Code: AE-06/ AC-04/ AT-04                                                     Subject: SIGNALS & SYSTEMS

Time: 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 best alternative in the following:                                         (2x10)

a. Let  be a unit step sequence. The sequence can be described as
(A) (B)

(C) (D) 

b. A continuous-time periodic signal , having a period T, is convolved with itself. The resulting signal is

(A) not periodic                                  (B)  periodic having a period T

(C) periodic having a period 2T           (D)  periodic having a period T/2

             c.   If the Fourier series coefficients of a signal are periodic then the signal must  be      

(A) continuous-time, periodic              (B)  discrete-time, periodic

(C)  continuous-time, nonperiodic        (D)  descrete-time, nonperiodic

             d.   The Fourier transform of a signal  is given by 

(A) .                                     (B)

(C) (D)

e. For the function , maximum value of group delay is

(A) 1                                                  (B) 1/2

(C) 2                                                   (D) 3

f. A continuous-time signal  is sampled using an impulse train. In terms of , the Fourier transform of , the spectrum of the sampled signal can be expressed as

(A) (B)

(C) (D)

g. The region of convergence of a causal finite duration discrete-time signal is

(A) the entire z-plane except 

(B) the entire z-plane except 

(C) the entire z-plane                         

(D) a strip in z-plane enclosing -axis

             h.   Let be the frequency response of a discrete-time LTI system, and  be the frequency response of its inverse. Then,

(A) (B)

(C)                     (D)

i.     The transfer function of a stable system is . Its

                                                                 impulse response will be

(A) (B)

(C) (D)

j.    The probability cumulative distribution function must be monotone and 

(A) increasing                                      (B) decreasing       

(C) nonincreasing                                 (D) nondecreasing

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Determine the fundamental frequency of the signal .                    (4)

b. A CT system is described by. Find if the system is time invariant and stable.                                                                     (6)

c. Let be a real signal and. Find a condition so that

(6)          

Q.3     a.   If ,  and  are impulse responses of three LTI systems, determine the impulse response of the system shown in Fig.1.                (6)

Fig. 1                                                              

             b.   Given that , determine  in terms of . Assume that a is real. For what values of a the system will be (i) causal, (ii) stable?                                                             (10)

Q.4           One period of a continuous-time periodic signal is as given below.

             a.   Determine Fourier series coefficients of , assuming its period to be 3.

b.   Determine Fourier series coefficients of the same signal but now, assuming its period to be 6. What is the relationship between the coefficients determined in parts a. and b.?          (10+6)

Q.5       a.    The Fourier transform of a signal x(t) is described as

    and  .

Determine whether x(t) is real or complex.                                                (4)

             b.   Determine the inverse Fourier transform of  using the convolution property of the Fourier transform.                                (4)

c.   A system is described by the difference equation . Find the impulse response of the inverse of this system. From the impulse response, find the difference equation of the inverse system.           (8)

  Q.6     a. Determine the autocorrelation of the sequence .                                     (8)

     b.   Determine the cross correlation of the processes

       and      ,

                   where  is an  independent random variable uniformly distributed over the interval .             (8)

Q.7     a.   A signal signal is sampled by  Determine and sketch the sampled signal and its Fourier transform.                     (8)

             b.   Determine the Fourier transform of                                                                     (8)

(i)        and       (ii)

Q.8     a.   Find the inverse Laplace transform of for all possible ROC.                        (8)

b.   Using Laplace transform, find the forced response and the natural    response of        the system described by . The initial conditions for the system are  and  . Determine the two responses for a step input.                          (8)

Q.9     a.   A causal system is described by. For what values of a and b will the system be (i) unstable, (ii) noncausal?                   (8)

b.   Determine the ROC of , given that .

                   For what relationship between a and b the ROC will be the largest?                    (8)