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1 TB 1-9 / Exam Style Questions 1 EXAM STYLE QUESTIONS Covering Chapters 1-9 of Telecommunication Breakdown 1. Clearly circle one answer for each part. (a) TRUE or FALSE: Absolute bandwidth is never less than 3-db power bandwidth. (b) TRUE or FALSE: The flatter the top of the pulse shape, the less sensitive the receiver is to small timing offsets. (c) TRUE or FALSE: A linear, time-invariant system exists that from zero initial conditions has input a cos(bt) and output c sin(dt) with a c and b d. (d) TRUE or FALSE: Sampling an analog signal with nonzero values in its magnitude spectrum at every frequency between B and B using a sampling period T with 1/T > B can result in a discrete-time signal with components in its spectrum at frequencies between B and 1/T. (e) TRUE or FALSE: Gradient descent of a unimodal single-variable cost function with only one point having a zero derivative with respect to that single variable and that point having a positive second derivative will converge to the global minimum from any initialization. (f) TRUE or FALSE: The bandwidth of x 4 (t) cannot be greater than that of x(t). (g) TRUE or FALSE: A second-order finite-impulse-response filter with its two zeros having positive real parts is a highpass filter. (h) TRUE or FALSE: A small, fixed phase offset in the receiver demodulating AM with suppressed carrier produces an undesirable low frequency modulated version of the analog message. (i) TRUE or FALSE: Quadrature amplitude modulation can combine two real, baseband messages of absolute bandwidth B in a radio-frequency signal of absolute bandwidth B. (j) TRUE or FALSE: Filtering a passband signal with absolute bandwidth B through certain fixed linear filters can result in an absolute bandwidth of the filter output greater than B. (k) TRUE or FALSE: In the downconversion of a 2-PAM passband signal, a fixed offset between the frequency of the transmitter oscillator and the frequency of the receiver oscillator will result in a fixed attenuation factor.

2 TB 1-9 / Exam Style Questions 2 (l) TRUE or FALSE: A linear, time-invariant, finite-impulse-response filter with a frequency response having unit magnitude over all frequencies and a straightline, slpoed phase curve has as its transfer function a pure delay. 2. Consider the message signal w(t) with magnitude spectrum shown in Figure 1. The transmitter in Figure 2 produces the transmitted signal x(t) which passes through Figure 1: Message Magnitude Spectrum the channel in Figure 3 which scales the signal and adds narrowband interferers to create the received signal r(t). The transmitter and channel parameters are φ 1 = 0.3 radians, f 1 = 24.1 khz, f 2 = 23.9 khz, f 3 = 27.5 khz, f 4 = 29.3 khz, and f 5 = 22.6 khz. The receiver processing r(t) is shown in Figure 4. All bandpass and lowpass Figure 2: Transmitter

3 TB 1-9 / Exam Style Questions 3 Figure 3: Channel Figure 4: Receiver filters are considered ideal with a gain of unity in the passband and zero in the stopband. (a) Sketch R(f) for 30kHz f 30kHz. Be certain to label clearly the amplitudes and frequencies of key points on your sketch. (b) Assume that φ 2 is chosen to maximize the magnitude of y(t) and reflects the value of φ 1 and the delays imposed by the two ideal bandpass filters between the transmitter and receiver mixers. Select the receiver parameters f 6, f 7, f 8, and f 9, so the receiver output y(t) is a scaled version of w(t). 3. The analog baseband message signal w(t) with all of its nonzero values in its magnitude expression between B and B Hz is upconverted into the transmitted passband signal x(t) via AM with suppressed carrier modulation x(t) = w(t) cos(2πf c t + φ c ) with the carrier frequency f c > 10B. The channel offers only a delay such that the

