## The Ideal Lowpass Filter

Consider the*ideal*lowpass filter, depicted in Fig.4.1.

An ideal lowpass may be characterized by a gain of 1 for all frequencies below some

*cut-off frequency*in Hz, and a gain of 0 for all higher frequencies.

^{5.2}The impulse response of the ideal lowpass filter is easy to calculate:

where denotes the normalized cut-off frequency in radians per sample. Thus, the impulse response of an ideal lowpass filter is a

*sinc*function. Unfortunately, we cannot implement the ideal lowpass filter in practice because its impulse response is

*infinitely long*in time. It is also

*noncausal*; it cannot be shifted to make it causal because the impulse response extends all the way to time . It is clear we will have to accept some sort of compromise in the design of any practical lowpass filter. The subject of

*digital filter design*is generally concerned with finding an

*optimal approximation*to the desired frequency response by minimizing some norm of a prescribed

*error criterion*with respect to a set of practical filter coefficients, perhaps subject also to some

*constraints*(usually linear equality or inequality constraints) on the filter coefficients, as we saw for optimal window design in §3.13.

^{5.3}In

*audio*applications, optimality is difficult to define precisely because

*perception*is involved. It is therefore valuable to consider also

*suboptimal*methods that are ``close enough'' to optimal, and which may have other advantages such as extreme simplicity and/or speed. We will examine some specific cases below.

**Next Section:**

Lowpass Filter Design Specifications

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Window Design by Linear Programming