Signal Representation in Communication Questions and Answers

Answer: A

Uncorrelated powers add directly:

\[ P = 2 + 2 = 4 \ \mathrm{\mu W} \]

so 4 microwatts, a rise of 3 dB. 2.83 microwatts is what would follow from adding the amplitudes in quadrature and then treating the result as a power, which double-counts the square root. 2 microwatts ignores one source. 8 microwatts would follow from adding amplitudes and squaring, which applies only to fully correlated sources in phase.

Answer: B

Crest factor is peak over root mean square, and it matters because a transmitter must handle the peaks while the useful power is set by the mean: a high crest factor forces the amplifier to be backed off and wastes efficiency. Root mean square over average is the form factor, a different measure. Bandwidth over centre frequency is fractional bandwidth. And a comparison with the transmitter rating is a system margin rather than a property of the signal.

Answer: C

For a sinusoid the peak is \( A \) and the root mean square value is \( A/\sqrt{2} \), so the ratio is

\[ \frac{A}{A/\sqrt{2}} = \sqrt{2} = 1.41 \]

giving 1.41, or about 3 dB. A crest factor of unity would describe a square wave of constant magnitude. Two and three correspond to waveforms with much peakier statistics, such as some multicarrier signals, and those are exactly the cases that trouble power amplifier designers.

Answer: D

The transform is invertible, so the two are completely equivalent and either can be computed from the other without loss, provided both amplitude and phase are retained. The common belief that a spectrum is somehow less informative comes from seeing only magnitude plots, which do discard the phase. Nothing is expressible in one domain and not the other, which is precisely why the choice of domain is a matter of convenience.

Answer: A

Modulation translates the spectrum up to the carrier, which is the whole purpose: it puts the information where an antenna of reasonable size can radiate it and where a channel allocation exists. Bandwidth is generally preserved or doubled rather than compressed. Phase information is carried through, and in some schemes is the information. And the continuous or discrete character of the spectrum is a property of the message's periodicity, unchanged by the translation.

Answer: B

Single sideband needs the message bandwidth, double sideband twice it, and wideband frequency modulation several times more still, so the order is single sideband, double sideband, wideband frequency modulation. The progression is the central trade of the subject: bandwidth bought in exchange for noise performance and simplicity. The reversed orders invert it, and the third places double sideband below single sideband, which would require the two sidebands to occupy less space than one.

Answer: C

A phase response that is not linear in frequency gives different components different delays, so a wideband signal spreads out in time even though every component arrives with the right amplitude. That is the dominant impairment on many wideband links and the reason equalisers exist. Power delivery depends on the amplitude response. Noise figure is a receiver property. And the data rate limit comes from bandwidth and signal-to-noise ratio through Shannon, with phase distortion acting as an additional penalty rather than the primary bound.

Answer: D

The construction is the signal plus \( j \) times its Hilbert transform, which cancels the negative-frequency half and leaves a complex signal whose magnitude is the envelope. Squaring and filtering is an envelope detector, which recovers the magnitude but not the analytic signal. Multiplying by a complex exponential shifts frequency, which is the next step towards a complex envelope rather than this one. And differentiation is unrelated.

Answer: A

The two are one step apart: multiplying the analytic signal by \( e^{-j2\pi f_c t} \) translates it to baseband and yields the complex envelope. They are therefore not the same, though they carry the same information. The magnitude of the analytic signal is the real envelope, a single real function rather than the complex one. And the last option inverts the relationship entirely.

Answer: B

A spectral mask bounds how much power may fall outside the assigned channel, which is what protects neighbouring users from interference and makes dense frequency reuse possible. It says nothing directly about the data rate achieved inside the channel. Noise figure is a receiver specification. And the modulation index is a transmitter parameter that the mask constrains indirectly, by limiting the spectral spreading it may cause.