**Important questions**

__Signals and Systems :-__(i) What is time varying and time invariant system?

(ii) What is superposition condition or theorem for systems?

(iii) Find total energy contained in the Impulse function.

(iv) Differentiate in brief

(a) Energy and Power signals (b) Even and Odd signals

(v) Find the fundamental period of the discrete time sinusoidal signal x[n] = 5cos[0.2Πn]

(vi) 9Find the fundamental period of each discrete time sinusoidal signal

x[n] = 2sin [6Πn/35]

(vii) A pair of sinusoidal signal with a common angular frequency is defined by

x1[n] = sin[5∏n] and x2[n] = 4 cos[5∏n] . Both signals are periodic. Find their fundamental periods and the fundamental period

of x[n]= x1[n] + x2[n]

(viii) Find the Laplace Transform and ROC of unit step function.

(ix) Find the ROC of LT of unit ramp function.

(x) Derive the relationship between Continuous Time Fourier Transform (CTFT) and Laplace Transform.

(xi) Explain Dirichlet’s conditions for the convergence of DTFT.

(xii) Explain Dirichlet’s conditions for the convergence of CTFT.

(xiii) State and prove time shifting property of z- transform.

(xiv) State the condition for stability and causality of Discrete Time LTI system in terms of ROC of its system function.

(xv) A discrete time signal x[n]={ 0, 2 ,3 ,4,5,6,7,8,1,2}

Draw i) x[2n] ii) x[n-4] iii) x[2-n]

(xvi) Establish the relationship between convolution and correlation function for CT system.

(xvii) Show that the system y[n] = 7x[n] + 5 is nonlinear system.

(xviii) Show that the given system is nonlinear system.

(xix) (a) y(t) = x2(t)

(xx) What do you mean by Group Delay?

(xxi) Write the mathematical expression for the sinc function.

(xxii) Write the condition for the convergence of DTFT.

(xxiii) Find Laplace Transform of unit step function.

Define the stability of the system.
(i) Check the linearity for the system equation y(t) = t x(t)

(ii) Find the impulse response of the system having gain A.

(iii) Check the causality of the system having input-output relation y(n) = x(n-1) + x(n-2).

(iv) Find the total energy and total power contained in the unit step signal u(t).

(v) Find the fundamental period of the signal cos10πt.

(vi) What will be the odd part of the signal cos2ωt.

(vii) Find the output of an LTI discrete time system for x(n)= (1/2)n u(n-2) whose impulse response is unit step sequence.

(viii) Signal x(t) is shown in the Figure 1.

(i) Find x(-4t+5) + 2x(t) for the signal given in figure 1.

Find the even and odd part of the signal given in figure 1.
(XXII) State and prove Parseval’s theorem for CTFT.

A triangular pulse signal is shown in above figure 2. Sketch each of the following derived from x(t)

i) x(3t) (ii) x(2(t+2))

(iii) x(3t+2) (iv) x(3t)+x(3t+2)

(XXII) Find the DTFT of the sequence x (n) = n a

^{n }u(n).
(XXIII) For the system specification y (t) = t x (2t) find whether the system is Linear or Nonlinear, static or dynamic, fixed or time varying, causal or non-causal.

(XXII) For the system specification y (t) = x (3t+2) find whether the system is Linear or Nonlinear, static or dynamic, fixed or time varying.

xxiv) Find the Laplace Transform of e

^{-7t }_{Sin ωt u(t).}
xxv) Find the Fourier Transform of Signum function.

xxvi) Find the Laplace Transform of the signal t sin ωt u(t).

xxvii) Derive any Four properties of DTFT.

xxiv) Determine the frequency response and impulse response of a causal Discrete time LTI system that is characterized by the difference equation given as

Y[n] - Ay [n-1] = x[n ] with |A| < 1

xxv) Determine the DTFT of the discrete time periodic signal X[n] = cos w

_{0}
with fundamental frequency w

_{0}= 2П / 5.
xxvi) State and prove Differentiation property in frequency domain for z-transform.

xxvii) Determine the impulse response of a continuous-time LTI system described by first order differential equation

a dy(t)/dt + y(t) = x(t)

xxviii) Two systems are described by the following input-output relations:

y(t) = {Cos(3t)} x(t) and

y[n]=x[n-2] +x[8-n]

Check the properties of linearity, time-invariance, Causality and Stability for these systems and give explanation for your answers.

xxix) Find the output of an LTI system having input x(t)=1 for 0 < t < 2 and impulse response h(t)= 2 for 0 < t < 5 .

xxx) Find the impulse response of the system whose input-output relation is given as y (n)-y (n-1) +3/16 y(n-2)= x(n)- ½ x(n-1).

xxxi) Find the DTFT of the sequence x (n) = n a

^{n}u(n).
xxxii) State and prove time shifting property of CTFT.

xxxiii) Derive the Differentiation in frequency domain and convolution properties for z- transform.

