(a) A particular discrete-time system has input x[n] and output y[n]. The Fourier transforms of these signals are related by the equation
(i) Is the system linear? Clearly justify your answer. (ii) Is the system time invariant? Clearly justify your answer. (iii) What is y[n] if x[n]=δ[n]? (b) Consider a discrete-time system for which the transform
of the output is related to the transform of the input through the relation
. Find an expression for y[n] in terms of x[n].
Step 11 of 6
a) i) Consider the signal:
where and are constants. Then using linearity property of Fourier transform, Fourier transform of is:
Let the response of the system to the signal is and to the signal is . Then:
and for the signal , Fourier transform is:
We proved that system is linear!