AMIETE – ET/CS/IT (NEW SCHEME)   –   Code: AE57/AC57/AT57

Subject: SIGNALS AND 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 the best alternative in the following:                                  (210)

             a.  Any signal x(t) can be represented in terms of its odd and even components as.  

(A) (B)

                  (C) (D)

             b. Find the type of system described by

(A) Linear and dynamic                        (B) Linear and static

(C) Non linear and dynamic                  (D) Non linear and static

             c.  The discrete LTI system is represented by impulse response

                 h(n) = u(n). Then the system is

(A) Anti-causal and Stable                    (B) Causal and Stable

(C) Causal and Unstable                       (D) Anti-causal and Unstable

             d.  Laplace transform of  is

(A) (B)

(C) (D)

             e.  The impulse response of the system having transfer function H(s) = is

(A) () u(t)                          (B) u(t)

(C) (D) t u(t)

             f.   If X() =  the x(t) is

(A) (B)

(C) (D)

             g. Fourier transform of  x(-n) is

(A) –X()                                         (B)

(C) (D)

             h.  The condition for events A and B to be statistically independent.

(A) P(A/B)=P(AB)P(A)                   (B) P(AB)=P(A)P(B)

(C) P(A/B)=P(A) and P(B/A)=P(B)     (D) P(A/B)=P(AB)

             i.   Inverse Z transform of X(z)=  is

(A) x(n)=a²u(n)                                     (B) x(n)=aⁿu(n)

(C) x(n)=2a²u(n)                                   (D) x(n)=naⁿu(n)

             j.   System function H(z) for the system described by difference equation y(n)=2x(n)+3x(n-1)-4y(n-1) is 

(A) (B)

                  (C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

  Q.2     a.   Evaluate the following integrals.                                                                              

                   (i)                             (ii) 

                   (iii) (23 = 6)

b.   Compute the convolution sum of .                             (4)

             c.   Determine whether the systems are Linear, Causal, Time-invariant, Stable and Memoryless                                                                                    (6)

(i) T[x(n)]=x(-n)                                  (ii) y(t)=x(t) cos

Q.3     a.   State and prove the scaling and duality property of continuous time fourier transform.                                                                  (8)

b.   Find the fourier transform of the following signals                                                    

                   (i) x(t) =                               (ii)       x(t) =

                  (iii)  x(t) = u(t)                             (iv)      x(t) = u(-t + 2)         (8)

Q.4     a.   Determine the discrete fourier series representation for each of the following sequence.

                  (a)  x(n) = 2cos                             (b) x(n) = cos

                  (c)  x(n) = 1+ 2cos (6)

             b.   For the signal shown in Fig. 4(b) Find the fourier series co-efficient.                   (10)

  Q.5     a.   Determine the frequency response and impulse response of the systems described by the following equations.

      (i)

                   (ii) 3y(n)-4y(n-1) + y(n-2) = 3x(n)                                                                     (8)

             b.   State and prove the Sampling theorem for Lowpass signals and also explain the reconstruction of the signal from its sample value.                           (8)

Q.6     a.   Find the Z Transform of the following sequences and mention their ROC.

(i)  x(n) =n u(n)                                (ii) x(n)=

                  (iii) (9)

             b.   Find the inverse Z transform of the following X(z)

                   (i)  X(z) =  log      ,            >

                   (ii) X(z) =       ,            >2

                  (iii) X(z) =                 ,           >1  (2+2+3)

Q.7       a.    Find the Laplace transform of the following signals and the associated ROC in each case.

(i) x(t) =             (ii) x(t) =

                   (iii) x(t) = u(t)                  (iv) x(t) = (8)

             b.   State and prove the initial and final value theorems in Laplace transform.              (8)

  Q.8     a.   If the probability density function of a random variable X is given by

,

                   find the mean, variance and standard deviation.                                                   (8)

      b.   Consider a sinusoidal signal with random phase designed by

                                        x(t)= A cos()   

                   Where A and f_c are constant and  is a random variable that is uniformly distributed over the interval [,  ], i.e

                   Find (i) Auto correlation function of x(t).

                         (ii) Power spectral density of x(t).   (8)

Q.9       a.   Let x[n] and h[n] be signals with the following Fourier transforms

                   Determine .                                                                             (8)

      b.   Find the discrete Time fourier Transform of the following:

            (i)                                        (ii) 

            (iii)                         (iv)  (8)