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Lyons, Richard G - Understanding Digital Signal Processing

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```A description of Parks and McClellan\u2019s revolutionary design method was eventually
published in reference
[17]. That story is reminiscent of when Decca Records auditioned a group of four young musicians in 1961.
Decca executives decided not to sign the group to a contract. You may have heard of that musical group\u2014they
were called the Beatles.
5.7 Half-band FIR Filters
There\u2019s a specialized FIR filter that\u2019s proved very useful in signal decimation and interpolation applications[
21\u201325]. Called a half-band FIR filter, its frequency magnitude response is symmetrical about the fs/4 point as
shown in Figure 5-33(a). As such, the sum of fpass and fstop is fs/2. When the filter has an odd number of taps,
this symmetry has the beautiful property that the filter\u2019s time-domain impulse response has every other filter
coefficient being zero, except the center coefficient. This enables us to avoid approximately half the number of
multiplications when implementing this kind of filter. By way of example, Figure 5-33(b) shows the
coefficients for a 31-tap half-band filter where \u394f was defined to be approximately fs/32 using the Parks-
McClellan FIR filter design method.
Figure 5-33 Half-band FIR filter: (a) frequency magnitude response [transition region centered at fs/4]; (b) 31-
tap filter coefficients; (c) 7-tap half-band filter structure.
Notice how the alternating h(k) coefficients are zero, so we perform 17 multiplications per output sample
instead of the expected 31 multiplications. Stated in different words, we achieve the performance of a 31-tap
filter at the computational expense of only 17 multiplies per output sample. In the general case, for an N-tap
half-band FIR filter, we\u2019ll only need to perform (N + 1)/2 + 1 multiplications per output sample. (
Section 13.7 shows a technique to further reduce the number of necessary multiplies for linear-phase tapped-
delay line FIR filters, including half-band filters.) The structure of a simple seven-coefficient half-band filter is
shown in Figure 5-33(c), with the h(1) and h(5) multipliers absent.
Be aware, there\u2019s a restriction on the number of half-band filter coefficients. To build linear-phase N-tap half-
band FIR filters, having alternating zero-valued coefficients, N + 1 must be an integer multiple of four. If this
restriction is not met, for example when N = 9, the first and last coefficients of the filter will both be equal to
zero and can be discarded, yielding a 7-tap half-band filter.
On a practical note, there are two issues to keep in mind when we use an FIR filter design software package to
design a half-band filter. First, assuming that the modeled filter has a passband gain of unity, ensure that your
filter has a gain of 0.5 (\u22126 dB) at a frequency of fs/4. Second, unavoidable numerical computation errors will
yield alternate filter coefficients that are indeed very small but not exactly zero-valued as we desire. So in our
filter modeling efforts, we must force those very small coefficient values to zero before we proceed to analyze
half-band filter frequency responses.
You might sit back and think, \u201cOK, these half-band filters are mildly interesting, but they\u2019re certainly not worth
writing home about.\u201d As it turns out, half-band filters are very important because they\u2019re widely used in
applications with which you\u2019re familiar\u2014like pagers, cell phones, digital receivers/televisions,
5.8 Phase Response of FIR Filters
Although we illustrated a couple of output phase shift examples for our original averaging FIR filter in
Figure 5-10, the subject of FIR phase response deserves additional attention. One of the dominant features of
FIR filters is their linear phase response which we can demonstrate by way of example. Given the 25 h(k) FIR
filter coefficients in Figure 5-34(a), we can perform a DFT to determine the filter\u2019s H(m) frequency response.
The normalized real part, imaginary part, and magnitude of H(m) are shown in Figures 5-34(b) and 5-34(c),
respectively.\u2020 Being complex values, each H(m) sample value can be described by its real and imaginary parts,
or equivalently, by its magnitude |H(m)| and its phase Hø(m) shown in Figure 5-35(a).
\u2020 Any DFT size greater than the h(k) width of 25 is sufficient to obtain H(m). The h(k) sequence was padded with 103 zeros to take a
128-point DFT, resulting in the H(m) sample values in Figure 5-34.
Figure 5-34 FIR filter frequency response H(m): (a) h(k) filter coefficients; (b) real and imaginary parts of H
(m); (c) magnitude of H(m).
Figure 5-35 FIR filter phase response Hø(m) in degrees: (a) calculated Hø(m); (b) polar plot of Hø(m)\u2019s first ten
phase angles in degrees; (c) actual Hø(m).
The phase of a complex quantity is, of course, the arctangent of the imaginary part divided by the real part, or ø
= tan \u22121(imag/real). Thus the phase of Hø(m) is determined from the samples in
Figure 5-34(b).
The phase response in Figure 5-35(a) certainly looks linear over selected frequency ranges, but what do we
make of those sudden jumps, or discontinuities, in this phase response? If we were to plot the angles of Hø(m)
starting with the m = 0 sample on a polar graph, using the nonzero real part of H(0), and the zero-valued
imaginary part of H(0), we\u2019d get the zero-angled Hø(0) phasor shown on the right side of Figure 5-35(b).
Continuing to use the real and imaginary parts of H(m) to plot additional phase angles results in the phasors
going clockwise around the circle in increments of \u221233.75°. It\u2019s at the Hø(6) that we discover the cause of the
first discontinuity in Figure 5-35(a). Taking the real and imaginary parts of H(6), we\u2019d plot our phasor oriented
at an angle of \u2212202.5°. But Figure 5-35(a) shows that Hø(6) is equal to 157.5°. The problem lies in the software
routine used to generate the arctangent values plotted in Figure 5-35(a). The software adds 360° to any negative
angles in the range of \u2212180° > ø \u2265 \u2212360°, i.e., angles in the upper half of the circle. This makes ø a positive
angle in the range of 0° < ø \u2264 180° and that\u2019s what gets plotted. (This apparent discontinuity between Hø(5) and
Hø(6) is called phase wrapping.) So the true Hø(6) of \u2212202.5° is converted to a +157.5° as shown in
parentheses in Figure 5-35(b). If we continue our polar plot for additional Hø(m) values, we\u2019ll see that their
phase angles continue to decrease with an angle increment of \u221233.75°. If we compensate for the software\u2019s
behavior and plot phase angles more negative than \u2212180°, by unwrapping the phase, we get the true Hø(m)
shown in Figure 5-35(c).
Notice that Hø(m) is, indeed, linear over the passband of H(m). It\u2019s at Hø(17) that our particular H(m)
experiences a polarity change of its real part while its imaginary part remains negative\u2014this induces a true
phase-angle discontinuity that really is a constituent of H(m) at m = 17. (Additional phase discontinuities occur
each time the real part of H(m) reverses polarity, as shown in Figure 5-35(c).) The reader may wonder why we
care about the linear phase response of H(m). The answer, an important one, requires us to introduce the notion
of group delay.
Group delay is defined as the negative of the derivative of the phase with respect to frequency, or G = \u2212dø/df.
For FIR filters, then, group delay is the slope of the Hø(m) response curve. When the group delay is constant, as
it is over the passband of all FIR filters having symmetrical coefficients, all frequency components of the filter
input signal are delayed by an equal amount of time G before they reach the filter\u2019s output. This means that no
phase distortion is induced in the filter\u2019s desired output signal, and this is crucial in communications signals.
For amplitude modulation (AM) signals, constant group delay preserves the time waveform shape of the signal\u2019
s modulation```
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