EE 341 A Few Review Problems Fall 2018 1. A linear, time-invariant system with input, xx(tt), and output, yy(tt), has the unit-step response, cc(tt), shown below. a. Find, and accurately sketch, the system impulse response, h(tt). b. Find, and accurately sketch, yy(tt) = h(tt) [uu(tt) uu(tt 2)]. 2. A discrete-time, linear, time-invariant system with input xx(nn) and output yy(nn) has the impulse response h(nn) = [2, 2, 1, 1], where the underline denotes the nn = 0 sample, and all signal values outside the range specified are zero. a. (2 points) Is the system causal? (You must provide correct justification to receive credit.) b. (3 points) Is the system BIBO stable? (You must provide correct justification to receive credit.) c. The system input is xx(nn) = [ 1, 1, 0, 1, 1], where the underline denotes the nn = 0 sample, and all signal values outside the range specified are zero. i. (2 points) yy(nn) begins at nn =? ii. (3 points) yy(nn) ends at nn =? iii. (10 points) Find yy(nn) for all nn. 1
3. A discrete-time system with input, xx(nn), and output, yy(nn), is described by yy(nn) = 2xx(nn) 1. a. (5 points) Find the system response to the (Kronecker) impulse input, δδ(nn). b. (5 points) Determine if the system is BIBO stable. Proper justification must be provided to receive credit. 4. A continuous-time system with input, xx(tt), and output, yy(tt), has the inputoutput description yy(tt) = 2 cos(tt) xx(tt). Provide yes/no answers to the following by circling the correct answer. Determine if the system is a. Memoryless? b. Causal? c. BIBO stable? d. Linear? e. Time-invariant? 2
5. A linear, time-invariant system with input, xx(tt), and output, yy(tt), has the impulse response, h(tt), and the unit-step response, cc(tt), shown below. Find, and accurately sketch, the response to the input xx(tt) = 2δδ(tt 1) uu(tt 3). 3
6. Consider the composite system below, constructed from distinct linear, time- invariant systems. a. (10 points) Determine the overall impulse response, gg(tt), in terms of h(tt) and rr(tt). b. (10 points) Let h(tt) = uu(tt) uu(tt 1) and rr(tt) = δδ(tt 1). Find yy(tt) if the system input is xx(tt) = uu(tt) 7. A discrete-time, linear, time-invariant system with input, xx(nn), and output, yy(nn), has impulse response h(nn). a. If h(nn) = (1, 2, 1), use convolution to find yy(nn) = h(nn) xx(nn), for xx(nn) = [uu(nn) uu(nn 6)]. b. If h(nn) = 1 2 nn uu(nn), use convolution to find yy(nn) = h(nn) xx(nn), for xx(nn) = [uu(nn) uu(nn 6)]. 4
5. A periodic signal, xx(tt), has period TT = ππ sec, fundamental frequency 5 ωω 0 = 2ππ = 10 rad/s, and exponential Fourier series coefficients TT 5, kk = 0 10, kk = 1, 1 7jj, kk = 2 cc kk = 7jj, kk = 2 3, kk = 3, 3 0, otherwise d. Find the power in the signal xx(tt). e. A truncated Fourier series is of the form NN xx NN (tt) = cc kk ee jj2ππππππ/tt. kk= NN Find the normalized mean-square truncation error for i. NN = 1. ii. NN = 5. f. Accurately sketch the magnitude spectrum for xx(tt), labeling all relevant amplitudes. g. The signal, xx(tt) is applied as input to a linear, time-invariant system with output, yy(tt). The system has impulse response h(tt) with Fourier transform (the system frequency response) HH(ωω), where 20, 5 ωω 25 HH(ωω) =. Find the Fourier series for yy(tt). Express yy(tt) in simplest form (e.g., 0, otherwise sines and cosines). 5
