Question

Consider a discrete time signal x(n) be a real and odd periodic signal with period N = 7 and Fourier series coefficients $X_K$ are given as $X_{15}=j,X_{16}=2j,X_{17}=3j$. Then $X_0+X_{-1}+X_{-2}+X_{-3}$ is

• A. 0
• B. 3j
• C. - 6j
• D. - 3j

Answer 1

The correct option is C - 6j

(d)
The discrete time Fourier series coefficient are periodic with period N.

$X_K=X_{K+N}=X_{K+2N}$

Therefore, for N=7, we have

$X_1=X_8=X_{15}=j$

$X_2=X_9=X_{16}=2j$

$X_3=X_{10}=X_{17}=3j$

Since the given signal x(n) is real and odd, the Fourier series coefficients $X_K$ will be purely imaginary and odd

$X_K=-X_{-K}$

$X_0=0$

$X_{-1}=-X_1=-j$

$X_{-2}=-X_2=-2j$

$X_{-3}=-X_3=-3j$

$\therefore X_0+X_1+X_2+X_{-3}=-6j$