4 TB 1-9 / Exam Style Questions 4 received signal r is r(t) = x(t d) where the delay is an integer multiple of the carrier period T c (= 1/f c ) plus a fraction of T c d = nt c + T c /α where n is nonnegative and α > 1. The receiver mixer is perfectly synchronized to the transmitter such that the mixer output y(t) is y(t) = r(t) cos(2πf c t + φ r ) where the receiver mixer phase need not match the transmitter mixer phase φ c. The receiver then lowpass filters y to produce v(t) = LPF{y(t)} where the lowpass filter is ideal with unity passband gain, linear passband phase with zero phase at zero frequency, and cutoff frequency 1.2B. (a) Write a formula for the receiver mixer output y(t) as a function of f c, φ c, d, α, φ r, and w(t) (without use of x, r, n, or T c ). (b) Determine the amplitude of the minimum and maximum values of y(t) for α = 4. (c) For α = 6, n = 42, φ c = 0.2 radians, and T c = 20µsec, determine φ r that maximizes the magnitude of the maximum and minimum values of v(t). 4. Consider performing iterative maximization of via J(x) = 8 6 x + 6 cos(6x) x(k + 1) = x(k) + µ J(x) x x=x(k) from the initialization x(0) = 0.7. (a) Assuming the use of a suitably small stepsize µ, determine the convergent value of x. (b) Is the convergent value of x in part (a) the global maximum of J(x)? Justify your answer. 5. The receiver processing u(t) is shown in Figure 5. The triangularly-shaped magnitude spectrum of the real message signal w(t) is shown in Figure 6 where B = 0.2MHz. The recevied signal u(t) is an attenuated version of the transmitted

5 TB 1-9 / Exam Style Questions 5 Figure 5: Receiver Figure 6: Message Magnitude Spectrum AM-with-suppressed-carrier signal u(t) = 0.15w(t) cos(2πf t) with f = 1.45MHz. With (1/(2B)) < T 1 < (1/f), select T 1, T 2, T 3, and β so magnitude spectrum of x[k] matches the magnitude spectrum of T 1 -spaced samples of w(t). Justify your answer by drawing the magnitude spectrum of x 1, x 2, and x 3 between f and f. 6. Consider the system described in Figure 7. The message with a bandwidth of 22kHz and a magnitude spectrum described in Figure 8 is upconverted by a mixer with carrier frequency f c. The channel adds an interferer n. The received signal r is downconverted to the IF signal x(t) by a mixer with frequency f r.

6 TB 1-9 / Exam Style Questions 6 Figure 7: System (a) With n(t) = 0, f r = 36 khz, and f c = 83 khz, mark each and every range below that includes any part of the IF passband signal x(t). (i) 0-20 khz, (ii) khz, (iii) khz, (iv) khz, (v) khz, (vi) khz, (vii) khz, (viii) khz, (ix) khz, (x) khz (b) With f r = 36 khz and f c = 83 khz, mark each and every range below that includes any frequency that causes a narrowband interferer n to induce an image in the nonzero portions of the magnitude spectrum of the IF passband signal x(t). (i) 0-20 khz, (ii) khz, (iii) khz, (iv) khz, (v) khz, (vi) khz, (vii) khz, (viii) khz, (ix) khz, (x) khz (c) With f r = 84 khz and f c = 62 khz, mark each and every range below that includes any frequency that causes a narrowband interferer n to induce an image in the nonzero portions of the magnitude spectrum of the IF passband signal x(t). (i) 0-20 khz, (ii) khz, (iii) khz, (iv) khz, (v) khz, (vi) khz, (vii) khz, (viii) khz, (ix) khz, (x) khz 7. In this problem you will be given a schematic and numerical specifications and asked to provide certain specifications of various signals. All bandpass and lowpass filters are considered ideal with a gain of unity in the passband and zero in the stopband. Write your answer in the space provided. There will be no partial credit. (a) Consider the schematic shown in Figure 9 with the absolute bandwidth of the baseband signal x 1 of 4 khz, f 1 = 28 khz, f 2 = 20 khz, and f 3 = 26 khz.

