EXAM STYLE QUESTIONS Covering Chapters 1 - 9 of Telecommunication Breakdown

TB 1-9 / Exam Style Questions 1

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 =6 c and |b|= 6 |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 x4(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. 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 straight- line, 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 φ1 =0.3 radians, f1 = 24.1 kHz, f2 = 23.9 kHz, f3 = 27.5 kHz, f4 = 29.3 kHz, and f5 = 22.6 kHz. The receiver processing r(t) is shown in Figure 4. All bandpass and lowpass

Figure 2: Transmitter 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 φ2 is chosen to maximize the magnitude of y(t) and reﬂects the value of φ1 and the delays imposed by the two ideal bandpass ﬁlters between the transmitter and receiver mixers. Select the receiver parameters f6, f7, f8, and f9, 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(2πfct + φc) with the carrier frequency fc > 10B. The channel oﬀers only a delay such that the 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 Tc (= 1/fc) plus a fraction of Tc d = nTc + Tc/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(2πfct + φr)

where the receiver mixer phase need not match the transmitter mixer phase φc. 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 fc, φc, d, α, φr, and w(t) (without use of x, r, n, or Tc). (b) Determine the amplitude of the minimum and maximum values of y(t) for α = 4.

(c) For α = 6, n = 42, φc = 0.2 radians, and Tc = 20µsec, determine φr that maximizes the magnitude of the maximum and minimum values of v(t).

4. Consider performing iterative maximization of

J(x)=8 − 6|x| + 6cos(6x)

via ∂J(x) x(k +1) = x(k)+ µ | ∂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 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(2πft) with f = 1.45MHz. With (1/(2B)) < T1 < (1/f), select T1, T2, T3, and β so magnitude spectrum of x[k] matches the magnitude spectrum of T1-spaced samples of w(t). Justify your answer by drawing the magnitude spectrum of x1, x2, and x3 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 fc. The channel adds an interferer n. The received signal r is downconverted to the IF signal x(t) by a mixer with frequency fr. TB 1-9 / Exam Style Questions 6

Figure 7: System

(a) With n(t)=0, fr = 36 kHz, and fc = 83 kHz, mark each and every range below that includes any part of the IF passband signal x(t). (i) 0-20 kHz, (ii) 20-40 kHz, (iii) 40-60 kHz, (iv) 60-80 kHz, (v) 80-100 kHz, (vi) 100-120 kHz, (vii) 120-140 kHz, (viii) 140-160 kHz, (ix) 160-180 kHz, (x) 180-200 kHz

(b) With fr = 36 kHz and fc = 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) 20-40 kHz, (iii) 40-60 kHz, (iv) 60-80 kHz, (v) 80-100 kHz, (vi) 100-120 kHz, (vii) 120-140 kHz, (viii) 140-160 kHz, (ix) 160-180 kHz, (x) 180-200 kHz

(c) With fr = 84 kHz and fc = 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) 20-40 kHz, (iii) 40-60 kHz, (iv) 60-80 kHz, (v) 80-100 kHz, (vi) 100-120 kHz, (vii) 120-140 kHz, (viii) 140-160 kHz, (ix) 160-180 kHz, (x) 180-200 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 x1 of 4 kHz, f1 = 28 kHz, f2 = 20 kHz, and f3 = 26 kHz. TB 1-9 / Exam Style Questions 7

Figure 8: Spectrum

Figure 9: System A

(i) The absolute bandwidth of x2(t) is

(ii) The absolute bandwidth of x3(t) is

(iii) The absolute bandwidth of x4(t) is

(iv) The maximum frequency in x2(t) is

(v) The maximum frequency in x3(t) is

(b) Consider the schematic shown in Figure 10 with the absolute bandwidth of the baseband signal x1 of 6 MHz and of the baseband signal x2(t) of 4 MHz, f1 = 164 MHz, f2 = 154 MHz, f3 = 148 MHz, f4 = 160 MHz, f5 = 80 MHz, φ = π/2, and f6 = 82 MHz. TB 1-9 / Exam Style Questions 8

Figure 10: System B

(i) The absolute bandwidth of x3(t) is

(ii) The absolute bandwidth of x5(t) is

(iii) The absolute bandwidth of x6(t) is

(iv) The maximum frequency in x3(t) is

(v) The maximum frequency in x5(t) is

8. Consider the PAM communication system in Figure 11. The input x1(t) has a triangular baseband magnitude spectrum shown in Figure 12. The frequency speciﬁ- cations are f1 = 100 kHz, f2 = 1720 kHz, f3 = 1940 kHz, f4 = 1580 kHz, f5 = 1720 kHz, f6 = 1880 kHz, and f7 = 1300 kHz.

