ECE341: Introduction to Communication Systems

Practice Questions for Midterm Exam - 2012 Spring

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Q1. An AM system operates with a modulation index a = 0.7. The (un-normalized) message signal is

m(t) = 12 cos(8πt).

(i). What is the average power m2n(t) of the normalized message signal mn(t)?

(ii). Compute the efficiency of this AM modulator?

Q2. Consider a sinusoidal message signal m(t) = 2 cos(8πt).

(a). Let xDSB(t) be the output signal when above message signal m(t) is modulated on to a carrier

c(t) = cos(300πt) using DSB modulation. Derive and plot the spectrum XDSB(f) of xDSB(t).

You only need to draw the amplitude spectrum. (Hint: Use a trigonometric identity to express

the output as a sum of cosines.)

(b). What is the Hilbert transform m̂(t) of m(t)?

(c). If the above message signal m(t) is the input to a lower-sideband SSB modulator that uses a

carrier c(t) = cos(300πt), derive an expression for the output signal yLSB(t). Then, derive and

plot the spectrum YLSB(f) of yLSB(t). You only need to draw the amplitude spectrum. (Hint: Use

a trigonometric identity to express the output as a single cosine.)

Q3. Recall that if x(t) =∏(

tT

), then Fourier transform X(f) of x(t) is given by X(f) = Tsinc(fT ).

Define z(t) = x(t − 2) cos(20πt). If Z(f) is the Fourier transform of z(t), use Fourier transform

properties to find Z(f).

Q4. The allowed peak frequency deviation for commercial FM broadcasting in North America is 75 kHz.

For FM transmissions, usually the bandwidth of an audio signal is taken to be 15 kHz. Use Carson’s

rule to estimate the approximate transmission bandwidth required for commercial FM radio.

Q5. An analog speech waveform is quantized and transmitted using a PCM system. Suppose that each

data sample at the receiving end of the system must be known to within ±0.25% of the peak-to-peak

full-scale value.

(a). How many binary symbols must each transmitted digital word contain?

(b). If the speech waveform has a bandwidth of 4 kHz, what should be the minimum bandwidth of

final PCM output?

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I. USEFUL FORMULAS

• Fourier transform of a signal x(t) is F{x(t)} = X(f) =∫∞−∞ x(t)e

−j2πftdt.

• If F{x(t)} = X(f), then F{x(t− t0)} = X(f)e−j2πft0 .

• If F{x(t)} = X(f), then F{x(t)ej2πf0t} = X(f − f0).• F {A} = Aδ(f).

• Euler’s formula: ejθ = cos θ + j sin θ.

• cos(θ − π2) = sin θ.

• cos(A+B) = cosA cosB − sinA sinB.

• cos(A−B) = cosA cosB + sinA sinB.

• sin(A+B) = sinA cosB + cosA sinB.

• sin(A−B) = sinA cosB − cosA sinB.

• Transfer function of the Hilbert transformer is H(f) = −jsgn(f).