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U. Glaeser

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FIGURE 24.2 Effects of windowing an FIR. Note that the windowed spectrum is smoothed and has an increased<br />

transition bandwidth.<br />

The frequency response of an N th -order IIR is given by<br />

evaluated over the normalized frequency range ϖ ∈[−π ≤ ω ≤ π), which defines the baseband frequency<br />

range bounded by ±f Nyquist.<br />

The traditional IIR design strategy is based on converting classic analog filter models into their digital<br />

filter equivalents. Throughout the first half of the 20th century, analog radio filter engineers created classic<br />

Bessel, Butterworth, Chebyshev, and Elliptic (Cauer) filter instantiations whose magnitude frequency<br />

response emulates that of an ideal filter. To standardize the analog filter design procedures, a set of<br />

normalized −1 dB or −3 dB lowpass filter models, having a 1.0 rad/s passband were created. These models<br />

were reduced to tables, charts, and graphs and are called analog prototype filters. The prototype lowpass<br />

filters can be the frequency scaled to define an analog lowpass, highpass, bandpass, and bandstop filters<br />

H(s), having desired frequency-domain attributes. The classic analog filter H(s) can then be converted into<br />

a digital filter model H(z) to define a classic digital filter (See Table 24.4). The basic domain conversion<br />

techniques (i.e., H(s) → H(z)) are (1) the impulse-invariant and (2) bilinear z-transform methods.<br />

The impulse invariance filter design method results in a digital filter having an impulse response h[k] that<br />

agrees with that of the parent analog filter’s impulse response h a(t), up to a scale factor (h d[k] = T s h a(kT s)).<br />

An impulse invariant design can be of significant value in applications, such as automatic control, where<br />

design objectives are defined in the time-domain (e.g., rise-time, overshoot, settling time). If the parent<br />

analog filter’s impulse response h a(t), or transfer function H a(s) are known, then the impulse invariant<br />

digital filter is defined by<br />

and in the frequency-domain by<br />

© 2001 by CRC Press LLC<br />

H e jϖ<br />

( ) bie −jϖi<br />

/ aie −jϖi<br />

M<br />

N<br />

jϖ N M<br />

∑ ∑ Ke – ( )<br />

( ) e jϖ<br />

= = ∏ –<br />

i=0<br />

N<br />

i=1<br />

a i<br />

i=0<br />

N<br />

i=1<br />

( zi)/ e jϖ<br />

M−1<br />

N−1<br />

i=0<br />

∏<br />

i=0<br />

( – )<br />

z<br />

ha() t ⇔ Ha() s ---------------- ⎛ 1<br />

∑<br />

---- ⎞ ai ( s+ pi) ⎝T⎠ s 1 e piT – s<br />

z 1 –<br />

------------------------------- ⎛ 1<br />

∑<br />

---- ⎞ 1<br />

= = H( z)<br />

⇔ ⎛---- ⎞ h[ k]<br />

⎝<br />

( + )<br />

T ⎠ ⎝ s<br />

T ⎠ s<br />

H e jΩ<br />

( )<br />

⎛ 1<br />

---- ⎞ H<br />

⎝ ⎠ a j Ω 2πk<br />

=<br />

⎛ ⎛---- – -------- ⎞⎞<br />

∑ ⎝ ⎝ T ⎠⎠<br />

T s<br />

∞<br />

k= – ∞<br />

T s<br />

p i

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