(Solved): (6 points, 2 points each) Properties of Convolution (a) First show that convolution in time-domain ...

(6 points, 2 points each) Properties of Convolution (a) First show that convolution in time-domain is equal to multiplication in frequency domain, prove that commutativity property for two real signals \( \mathrm{x} \) and \( \mathrm{y} \). That is to say, prove \[ x[n] \circledast y[n]=y[n] \circledast x[n] \] (b) Prove the same commutativity property using the DTFT, i.e., prove \[ \operatorname{DTFT}\{x[n] \circledast y[n]\}=\operatorname{DTFT}\{y[n] \circledast x[n]\} \] (c) Convolution is also a linear operator on each of its elements. Prove that for purely real signals \( \mathrm{x}, \mathrm{y} \) and \( \mathrm{z} \), \[ (x+y) \circledast z=(x \circledast z)+(y \circledast z) \]