1. Clearly circle one answer for each part.

Transcription

1 TB / Exam Style Questions 1 EXAM STYLE QUESTIONS Covering Chapters of Telecommunication Breakdown 1. Clearly circle one answer for each part. (a) TRUE or FALSE: For two rectangular impulse responses with the same maximum magnitude but different time widths T 1 and T 2 with T 1 > T 2, the halfpower bandwidth of the frequency response of the pulse with width T 1 exceeds that of the pulse with width T 2. (b) TRUE or FALSE: For the PAM baseband signals created by a rectangular pulse and a triangular pulse with the same time width and the same maximum amplitude, the half-power bandwidth of the sequence using the triangular pulse exceeds that of the rectangular pulse. (c) TRUE or FALSE: The impulse response of a series combination of any α-secondwide pulse shape filter and its matched filter form a Nyquist pulse shape for a T-spaced symbol sequence for any T α. (d) TRUE or FALSE: Decision-directed phase recovery can exhibit local minima of different depths. (e) TRUE or FALSE: Implementing a Costas loop phase recovery scheme on the preprocessed version (i.e. squared and narrowly bandpass filtered at twice the carrier frequency) of a received PAM signal results in one and only one local minimum in any 179 window of the adjusted phase. (f) TRUE or FALSE: The optimum settings of timing recovery via output power maximization with and without intersymbol interference in the analog channel are the same. (g) TRUE or FALSE: The optimum settings of phase recovery via the phase-locked loop operating on a preprocessed (i.e. squared and narrowly bandpass filtered at twice the carrier frequency) received PAM signal are unaffected by the channel transfer function outside a narrow band around the carrier frequency. (h) TRUE or FALSE: The causal, minimum-delay, matched (for a spectrally flat channel noise) filter for a τ-second wide pulse shape with an even-symmetric impulse response has its optimum sampling time at τ seconds after the start of an undistorted pulse at the matched filter input.

2 TB / Exam Style Questions 2 (i) TRUE or FALSE: Processing a bandlimited signal through a linear, timeinvariant filter can increase its half-power bandwidth. (j) TRUE or FALSE: The linear, time-invariant filter with impulse response f[k] = βδ[k j] is a highpass filter. j= 2. Consider the sampled version x(kt + τ) of the baseband signal x(t) recovered by the receiver in a 2-PAM (±1) communication system. In ideal circumstances, the baud-timing variable τ could be set so x(kt + τ) exactly matched the source symbol sequence transmitted by this PAM system. We choose to select τ as the value that optimizes for a suitably large N. J DM (τ) = 1 N k 0 +N k=k 0 +1 (1 x 2 (kt s + τ)) 2 (a) Should J DM be minimized or maximized? Clearly justify your answer. (Equations and diagrams without prose commentary do not qualify as an explanation.) (b) Using your choice in part (a) and the approximation d(x(kt s + τ)) x(kt s + τ) x(kt s + τ ɛ) dτ ɛ for a positive ɛ, derive an approximate gradient following algorithm that optimizes J DM. (To receive any partial credit for an incorrect answer, your work must be clearly explained. Equations and diagrams without prose commentary do not qualify as an explanation.) 3. Consider the 1.2 msec wide pulse shape p(t) shown in Figure 1. (a) Is p(t) in Figure 1 a Nyquist pulse for the symbol period T = 0.35 msec? Clearly justify your answer. (Equations and diagrams without prose commentary do not qualify as an explanation.) (b) Is p(t) in Figure 1 a Nyquist pulse for the symbol period T = 0.70 msec? Clearly justify your answer. (Equations and diagrams without prose commentary do not qualify as an explanation.) 4. Consider the baseband communication system in Figure 2. The difference equation relating the symbols m[k] to the T-spaced equalizer input u[k] for the chosen baud-timing factor ɛ is u[k] = 0.04m[k ρ] m[k 1 ρ] m[k 2 ρ] m[k 3 ρ]

3 TB / Exam Style Questions 3 Figure 1: Pulse Shape where ρ is a nonnegative integer. The finite-impulse-response equalizer is described by the difference equation y[k] = u[k] + αu[k 1] relating its input u to its output y. Figure 2: Baseband Communication System (a) With α = 0.4 and a binary (±1) source is the system from the source symbols m[k] to the equalizer output y[k] open-eye? Clearly justify your answer. (Equations and diagrams without prose commentary do not qualify as an explanation.) (b) Is it possible for the system from 4-PAM (±1, ±3) source symbols m[k] to the equalizer output y[k] to be made open-eye by selection of α? If so, provide a successful value of α. If not, explain why not. (To receive any partial credit for an incorrect answer, your work must be clearly explained. Equations and diagrams without prose commentary do not qualify as an explanation.) 5. Consider the passband communication system shown in Figure 3. The message signal m(t) is a sinusoid with frequency f 0 Hz, i.e. m(t) = cos(2πf 0 t). The carrier

