Here are some optional questions for those who want more practice.

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ECS 332 HW 7 — Due: November 8, 4 PM 2019/1

Extra Questions

Problem 3. You are asked to design a DSB-SC modulator to generate a modulated signal km(t) cos(2πfct), where m(t) is a signal band-limited to B Hz. Figure 7.2 shows a DSB-SC modulator available in the stockroom. Note that, as usual, ω_c= 2πf_c. The carrier generator available generates not cos (2πf_ct), but cos³(2πf_ct). Explain whether you would be able to generate the desired signal using only this equipment. You may use any kind of filter you like. [Lathi and Ding, 2009, Q4.2-3]

3. W. L. Smith, "Frequency and Time in Communications," Proc. IEEE, vol. 60, pp. 589-594, May 1972.

4. B. P. Lathi, Linear Systems and Signals, Oxford University Press, New York, 2000.

5. A. J. Viterbi, Principles of Coherent Communication, McGraw-Hill, New York, 1966.

6. F. M. Gardner, Phaselock Techniques, 3rd ed., Wiley, Hoboken, NJ, 2005.

7. W. C. Lindsey, Synchronization Systems i11. Communicmion and Control, Prentice-Hall, Englewood Cliffs, NJ, 1972.

8. J.P. Costas, "Synchronous Communication," Proc. IRE, vol. 44, pp. 1713-1718, Dec. 1956.

9. L. H. Hansen, Introduction to Solid-State Television Systems, Prentice-Hall, Englewood Cliffs, NJ, 1969.

PROBLEMS

Figure P.4.2-3

4.2-1 You are given the baseband signals (i) m(t) = cos lOOOm; (ii) m(t) = 2cos lOOOm +

sin 2000m; (iii) m.(t) = cos IOOOm cos 3000.rrt; (iv) m(t) = exp( - lOlLI). For each one, do the following.

(a) Sketch the spectrum of m(c).

(b) Sketch the spectrum of the DSB-SC signal m(c) cos IO,OOO.rrt.

(c) Identify the upper sideband (USB) and the lower sideband (LSB) spectra.

4.2-2 Repeat Prob. 4.2-1 if (i) m(t) = sine (lOOm); (ii) m(t) = (l + ^t^2)-¹^;⁽ⁱⁱⁱ⁾m(t) = e-IOit-ll_ Observe that e- 10lt-1₁is e-IOirl delayed by l second. For the last case you need to consider both the amplitude and the phase spectra.

4.2-3 You are asked to design a DSB-SC modulator to generate a modulated signal km(l) cos (wet +8), where m(t) is a signal band-limited to B Hz. Figure P4.2-3 shows a DSB-SC modulator available in the stockroom. The catTier generator available generates not cos Wet, but cos³Wei. Explain whether you would be able to generate the desired signal using only this equipment. You may use any kind of filter you lik~.

m(t)

(a) What kind of filter is required in Fig. P4.2-3?

(b) Determine the signal spectra at points band c, and indicate the frequency bands occupied by these spectra.

(c) What is the minimum usable value of eve?

(d) Would this scheme work if the carrier generator output were sin3 _cvc_l')Explain.

(f) Would this scheme work if the carrier generator output were cos" cvet for any integer n ::: 2.?

., I

@.._ _ __, ^Filter (a)

-8

M(.f)

(b)

B J- -

4.2-4 You are asked to design a DSB-SC modulator to generate a modulated signal km(l) cos wet with the carrier frequency fe = 500kHz (we = 2.rr x 500, 000). The following equipment is available in the stockroom: (i) a signal generator of frequency I 00 k.Hz; (ii) a ring modulator; (iii) a bandpass fi Iter tuned to 500 k.Hz.

Figure 7.2: Problem 3

(a) We know that a real-valued signal r(t) that is even and periodic with period T0 can be expanded using Fourier series into

r(t) = c₀+ a₁cos(2πf₀t) + a₂cos(2π(2f₀)t) + a₃cos(2π(3f₀)t) + · · · (7.1) where f₀ = _T¹

0. Consider the signal r(t) = cos³(2πf_ct).

(i) Is it periodic?

(ii) Is it even?

(iii) Expand r(t) = cos³(2πf_ct) into a linear combination of cos(2π(nf_c)t) as in (7.1) above.

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(b) What kind of filter is required in Figure 7.2?

(c) Determine the signal spectra at points (b) and (c) in Figure 7.2, and indicate the frequency bands occupied by these spectra.

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(d) What is the minimum usable value of f_c?

(e) Would this scheme work if the carrier generator output were cos²(2πf_ct)? Explain.

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Problem 4. Consider the basic DSB-SC transceiver with time-delay channel presented in class. Recall that the input of the receiver is

x (t − τ ) = m (t − τ )√

2 cos (ω_c(t − τ )) where m(t) −−*)−−^F

F⁻¹ M (f ) is bandlimited to B, i.e., |M (f )| = 0 for |f | > B. We also assume that f_c B.

(a) Suppose that, at the receiver, we multiply by√

2 cos ((ω_ct) − θ) instead of √

2 cos (ω_ct) as illustrated in Figure 7.3. Assume

H_LP(f ) = 1, |f | ≤ B 0, otherwise.

