We describe error correction using symmetry of the discrete

We describe error correction using symmetry of the discrete Fourier transform (DFT). You wish to send a real-valued message, X[k], of length 12 to your friend. To do this, take the following steps: First, zero pad the message with four zeros to length 16. Take the inverse DFT of the zero-padded (X[k])_k = 0^15 to obtain (x[n])_n = 0^15. Then, send (x[n])_n = 0^15. across a communication channel to your friend. The channel corrupts x[n] with additive noise at exactly two values of n, i.e., (y[n])_n = 0^15 = x[n] + e[n] where e[n] is a length-16 signal that is non-zero at only two locations. Is it possible for your friend to recover the original message, X[k], from the received sequence y[n]? If

Solution

Yes. It is possible to recover the original message X[k] from the received sequence y[n].