Signals and Systems in Communications

Twenty-seven chapters have built an apparatus for describing signals and the systems that act on them. This chapter spends that apparatus on the application that motivated most of it in the first place. Communication engineering is the reason Fourier analysis left mathematics and entered the electrical engineering syllabus, and almost everything in Parts 3 to 6 of this book has a communication problem sitting behind it.

Start with the problem in its rawest form. Speech occupies roughly 300 Hz to 3.4 kHz; music, up to about 20 kHz; a video signal, several megahertz. These are baseband signals — their energy sits in a band beginning at or near zero frequency. Suppose we simply connect a microphone to an antenna and hope. Three separate things go wrong, and each one is a spectrum problem in disguise.

The first is antenna size. An antenna radiates efficiently only when its physical length is a respectable fraction of the wavelength it is asked to carry — a quarter wavelength is the usual rule. At 3 kHz the wavelength is \(c/f = 3\times10^8/3\times10^3 = 100\) km, so a quarter-wave antenna would be 25 km tall. At 100 MHz the wavelength is 3 m and the antenna is 75 cm. Nothing about the message has changed; only the frequency band it occupies has, and the antenna does not care about the message.

The second is channel allocation. A physical channel passes a particular band and rejects the rest — an optical fibre near \(2\times10^{14}\) Hz, a satellite link in the gigahertz, a copper pair up to a few megahertz. A signal must be delivered to the channel already sitting inside the band the channel supports.

The third, and the one with the most far-reaching consequences, is multiplexing. If two speakers are placed on the same channel at baseband their spectra overlap completely and no filter can ever separate them again. Chapter 15 was explicit about this: a linear filter can only separate signals that occupy different frequency bands. Give each speaker a different band and separation becomes trivial.

Chapter 14 established the frequency-shifting property: multiplying a signal by \(e^{j\omega_c t}\) translates its entire spectrum by \(\omega_c\). A complex exponential is not a voltage anyone can generate, but a cosine is, and a cosine is the sum of two of them. Applying the shift twice and adding gives the result on which the rest of this chapter rests.

Read the right-hand side literally. The message spectrum appears twice, once centred at \(+\omega_c\) and once at \(-\omega_c\), each at half its original height. Nothing is distorted and nothing is lost; the spectrum has simply been copied to two new places. A signal formed this way is called double-sideband suppressed-carrier — double-sideband because both the portion above \(\omega_c\) and the portion below it are transmitted, suppressed-carrier because there is no discrete spectral line at \(\omega_c\) itself.

The assumption \(\omega_c \gt W\) is doing quiet work here. If the carrier were lower than the message bandwidth, the two shifted copies would overlap around the origin, they would add, and no filter could ever pull them apart again. This is the same non-overlap condition that Chapter 20 will have already made familiar under a different name: it is exactly the condition that the shifted spectral replicas produced by sampling do not alias into one another. In practice \(\omega_c\) is larger than \(W\) by orders of magnitude, so the condition is never in doubt.

Two features of DSB-SC are worth noticing before we improve on it. It is efficient in power, because every watt transmitted carries message information — there is no carrier line consuming power while conveying nothing. It is wasteful in bandwidth, because the upper and lower sidebands are mirror images of one another and therefore carry the same information twice. Sections 28-4 and 28-5 trade these two properties against each other in opposite directions.

Having slid the spectrum up, the receiver must slide it back. The natural move is to do the same thing again: multiply the received signal by a locally generated cosine at the same frequency. Multiplying by a cosine shifts the spectrum both up and down, so one of the two shifted copies of the message lands back at the origin while the rest is thrown out to \(2\omega_c\), where a lowpass filter disposes of it.

The first term is the message, halved. The second is the message translated to \(2\omega_c\), which occupies the band \(2\omega_c \pm W\) and is therefore entirely separated from the first term whenever \(\omega_c \gt W\). A lowpass filter with cutoff anywhere between \(W\) and \(2\omega_c - W\) recovers \(\tfrac12 x(t)\), and a gain of two finishes the job. This is coherent or synchronous detection, and the word coherent is the entire difficulty.

Suppose the local oscillator is at the right frequency but the wrong phase, producing \(\cos(\omega_c t + \theta)\). Repeating the product with the general identity \(\cos A\cos B = \tfrac12\cos(A-B) + \tfrac12\cos(A+B)\):

The quadrature null is not merely a nuisance to be avoided; it is a resource. Two entirely separate messages can be sent on the same carrier frequency, one on \(\cos\omega_c t\) and one on \(\sin\omega_c t\), and each is invisible to a detector locked to the other. That is quadrature amplitude multiplexing, and it is why every modern digital radio describes its signal as a pair \((I, Q)\) — a point Section 28-9 returns to.

