Question

A discrete-time signal x[n] = \(x\left[ n \right] = {e^{j\left( {\frac{{5\pi }}{3}} \right)n}} + {e^{j\left( {\frac{{\pi }}{4}} \right)n}}\) is down-sampled to the signal x_d [n] such that x_d [n] = x[4n]. The fundamental period of the down-sampled signal x_d [n] is __________.

Answer 1

\(x\left[ n \right] = {e^{j\left( {\frac{{5\pi }}{3}} \right)n}} + {e^{j\left( {\frac{\pi }{4}} \right)n}}\)

x_d[n] = x[4n]

\({x_d}\left[ n \right] = {e^{j\left( {\frac{{20\pi }}{3}} \right)n}} + {e^{j\pi n}}\)

\({n_1} = \frac{{20\pi }}{3},{n_2} = \pi\)

ω₀ = GCD (n₁, n₂)

\(= GCD\left( {\frac{{20\pi }}{3},\frac{{3\pi }}{3}} \right)\)

\(= \frac{\pi }{3}\)

Fundamental period (N) \(= \frac{{2\pi }}{{{\omega _0}}} = \frac{{2\pi }}{{\pi /3}} = 6\)