(a) Draw the magnitude spectrum |X5(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 f8 and f9 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(2πfct + φ) 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 fc/2. The signals x and y are generated via

x(t) = LPF[r(t) cos(2πfct + θ)]

y(t) = LPF[r(t) sin(2πfct + θ)] where the LPF cutoﬀ frequency is fc/2.

(a) Determine x(t) in terms of w(t), fc, 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 ∂ 1 { x2(t)} = −x(t)y(t) ∂θ 2 You must show your work clearly to receive partial credit for an incorrect answer. (c) Determine the values of θ maximizing x2(t). You must show your work clearly to receive partial credit for an incorrect answer. 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 p1(t), p2(t), and p3(t) has the largest baseband power bandwidth? Justify your answer.

(b) Which pulse shape among p1(t), p2(t), and p3(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 |X1(f)| of

x1(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. TB 1-9 / Exam Style Questions 11

Figure 14: A Communication System Segment

Figure 15: Input Magnitude Spectrum

(b) Draw the magnitude spectrum |X2(f)| of

x2(t)= w(t)x1(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 -3750 Hz and 3750 Hz, draw the magnitude spectrum |X3(f)| of

∞

x3(t)= x2(t) X δ(t − kTs) (3) k=−∞ from Figure 14 for Ts = 400µsec. Be certain to give speciﬁc values of frequency and magnitude at all breakpoints and local maxima of the resulting curve. 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(2πfit) (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 fL and fU (and between −fU and −fL) with fU > fL > 0 where it has a unit gain. The lowpass ﬁlter is also ideal with a rectangular magnitude spectrum of gain zero at frequencies below −fC and above fC 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 fL = 12.4 kHz and fC = 9.8 kHz, select f1, f2, and fU to produce |Y (f)| (i.e. the single sideband intermediate frequency version of u1(t)) in Figure 17(c) given |U1(f)| and |U2(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 f1, f2, and fU selected in part (a), sketch the magnitude spectrum of x1(t), which is the output of the summer in Figure 16.

(c) For the design variables f1, f2, and fU selected in part (a), sketch the magnitude spectrum of x2(t), which is the output of the bandpass ﬁlter in Figure 16.

13. 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

y(t)= u(t)cos(2πfct) (4)

and carrier frequencies fc of 12 MHz and 50 MHz

• two linear bandpass ﬁlters with ideal rectangular magnitude spectrum of gain one between −fU and −fL and between fL and fU and zero elsewhere with (fL, fU ) of (12MHz, 32MHz) and (35MHz, 50MHz). TB 1-9 / Exam Style Questions 13

Figure 17: Magnitude Spectra of (a) u1(t), (b) u2(t), and (c) desired y(t) TB 1-9 / Exam Style Questions 14

• two impulse samplers with input u and output y related by

∞

y(t)= X u(t)δ(t − kTs) (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)= u2(t) (6)

• and three summers with inputs u1 and u2 and output y related by

y(t)= u1(t)+ u2(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. TB 1-9 / Exam Style Questions 15

14. 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 fC . The channel adds a narrowband interferer at frequency fI . The received signal is sampled with period Ts. The sampled signal is demodulated by mixing with a cosine of frequency f1 and ideal lowpass ﬁltering with a cutoﬀ frequency of f2.

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 x4 is the same (up to a scale factor) as the magnitude spectrum of the sampled w(t) with a sample period of Ts. You must clearly justify your answer to receive any credit. (Equations and diagrams without prose commentary do not qualify as an explanation.) 1 (a) Candidate System A: B = 7 kHz, fC = 34 kHz, fI = 49 kHz, Ts = 34 msec, f1 = 0, and f2 = 16kHz. 1 (b) Candidate System B: B = 11 kHz, fC = 39 kHz, fI = 130 kHz, Ts = 52 msec, f1 = 13kHz, and f2 = 12kHz.

