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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
$$ IFFT\left\{FFT[Y] \times conj(FFT[X])\right\}$$
$$IFFT\left\{FFT[Y] \times FFT[X]\right\}$$
$$IFFT\left\{FFT[Y] \times (FFT[-X])\right\}$$
$$IFFT\left\{FFT[-Y] \times (FFT[X])\right\}$$
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