4 TB / Exam Style Questions 4 frequency used in the transmitter and receiver is the same, i.e. f C Hz, where f C > 5f 0. The transmitter mixer phase is θ radians. The receiver mixer phase is φ radians. The frequency-selective, linear, time-invariant channel has impulse response h(t). The linear, time-invariant bandpass filter at the front-end of the receiver has impulse response g(t). Their convolution forms the impulse response p(t) (= h(t) g(t)). The frequency response P(f) of p(t) has the special symmetry property that and P(f) f=fc f 0 = β (γ + α) P(f) f=fc +f 0 = β (γ α) where a b indicates a complex number with magnitude a and phase b. The ideal lowpass filter with output x 2 (t) has a cutoff frequency greater than f 0 but less than f C. Figure 3: Passband Communication System (a) Write the output x 1 (t) of the bandpass filter g(t) as a sum of two scaled cosines at different frequencies. (To receive any partial credit for an incorrect answer, your work must be clearly explained. Equations and diagrams without prose commentary do not qualify as an explanation.) (b) Compute x 2 (t) as a single scaled cosine. (To receive any partial credit for an incorrect answer, your work must be clearly explained. Equations and diagrams without prose commentary do not qualify as an explanation.) (c) Determine the value of the receiver mixer phase φ that maximizes the amplitude of x 2 (t). (To receive any partial credit for an incorrect answer, your work must be clearly explained. Equations and diagrams without prose commentary do not qualify as an explanation.) 6. Consider the generation of a PAM transmitter baseband signal using the following Matlab code Ts=1/10000;time=2; t=ts:ts:time; M=8; N=length(t)/M;

5 TB / Exam Style Questions 5 m=pam(n,4,5); mup=zeros(1,n*m); mup(1:m:end)=m; s=filter(ps,1,mup); ssf=(-n*m/2:n*m/2-1)/(ts*n*m); fs=fft(s); fss=fftshift(fs); plot(ssf,abs(fss)) preceded by definition of the pulse shape impulse response vector ps as one of the following six choices: 1. [1, 1, 1, 1, 1, 1, 1, 1] 2. [0.15, 0.45, 0.75, 1, 1, 0.75, 0.45, 0.15] 3. [1, 1, 1, 1, 1, 1, 1, 1] 4. [1, 1, 1, 1, 0, 0, 0, 0] 5. [0, 0, 1, 0, 0, 1, 0, 0] 6. [0, 0, 0, 1, 1, 0, 0, 0] Only four of these are plotted in Figure 4. You are to indicate with reasons which pulse shape created each of these 4 figures. (a) Which pulse shape among the six choices results in Figure A? Justify your answer. (b) Which pulse shape among the six choices results in Figure B? Justify your answer. (c) Which pulse shape among the six choices results in Figure C? Justify your answer. (d) Which pulse shape among the six choices results in Figure D? Justify your answer. 7. Consider a resampler with input x(kt s ) and output x(kt s + τ). Using the approximation d(x(kt s + τ)) x(kt s + τ) x(kt s + τ ɛ) dτ ɛ for a positive ɛ, derive an approximate gradient descent algorithm that minimizes J FP (τ) = 1 N k 0 +N k=k 0 +1 x 4 (kt s + τ)

6 TB / Exam Style Questions 6 Figure 4: Various PAM Baseband Transmitted Signal Spectra for a suitably large N. 8. Consider the multiuser system shown in Figure 5. Both users are transmitting PAM signals for binary (±1), independent, equally probable symbols. The signal from the desired user (user 1) is distorted by a frequency selective channel with impulse response h 1 [k] = δ[k] + bδ[k 1] The signal from the interfering user (user 2) is simply scaled by h 2 [k] = c Thus, the difference equation generating the received signal r is r[k] = s 1 [k] + bs 1 [k 1] + cs 2 [k] The difference equation describing the equalizer input-output behavior is x[k] = r[k] + dr[k 1]

7 TB / Exam Style Questions 7 For the binary alphabet, the decision device is simply a sign operator. Figure 5: Baseband Communication System with Multiuser Interference (a) Write out the difference equation relating s 1 [k] and s 2 [k] to x[k] without using r by specifying the ρ i as functions of b, c, and d in x[k] = s 1 [k] + ρ 1 s 1 [k 1] + ρ 2 s 1 [k 2] + ρ 3 s 2 [k] + ρ 4 s 2 [k 1] + ρ 5 s 2 [k 2] ρ 1 = ρ 2 = ρ 3 = ρ 4 = ρ 5 = (b) Consider the cost function J MSE = 1 N k 0 +N k=k 0 +1 (s 1 [k] x[k]) 2 For the uncorrelated source symbol sequences, J MSE is minimized by minimizing ρ 2 i. For b = 0.8 and c = 0.3 compute the equalizer setting d that minimizes J MSE. You must provide a clear explanation of your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) (c) For b = 0.8, c = 0.3, and d = 0, (i.e. equalizer-less reception) does the system exhibit no decision errors for any input sequence or does it exhibit some decision errors with particular input sequences with regard to recovery of user 1 s symbol