Prove that for an energy signal, its auto-correlation function and its energy spectral density (ESD) are Fourier transform pairs.
xxiv) For following second order differential equations for causal and stable LTI system, describe whether the corresponding impulse response is under damped, critically damped or over damped?

xxv) d

^{2 }y (t)/dt^{2}+ 4dy (t)/dt + 4y (t) = x (t).
xxvi) What are the ideal frequency selective filters? Explain

xxvii) The discrete-time signal x[n] is defined as below

x[n] = 1, when n = 1, 2

-1, when, n = -1, -2

0, when n=0, │n│>2

Sketch the x[n] and the time shifted y[n] =x [n+3]

xxviii) Show graphically that δ[n]= u[n] – u[n-1]

xxix) Sketch signal x(t) = A[u(t+ a) – u(t – a)]. Identify whether it is power or energy signal and accordingly calculate suitable quantity.

xxx) Find the Laplace Transform of t x(t), if x(t) is having Laplace Transform X(s).

xxxi) Find inverse Laplace transform of the following

xxxii) X(S) = 10/ (S+1) (S+5), ROC : Re{S} < -5

xxxiii) Consider the rectangular pulse (gate pulse) signal defined as

x(t) = A rect (t/2T) = A; │t│< T

0; │t│> T

xxxiv) Find the Fourier transform of x(t).

xxxv) For a DT system, H(z) is given below ,

H(z) = 3 - 4Z-1/ 1 - 3.5Z-1 + 1.5Z-2

Specify the ROC and determine h[n], when (i) System is stable (ii) System is causal.

xxxvi) Find out Z- transform of signal x[n] = an u[n] and its ROC.

xxxvii) Find the impulse response of the system whose input-output relation is given as

y (n) - y (n-1) + 3/16 y(n-2)= x(n) – ½ x(n-1).

xxxviii) Find the convolution of following two sequences:

x[n]= u[n], h[n]= 2n u[n].

xxxix) Explain Invertibility and causality properties of any system.

xl) Derive the condition of mapping from s-plane to z plane and also correlate the ROCs of LT and ZT.

xli) For following second order differential equations for causal and stable LTI system, describe whether the corresponding impulse response is under damped, critically damped or over damped?

4d

^{2 }y (t)/dt2 + 5dy (t)/dt + 4y (t) = 7 x (t)
xlii) Define distortion less transmission through a filter. Derive the frequency response of a filter that will not distort any signal.

xliii) Write the expression for Rxy(τ) and Ryx(τ) and give all the relationship for real valued and complex valued signals.

xliv) Find the step response of the RC high pass filter.

xlv) Identify whether signal x(t)= e

^{-5t}u(t) is energy signal or power signal? Also calculate energy and power of signal.
xlvi) The discrete-time signal x[n] is defined as below

x[n] = 1, when n = -1,1

0, when n=0, │n│>1

Sketch x[n] and find y[n]= x[n] + x[-n] with suitable sketch.

liv) Explain causal and anti-causal signals with suitable examples.

xlvii) Find the DTFT of the sequence x(n)= a

^{|n| }and also draw the magnitude spectrum.
xlviii) Find the output of an LTI discrete time system for x(n)= (1/2)

^{n}u(n-2) whose impulse response is unit step sequence.
xlix) Find Z-transform and ROC of Ramp function.

l) Find the Energy Spectral Density of the function x(t)= e

^{-│t│}
li) Find the auto correlation of the sequence x(n)= a

^{n }u(n) for 0<a<1.
lii) Explain Invertibility and stability properties of any system with the help of examples.

liii) Find the Laplace Transform of the function x(t) = t

^{2 }e^{-2t}cos ωt u(t).
liv) Find inverse Fourier Transform ofδ(ω).

lv) Find the Fourier Transform of the constant signal ‘1’ which extends over entire time interval.

lvi) Find inverse Laplace transform of following

X(S) = 10/ (S+1) (S+5), ROC : Re{S} > -1

lvii) Find the step and impulse response of the RC low pass filter.

lviii) Define distortion less transmission through a filter. Derive the frequency response of a filter that will not distort any signal.

lix) Determine the impulse response and step response of a continuous-time LTI system described by first order differential equation

a dy(t)/dt + y(t) = x(t)

(lxxv) Find the Laplace Transform of the function x(t) = t

^{2}cos ωt u(t).
(lxxvi) Find the Fourier Transform of u(t) using Signum function.

(lxxvii) Find the convolution of two continuous time signals:

**x(t)= 3 cos2t,**for all

**t**and

**y(t)= e**

^{t}; t < 0**e**

^{-t}; t ≥ 0
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