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U. Glaeser

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TABLE 24.4 Comparisons of Nth-Order Classic IIR Lowpass Filters Having f s = 50 kHz, a − 3 dB 15 kHz<br />

Passband, 5 kHz Transition Band, and −50 dB Stopband<br />

Type Order Passband Stopband Magnitude Frequency Response<br />

Butterworth High<br />

(N = 8)<br />

Chebyshev I Medium<br />

(N = 5)<br />

Chebyshev II Medium<br />

(N = 5)<br />

Elliptic Low<br />

(N = 4)<br />

This equation exhibits a weakness of the impulse invariant design method. For any physically meaningful<br />

sampling rate f s = 1/T s, aliasing errors can occur whenever the analog filter passes components at frequencies<br />

greater that 2f s. Typically, analog filters have a gain that is finite for all frequencies. The aliased<br />

filter energy can be mapped back into the baseband and can distort (sometime significantly) the frequency<br />

response of an impulse invariant filter. As a result, the impulse invariant method is generally only used<br />

to design frequency selective filters that are decidedly lowpass.<br />

When meeting frequency domain specifications is the design objective, the bilinear z-transform method<br />

is normally used. The bilinear z-transform maps a classic analog filter H a(s) into a digital filter H(z)<br />

without introducing aliasing errors. The bilinear z-transform establishes a relationship between the s- and<br />

z-domain, given by s = (2/T s)(z + 1)/(z − 1). The bilinear z-transform also defines an algebraic connection<br />

between the analog and digital frequency axis given by ω = (2/T s)tan(ϖ/2), where ω is the analog<br />

frequency, |ω| < ∞, and ϖ is the normalized digital frequency range −π/2 ≤ ϖ < π, corresponding to the<br />

© 2001 by CRC Press LLC<br />

Smooth Smooth<br />

Ripple Smooth<br />

Smooth Ripple<br />

Ripple Ripple

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