Signal Representation in Communication Questions and Answers

Answer: A

The Wiener-Khinchin relation gives the power spectral density as the Fourier transform of the autocorrelation function, which is how a random signal, having no deterministic spectrum, is nonetheless described in the frequency domain. Squaring the transform of one record gives a periodogram, a noisy estimate rather than the quantity itself. The distribution function describes amplitude statistics, not spectral content. And the last option reverses the operations.

Answer: B

White noise has a flat power spectral density, by analogy with white light containing all colours equally, and its autocorrelation is therefore an impulse: samples taken at any two distinct instants are uncorrelated. A single frequency is a tone. A density proportional to frequency is sometimes called blue noise, and an inverse dependence is pink or flicker noise, which is what semiconductor devices show at low frequency. Truly flat noise over infinite bandwidth would carry infinite power, so the flatness is always stated over a band.

Answer: C

The Hilbert transform is an all-pass network that applies a uniform minus ninety degree phase shift to every positive-frequency component while leaving amplitudes alone. That property makes it the tool for constructing the analytic signal and for generating single-sideband signals by the phasing method. A shift of one hundred and eighty degrees is a simple inversion. Zero shift would be an identity. And a shift proportional to frequency is a pure time delay, which is a quite different operation.

Answer: D

Adding \( j \) times the Hilbert transform to a real signal cancels the negative-frequency half, so the analytic signal is one-sided in frequency. It is the natural representation for a modulated signal, since its magnitude is the envelope and its phase the instantaneous phase, both of which are exactly what a demodulator recovers. Symmetry about zero is the property of the original real signal. A negative-only spectrum would follow from subtracting rather than adding. And the whole point is that the spectrum changes.

Answer: A

Writing a passband signal as a complex envelope times a carrier exponential strips out the carrier and leaves a low-frequency complex signal carrying all the information, which is why simulation and digital signal processing work almost entirely in that representation: the sampling rate need only suit the bandwidth, not the carrier frequency. It changes no signal-to-noise ratio, since it is a representation rather than a processing gain. Filtering is still required. And nothing about it alters the statistical character of a signal.

Answer: B

Baseband means the spectrum sits at the low end, from near direct current up to the highest frequency the message contains. Speech from 300 Hz to 3.4 kHz is the classic example. A spectrum centred on a carrier is passband, which is what modulation produces. A single frequency is a tone. Flat power to the channel limit describes white noise filtered by the channel rather than a baseband message.

Answer: C

The Nyquist rate is twice the highest frequency present:

\[ f_s \geq 2 \times 4 = 8 \ \mathrm{kHz} \]

so 8 kHz, and telephony indeed samples at 8 kHz for a 3.4 kHz channel with guard band. 4 kHz would alias the whole band. 2 kHz is worse still. 16 kHz is four times the highest frequency, which is perfectly usable and is oversampling by a factor of two, but it is not the minimum the question asks for.

Answer: D

Under-sampling folds the spectrum, so components above half the sampling rate reappear as spurious lower frequencies that cannot afterwards be separated from genuine ones. That irreversibility is why an anti-alias filter precedes every sampler. A loss of signal-to-noise ratio is what quantisation causes. A delay comes from filtering or processing. And the reconstructed bandwidth is limited by half the sampling rate rather than increased.

Answer: A

It must come before the sampler, because once the spectrum has folded no later filter can unfold it. A filter after the sampler cannot distinguish an alias from a genuine component, since they occupy the same frequency. Placing it after the quantiser is later still. A reconstruction filter is also needed at the output, but that is a separate device with a different job: removing the images the sampling process created.

Answer: B

A linear channel can change the amplitude and phase of each component but cannot create new frequencies, so linear distortion is amplitude or phase distortion across the band. Generating new frequencies is the signature of nonlinearity, which produces harmonics and intermodulation. Passing only the fundamental would be extreme filtering, which is one form of linear distortion rather than a category of its own. And a pure delay is distortionless, since it affects only the phase and does so linearly with frequency.