8 TB / Exam Style Questions 8 sequence? You must provide a clear explanation of your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) (d) For b = 0.8, c = 0.3, and your d from part (b) does the system exhibit no decision errors for any input sequence or does it exhibit some decision errors with particular input sequences with regard to recovery of user 1 s symbol sequence? Does this system suffer more severe decision errors than the equalizer-less system of part (c)? You must provide a clear explanation of your answers to receive any credit. (Equations without prose commentary do not qualify as an explanation.) 9. Consider the baseband communication system segment shown in Figure 6. where Figure 6: Baseband Communication System Segment the message signal is a scaled impulse train m(t) = s[i]δ(t it + α) (1) i= with the scale factors, i.e. the symbols s[i], drawn from a finite alphabet. The channel noise, if presumed present, has a flat power spectral density. The objective is to recover the s[i] as the x 4 (kt) by proper selection of F(f) and T at the transmitter and (given knowledge of the transmitter-selected T) G(f) and τ at the receiver. (a) First we will examine a noise-free circumstance, where we assume n(t) = 0 for all t. Consider the transmit and receive filter spectra shown in Figure 7 and 8. With α and τ set to zero, is there a symbol period T for which x 4 (kt) = γm(k δt) for some integer δ? If your answer is yes, find this T. If your answer is no, provide the spectrum of a different G(f) that in combination with the original F(f) would prove satisfactory. You must persuasively and clearly explain your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) (b) Now we will consider the situation in Figure 6 where n(t) has a flat power spectral density of nonzero magnitude and α = 0.41 msec in m(t). We will consider the transmit filter with impulse response in Figure 9. (i) Draw the impulse response of the minimum-delay, causal receive filter that maximizes the SNR of the m-driven and n-driven components of x 4 (kt). Be certain to

9 TB / Exam Style Questions 9 Figure 7: Spectrum of Transmit Filter Figure 8: Spectrum of Receive Filter give specific values of time and amplitude at all breakpoints and local maxima of the resulting curve. Also, determine the associated optimum choice for receiver sampler timing offset τ. You must briefly explain your answers to receive any credit. (Equations without prose commentary do not qualify as an explanation.) (ii) For the optimal receive filter from part (i), determine the the minimum symbol interval T (in msec) at which no intersymbol interference appears with the optimum τ. You must briefly explain your answers to receive any credit. (Equations without prose commentary do not qualify as an explanation.) 10. The trained adaptive equalizer updates its impulse response coefficients f i [k] via f i [k + 1] = f i [k] + µe[k]r[k i] (2) where

10 TB / Exam Style Questions 10 Figure 9: Transmit Filter Impulse Response µ is the positive, small stepsize, e[k] is the prediction error, which is the difference s[k δ] y[k] between the equalizer output y[k] = N 1 i=0 f i [k]r[k i] (3) and the known transmitted training signal s with suitable delay δ, and r is the symbol-spaced output of the baud-timing and downsampling component of the receiver and therefore the input to the symbol-spaced equalizer. The equalizer parameter adaptation law of (2) is based on a stochastic gradient descent of the cost function J 1 [k] = (1/2)e 2 [k] (4) With channels with deep nulls in their frequency response and a very high SNR in the received signal, the minimization of (the average) of J 1 by (2) will result in a very large spike in the equalizer frequency response, in order for their product to be unity at the frequency of the channel null as desired at all frequencies. Such large spikes in the equalizer frequency response will undesirably amplify channel noise present in their sub-band. To inhibit the resulting large values of f i needed to create a frequency response with segments of high gain, consider a cost function that also penalizes the sum of the squares of the f i, e.g. J 2 [k] = (1/2) ( e 2 [k] + λ N 1 i=0 f 2 i [k] ) (5)

