When a periodic pseudorandom sequence of length N is used to adjust the coefficients of...

When a periodic pseudorandom sequence of length N is used to adjust the coefficients of an N-tap linear equalizer, the computations can be performed efficiently in the frequency domain by use of the discrete Fourier transform (DFT). Suppose that {yn} is a sequence of N received samples (taken at the symbol rate) at the equalizer input. Then the computation of the equalizer coefficients is performed as follows.

a. Compute the DFT of one period of the equalizer input sequence {yn}, i.e.,

b. Compute the desired equalizer spectrum

where {Xi } is the precomputed DFT of the training sequence.

c. Compute the inverse DFT of {Ck } to obtain the equalizer coefficients {cn}. Show that this procedure in the absence of noise yields an equalizer whose frequency response is equal to the frequency response of the inverse folded channel spectrum at the N uniformly spaced frequencies fk = k/NT, k = 0, 1, . . . , N −1.