SIGNALS AND SYSTEMS - B.Tech 5th Semester Exam., 2019

2019•Semester 3•Civil-CA•End Semester
Bihar Engineering University, Patna
B.Tech 5th Semester Exam., 2019

SIGNALS AND SYSTEMS

Time: 03 HoursCode: 031510Full Marks: 70

Instructions:

  1. The marks are indicated in the right-hand margin.
  2. There are NINE questions in this paper.
  3. Attempt FIVE questions in all.
  4. Question No. 1 is compulsory.
Q.1 Answer any seven of the following questions:[14]

  1. Find the fundamental period and frequency of the signal \( x(t)=\cos 18\pi t+\sin 12\pi t \)

  2. With an example, prove that cascade combination of an LTI system and its inverse system results an identity system.

  3. Check whether the system \( y(t)=tx(t) \) is causal and stable.

  4. What is the physical significance of convolution?

  5. What do you mean by convergence of Fourier series?

  6. Prove that \( x_{even}(t)\leftrightarrow Re\{a_{k}\} \) where \( x(t) \) is real and \( x(t)\leftrightarrow a_{k} \).

  7. Determine the Laplace transform for the signal \( x(t)=e^{-5t}u(t-1) \).

  8. For an LTI system to be causal and stable, what should be the condition on ROC and locations of poles?

  9. Find the z-transform and ROC of \( x[n]=(\frac{1}{5})^{n}u(n-3) \)

  10. State and prove time reversal property of z-transform.

Q.2 Solve both questions :[14]

  1. Compute and plot the even and odd parts of the given discrete and continuous signals from the figures.

    Question Diagram

  2. Let \( x(t)=u(t-3)-u(t-5) \) and \( h(t)=e^{-3t}u(t) \). Compute:
    (i) \( y(t)=x(t)*h(t) \)
    (ii) \( g(t)=(dx(t)/dt)*h(t) \)

Q.3 Solve both questions :[14]

  1. For each of the following systems, check the properties of linearity, time-invariance, causality and BIBO stability:
    (i) \( y(t)=x(t/2) \)
    (ii) \( y[n]=nx[n] \)

  2. Compute and plot the convolution \( y[n]=x[n]*h[n] \) where \( x[n]=(\frac{1}{2})^{n-2}u[n-2] \) and \( h[n]=u[n+2] \).

Q.4 Solve both questions :[14]

  1. Obtain the differential equations for the given systems.

    Question Diagram

  2. Explain the 'property of sifting' of discrete-time unit impulse.

Q.5 Solve both questions :[14]

  1. State and prove the following properties of continuous-time Fourier series:
    (i) Frequency shifting
    (ii) Periodic convolution in time-domain
    (iii) Time scaling
    (iv) Differentiation in time-domain

  2. Calculate the Fourier series coefficients for the periodic signal
    \( x(t)=\begin{cases}2&,&0\le t< 2\\ -2,&2\le t< 4\end{cases} \)
    with fundamental frequency \( \omega_{0}=\pi/2 \).

Q.6 Solve both questions :[14]

  1. State and prove the following properties of discrete-time Fourier transform:
    (i) Time shifting
    (ii) Time expansion
    (iii) Multiplication of two signals in time-domain
    (iv) Differentiation in frequency

  2. Compute the Fourier transform of \( x[n]=(\frac{1}{4})^{|n-1|} \).

Q.7 Solve both questions :[14]

  1. Consider an LTI system whose response to the input \( x(t)=(e^{-t}+e^{-3t})u(t) \) is \( y(t)=(2e^{-t}-2e^{-4t})u(t) \). Using Laplace transforms, find the impulse response of the system. Also compute its frequency response.

  2. Determine the inverse Laplace transform of \( X(s)=\frac{(s+1)}{(s+1)^{2}+4} \), \( Re\{s\}>-1 \).

Q.8 Solve both questions :[14]

  1. Determine the impulse response of the system described by the difference equation \( y[n]=\frac{1}{2}[x[n]+x[n-1]+y[n-1]] \). Assume that the system is initially relaxed.

  2. Solve the following linear difference equation:
    \( y[n]+\frac{1}{2}y[n-1]-\frac{1}{4}y[n-2]=0 \)
    Given that \( y[-1]=y[-2]=1 \).

Q.9 Solve both questions :[14]

  1. Determine all possible signals of \( x(n) \) associated with the following z-transforms:
    (i) \( X(z)=\frac{1}{1-\frac{3}{2}z^{-1}+\frac{1}{2}z^{-2}} \)
    (ii) \( X(z)=\frac{5}{1-\frac{1}{2}z^{-1}+\frac{1}{4}z^{-2}} \)

  2. Obtain the z-transform and ROC of the following sequence:
    \( x(n)=(\frac{1}{2})^{n}[u[n]-u[n-10]] \)