Lowpass Filter Design

We have discussed in detail (Chapter 1) the simplest lowpass filter, $y(n) = x(n) + x(n - 1)$ having the transfer function $H(z)=1+z^{-1}$ with one zero at $z = -1$ and one pole at $z=0$ . From the graphical method for visualizing the amplitude response (§8.2), we see that this filter totally rejects signal energy at half the sampling rate, while lower frequencies experience higher gains, reaching a maximum at $\omega=0$ . We also see that the pole at $z=0$ has no effect on the amplitude response.

A high quality lowpass filter should look more like the ``box car'' amplitude response shown in Fig.1.1. While it is impossible to achieve this ideal response exactly using a finite-order filter, we can come arbitrarily close. We can expect the amplitude response to improve if we add another pole or zero to the implementation.

Perhaps the best known ``classical'' methods for lowpass filter designs are those derived from analog Butterworth, Chebyshev, and Elliptic Function filters [64]. These generally yield IIR filters with the same number of poles as zeros. When an FIR lowpass filter is desired, different design methods are used, such as the window method [68, p. 88] (Matlab functions fir1 and fir2), Remez exchange algorithm [68, pp. 136-140], [64, pp. 89-106] (Matlab functions firpm, firpmord, and cfirpm), linear programming [93], [68, p. 140], and convex optimization [67]. This section will describe only Butterworth IIR lowpass design in some detail. For the remaining classical cases (Chebyshev, Inverse Chebyshev, and Elliptic), see, e.g., [64, Chapter 7] and/or Matlab/Octave functions butter, cheby1, cheby2, and ellip.