Recovering the carrier phase to the accuracy this demands is the price of DSB-SC. A Costas loop or a squaring loop can extract it from the received signal itself, but both add circuitry, and in the 1920s neither existed. That commercial pressure produced the scheme in the next section, which throws away power in order to make the receiver almost free.

The insight behind conventional amplitude modulation is that if the transmitted amplitude never goes negative, the message can be read off the outline of the waveform with a diode, a capacitor and a resistor — no oscillator, no phase lock, no multiplier. To guarantee that, add a constant \(A\) to the message before modulating.

Here \(m(t) = x(t)/\max|x(t)|\) is the message normalised to unit peak, and \(\mu\) is the modulation index. The envelope of \(s(t)\) — the slowly varying outline that the carrier oscillates inside — is \(|A + x(t)|\). If \(\mu \le 1\) then \(A + x(t) \ge 0\) always, the modulus sign is redundant, and the envelope is \(A + x(t)\): the message plus a constant, which a capacitor removes. If \(\mu \gt 1\) the bracket goes negative on the deepest troughs, the modulus folds those excursions back up, and the envelope no longer resembles the message at all. This is overmodulation, and it is irreversible distortion.

What does the added carrier cost? Expand the AM signal for a single-tone message \(x(t) = A\mu\cos\omega_m t\):

Three discrete frequencies: the carrier and the two sidebands. Their powers into a unit resistance are \(A^2/2\) for the carrier and \(A^2\mu^2/8\) for each sideband. Only the sidebands carry the message; the carrier is a constant tone conveying nothing but the receiver's convenience. Forming the ratio gives the power efficiency.

The envelope detector itself is a diode feeding a parallel \(RC\) network. On each positive carrier peak the diode conducts and charges the capacitor to the peak value; between peaks the diode is reverse-biased and the capacitor discharges through \(R\). The time constant must therefore be long enough to hold its value between carrier peaks but short enough to follow the envelope down. Both conditions can be made quantitative.

The upper limit comes from asking the capacitor to fall at least as fast as the envelope does. Discharging from \(E(t)\), the capacitor's initial rate of fall is \(E(t)/RC\). The envelope \(E(t) = A(1+\mu\cos\omega_m t)\) falls fastest at a rate \(A\mu\omega_m\sin\omega_m t\). Requiring the first to beat the second at every instant:

The right-hand side is minimised where its derivative vanishes, which happens at \(\cos\omega_m t = -\mu\). Substituting that value, \(1+\mu\cos\omega_m t = 1-\mu^2\) and \(\sin\omega_m t = \sqrt{1-\mu^2}\), so the minimum is \(\sqrt{1-\mu^2}/(\mu\omega_m)\). Combining with the lower limit gives the design rule.

Return to the DSB-SC spectrum. For a real message, \(X(-j\omega) = X^*(j\omega)\), so the upper sideband and the lower sideband are conjugate reflections of each other about \(\omega_c\). Knowing one determines the other completely. Transmitting both is transmitting the same information twice, and the obvious repair is to filter one away and send only the other. That is single-sideband modulation, and it fits a message of bandwidth \(W\) into a channel of bandwidth \(W\).

Filtering it away is harder than it sounds. The two sidebands meet at \(\omega_c\) with no gap between them, so the filter must fall from passband to stopband across the region where the message has its lowest-frequency content. Speech is convenient here because it has almost nothing below 300 Hz, leaving a 600 Hz transition band; a signal with significant DC content cannot be sent by SSB at all. The alternative is to build the sideband directly, and doing so introduces an operator that is worth knowing in its own right.

Define the Hilbert transform \(\hat{x}(t)\) as the output of an all-pass filter that shifts every positive-frequency component by \(-90^\circ\) and every negative-frequency component by \(+90^\circ\).

Its magnitude is 1 everywhere, so it changes no amplitudes — only phases. Now form the analytic signal \(x_+(t) = x(t) + j\hat{x}(t)\). Its spectrum is \(X(j\omega)\big[1 + \operatorname{sgn}(\omega)\big]\), which is \(2X(j\omega)\) for \(\omega \gt 0\) and zero for \(\omega \lt 0\): the negative-frequency half has been cancelled exactly. Shifting this one-sided spectrum up to \(\omega_c\) and taking the real part produces a signal occupying only the band above \(\omega_c\).

