EXAM STYLE QUESTIONS Covering Chapters 1 - 9 of ... · EXAM STYLE QUESTIONS Covering Chapters 1 - 9 of Telecommunication Breakdown 1. Clearly circle one answer for each part. (a)

TB 1-9 / Exam Style Questions 1

EXAM STYLE QUESTIONSCovering 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 thereceiver is to small timing offsets.

(c) TRUE or FALSE: A linear, time-invariant system exists that from zero initialconditions 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 magni-tude spectrum at every frequency between −B and B using a sampling periodT with 1/T > B can result in a discrete-time signal with components in itsspectrum at frequencies between B and 1/T .

(e) TRUE or FALSE: Gradient descent of a unimodal single-variable cost functionwith only one point having a zero derivative with respect to that single variableand that point having a positive second derivative will converge to the globalminimum 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 twozeros having positive real parts is a highpass filter.

(h) TRUE or FALSE: A small, fixed phase offset in the receiver demodulatingAM with suppressed carrier produces an undesirable low frequency modulatedversion 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 ab-solute bandwidth B.

(j) TRUE or FALSE: Filtering a passband signal with absolute bandwidth Bthrough certain fixed linear filters can result in an absolute bandwidth of thefilter output greater than B.

(k) TRUE or FALSE: In the downconversion of a 2-PAM passband signal, a fixedoffset between the frequency of the transmitter oscillator and the frequency ofthe receiver oscillator will result in a fixed attenuation factor.


(l) TRUE or FALSE: A linear, time-invariant, finite-impulse-response filter with afrequency 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 tocreate the received signal r(t). The transmitter and channel parameters are φ1 = 0.3radians, f1 = 24.1 kHz, f2 = 23.9 kHz, f3 = 27.5 kHz, f4 = 29.3 kHz, and f5 = 22.6kHz. The receiver processing r(t) is shown in Figure 4. All bandpass and lowpass

Figure 2: Transmitter


Figure 3: Channel

Figure 4: Receiver

filters are considered ideal with a gain of unity in the passband and zero in thestopband.

(a) Sketch |R(f)| for −30kHz ≤ f ≤ 30kHz. Be certain to label clearly the amplitudesand frequencies of key points on your sketch.

(b) Assume that φ2 is chosen to maximize the magnitude of y(t) and reflects thevalue of φ1 and the delays imposed by the two ideal bandpass filters between thetransmitter and receiver mixers. Select the receiver parameters f6, f7, f8, and f9, sothe 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 itsmagnitude expression between −B and B Hz is upconverted into the transmittedpassband 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


received signal r isr(t) = x(t − d)

where the delay is an integer multiple of the carrier period Tc (= 1/fc) plus a fractionof Tc

d = nTc + Tc/α

where n is nonnegative and α > 1. The receiver mixer is perfectly synchronized tothe 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. Thereceiver then lowpass filters y to produce

v(t) = LPF{y(t)}

where the lowpass filter is ideal with unity passband gain, linear passband phase withzero 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 thatmaximizes the magnitude of the maximum and minimum values of v(t).

4. Consider performing iterative maximization of

J(x) = 8 − 6|x| + 6 cos(6x)

via

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 valueof x.

(b) Is the convergent value of x in part (a) the global maximum of J(x)? Justify youranswer.

5. The receiver processing u(t) is shown in Figure 5. The triangularly-shapedmagnitude spectrum of the real message signal w(t) is shown in Figure 6 whereB = 0.2MHz. The recevied signal u(t) is an attenuated version of the transmitted


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 somagnitude spectrum of x[k] matches the magnitude spectrum of T1-spaced samplesof 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 of22kHz and a magnitude spectrum described in Figure 8 is upconverted by a mixerwith carrier frequency fc. The channel adds an interferer n. The received signal r isdownconverted to the IF signal x(t) by a mixer with frequency fr.


Figure 7: System

(a) With n(t) = 0, fr = 36 kHz, and fc = 83 kHz, mark each and every range belowthat 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 thatincludes any frequency that causes a narrowband interferer n to induce an image inthe nonzero portions of the magnitude spectrum of the IF passband signal x(t).


