AMIETE – ET/CS/IT (OLD SCHEME)

 

Code: AE06/ AC04/ AT04                                                         Subject: SIGNALS & SYSTEMS

Flowchart: Alternate Process: DECEMBER 2009Time: 3 Hours                                                                                                    Max. Marks: 100

 

 

 

NOTE: There are 9 Questions in all.

·      Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

·      Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

·      Any required data not explicitly given, may be suitably assumed and stated.

 

 

Q.1       Choose the correct or the best alternative in the following:                                  (210)

 

a.        dt is
(A)                                                (B)  1

(C)                                                                                                                     (D)  0

 

b.      The signal  is

 

(A)    Linear, causal, stable                    (B)  Non linear, causal, stable

(C) Non linear, non causal, unstable     (D)  Linear, non causal, stable

     

             c.   The even part of the signal  is                  

(A)                                             (B) 

(C)                                              (D) 

 

             d.   The area under Gaussian pulse  is  

 

(A)    unity                                            (B)  infinity

(C)  pulse                                           (D) Gaussian pulse

 

e.       A sequence x(n) is said to be causal if ROC of its z-transform X(z) is

 

 

                                                    

(A)     outside the unit circle.                   (B)  within the unit circle.

(C)  on the unit circle.                          (D) ROC cannot be defined for causal   

                                                                        systems.

 

f.        The energy of the  sequence  is

(A)                                                     (B) 

(C)                                                  (D) 

 

g.       The Laplace transform of  is equal to

  

(A)                                            (B)                                 

(C)                                       (D) 

 

             h.   The autocorrelation function of an energy signal has

                                                                                                                                                                   

(A)    no symmetry                                 (B) conjugate symmetry                                              

(C) odd symmetry                               (D) even symmetry            

i.    The inverse z transform of the function  whose ROC is  is

                                                                  

(A)    u(n)                                             (B)  u(n–1)

(C)  –u(–n–1)                                      (D)  –u(n–1)

 

j.    A Continuous Random Variable X has a pdf .  Find the value of k.  

(A)  1                                                  (B)      

(C)                                                (D)  3

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

  Q.2     a.   Find the energy or power of the following signals

                  

            

                                                                                     (7)

 

             b.   Check the stability of the following systems whose impulse response is given as

                   (i) 

                   (ii)                                                                                (5)

       

c.       Check the linearity, causality of the systems:

(i)                                    (ii)                                        (4)          

 

  Q.3     a.   Find the exponential Fourier series of the following function:                                (8)
 
 

 

 

 

 

 

 


       

             b.   Find the Fourier series of a periodic impulse train whose magnitude is A and period .                 (8)

                                                                                                                                                                                                 

  Q.4     a.   Find the Fourier transform of the following functions:                                           (8)

 
 
                                                                                                                                                  

 

 

 

                   (i)   

 

 

 
 
 

 

 


                   (ii)      

 

 

 

 

 

             b.   Show that the Fourier transform of a periodic impulse train is another impulse train.               (8)  

                                                                             

Q.5       a.    State and prove the following properties of Discrete Time Fourier Transform:

 

(i)                  Time shifting and Frequency shifting.

(ii)                Conjugate symmetry.

(iii)               Time reversal.                                                                         (12)

            

             b.   A causal LTI system is characterised by the difference equation

                   .  Find the impulse response of the system using DTFT.                                                                  (4)

 

  Q.6     a.   State and prove the Sampling Theorem.                                                            (10)

 

             b.   Derive the step response of a first order discrete time system described by the difference equation , with .   (6)

                                      

  Q.7     a.   Find the Laplace transform of the following functions:                                             

                   (i)                                  (ii)                            (8)

 

             b.   State and explain the Initial Value Theorem in Laplace transform. Using initial value theorem find the initial value of the signal corresponding to the Laplace transform  and verify the same.      (8)

                  

  Q.8     a.   Determine the inverse z-transform of the function  if the ROCs are                  

                   (i)                                              (ii) 

                   (iii)                                                                                                    (8)

 

b.   Consider an LTI system for which the input x[n] and output y[n] satisfy the linear constant coefficient difference equation

                                                                          

      State whether the system is stable, causal by finding the impulse response by using z transform method.                                                                             (8)

 

  Q.9     a.   Consider a random variable X defined by (assuming b > a)                                     

                  

                   Evaluate mean and variance of X.                                                                       (8)

 

             b.   For a given sinusoidal wave , calculate the power spectral density, average power and autocorrelation.                             (8)