of the extremal frequencies (frequencies of maximum error in the DTFT of the window), then the unique Chebyshev window for the specified main-lobe width must be obtained. Iterating to find the extremal frequencies is the heart of the Remez multiple exchange algorithm, discussed in the next section.

Figure 3.37: Normal Chebyshev Window
\includegraphics[width=\twidth,height=6.5in]{eps/print_normal_chebwin}


Remez Exchange Algorithm

The Remez multiple exchange algorithm works by moving the frequency samples each iteration to points of maximum error (on a denser grid). Remez iterations could be added to our formulation as well. The Remez multiple exchange algorithm (function firpm [formerly remez] in the Matlab Signal Processing Toolbox, and still remez in Octave) is normally faster than a linear programming formulation, which can be regarded as a single exchange method [224, p. 140]. Another reason for the speed of firpm is that it solves the following equations non-iteratively for the filter exhibiting the desired error alternation over the current set of extremal frequencies:

$\displaystyle \left[ \begin{array}{c} H(\omega_1) \\ H(\omega_2) \\ \vdots \\ H(\omega_{K}) \end{array} \right] = \left[ \begin{array}{cccccc} 1 & 2\cos(\omega_1) & \dots & 2\cos(\omega_1L) & \frac{1}{W(\omega_1)} \\ 1 & 2\cos(\omega_2) & \dots & 2\cos(\omega_2L) & \frac{-1}{W(\omega_2)} \\ \vdots & & & \\ 1 & 2\cos(\omega_{K}) & \dots & 2\cos(\omega_{K}L) & \frac{(-1)^{K}}{W(\omega_{K})} \end{array} \right] \left[ \begin{array}{c} h_0 \\ h_1 \\ \vdots \\ h_{L} \\ \delta \end{array} \right]$ (4.76)

where $ W(\omega_k)\delta$ is the weighted ripple amplitude at frequency $ \omega_k$ . ( $ W(\omega_k)$ is an arbitrary ripple weighting function.) Note that the desired frequency-response amplitude $ H(\omega_k)$ is also arbitrary at each frequency sample.

Convergence of Remez Exchange

According to a theorem of Remez, $ \delta $ is guaranteed to increase monotonically each iteration, ultimately converging to its optimal value. This value is reached when all the extremal frequencies are found. In practice, numerical round-off error may cause $ \delta $ not to increase monotonically. When this is detected, the algorithm normally halts and reports a failure to converge. Convergence failure is common in practice for FIR filters having more than 300 or so taps and stringent design specifications (such as very narrow pass-bands). Further details on Remez exchange are given in [224, p. 136].

As a result of the non-iterative internal LP solution on each iteration, firpm cannot be used when additional constraints are added, such as those to be discussed in the following sections. In such cases, a more general LP solver such as linprog must be used. Recent advances in convex optimization enable faster solution of much larger problems [22].


Monotonicity Constraint

We can constrain the positive-time part of the window to be monotonically decreasing:

$\displaystyle \Delta h_{i}=h\left(i+1\right)-h\left(i\right)\leq 0\qquad i=1,\ldots ,L-1$ (4.77)

In matrix form,
$\displaystyle \left[\begin{array}{ccccc}
-1 & 1 & & & 0\\
& -1 & 1 & & \\
& & \ddots & \ddots & \\
0 & & & -1 & 1\end{array}\right]h$ $\displaystyle \le$ $\displaystyle 0,$  

or,

$\displaystyle \zbox {\mathbf{D}\,h \le 0.}$ (4.78)

See Fig.3.38.

Figure 3.38: Monotonic Chebyshev Window
\includegraphics[width=\twidth,height=6.5in]{eps/print_monotonic_chebwin}


L-Infinity Norm of Derivative Objective

We can add a smoothness objective by adding $ L-infinity$ -norm of the derivative to the objective function.

$\displaystyle \mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _{\infty }.$ (4.79)

The $ L-infinity$ -norm only cares about the maximum derivative. Large $ \eta $ means we put more weight on the smoothness than the side-lobe level.

This can be formulated as an LP by adding one optimization parameter $ \sigma $ which bounds all derivatives.

$\displaystyle -\sigma \leq \Delta h_{i}\leq \sigma \qquad i=1,\ldots ,L-1.$ (4.80)

In matrix form,
$\displaystyle \left[\begin{array}{r}
-\mathbf{D}\\
\mathbf{D}\end{array}\right]h-\sigma \mathbf1$ $\displaystyle \le$ $\displaystyle 0.$  

Objective function becomes

$\displaystyle \mathrm{minimize}\quad \delta +\eta \sigma .$ (4.81)

The result of adding the Chebyshev norm of diff(h) to the objective function to be minimized ($ \eta =1$ ) is shown in Fig.3.39. The result of increasing $ \eta $ to 20 is shown in Fig.3.40.

Figure: Chebyshev norm of diff(h) added to the objective function to be minimized ($ \eta =1$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_linf_chebwin_1}

Figure: Twenty times the norm of diff(h) added to the objective function to be minimized ($ \eta =20$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_linf_chebwin_2}


L-One Norm of Derivative Objective

Another way to add smoothness constraint is to add $ L1$ -norm of the derivative to the objective:

$\displaystyle \mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _1$ (4.82)

Note that the $ L1$ norm is sensitive to all the derivatives, not just the largest.

We can formulate an LP problem by adding a vector of optimization parameters $ \tau$ which bound derivatives:

$\displaystyle -\tau _{i}\leq \Delta h_{i}\leq \tau _{i}\qquad i=1,\ldots ,L-1.$ (4.83)

In matrix form,

$\displaystyle \left[\begin{array}{r} -\mathbf{D}\\ \mathbf{D}\end{array} \right]h-\left[\begin{array}{c} -\tau \\ -\tau \end{array} \right]\le 0.$ (4.84)

The objective function becomes

$\displaystyle \mathrm{minimize}\quad \delta +\eta \mathbf1^{T}\tau .$ (4.85)

See Fig.3.41 and Fig.3.42 for example results.

Figure: $ L1$ norm of diff(h) added to the objective function ($ \eta =1$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_1}

Figure: Six times the $ L1$ norm of diff(h) added to the objective function ($ \eta =6$ )
\includegraphics[width=\twidth,height=6.5in]{eps/print_lone_chebwin_2}


Summary

This section illustrated the design of optimal spectrum-analysis windows made using linear-programming (linprog in matlab) or Remez multiple exchange algorithms (firpm in Matlab). After formulating the Chebyshev window as a linear programming problem, we found we could impose a monotonicity constraint on its shape in the time domain, or various derivative constraints. In Chapter 4, we will look at methods for FIR filter design, including the window method4.5) which designs FIR filters as a windowed ideal impulse response. The formulation introduced in this section can be used to design such windows, and it can be used to design optimal FIR filters. In such cases, the impulse response is designed directly (as the window was here) to minimize an error criterion subject to various equality and inequality constraints, as discussed above for window design.4.16


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FIR Digital Filter Design
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Fourier Transforms for Continuous/Discrete Time/Frequency