Find ˆm(t) (the output of the LPF).

LP

 

H f

 

x t



 



2 cos _ct 



^{v t}

^{ }

^{m t}^ˆ

^{ }



Figure 7.3: Receiver for Problem 4a

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HWR HWR

Figure 7.4: Receiver for Problem 4b

(b) Use the same assumptions as part (a). However, at the receiver, instead of multiplying by√

2 cos ((ωct) − θ), we pass x(t − τ ) through a half-wave rectifier (HWR) as shown in Figure 7.4b.

Make an extra assumption that m(t) ≥ 0 for all time t and that the half-wave rectifier input-output relation is described by a function f (·):

f (x) = x, x ≥ 0, 0, x < 0.

Find ˆm(t) (the output of the LPF).

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Problem 5 (M2011Q7). Suppose m (t) −−*)−−^F

F⁻¹ M (f ) is bandlimited to W , i.e., |M (f )| = 0 for |f | > W . Consider a DSB-SC transceiver shown in Figure 7.5.

Name_____________________________________ ID_______________________

Page 5 of 12

b. (5 pt) Sketch the spectrum of the DSB-SC signal ^{m t}

  

^{cos 5, 000}^^t



^{. No}

explanation is needed.

7. (8 pt) Suppose ^{m t}

 

^^{M f}

 

is bandlimited to W, i.e., ^{M f}

 

^⁰^for

f W. Consider the following DSB-SC transceiver.

Also assume that f_c W and that

 

^1,

0, otherwise.

LP

f W

H f  

 

Make an extra assumption that ^{m t}

 

^⁰ for all time t and that the full-wave rectifier (FWR) input-output relation is described by a function fFWR

 

:

 

^, ^0,

, 0.

FWR

x x

f x

x x

 

   (Questions start on the next page.)

 

m t

 

cos _ct



Delayed by 

 

x t

LP

 

H f

 

x t

 

y t FWR v t

Transmitter (modulator)

Receiver (demodulator)

Figure 7.5: A DSB-SC transceiver

Also assume that f_c W and that H_LP(f ) = 1, |f | ≤ W 0, otherwise.

Make an extra assumption that m (t) ≥ 0 for all time t and that the full-wave rectifier (FWR) input-output relation is described by a function f_{F W R}(·):

f_{F W R}(x) = x, x ≥ 0,

−x, x < 0.

(a) Recall that the half -wave rectifier input-output relation is described by a function fHW R(·) : fHW R(x) = x, x ≥ 0,

0, x < 0. We have seen in Problem 4b that when the 7-8

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receiver uses half -wave rectifier,

v (t) = x (t − τ ) × g_{HW R}(t − τ ) where g_{HW R}(t) = 1 [cos (ω_ct) ≥ 0].

(i) The receiver in this question uses full-wave rectifier. Its v(t) can be described in a similar manner; that is

v (t) = x (t − τ ) × g_{F W R}(t − τ ) .

Find g_{F W R}(t). Hint: g_{F W R}(t) = c₁× g_{HW R}(t) + c₂ for some constants c₁ and c₂. Find these constants.

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(ii) Recall that the Fourier series expansion of g_{HW R}(t) is given by

g_{HW R}(t) = 1 2 + 2

π

cos ω_ct − 1

3cos 3ω_ct + 1

5cos 5ω_ct − 1

7cos 7ω_ct + . . .

.

Find the Fourier series expansion of gF W R(t).

(b) Find y(t) (the output of the LPF).

Problem 6. Would the scheme in Problem 3 work if the carrier generator output were cosⁿω_ct for any integer n ≥ 2?

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Problem 7 (M2011Q5). In this question, you are provided with a partial proof of an im- portant result in the study of Fourier transform. Your task is to figure out the quantities/- expressions inside the boxes labeled a,b,c, and d.

We start with a function g(t). Then, we define x (t) =

∞

P

`=−∞

g (t − `T ). It is a sum that involves g(t). What you will see next is our attempt to find another expression for x(t) in terms of a sum that involves G(f ).

To do this, we first write x(t) as x (t) = g (t) ∗

∞

P

`=−∞

δ (t − `T ). Then, by the convolution- in-time property, we know that X(f ) is given by

X (f ) = G (f ) × a

∞

X

`=−∞

δ

f + b

We can get x(t) back from X(f ) by the inverse Fourier transform formula: x (t) =

∞

R

−∞

X (f ) e^{j2πf t}df . After plugging in the expression for X(f ) from above, we get

x (t) =

∞

Z

−∞

e^{j2πf t}G (f ) a

∞

X

`=−∞

δ

f + b

df

= a

∞

Z

−∞

∞

X

`=−∞

e^{j2πf t}G (f ) δ

f + b

df.

By interchanging the order of summation and integration, we have

x (t) = a

∞

X

`=−∞

∞

Z

−∞

e^{j2πf t}G (f ) δ

f + b

df.

We can now evaluate the integral via the sifting property of the delta function and get

x (t) = a

∞

X

`=−∞

e c G

d

.

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