7 TB 1-9 / Exam Style Questions 7 Figure 8: Spectrum Figure 9: System A (i) The absolute bandwidth of x 2 (t) is (ii) The absolute bandwidth of x 3 (t) is (iii) The absolute bandwidth of x 4 (t) is (iv) The maximum frequency in x 2 (t) is (v) The maximum frequency in x 3 (t) is (b) Consider the schematic shown in Figure 10 with the absolute bandwidth of the baseband signal x 1 of 6 MHz and of the baseband signal x 2 (t) of 4 MHz, f 1 = 164 MHz, f 2 = 154 MHz, f 3 = 148 MHz, f 4 = 160 MHz, f 5 = 80 MHz, φ = π/2, and f 6 = 82 MHz.

8 TB 1-9 / Exam Style Questions 8 Figure 10: System B (i) The absolute bandwidth of x 3 (t) is (ii) The absolute bandwidth of x 5 (t) is (iii) The absolute bandwidth of x 6 (t) is (iv) The maximum frequency in x 3 (t) is (v) The maximum frequency in x 5 (t) is 8. Consider the PAM communication system in Figure 11. The input x 1 (t) has a triangular baseband magnitude spectrum shown in Figure 12. The frequency specifications are f 1 = 100 khz, f 2 = 1720 khz, f 3 = 1940 khz, f 4 = 1580 khz, f 5 = 1720 khz, f 6 = 1880 khz, and f 7 = 1300 khz. (a) Draw the magnitude spectrum X 5 (f) between ±3000 khz. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve. You must show your work clearly to receive partial credit for an incorrect answer. (b) Specify values of f 8 and f 9 that recover the original message without corruption with M = 2. You must show your work clearly to receive partial credit for an incorrect answer. 9. Consider the modulated signal r(t) = w(t) cos(2πf c t + φ)

9 TB 1-9 / Exam Style Questions 9 Figure 11: PAM System where the absolute bandwidth of the baseband message waveform w(t) is less than f c /2. The signals x and y are generated via where the LPF cutoff frequency is f c /2. x(t) = LPF[r(t) cos(2πf c t + θ)] y(t) = LPF[r(t) sin(2πf c t + θ)] (a) Determine x(t) in terms of w(t), f c, φ, and θ, i.e. without using r(t) or the LPF operator included in its definition above. You must show your work clearly to receive partial credit for an incorrect answer. (b) With {LPF[x(α, t)]} = LPF[ {x(α, t)}] α α show that θ {1 2 x2 (t)} = x(t)y(t) You must show your work clearly to receive partial credit for an incorrect answer. (c) Determine the values of θ maximizing x 2 (t). You must show your work clearly to receive partial credit for an incorrect answer.

10 TB 1-9 / Exam Style Questions 10 Figure 12: Input Magnitude Spectrum 10. Consider the three pulse shapes sketched in Figure 13 for a T-spaced PAM system. Figure 13: Pulse Shapes (a) Which pulse shape among p 1 (t), p 2 (t), and p 3 (t) has the largest baseband power bandwidth? Justify your answer. (b) Which pulse shape among p 1 (t), p 2 (t), and p 3 (t) has the smallest baseband power bandwidth? Justify your answer. 11. Consider the communication system segment shown in Figure 14. The magnitude spectrum of the input w(t) is shown in Figure 15. (a) Draw the magnitude spectrum X 1 (f) of x 1 (t) = w(t)cos(1500πt) (1) from Figure 14. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve.

11 TB 1-9 / Exam Style Questions 11 Figure 14: A Communication System Segment Figure 15: Input Magnitude Spectrum (b) Draw the magnitude spectrum X 2 (f) of x 2 (t) = w(t)x 1 (t) (2) from Figure 14. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve. (c) Between Hz and 3750 Hz, draw the magnitude spectrum X 3 (f) of x 3 (t) = x 2 (t) k= δ(t kt s ) (3) from Figure 14 for T s = 400µsec. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve.

