Question

Which of the following is the correct expression for h(n) in terms of H(k+α)?

(a) \(\frac{1}{M} \sum_{k=0}^{M-1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M-1

(b) \(\sum_{k=0}^{M-1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M-1

(c) \(\frac{1}{M} \sum_{k=0}^{M+1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M+1

(d) \(\sum_{k=0}^{M+1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M+1

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Answer 1

Correct choice is (a) \(\frac{1}{M} \sum_{k=0}^{M-1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M-1

The best explanation: We know that

H(k+α)=\(\sum_{n=0}^{M-1} h(n)e^{-j2π(k+α)n/M}\); k=0,1,2…M-1

If we multiply the above equation on both sides by the exponential exp(j2πkm/M), m=0,1,2….M-1 and sum over k=0,1,….M-1, we get the equation

h(n)=\(\frac{1}{M} \sum_{k=0}^{M-1}H(k+α)e^{j2π(k+α)n/M}\); n=0,1,2…M-1