(c) With fr = 84 kHz and fc = 62 kHZ, mark each and every range below thatincludes any frequency that causes a narrowband interferer n to induce an image inthe nonzero portions of the magnitude spectrum of the IF passband signal x(t).


7. In this problem you will be given a schematic and numerical specifications andasked to provide certain specifications of various signals. All bandpass and lowpassfilters are considered ideal with a gain of unity in the passband and zero in thestopband. 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 thebaseband signal x1 of 4 kHz, f1 = 28 kHz, f2 = 20 kHz, and f3 = 26 kHz.


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 thebaseband signal x1 of 6 MHz and of the baseband signal x2(t) of 4 MHz, f1 = 164MHz, f2 = 154 MHz, f3 = 148 MHz, f4 = 160 MHz, f5 = 80 MHz, φ = π/2, andf6 = 82 MHz.


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 atriangular 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 = 1720kHz, f6 = 1880 kHz, and f7 = 1300 kHz.

(a) Draw the magnitude spectrum |X5(f)| between ±3000 kHz. Be certain to givespecific values of frequency and magnitude at all breakpoints and local maxima ofthe resulting curve. You must show your work clearly to receive partial credit for anincorrect answer.

(b) Specify values of f8 and f9 that recover the original message without corruptionwith M = 2. You must show your work clearly to receive partial credit for an incorrectanswer.

9. Consider the modulated signal

r(t) = w(t) cos(2πfct + φ)


Figure 11: PAM System

where the absolute bandwidth of the baseband message waveform w(t) is less thanfc/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 LPFoperator included in its definition above. You must show your work clearly to receivepartial credit for an incorrect answer.

(b) With∂

∂α{LPF[x(α, t)]} = LPF[

∂

∂α{x(α, t)}]

show that∂

∂θ{1

2x2(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 x2(t). You must show your work clearly toreceive partial credit for an incorrect answer.


Figure 12: Input Magnitude Spectrum

10. Consider the three pulse shapes sketched in Figure 13 for a T -spaced PAMsystem.

Figure 13: Pulse Shapes

(a) Which pulse shape among p1(t), p2(t), and p3(t) has the largest baseband powerbandwidth? Justify your answer.

(b) Which pulse shape among p1(t), p2(t), and p3(t) has the smallest baseband powerbandwidth? Justify your answer.

11. Consider the communication system segment shown in Figure 14. The mag-nitude 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 allbreakpoints and local maxima of the resulting curve.


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 allbreakpoints 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)

∞∑

k=−∞

δ(t − kTs) (3)

from Figure 14 for Ts = 400µsec. Be certain to give specific values of frequency andmagnitude at all breakpoints and local maxima of the resulting curve.


Figure 16: Another Communication System Segment

12. Consider the communication system segment shown in Figure 16. Each mod-ulator is described by the product of its input with cos(2πfit) (where the subscript iindicates the index of the modulator). The bandpass filter is ideal with a rectangularmagnitude spectrum of gain zero at all frequencies except between fL and fU (andbetween −fU and −fL) with fU > fL > 0 where it has a unit gain. The lowpass filteris 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 thedesired output spectra are provided, respectively in parts (a), (b), and (c) of Figure17.

(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 anincorrect 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 magnitudespectrum 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 magnitudespectrum 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 compo-nents. 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 gainone between −fU and −fL and between fL and fU and zero elsewhere with (fL,fU) of (12MHz, 32MHz) and (35MHz, 50MHz).


Figure 17: Magnitude Spectra of (a) u1(t), (b) u2(t), and (c) desired y(t)


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

y(t) =∞∑

k=−∞

u(t)δ(t − kTs) (5)

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 centeredat 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 producesthe desired baseband signal. Draw its block diagram.

If you can build the desired receiver with one extra part of one of the typesdescribed but with different specifications (such as carrier frequency for a productdemodulator) than those in stock, you may do so, but your maximum score will beonly 2/3 of that possible.

To receive credit, you must clearly explain how you determined your answer. Arecommended method of validating your design is to sketch the magnitude spectrumof the output of each part of your receiver as the input spectrum is modified on itspassage through the system.