15. Consider the communication system segment shown in Figure 20. In this

Figure 20: Another Communication System Segment 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(2πfot) (8)

and oscillator frequencies fo of 1 MHz, 1.5 MHz, 2 MHz, and 4 MHz • four linear bandpass ﬁlters with ideal rectangular magnitude spectra of gain one between −fU and −fL and between fL and fU (with fL < fU ) and zero elsewhere with (fL, fU ) 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)= X u(t)δ(t − kTs) (9) k=−∞ 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 (fL, fU ) in MHz:

• sampler period Ts in µsec: TB 1-9 / Exam Style Questions 17

• mixer oscillator frequency fo 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(2πfT t + θT ) and translates it to another center frequency. The message signal s(t) has a bandwidth of fT /100 Hz. We wish to achieve a new center frequency greater than that of the original transmitter carrier fT (e.g. for subsequent retransmission).

(a) Write X(f) in terms of S(f), fT , fR, θT , and θR.

(b) With the transmitter frequency of fT = 41 kHz and the transmitter phase of ◦ θT = 30 , select the downconverter’s local oscillator frequency fR and phase θR and the ideal (i.e. unit gain in the passband, zero otherwise) bandpass ﬁlter’s maximum lower fL and minimum upper fU 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). TB 1-9 / Exam Style Questions 18

Figure 22: Carrier Conversion Answers

• 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 2.

Figure 23: Magnitude Spectrum

• 3. (a) y(t) = (1/2)w(t − d)[cos(φc − 2πfcd − φr) + cos(4πfct + φc − 2πfcd + φr)], (b) v(t) = 0, (c) φr = −0.847 ± 2π` for ` = ±{0, 1, 2, ...} • 4. (a) x = 1.0193, (b) No, global maximum at x = 0 with J(0) = 14 > J(1.0193) ≈ 7.92.

−6 • 5. One choice: T1 = 1.38 × 10 , T2 = 0, 1/T3 >B, and β = 2/0.3. Another choice: 1/T1 = 1.25 MHz, T2 = 0.2 MHz, 1/T3 = 0.2 MHz, and β = 2/0.3. Note that T2 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) TB 1-9 / Exam Style Questions 19

• 7. (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) f8 = 420 kHz or 880 kHz or 1720 kHz or 2180 kHz,

Figure 24: Magnitude Spectrum

100

• 10. (a) p2(t), (b) p3(t). • 11. (a) See Figure 25. (b) See Figure 26. (c) See Figure 27.

Figure 25: Magnitude Spectrum of x1(t)

• 12. (a) fU = f1 = 15.2 kHz, f2 = 6.9 kHz or fU = 15.2 kHz, f1 = 12.4 kHz, f2 = 20.7 kHz, (b) See Figure 28 or 29. (c) See Figure 30 or 31.

• 13. See Figures 32 and 33.

• 14. (a) Not successful, (b) Successful

• 15. (a) (fL,fU ) = (3.4, 7.2) MHz or (4.2, 8.3) MHz, Ts = 0.25 µsec, fo = 2 Mhz, (b) See Figure 34. (c) See Figure 35. TB 1-9 / Exam Style Questions 20

Figure 26: Magnitude Spectrum of x2(t)

Figure 27: Magnitude Spectrum of x3(t)

1 j(θR+θT ) j(θR−θT ) −j(θR−θT ) • 16. (a) X(f)= 4 [e S(f −fT −fR)+e S(f +fT −fR)+e S(f − −j(θR+θT ) ◦ fT + fR)+ e S(f + fT + fR)], (b) fR = 71 kHz, θR = −30 , fU = 112.41 kHz, ◦ and fL = 111.59 kHz or fR = 153 kHz, θR = 30 , fU = 112.41 kHz, and fL = 111.59 kHz (c) 185 kHz for fR = 71 kHz or 265 kHz for fR = 153 kHz.

Figure 28: Magnitude Spectrum of x1(t) with f1 = fU = 15.2, f2 =6.9 TB 1-9 / Exam Style Questions 21

Figure 29: Magnitude Spectrum of x1(t) with fU = 15.2, f1 = 12.4, f2 = 20.7

Figure 30: Magnitude Spectrum of x2(t) with f1 = fU = 15.2, f2 =6.9 TB 1-9 / Exam Style Questions 22

Figure 31: Magnitude Spectrum of x2(t) with fU = 15, 2, f1 = 12.4, f2 = 20.7

Figure 32: Block Diagram TB 1-9 / Exam Style Questions 23

Figure 33: Magnitude Spectrum of x1(t), x2(t), and x3(t)

Figure 34: Magnitude Spectrum of Sampler Output TB 1-9 / Exam Style Questions 24

Figure 35: Magnitude Spectrum of Sampler Output