Recall from Section 7.5 in Chapter 7 on the "Frequency Sampling Design Method":

(E.1)

where h( n), 0 ? n ? N ? 1, is the causal impulse response that approximates the finite impulse response (FIR) filter, and H( k), 0 ? k ? N ? 1, represents the corresponding coefficients of the discrete Fourier transform (DFT), and W _N = . We further write DFT coefficients H( k), 0 ? k ? N ? 1, into the polar form:

(E.2)

where H _k and ? _k are the kth magnitude and the phase angle, respectively. The frequency response of the FIR filter is expressed as

(E.3)

Substituting (E.1) into (E.3) yields

(E.4)

Interchanging the order of the summation in Equation (E.4) leads to

(E.5)

Since ,

and using the identity ,

we can write the second summation in Equation (E.5) as

(E.6)

Using the Euler formula leads Equation (E.6) to

(E.7)

Substituting Equation (E.7) into Equation (E.5) leads to

(E.8)

Let ? = ? _m = , and substituting it into Equation (E.8) we get

(E.9)

Clearly, when m ? k, the last term of the summation in Equation (E.9) becomes

When m = k, and using L'Hospital's rule, we have

Then Equation (E.9) is simplified to

(E.10)

where ? _k = , corresponding to the kth DFT frequency component. The fact is that if we specify the desired frequency response,