14. Consider the RF link with a digital receiver shown in Figure 19. The basebandsignal w(t) has absolute bandwidth B. The carrier frequency is fC . The channel addsa narrowband interferer at frequency fI . The received signal is sampled with periodTs. The sampled signal is demodulated by mixing with a cosine of frequency f1 andideal 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 ornot the magnitude spectrum of the lowpass filter output x4 is the same (up to a scalefactor) 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 diagramswithout prose commentary do not qualify as an explanation.)

(a) Candidate System A: B = 7 kHz, fC = 34 kHz, fI = 49 kHz, Ts = 134

msec,f1 = 0, and f2 = 16kHz.

(b) Candidate System B: B = 11 kHz, fC = 39 kHz, fI = 130 kHz, Ts = 152

msec,f1 = 13kHz, and f2 = 12kHz.

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

Figure 20: Another Communication System Segment


problem you are to build a receiver from a limited number of choices for the specifi-cations of the 3 major components: a bandpass filter, a sampler, and a mixer. Thespecific 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 gainone between −fU and −fL and between fL and fU (with fL < fU) and zeroelsewhere 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) =∞∑

k=−∞

u(t)δ(t − kTs) (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. Theobjective is to specify the bandpass filter, sampler, and mixer so that the “M”-shapedmagnitude spectrum segment is centered at f = 0 in |Y (f)| with no other signalswithin ±1.5 MHz of upper and lower edges of the baseband segment of the magnitudespectrum.

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:


• mixer oscillator frequency fo in MHz:

(There will be no partial credit for this part of this problem. Therefore, there is noneed for an explanation of your answer, which will be “described” in the followingparts of this problem. Partial credit for this problem will be available in the remainingparts, which will be graded independently of whether or not the answer to this partis correct.)

(b) For the three components selected in part (a), sketch the magnitude spectrum ofthe sampler output between −20 and +20 MHz. Be certain to give specific valuesof frequency and magnitude at all breakpoints and local maxima of the resultingcurve. You should provide an explanation of your answer if you expect to receivepartial credit for an answer that is only partially correct. (Equations without prosecommentary do not qualify as an explanation.)

(c) For the three components selected in part (a), draw the magnitude spectrum ofy(t) between between the frequencies −12 and +12 MHz for your design. Be certain togive specific values of frequency and magnitude at all breakpoints and local maxima ofthe resulting curve. You should provide an explanation of your answer if you expect toreceive partial credit for an answer that is only partially correct. (Equations withoutprose 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 aclear explanation of your answer to receive any credit. (Equations without prosecommentary do not qualify as an explanation.)

16. Consider the receiver front end illustrated in Figure 22 that receives the radiofrequency (RF) signal

r(t) = s(t)cos(2πfT t + θT )

and translates it to another center frequency. The message signal s(t) has a bandwidthof fT /100 Hz. We wish to achieve a new center frequency greater than that of theoriginal 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 ofθT = 30◦, select the downconverter’s local oscillator frequency fR and phase θR andthe ideal (i.e. unit gain in the passband, zero otherwise) bandpass filter’s maximumlower fL and minimum upper fU cutoff frequencies to pass the largest magnitudereplica 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 inthe spectral bands occupied by R(f), if added to r(t) ends up in y(t).


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 < f6 < 24.4, 26.1 < f7 < 29.3, f8 = 24.1, f9 > 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.

• 5. One choice: T1 = 1.38 × 10−6, 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 afrequency 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)


• 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 < f9 < 360 kHz

• 9. (a) x(t) = w(t)2 cos(φ−θ), (b) identity because y(t) = −w(t)

2 sin(φ−θ), (c) θ = φ+nπfor integer n.

• 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 = 20.7kHz, (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.




• 16. (a) X(f) = 14 [ej(θR+θT )S(f−fT −fR)+ej(θR−θT )S(f +fT −fR)+e−j(θR−θT )S(f−

fT +fR)+ e−j(θR+θT )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 = 111.59kHz (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


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


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

Figure 32: Block Diagram


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

Figure 34: Magnitude Spectrum of Sampler Output


Figure 35: Magnitude Spectrum of Sampler Output

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EXAM STYLE QUESTIONS Covering Chapters 1 - 9 of ... · EXAM STYLE QUESTIONS Covering Chapters 1 - 9 of Telecommunication Breakdown 1. Clearly circle one answer for each part. (a)