Question

What is the energy of a discrete time signal in terms of X(ω)?

(a) \(2π\int_{-π}^π |X(ω)|^2 dω\)

(b) \(\frac{1}{2π} \int_{-π}^π |X(ω)|^2 dω\)

(c) \(\frac{1}{2π} \int_0^π |X(ω)|^2 dω\)

(d) None of the mentioned

I have been asked this question in an interview.

My question is based upon Frequency Analysis of Discrete Time Signal in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

The correct choice is (b) \(\frac{1}{2π} \int_{-π}^π |X(ω)|^2 dω\)

For explanation I would say: We know that, Ex=\(\sum_{n=-∞}^∞ |x(n)|^2\)

=\(\sum_{n=-∞}^∞ x(n).x^*(n)\)

=\(\sum_{n=-∞}^∞ x(n)\frac{1}{2π} \int_{-π}^π X^*(ω) e^{-jωn} dω\)

=\(\frac{1}{2π} \int_{-π}^π|X(ω)|^2 dω\)