12 TB 1-9 / Exam Style Questions 12 Figure 16: Another Communication System Segment 12. Consider the communication system segment shown in Figure 16. Each modulator is described by the product of its input with cos(2πf i t) (where the subscript i indicates the index of the modulator). The bandpass filter is ideal with a rectangular magnitude spectrum of gain zero at all frequencies except between f L and f U (and between f U and f L ) with f U > f L > 0 where it has a unit gain. The lowpass filter is also ideal with a rectangular magnitude spectrum of gain zero at frequencies below f C and above f C and unit gain in between. The spectra for the two inputs and the desired output spectra are provided, respectively in parts (a), (b), and (c) of Figure 17. (a) Given f L = 12.4 khz and f C = 9.8 khz, select f 1, f 2, and f U to produce Y (f) (i.e. the single sideband intermediate frequency version of u 1 (t)) in Figure 17(c) given U 1 (f) and U 2 (f) in Figures 17(a) and 17(b). To receive any partial credit for an incorrect answer, you must clearly explain how you determined your answer. (b) For the design variables f 1, f 2, and f U selected in part (a), sketch the magnitude spectrum of x 1 (t), which is the output of the summer in Figure 16. (c) For the design variables f 1, f 2, and f U selected in part (a), sketch the magnitude spectrum of x 2 (t), which is the output of the bandpass filter in Figure In this problem you are to build a receiver from a limited number of components. The parts available are: two product modulators with input u and output y related by and carrier frequencies f c of 12 MHz and 50 MHz y(t) = u(t)cos(2πf c t) (4) two linear bandpass filters with ideal rectangular magnitude spectrum of gain one between f U and f L and between f L and f U and zero elsewhere with (f L, f U ) of (12MHz, 32MHz) and (35MHz, 50MHz).

13 TB 1-9 / Exam Style Questions 13 Figure 17: Magnitude Spectra of (a) u 1 (t), (b) u 2 (t), and (c) desired y(t)

14 TB 1-9 / Exam Style Questions 14 two impulse samplers with input u and output y related by y(t) = u(t)δ(t kt s ) (5) k= with sample periods of 1/15 and 1/12 microseconds one square law device with input u and output y related by y(t) = u 2 (t) (6) and three summers with inputs u 1 and u 2 and output y related by y(t) = u 1 (t) + u 2 (t) (7) The spectrum of the received signal is illustrated in Figure 18. The desired baseband Figure 18: Magnitude Spectra of Received Signal r(t) output of your receiver should be a scaled version of the triangular portion centered at zero frequency with no other signals in the range of -8 to 8 MHz. Using no more than four parts from the 10 available, build a receiver that produces the desired baseband signal. Draw its block diagram. If you can build the desired receiver with one extra part of one of the types described but with different specifications (such as carrier frequency for a product demodulator) than those in stock, you may do so, but your maximum score will be only 2/3 of that possible. To receive credit, you must clearly explain how you determined your answer. A recommended method of validating your design is to sketch the magnitude spectrum of the output of each part of your receiver as the input spectrum is modified on its passage through the system.

15 TB 1-9 / Exam Style Questions Consider the RF link with a digital receiver shown in Figure 19. The baseband signal w(t) has absolute bandwidth B. The carrier frequency is f C. The channel adds a narrowband interferer at frequency f I. The received signal is sampled with period T s. The sampled signal is demodulated by mixing with a cosine of frequency f 1 and ideal lowpass filtering with a cutoff frequency of f 2. Figure 19: RF Link with Digital Receiver For the following designs you are to decide if they are successful, i.e. whether or not the magnitude spectrum of the lowpass filter output x 4 is the same (up to a scale factor) as the magnitude spectrum of the sampled w(t) with a sample period of T s. You must clearly justify your answer to receive any credit. (Equations and diagrams without prose commentary do not qualify as an explanation.) (a) Candidate System A: B = 7 khz, f C = 34 khz, f I = 49 khz, T s = 1 34 msec, f 1 = 0, and f 2 = 16kHz. (b) Candidate System B: B = 11 khz, f C = 39 khz, f I = 130 khz, T s = 1 52 msec, f 1 = 13kHz, and f 2 = 12kHz. 15. Consider the communication system segment shown in Figure 20. In this Figure 20: Another Communication System Segment

