Question

Which of the following is true?

(a) \(W_N^*=\frac{1}{N} W_{N^{-1}}\)

(b) \(W_N-1=\frac{1}{N} W_{N^*}\)

(c) \(W_N-1=W_{N^*}\)

(d) None of the mentioned

This question was posed to me in final exam.

This is a very interesting question from Sampling Rate Conversion by a Rational Factor I/D in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Right option is (b) \(W_N-1=\frac{1}{N} W_{N^*}\)

For explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that XN = WN.xN

By pre-multiplying both sides by WN^-1, we get

xN=WN-1.XN

But we know that the inverse DFT of XN is defined as

xN=1/N*XN

Thus by comparing the above two equations we get

WN-1=1/N WN*