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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 response8.2), 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 remez and cremez), 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.

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