What if the frequencies are equally spaced between

Then

Inside Collection (Course): Digital Signal Processing: A User's Guide

Given a desired frequency response, the frequency sampling
design method designs a filter with a frequency response
*exactly* equal to the desired response at a
particular set of frequencies

Desired Response must incluce linear phase shift (if linear
phase is desired)

What is

This set of linear equations can be written in matrix form

or

What if the frequencies are equally spaced between

Then

Removing linear phase from both sides yields
*
real-valued
*
simultaneous linear equations to solve for

To yield real coefficients,

Simlar equations exist for even lengths, anti-symmetric, and

This method is simple conceptually and very efficient for
equally spaced samples, since

*exactly* through the sample points, but it
may be very far off from the desired response for

Possible solution to this problem: specify more frequency samples than degrees of freedom, and minimize the total error in the frequency response at all of these samples.

For the samples

For

Here we will consider the

To minimize the

The minimum error norm solution is well known to be

Extended frequency sampled design discourages radical
behavior of the frequency response between samples for
sufficiently closely spaced samples. However, the actual
frequency response may no longer pass exactly through
*any* of the
H
d
ω
k
H
d
ω
k
.

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