SSB is recovered by coherent detection exactly as DSB-SC is, and it is if anything more demanding: a phase error in SSB does not merely attenuate the output, it phase-shifts every message component by the same angle, which speech tolerates but data does not, while a frequency error shifts every component by the same number of hertz, turning a voice into the familiar comical warble of a mistuned shortwave receiver.

Vestigial sideband is the engineering compromise between the two. One sideband is transmitted in full and a small vestige of the other is retained, with the filter's transition shaped so that the two overlapping vestiges add to a constant across the crossover region. The bandwidth is a little more than \(W\), the filter is realisable, and DC content survives — which is why analogue television, whose video signal runs down to DC, used VSB for its picture carrier.

The third motivation of Section 28-1 now pays off. Give each message its own carrier, spaced far enough apart that the translated spectra do not overlap, add the modulated signals together and transmit the sum. At the receiver a bandpass filter selects one band and a coherent or envelope detector brings it back to baseband. This is frequency-division multiplexing, and it is the organising principle of the entire radio spectrum.

Guard bands exist because real filters are not brick walls. Chapter 15 showed that an ideal brick-wall filter is non-causal and therefore unbuildable; a practical filter rolls off over a finite transition, so adjacent channels must be spaced a little more than \(2W\) apart to keep the rolloff from eating into the neighbour. The classical telephone hierarchy did exactly this: twelve SSB voice channels, each nominally 4 kHz wide although the speech occupied only 3.4 kHz, stacked into a 48 kHz group.

Building a receiver whose bandpass filter can be retuned across a whole broadcast band is unpleasant — the centre frequency must move while the bandwidth stays fixed, so the fractional bandwidth changes as you tune. The superheterodyne receiver avoids the problem entirely. A tunable local oscillator first translates whatever station you want down to a fixed intermediate frequency, and all the selective filtering and most of the gain happen there, in a filter that never has to be retuned.

The image arises because a multiplier is blind to the sign of a frequency difference. Both \(f_{\text{LO}} - f_{\text{IF}}\) and \(f_{\text{LO}} + f_{\text{IF}}\) land on the intermediate frequency, so a station at either one reaches the IF filter. Only a coarse filter ahead of the mixer can reject the unwanted one, and since it need only separate two frequencies \(2f_{\text{IF}}\) apart it can be a broad, easily tuned circuit. This is why the AM broadcast IF of 455 kHz was chosen: it is small enough for good IF selectivity and large enough that a simple front end can reject an image sitting 910 kHz away.

Every scheme so far has written the message into the amplitude of a carrier, which makes the message vulnerable to anything that disturbs amplitude — and almost every channel impairment does. The alternative is to hold the amplitude rigidly constant and write the message into the angle instead.

The definition of instantaneous frequency as the derivative of the total angle is the key idea, and it is consistent with everything before it: for an unmodulated carrier \(\theta = \omega_c t\) and \(\omega_i = \omega_c\), as it should be. Two choices then present themselves. Make the phase deviation proportional to the message and you have phase modulation; make the frequency deviation proportional to it and you have frequency modulation.

The two are the same operation applied to different signals: FM of \(x(t)\) is PM of the integral of \(x(t)\), and PM of \(x(t)\) is FM of its derivative. A single modulator therefore serves both, with an integrator or differentiator in front — which is exactly how pre-emphasis and de-emphasis in broadcast FM are built.

Now the spectrum, and here angle modulation parts company with everything in this chapter so far. Take the single tone \(x(t) = A_m\cos\omega_m t\) into an FM modulator. Integrating gives a phase deviation \(\beta\sin\omega_m t\) where \(\beta = k_f A_m/\omega_m = \Delta\omega/\omega_m = \Delta f/f_m\), the modulation index, with \(\Delta f\) the peak frequency deviation. The transmitted signal is \(A\cos(\omega_c t + \beta\sin\omega_m t)\), and the message now sits inside a cosine of a cosine. The relation between message and signal is not linear, so the superposition arguments used everywhere else in this book simply do not apply, and the spectrum cannot be obtained by shifting anything.

Infinitely wide is not the same as unusable, because \(J_n(\beta)\) becomes negligible once \(n\) exceeds \(\beta\) by a comfortable margin. Counting the significant pairs of sidebands as roughly \(\beta + 1\), each spaced \(f_m\) apart, and doubling for both sides of the carrier gives the standard engineering estimate.

