Question Paper Code : AEC11T02

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1 Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank (Regulations: VCE-R11) SIGNALS AND SYSTEMS (Electronics and Communication Engineering) PART-A Unit-I 1. A continuous time signal is shown in the figure. Sketch and label the following signals 2) 3) 4) 2. Find the even and odd components of 3. Show that the complex exponential sequence is periodic only if is a rational number. 4. Check whether the system shown is stable, causal, time invariant and linear y [n] = x [n] 5. Determine whether the signal x(n) = 2 sin(3n) is periodic or not 6. Define unit step function for both continuous and discrete time signal 7. Define unit impulse function for both continuous and discrete time signal 8. Define unit ramp function for both continuous and discrete time signal 9. What is a magnitude spectrum? 10. What is a phase spectrum? Unit-II 1. Find the Fourier transform of a one sided exponential function x(t)=e -at 2. Properties of Fourier transform 3. Properties of discrete time Fourier transform 4. Fourier transform of standard functions 5. What is Hilbert transform 6. Properties of Hilbert transform 7. State convolution theorem 8. State modulation property 9. Find the Fourier transform of a one sided exponential function x(t)=e -at 10. Properties of Fourier transform Unit-III 1. Show that x(n)*δ(n) = x(n) 2. Show that x(n)*u(n) = 3. Convolve x(t) and h(t) where x(t)=u(t) and h(t)=u(t-1) 4. Compute the convolution of the two signals and given by and 5. State and prove the commutative property of convolution sum 6. State and prove the associate property of convolution sum 7. State and prove the distributive property of convolution sum 8. State and prove the commutative property of convolution integral 9. State and prove the associate property of convolution integral 10. State and prove the distributive property of convolution integral Unit-IV

2 Unit-V 1. State and prove sampling theorem for low pass signals 2. Find the Z Transform of the signal x(n)=3 (2) n u(-n) 3. List the properties of ROC 4. Find the Z-transform of 5. Find the inverse Z-transform of 6. Find the Z-Transform of a n u(-n-1) and find ROC 7. Find the inverse Z-Transform of X(Z)= for Z < 8. Find the Z-transform and the associated ROC for the sequence 9. Explain the sampling of band pass signals. 10. Explain the sampling theorem in frequency domain 11. Explain natural sampling and flat sampling with equations and waveforms. PART-B Unit-I 1. A discrete time signal x[n] is defined as n 1 + ; 3 n 1 3 x[n] = 1 ; 0 n 3 0 ; elsewhere Determine its values and sketch the signal x [n]. Sketch the signal that result if we 1) First fold x[n] and then delay the resulting signal by four samples. 2) First delay x[n] by four samples and then fold the resulting sequence Sketch the signal x [-n + 4]. Express the signal x [n] in terms of δ [n] and u [n] 2. Determine whether the following signals are energy signals, power signals or neither A) B) C) 3. A continuous time signal is shown in the figure. Sketch and label each of the following signals. b) c) 4. Consider the sequence 3 n x[ n] = ( ) u[ n] 2 Find the numerical value of A = n = x [ n ].Compute the power in x[n] 5. Consider a square law device y ( t) = x 2 ( t) and input x( t) = A1 cos( ω 1t + φ1) A2 cos( ω2t + φ2 ) Determine the corresponding output y (t). Show that y(t) contains new components with following frequencies 0, 2ω 1, 2ω 2, ω 1 - ω 2. What are their respective amplitudes and phase shifts. Plot amplitude and phase spectra. 6. For the following signals, (i) determine analytically which are periodic (if periodic, give the period) and (ii) sketch the signals. (Scale your time axis so that a sufficient amount of the signal is being plotted) o x(t) = 4 cos(5 t - /4) o x(t) = 4u(t) + 2sin(3t)