11 TB / Exam Style Questions 11 Derive the associated stochastic gradient descent based adaptive equalizer parameter update law for J 2. You must provide a clear explanation of your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) 11. You are attempting to build a frequency-division multiplexed (FDM) system with a square root raised cosine as the transmitter pulse shape. The symbol period is T (= 137 msec). Your design uses T/4 sampling, pulse lengths of 8T, and a rolloff factor of 0.9, but it s not working. You currently can only fit 3 modulated carrier signals without multiuser interference into the allotted bandwidth, but you need to fit 5. What parameters in your design would you change and why? You must provide a clear explanation of your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) 12. Consider two waveforms s 1 (t) and s 2 (t) neither of which is a Nyquist pulse. (a) Can the product s 1 (t)s 2 (t) be a Nyquist pulse? Explain your answer. (b) Can the convolution s 1 (t) s 2 (t) be a Nyquist pulse? Explain your answer. 13. Consider the baseband communication system with a symbol-scaled impulse train input s(t) = s i δ(t it ɛ) i= where T is the symbol period in seconds and 0.25msec > ɛ > 0 passing through (in sequence) a pulse shaping filter P(f), a channel transfer function C(f) with additive noise n(t), and a receive filter V (f), as shown in Figure 10. In addition consider the Figure 10: Baseband Communication System time signal x(t) provided in Figure 11. Each of the arcs is a semi-circle. (a) If x(t) is the inverse Fourier transform of P(f)C(f)V (f), what is the highest symbol frequency with no intersymbol interference supported by this communication system when = 0? Explain your answer.

12 TB / Exam Style Questions 12 Figure 11: Time Signal (b) With x(t) the inverse Fourier transform of P(f)C(f)V (f), select a sampler time offset T/2 > > T/2 in Figure?? to achieve the highest symbol frequency with no intersymbol interference supported by this communication system. What is this symbol frequency? Explain your answer. (c) Consider designing the receive filter under the assumption that a symbol period T can be (and will be) chosen so that samples y k of the receive filter output suffer no intersymbol interference. If x(t) represents the inverse Fourier transform of P(f)C(f) and the power spectral density (PSD) of n(t) is a constant, plot the impulse response of the causal, minimum delay, matched receive filter for V (f). 14. Consider the passband communication system shown in Figure 12. The mag- Figure 12: Passband Communication System nitude spectrum of the input signal s(t) is provided in Figure 13. Design the receiver to recover s(t) by selecting the maximum possible sampler period T s and the minimum possible cutoff frequency f c of the ideal lowpass filter. Explain why the sampling frequency 1/T s and the cutoff frequency f c selected are the smallest possible values.

13 TB / Exam Style Questions 13 Answers Figure 13: Source Sequence Magnitude Spectrum 1. (a) False, (b) True, (c) False, (d) True, (e) False, (f) False, (g) True, (h) True, (i) True, (j) False 2. (a) minimized, (b) τ[i + 1] = τ[i] + 4µ ɛn i k=i N+1 (1 x2 (kt s + τ[i]))(x(kt s + τ[i]))(x(kt s + τ) x(kt s + τ ɛ)) 3. (a) yes, examine p(kt + τ), which should be nonzero for only one value of k, for τ = 0.1 or 0.6, (b) yes, for p(kt + τ) consider τ = 0.15 or (a) yes, m[k] to y[k] is open-eye, (b) yes, e.g. α = (a) x 1 (t) = β 2 [cos(2π(f c f 0 )t + θ + γ + α) + cos(2π(f c + f 0 )t + θ + γ α)], (b) x 2 (t) = β 2 cos(2πf 0t + ξ α)cos(θ φ + γ) where ξ is the phase shift of LPF at f 0 (and f 0 ), (c) φ = θ + γ 6. (a) 2, (b) 3, (c) 4, (d) 6 7.τ[i + 1] = τ[i] 4µ ɛ (x3 (kt s + τ[i]))(x(kt s + τ[i]) x(kt s + τ[i] ɛ)) 8. (a) ρ 1 = b + d, ρ 2 = bd, ρ 3 = c, ρ 4 = cd, ρ 5 = 0, (b) d = , (c) some errors, (d) some errors 9. (a) yes, T = µsec, (b) (i) g(t) equals f(t) and τ = 2.59 msec, (ii) T = 3 msec 10. f j [k + 1] = (1 µλ)f j [k] + µe[k]r[k j] 11. Make rolloff factor small, i.e. 0.1, and width of pulse spectrum cut in half from rolloff of 0.9 or double T to halve bandwidth for small rolloff factor. 12. (a) yes, e.g. s 1 (t) = sin(x) and s 2 (t) = 1 x so s 1(t)s 2 (t) = sinc(x), (b) yes, e.g. s 1 (t) and s 2 (t) same square root raised cosine so s 1 (t) s 2 (t) is raised cosine 13. (a) f = 1 T = ɛ khz, (b) with = ɛ, T = 1, (c) flip Figure 11 about origin and slide to right until leading edge is at t = 4.5 msec and all nonzero values are on the nonnegative portion of the axis from 0 to 4.5 msec

14 TB / Exam Style Questions T s = 2 msec and f c = 0.2 khz,