Topic: Signals and Systems
Consider the system $R[m] = \sum_{n=0}^{N-1} y[n]x[n-m]$ where $y[n]$ and $x[n]$ are real periodic signals with period N. The above output can be obtained using
- A. $IFFT\{FFT[Y] \times conj(FFT[X])\}$
- B. $IFFT\{FFT[Y] \times FFT[X]\}$
- C. $IFFT\{FFT[Y] \times (FFT[-X])\}$
- D. $IFFT\{FFT[-Y] \times (FFT[X])\}$
Answer: A. $IFFT\{FFT[Y] \times conj(FFT[X])\}$
Explanation: The given expression defines the circular cross-correlation of two periodic signals $y[n]$ and $x[n]$. In the frequency domain, circular cross-correlation corresponds to the multiplication of the DFT of the first signal by the complex conjugate of the DFT of the second signal. Thus, $R(m) = IFFT\{FFT[Y] \times conj(FFT[X])\}$.