## 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