Nepal Engineering College Department of Electrical & Electronics Engineering
ELX 482.3 Digital Signal Processing Tutorial I Date of Distribution: September 23, 2013 Due Date: October 2, 2013.
Signals and Systems Basics: 1. Determine whether or not each of the following signals is periodic. In case a signal is periodic, specify its fundamental period. (a) xa(t)=3cos(5t+π/6) (b) x[n] = 3cos[5n + π/6] (d) x[n] = cos[πn/2] – sin[πn/8] + 3cos[πn/4+π/3]
(c) x[n] = cos[n/8]cos[πn/8] (e) x[n] = 1 + e(j4πn)/7 - e(j2πn)/5
2. Let x[n] be an arbitrary discrete-time signal with x[n] = 0 for n < -2 and n > 4. Plot each signals given below: (a) x[n-3]
(b) x[n+4]
(c) x[-n]
(d) x[-n+2]
(e) x[2n-1]
3. The periodic square wave, defined over one period as | | 1, | | 0, 2 With fundamental period T and fundamental frequency ω0 = 2π/T. a. Determine and plot the Fourier Series Coefficients for x(t). b. Determine Fourier Transform of an aperiodic signal x(t). Plot magnitude and phase response. 4. Determine whether the following signals is an energy signal or power signal. (a) x[n] = (1/2)n u[n] (b) x[n] = cos[πn/4]
Chapter 1 1. An analog signal xa(t) = sin(480πt) + 3sin(720πt) is sampled 600 times per second. a. Determine the Nyquist sampling rate for xa(t). b. Determine the folding frequency. c. What are the frequencies, in radians, in the resulting discrete time signal x[n] ? d. What is the analog signal ya(t) you can reconstruct from the samples if the ideal interpolation is used ? 2. A digital communication link carries binary-coded words representing samples of an input signal xa(t) = 2sin2300πt + 5sin500πt. The link is operated at 10,000 bits/sec and each input sample is quantized into 1024 different voltage levels. a. What is the sampling frequency and the folding frequency? b. What is the Nyquist rate for the signal xa(t) ? c. What are the different frequencies in the resulting discrete-time signals x[n] ? d. What is the resolution ∆ ?
Chapter 2 1. Check for the memory, linearity, time invariance, causality and stability of following systems: (a) y[n] = 4nx[n] (b) y[n] = x[n2] (c) y[n] = x2[n] (d) y[n] = ax[n] + b (g) y[n] = n[x[n]]2
(e) y[n] = ex[n] (f) y[n] = x[-n] (h) y[n] = 3x[n-2] + 3x[n+2]
2. The impulse response h[n] of an LTI system is known to be zero, except in the interval N0≤ n ≤ N1 . The input x[n] is known to be zero, except in the interval N2 ≤ n ≤ N3 . As a result, the output is constrained to be zero, except in some interval N4 ≤ n ≤ N5. Determine N4 and N5 in terms of N0, N1, N2, and N3 . 3. Consider an LTI system with impulse response h[n] and input x[n]. Determine the output of the system, y[n] for the following pairs of inputs and impulse responses. a. x[n] = {-2, 1, 2, 3} ; h[n] = {3, -2, 1, 3} (Bold no. is the position of n=0) b. x[n] = αnu[n], |α| < 1; h[n] = u[n] – u[n-7] c. x[n] = αn-1u[n-1]; h[n] = α-n+2u[-n+2], |α| < 1. 4. Given h1[n] = (1/2)nu[n] and h2[n] = (1/4)n u[n]. Find the overall response h[n].
5. The impulse response of an LTI system is as shown in fig 1:
Fig2 : QN. 12
Determine and carefully sketch the response of this system to input x[n] = u[n-4]. 6. Determine the direct form I & II realization for each of the following LTI systems. a. 2y[n] + y[n-1] - 4y[n-3] = x[n] + 3x[n-5]. b. y[n] = x[n] – x[n-1] + 2x[n-2] - 3x[n-4].
Additional Problems from DSP by Proakis and Manolakis. 1) Problems 1.10, 1.11, 1.13, 1.14 2) Problems 2.3, 2.8, 2.17, 2.44
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