Broadcast FM uses \(\Delta f = 75\) kHz and \(W = 15\) kHz, giving \(\beta = 5\) and \(B = 180\) kHz — six times what AM would need for the same audio. That bandwidth is not wasted. It buys a signal-to-noise improvement that grows as \(\beta^2\), which is why FM sounds better than AM despite using less transmitter power, and it is the first clear example in this book of bandwidth being deliberately traded for noise performance. Making that trade precise requires a description of noise itself, and that is Chapter 29.

Narrowband FM deserves one more line because it can be built from the linear tools of this chapter. For \(\beta \ll 1\), \(\cos(\beta\sin\omega_m t)\approx 1\) and \(\sin(\beta\sin\omega_m t)\approx\beta\sin\omega_m t\), so \(s(t) \approx A\cos\omega_c t - A\beta\sin\omega_m t\sin\omega_c t\). Compare this with the AM expansion of Section 28-4: the same three lines, the same spacing, but the sidebands enter in quadrature with the carrier rather than in phase with it. That single change of phase is the entire difference between a signal whose amplitude varies and one whose angle does.

Everything so far has been continuous in time. Chapter 20 proved that a signal bandlimited to \(W\) hertz is completely determined by samples taken at \(2W\) per second, and Chapter 21 showed how to rebuild it. That theorem has a consequence for communications that is easy to state and enormous in effect: between the samples, the channel is idle, and the idle time can be sold to somebody else.

The simplest pulse scheme is pulse-amplitude modulation. The message is sampled and each sample sets the height of a short pulse. Chapter 20 treated ideal impulse sampling; a real system uses pulses of finite width \(\tau\), which is the difference between impulse sampling and what is usually called flat-top sampling. The distinction matters in the frequency domain.

A flat-top pulse train is impulse sampling convolved with the pulse shape, so its spectrum is the sampled spectrum multiplied by \(P(j\omega)\), which for a rectangular pulse of width \(\tau\) is \(\tau\operatorname{sinc}(\omega\tau/2\pi)\). Within the baseband copy this sinc is not flat, so the recovered message is slightly tilted in frequency — the aperture effect. It is a fixed, known distortion and is undone by an equaliser with the reciprocal response, exactly the equaliser Chapter 22 used to correct a zero-order hold.

With the channel occupied only for \(\tau\) out of every \(T\) seconds, interleaving is immediate. Sample message 1, then message 2, then message 3, then return to message 1 before its next sample is due. This is time-division multiplexing, the time-domain twin of Section 28-6's frequency-domain stacking, and the two are related by exactly the duality Chapter 14 established between the domains.

One representation ties the chapter together and is the form in which communication signals are actually handled in software. Every scheme above produces a passband signal: real, and with its energy confined to a narrow band around \(\omega_c\). Any such signal can be written with the carrier factored out.

The complex envelope \(\tilde{s}(t)\) is a lowpass signal of bandwidth \(W\) that carries everything the passband signal carries. Its real part \(s_I\) is the in-phase component, its imaginary part \(s_Q\) the quadrature component, and the modulation schemes of this chapter are now distinguished by nothing more than the shape of \(\tilde{s}\).

Two practical consequences follow. First, the envelope and the instantaneous phase of any passband signal are \(|\tilde{s}(t)|\) and \(\arg\tilde{s}(t)\), so an envelope detector computes a modulus and an FM discriminator computes the derivative of an argument — the same two operations for every scheme. Second, and more useful, a passband system can be simulated entirely at baseband. If a bandpass channel has impulse response \(h(t)\) with complex envelope \(\tilde{h}(t)\), the passband convolution \(s * h\) corresponds to \(\tfrac12\tilde{s} * \tilde{h}\) at baseband. A radio at 2.4 GHz can therefore be simulated with a sampling rate set by \(W\), not by \(\omega_c\) — a saving of six orders of magnitude, and the reason software-defined radio is possible at all.

This also closes the loop with Chapter 22. Sampling the complex envelope at a rate above \(2W\) rather than above \(2(f_c+W)\) is legitimate for exactly the reason bandpass sampling is legitimate: what matters is the width of the occupied band, not its distance from the origin. The I/Q pair is bandpass sampling with the down-conversion done in hardware first.

Problems 1 to 3 exercise the spectra of the amplitude schemes; 4 and 5 concern detection; 6 and 7 treat angle modulation; 8 is a multiplexing budget. Sketch the spectrum before computing anything — most errors in this material are geometric, not algebraic.