16 TB 1-9 / Exam Style Questions 16 problem you are to build a receiver from a limited number of choices for the specifications of the 3 major components: a bandpass filter, a sampler, and a mixer. The specific parts available are: four mixers with input u and output y related by y(t) = u(t)cos(2πf o t) (8) and oscillator frequencies f o of 1 MHz, 1.5 MHz, 2 MHz, and 4 MHz four linear bandpass filters with ideal rectangular magnitude spectra of gain one between f U and f L and between f L and f U (with f L < f U ) and zero elsewhere with (f L, f U ) of (0.5MHz, 6MHz), (1.2 MHz, 6.2MHz), (3.4 MHz, 7.2MHz), and (4.2 MHz, 8.3MHz) four impulse samplers with input u and output y related by y(t) = k= u(t)δ(t kt s ) (9) with sample periods of 1/7, 1/5, 1/4, and 1/3.5 microseconds The received signal r(t) magnitude spectrum R(f) is shown in Figure 21. The objective is to specify the bandpass filter, sampler, and mixer so that the M -shaped magnitude spectrum segment is centered at f = 0 in Y (f) with no other signals within ±1.5 MHz of upper and lower edges of the baseband segment of the magnitude spectrum. Figure 21: Received Signal Magnitude Spectrum (a) Specify the three parts from the 12 provided that you propose for your receiver: bandpass filter passband range (f L, f U ) in MHz: sampler period T s in µsec:

17 TB 1-9 / Exam Style Questions 17 mixer oscillator frequency f o in MHz: (There will be no partial credit for this part of this problem. Therefore, there is no need for an explanation of your answer, which will be described in the following parts of this problem. Partial credit for this problem will be available in the remaining parts, which will be graded independently of whether or not the answer to this part is correct.) (b) For the three components selected in part (a), sketch the magnitude spectrum of the sampler output between 20 and +20 MHz. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve. You should provide an explanation of your answer if you expect to receive partial credit for an answer that is only partially correct. (Equations without prose commentary do not qualify as an explanation.) (c) For the three components selected in part (a), draw the magnitude spectrum of y(t) between between the frequencies 12 and +12 MHz for your design. Be certain to give specific values of frequency and magnitude at all breakpoints and local maxima of the resulting curve. You should provide an explanation of your answer if you expect to receive partial credit for an answer that is only partially correct. (Equations without prose commentary do not qualify as an explanation.) (d) Is the magnitude spectrum of y(t) identical to the the M-shaped segment of R(f) first downconverted to baseband and then sampled? You must provide a clear explanation of your answer to receive any credit. (Equations without prose commentary do not qualify as an explanation.) 16. Consider the receiver front end illustrated in Figure 22 that receives the radio frequency (RF) signal r(t) = s(t)cos(2πf T t + θ T ) and translates it to another center frequency. The message signal s(t) has a bandwidth of f T /100 Hz. We wish to achieve a new center frequency greater than that of the original transmitter carrier f T (e.g. for subsequent retransmission). (a) Write X(f) in terms of S(f), f T, f R, θ T, and θ R. (b) With the transmitter frequency of f T = 41 khz and the transmitter phase of θ T = 30, select the downconverter s local oscillator frequency f R and phase θ R and the ideal (i.e. unit gain in the passband, zero otherwise) bandpass filter s maximum lower f L and minimum upper f U cutoff frequencies to pass the largest magnitude replica possible of a scaled version of S(f) centered at the frequency of 112 khz. (c) Specify the center frequency of a very narrowband interferer that, though not in the spectral bands occupied by R(f), if added to r(t) ends up in y(t).

