Topic: Networks, Signals and Systems
Which of the following is NOT TRUE, given that "$\rightarrow$" represents fourier transform, "*" represents convolution, $x(n)\rightarrow X(e^{j\omega})$ and $y(n)\rightarrow Y(e^{j\omega})$
- A. $ax(n)+by(n)\rightarrow aX(e^{j\omega})+bY(e^{j\omega})$
- B. $x(n-n_{d})\rightarrow e^{j\omega n_{d}}(e^{j\omega})$
- C. $e^{jw_{0}}x(n)\rightarrow X(e^{j(w-w_{0})})$
- D. $x(n)*y(n)\rightarrow X(e^{j\omega}).Y(e^{j\omega})$
Answer: B. $x(n-n_{d})\rightarrow e^{j\omega n_{d}}(e^{j\omega})$
Explanation: According to the time-shifting property of the Discrete-Time Fourier Transform (DTFT), shifting a signal in time by $n_d$ samples introduces a linear phase shift in the frequency domain. The correct property is $x(n - n_d) \leftrightarrow e^{-j\omega n_d} X(e^{j\omega})$. Option (b) has a positive exponent and is notationally incorrect, making it the false statement.