Assignment
1.
a) Obtain the y[n] = x1[n] ⊗ x2[n] for x2[n]= {-2, 1, -3, 7}, x1[n] = {10, 2, -5, 8} where ⊗ denotes the circular convolution.
b) The DTFT of unknown signal sampled by fs = 300 Hz is given in the first row of the following figure, where two peaks belong to 0.4 and 1.6 in the normalized frequency (i.e., normalized by n). What is the frequency of the unknown signal?
c) The 64-points OFT of unknown signal sampled by fs = 100 Hz is given in the second row of above figure, where two peaks belong to 16 and 48 in the k index. What is the frequency of the unknown signal?
2.
y[n] - 3y[n -1] - 10y[n - 2] = x[n]
where x[n] = (1/10)n u[n]
a) Obtain the transfer function H(z).
b) Obtain the impulse response, h[n], of the given system. (Assume |z|>5)
c) Is the system stable? Explain your answer.
d) Is the system causal? Explain your answer.
3.
a) For the given x[n], obtain the X(Ω), which is the DTFT of x[n].
b) For the given x[n] and h·[n], obtain the system output Y·(Ω), which is the DTFT of y[n].
4. The Z-transform of a system in discrete time is given below. Answer to the given questions
X(z) = (1 +z-1) / (2 + z-2)
a) Assuming that the ROC of X(z) includes the unit circle, obtain X(Ω), where X(Ω) is the DTFT of x[n].
b) Calculate X[k] for k=4, where x(11 is the 8-points DFT of x[n].
c) Suppose the given x(n] is a periodic signal of 8 samples, then obtain X‾[k] for k=12 using the relationship between DFT and DFS, where X‾[k] is the DFS of x[n].