U. Glaeser

frequency range ±f Nyquist = ±f s/2. The mapping from analog frequencies ω to digital frequencies ϖ is called
warping, and pre-warping in the opposite direction. The bilinear z-transform design paradigm is a
multistep process consisting of the following steps:
1. Define the digital filter’s frequency-domain attributes (gains at critical frequencies).
2. Prewarp the critical digital frequencies ϖ into analog frequencies ω.
3. Design a prewarped classic analog filter H a(s) that meets specified passband and stopband gain
requirements.
4. Apply the bilinear z-transform to convert H a(s) into a digital filter H(z). In the process, the
prewarped analog filter frequencies ω will be warped back to their original locations ϖ.
Multirate Systems
DSP systems that contain multiple sample rates are called multirate systems. A signal x[k], sampled at a
rate f in, is said to be decimated by M if it is exported at a rate f out = f in/M, where M > 1. Mathematically,
the decimated signal x d[k] can be expressed as x d[k] = x[Mk], indicating that only every Mth sample of
the fast sampled time-series x[k] is retained in the decimated signal x d[k]. Decimation can also be modeled
in the z-transform domain as X d(z) = X(z M ) and X d(e jφ ) = X d(e jMφ ) in the frequency domain, as suggested
in Fig. 24.3. In order to insure that a signal x[k] can be reconstructed from its decimated samples of
x d[k], Shannon’s sampling theorem must be obeyed. Specifically, if the minimum sampling frequency is
bounded by f s > 2B Hz, the maximum decimation rate must be bounded by M ≤ f s/2B. Decimation is
routinely found in audio signal and video data transmission and signal compression applications, and
interfacing equipment with dissimilar fixed sample rates. By reducing the system’s sample rate by a factor
M, arithmetic bandwidth requirements can often be reduced by a similar amount.
Interpolation is the antithesis of decimation. While decimation is used to reduce the sampling rate,
interpolation is used to increase the sample rate. A signal x[k], sampled at a rate f in, is said to be interpolated
by N if x i[k] = x[k] whenever k ≡ 0 modulo (N), and zero elsewhere. The interpolated signal
x i[k] is a time-series consisting of N − 1 zeros separated by the sample values x[k] and x[k + N] and is
clocked at a rate f out = Nf in N > 1. In the z-transform domain, X i(z) = X(z N ), and X i(e jφ ) = X(e jNφ ) in the
frequency-domain, as shown in Fig. 24.3. It can be noted that the interpolated spectrum contains
multiple copies of the baseband spectrum X(e jφ ), where the unwanted copies can be removed using a
lowpass filter.
FIGURE 24.3 Multirate system elements showing decimation (top) and interpolation (bottom).
© 2001 by CRC Press LLC