18 TB 1-9 / Exam Style Questions 18 Answers Figure 22: Carrier Conversion 1. (a) True, (b) True, (c) False, (d) True, (e) True, (f) False, (g) True, (h) False, (i) True, (j) False, (k) False, (l) True 2. (a) See Figure 23. (b) 22.6 < f 6 < 24.4, 26.1 < f 7 < 29.3, f 8 = 24.1, f 9 > 2. Figure 23: Magnitude Spectrum 3. (a) y(t) = (1/2)w(t d)[cos(φ c 2πf c d φ r ) + cos(4πf c t + φ c 2πf c d + φ r )], (b) v(t) = 0, (c) φ r = ± 2πl for l = ±{0,1,2,...} 4. (a) x = , (b) No, global maximum at x = 0 with J(0) = 14 > J(1.0193) One choice: T 1 = , T 2 = 0, 1/T 3 > B, and β = 2/0.3. Another choice: 1/T 1 = 1.25 MHz, T 2 = 0.2 MHz, 1/T 3 = 0.2 MHz, and β = 2/0.3. Note that T 2 is a frequency and not a time variable. 6. (a) (ii), (iii), (iv), (v), (vi), (vii), (viii), (b) (i), (ii), (iv), (v), (vii), (viii), (ix), (c) (i), (ii), (iii), (iv), (v), (vi), (x)

19 TB 1-9 / Exam Style Questions (a): (i) 8 khz, (ii) 64 khz, (iii) 0, (iv) 32 khz, (v) 64 khz; (b): (i) 20 MHz, (ii) 170 MHz, (iii) 10 MHz, (iv) 170 MHz, (v) 240 MHz 8. (a) See Figure 24. (b) f 8 = 420 khz or 880 khz or 1720 khz or 2180 khz, Figure 24: Magnitude Spectrum 100 < f 9 < 360 khz 9. (a) x(t) = w(t) w(t) 2 cos(φ θ), (b) identity because y(t) = 2 sin(φ θ), (c) θ = φ+nπ for integer n. 10. (a) p 2 (t), (b) p 3 (t). 11. (a) See Figure 25. (b) See Figure 26. (c) See Figure 27. Figure 25: Magnitude Spectrum of x 1 (t) 12. (a) f U = f 1 = 15.2 khz, f 2 = 6.9 khz or f U = 15.2 khz, f 1 = 12.4 khz, f 2 = 20.7 khz, (b) See Figure 28 or 29. (c) See Figure 30 or See Figures 32 and (a) Not successful, (b) Successful 15. (a) (f L,f U ) = (3.4,7.2) MHz or (4.2,8.3) MHz, T s = 0.25 µsec, f o = 2 Mhz, (b) See Figure 34. (c) See Figure 35.

20 TB 1-9 / Exam Style Questions 20 Figure 26: Magnitude Spectrum of x 2 (t) Figure 27: Magnitude Spectrum of x 3 (t) 16. (a) X(f) = 1 4 [ej(θ R+θ T ) S(f f T f R )+e j(θ R θ T ) S(f +f T f R )+e j(θ R θ T ) S(f f T +f R )+e j(θ R+θ T ) S(f +f T +f R )], (b) f R = 71 khz, θ R = 30, f U = khz, and f L = khz or f R = 153 khz, θ R = 30, f U = khz, and f L = khz (c) 185 khz for f R = 71 khz or 265 khz for f R = 153 khz. Figure 28: Magnitude Spectrum of x 1 (t) with f 1 = f U = 15.2, f 2 = 6.9

21 TB 1-9 / Exam Style Questions 21 Figure 29: Magnitude Spectrum of x 1 (t) with f U = 15.2, f 1 = 12.4, f 2 = 20.7 Figure 30: Magnitude Spectrum of x 2 (t) with f 1 = f U = 15.2, f 2 = 6.9

22 TB 1-9 / Exam Style Questions 22 Figure 31: Magnitude Spectrum of x 2 (t) with f U = 15, 2, f 1 = 12.4, f 2 = 20.7 Figure 32: Block Diagram

23 TB 1-9 / Exam Style Questions 23 Figure 33: Magnitude Spectrum of x 1 (t), x 2 (t), and x 3 (t) Figure 34: Magnitude Spectrum of Sampler Output

24 TB 1-9 / Exam Style Questions 24 Figure 35: Magnitude Spectrum of Sampler Output

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