Question

The N^th root of unity WN is given as _____________

(a) e^j2πN

(b) e^-j2πN

(c) e^-j2π/N

(d) e^j2π/N

The question was posed to me by my school teacher while I was bunking the class.

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

Answer 1

The correct choice is (c) e^-j2π/N

To explain I would say: We know that the Discrete Fourier transform of a signal x(n) is given as

X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1} x(n) W_N^{kn}\)

Thus we get Nth rot of unity WN= e^-j2π/N