EE3660-Homework 1 Solved

1.  Explain whether the following systems are linear, time-invariant, causal, and stable.

(a)      y[n] = x[-n]

(b)      y[n] = cos(πn)𝑥[𝑛]

                   

 

2.  A linear system L generates output signals: y1[n], y2[n] in response to the input signals x1[n], x2[n] respectively. Explain whether the system L is time invariant.

  

 

                                                                       x1[n] y1[n] 

 

 

 

                                                                       x2[n] y2[n] 

 

 

3.  (12%) Given the z-transform pair x[n] ↔ X(z) =   with ROC: |z| < 2, use the z-transform properties to determine the z-transform of the following sequences:

 

(a)      y[n] = (1)n x[n]

3

(b)      y[n] = x[n]*x[-n] ( * denotes convolution) 

(c)      y[n] = nx[n]

4.  A causal LTI system has impulse response h[n], for which the z-transform is

 

1 + 𝑧−1

 

𝐻(𝑧) = (1 − 0.5𝑧−1)(1 + 0.25𝑧−1)

     

(a)      Draw the pole-zero plot of H(z) and specify its ROC.

(b)      Explain whether the system is stable?

(c)      Find the impulse response h[n] of the system.

 

5.  Use the method of partial fraction expansion to determine the sequences corresponding to the following z-transforms:

 

(a)       X(z) = 𝑧3+ 2𝑧2𝑧+ 5𝑧+ 1 , |z| 1.

                                                           4       4

(b)      X(z) = (𝑧2−𝑧 1)2 , |z| <0.5.

3

 

6.  A function called autocorrelation for a real-valued, absolutely summable sequence x[n], is defined as

 

𝑟𝑥𝑥[ℓ] ≜ ∑𝑛 𝑥[𝑛]𝑥[𝑛 − ℓ].  

 

Let X(z) be the z-transform of x[n] with ROC α < |z| < β.

 

(a)      Show that the z-transform of rxx[ℓ] is given by Rxx(z) = X(z)X(z−1).

(b)      Let x[n] = anu[n], |a| < 1. Determine Rxx(z) and sketch its pole-zero plot and the

ROC.

 

7.  Determine the DTFT of following signals:

 

                                              1    𝑛                𝜋𝑛

(a)      x1[n] = ( )          𝑐𝑜𝑠 (     ) 𝑢[𝑛 − 2]

                                              4                        4

(b)      x3[n] = 𝑠𝑖𝑛(0.1𝜋𝑛)(𝑢[𝑛] − 𝑢[𝑛 − 10])

8.  Let x[n] and y[n] denote complex sequences and X(ejω) and Y(ejω) their respective

Fourier transforms

 

(a)      Determine, in terms of x[n] and y[n], the sequence whose Fourier transform is  

X(ejω)Y*(ejω).  

 

(b)      Using the result in part (a), show that   

 .    (eq.7b)

                                                                               

 

(c)      Using (eq.7b), determine the value of the sum

 𝜋𝑛/3)

 𝜋𝑛