3 o o x[n] = 4cos( n-2) x[n] = 2sin(3n) 7. Determine if the following signals are periodic; if periodic, give the period. o x(t) = cos(4t) + 2sin(8t) o x(t) = 3cos(4t) + sin( t) o x(t) = cos(3 t) + 2cos(4 t) 8. Find the exponential Fourier series of a half wave rectified sine wave form with amplitude A and fundamental time period 1 sec. 9. Consider the sequence x(n)= Sketch x(n) Determine the Fourier coefficients 10. Verify Parseval s identity for the discrete Fourier series, that is = 11. Consider the following CT periodic signal Where is integer. o What is the fundamental periodic of this signal? o Does the signal have even or odd symmetry? o Solve the Fourier series coefficients. o What type of symmetry do the coefficients have? 12. Determine the Fourier series coefficients for the periodic signal x[n] depicted in figure1. Plot the magnitude of these coefficients. Figure Determine the discrete time Fourier series representation for x(n)=cos ( πn/4) Unit-II 1. A discrete time signal described by x(n)=sin(πn/8). Sketch the magnitude and phase of discrete time Fourier Transform x(n-2) 2. Determine the Fourier transform of a signal x(n)=cos (w 0 n) u(n) 3. Using convolution theorem, find the inverse Fourier transform of X(W)= 1/(a+jw) 2 4. Find the Fourier Transform of the following signals? Which of these signals have Fourier Transform that converge? Which of these signals have Fourier Transform that are real? Imaginary? 5. Find the time domain signal corresponding to the DTFT s of the following: 6. The following are the Fourier Transforms of discrete time signals. Determine the signal corresponding to each transform. 7. Find the Fourier series representation and sketch the amplitude and phase spectrum for the signal

4 8. Find the Fourier Transform of the signal 9. Determine the DTFT of the signal 10. A discrete LTI system is defined by Where and are the input and output of the system. If the input is, find the response using DTFT. 11. Find the Fourier Transform of the signal 12. Using the time convolution theorem, find the inverse Fourier transform of 13. Prove the time shifting, time reversal and convolution properties for discrete time Fourier transform. 14. Find the inverse Fourier Transform x[n] of the rectangular pulse spectrum X(Ω) defined by Plot x[n] for 1. Compute where Unit-III 2. Compute where, 0 < 3. Compute the convolution sum of the following sequences x(n)=u(n+4) u(n-1) and h(n)= 2 n u(2-n) 4. Compute the convolution integral of the following sequence x(t)=e -t u(t+1) and h(t)= e 2t u(-t) 5. Find the output of a system whose input-output is related by y(n) = 7 y(n-1)-12 y(n-2) + 2x(n)-x(n-2) for an input x(n) = u(n) 6. A discrete time LTI system has impulse response and input Find y(n) where y(n)=x(n) * h(n) Unit-V 1. The signal g(t)=10 cos (20πt) cos (200πt) is sampled at the rate of 250 samples/sec. Determine the spectrum of the resulting samples signal. Specify the cutoff frequency of the ideal reconstruction filter so as to recover g(t) from its Sampled version. What is the Nyquist rate for g(t)? 2. A low pass signal x(t) has a spectrum X(f) given by Sketch the spectrum X δ (t) for f <200 Hz if x(t) is ideally samples at fs=300 Hz. Repeat part 1 for f s =400 Hz 3. Convolve x 1 (n) and x 2 (n) using Z Transforms: x 1 (n)=(1/3) n u(n) and x 2 (n)=(1/5) n u(n) 4. A causal discrete time LTI system is implemented using the difference equation What is the transfer function of this system? Sketch the pole-zero diagram of the system. Find the impulse response 5. The discrete time signal is shown in the figure. What is the Z-transform of the signal Define, sketch the signal Define, sketch the signal Define, sketch the signal

5 6. Find the inverse Z-transform of for Z > 2 Figure 7. Obtain the convolution of α n u(n) and β n u(n) using Z-Transforms where α<1 and β<1 8. Using the power series expansion method, find the inverse Z-transform of Z < ½ 9. A causal discrete-time LTI system is described by Where x[n] and y[n] is the input output of the system, respectively. Determine the system function H(Z) Find the impulse response h[n] of the system

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) 1. For the signal shown in Fig. 1, find x(2t + 3). i. Fig. 1 2. What is the classification of the systems? 3. What are the Dirichlet s conditions of Fourier