6. A linear, time-invariant (LTI) system with overall impulse response h(tt) is constructed from three simpler LTI subsystems, as shown below, with respective impulse responses gg 1 (tt), gg 2 (tt), and gg 3 (tt). a. ) Find h(tt) in terms of gg 1 (tt), gg 2 (tt), and gg 3 (tt). b. If gg 1 (tt) = δδ(tt 2), gg 2 (tt) = 6δδ(tt 3), and gg 3 (tt) = 2δδ(tt 1), find h(tt). c. Suppose that h(tt) = δδ(tt + 1) + 2δδ(tt 3). Find, and accurately sketch yy(tt) = xx(tt) h(tt) for xx(tt) = [uu(tt + 2) uu(tt 2)]. d. Suppose that h(tt) = 2ee 2tt uu(tt). Find, and sketch, yy(tt) = xx(tt) h(tt) for the same xx(tt) as in part c), xx(tt) = [uu(tt + 2) uu(tt 2)]. 6
7. A periodic signal, xx(tt), is shown below. i. Find the exponential Fourier series, xx(tt) = kk= cc kk ee jj2ππππππ/tt i. Period TT = ii. cc 0 = iii. cc kk = ii. Let zz(tt) = 3xx(tt) + 6. Find the exponential Fourier series for zz(tt) in terms of the Fourier series coefficients for xx(tt). iii. The signal xx(tt) in part a) is applied as the input to a linear, time-invariant system with 0.2 ππ output yy(tt) and transfer function HH(ss) =. ss+0.4 ππ i. Find the Fourier series for yy(tt). ii. Find the ratio of the amplitude of the 1 st harmonic of the Fourier series for yy(tt) to the amplitude of the 1 st harmonic of the Fourier series for xx(tt). 7
8. The square wave signal, xx(tt), shown below, has period T sec and exponential Fourier series xx(tt) = cc kk ee jj2ππππππ/tt, kk= where the Fourier series coefficients are given by 1, kk = 0 cc kk = sinc kk, kk 0 2 sin (ππππ) with sinc(tt) =, and the fundamental frequency ωω 0 = 2ππ rad/s. TT ππππ iv. Find the Fourier transform of xx(tt), and plot the magnitude spectrum, XX(ωω), carefully labeling important amplitudes and frequency axis intercepts. v. Let TT = 0.0625 sec (so ωω 0 = 2ππ = 32ππ rad/s). The signal xx(tt) is applied as input to a TT linear, time-invariant system with output yy(tt). The system frequency response is shown below. Find an explicit expression for yy(tt). 8
9. Analog signal xx(tt) = 2 cos(32ππππ) 4 sin (7ππππ). vi. Find XX(ωω) and accurately sketch XX(ωω), carefully labeling important amplitudes and frequency axis intercepts. vii. The signal xx(tt) is sampled at rate FF ss = 1 samples/sec, and the analog signal yy(tt) is formed TT as shown in the block diagram below, yy(tt) = nn= xx(nnnn)h(tt nnnn), where h(tt) FF HH(ωω) and HH(ωω) = TT PPππ(ωω) = TT, ωω ππ TT TT 0, otherwise. i. From the sampling theorem, what is the smallest rate at which xx(tt) can be sampled and the reconstructed signal is yy(tt) = xx(tt). ii. Let FF ss = 1 = 25 samples/sec. Sketch the spectrum at the output of the sampler. Find an TT explicit (time-domain) expression for yy(tt). 10. viii. ix. Use Fourier transform pairs and properties to find the Fourier transform of the signal xx(tt) = 10ee 3 tt cos (30ππππ), where uu(tt) is the unit step function. Sketch XX(ωω), carefully indicating important amplitudes and frequencies. 9
11. A discrete source is modeled as a discrete random variable, XX, with 6 outcomes in the sample space SS = {0,1,2,3,4,5}. a. If PP(xx) = 1, for xx = 0,,5, determine the entropy, HH(XX). (Note that, log 6 22 = 1.0, log 2 3 = 1.5850, and log 2 6 = 2.5850.) b. Let the symbol probabilities be PP(xx) = (0.35, 0.25, 0.12, 0.12, 0.08, 0.08) for xx = 0,, 5, with the source entropy, HH(XX) = 6 xx=0 PP(xx)log 2 PP(xx) = 2.3473 bits. Design a Huffman code for the source. List all codewords, determine the average codeword length, and compare the average codeword length to the entropy. Codeword X P(x) 0 0.35 1 0.25 2 0.12 3 0.12 4 0.08 5 0.08 1