Frequency Spectra and System Response

By the end of Chapter 11 we could compute the Fourier coefficients of almost any periodic signal without ever performing the analysis integral. What we had not yet asked is the obvious question: having gone to the trouble of producing the numbers \(a_k\), what are they good for?

The first answer is that they are a picture. The synthesis equation \(x(t) = \sum_k a_k e^{jk\omega_0 t}\) says the signal is completely determined by the sequence \(\{a_k\}\); the analysis equation says the sequence is completely determined by the signal. Neither description is more fundamental than the other. They are two coordinate systems for the same object, and moving between them destroys nothing — the same way that giving a point by its Cartesian coordinates and by its polar coordinates are equally complete.

What changes is what is easy to see. In the time-domain picture you read off when things happen. In the frequency-domain picture you read off which oscillations are present and how strongly, and that is precisely the information a filter, an amplifier, a transmission line or a loudspeaker acts on. A plot of \(|a_k|\) against frequency is called the line spectrum of the signal, because a periodic signal contains energy only at the isolated frequencies \(0, \pm\omega_0, \pm 2\omega_0, \dots\) — its spectrum is a picket fence of lines, never a continuum.

Each \(a_k\) is a complex number, so it takes two real plots to display the whole sequence. Writing \(a_k = |a_k|e^{j\angle a_k}\), we plot \(|a_k|\) against \(k\omega_0\) as the magnitude spectrum and \(\angle a_k\) against \(k\omega_0\) as the phase spectrum. Both are needed. The magnitude spectrum alone tells you how much of each oscillation is present but not how the oscillations are lined up in time, and it is the alignment that decides the shape of the waveform.

For a real signal the two plots are not independent, because Chapter 11 established the conjugate symmetry \(a_{-k} = a_k^{*}\). Taking modulus and argument of that statement gives the symmetry that every spectrum in this book obeys.

That redundancy is what licenses the one-sided or amplitude–phase form. Pair the term at \(+k\omega_0\) with its partner at \(-k\omega_0\) and use \(z + z^{*} = 2\,\mathrm{Re}\{z\}\):

Summing over \(k \ge 1\) and keeping the DC term separately gives the form an instrument would report:

So the physical amplitude of the \(k\)-th harmonic is \(2|a_k|\), not \(|a_k|\): the two-sided spectrum splits every real oscillation into a conjugate pair and gives half of it to each. Forgetting the factor of two is the commonest slip in spectrum questions, and it always shows up as an answer that is exactly half the right size.

The phase spectrum has one behaviour worth memorising, because it recurs in every chapter from here to the end of the book. Chapter 11's time-shifting property says the coefficients of \(x(t-t_0)\) are \(a_k e^{-jk\omega_0 t_0}\). The magnitudes are untouched; the phases each pick up \(-k\omega_0 t_0\), a term proportional to the harmonic number. A pure delay is therefore invisible in the magnitude spectrum and appears in the phase spectrum as a straight line through the origin whose slope is the delay. Hold on to that; Section 12-7 is built on it.

Chapter 11 proved Parseval's relation for periodic signals: the average power of \(x(t)\) equals \(\sum_k |a_k|^2\). Read as a picture rather than an equation, it says the total power is a sum of contributions, one per spectral line, and that each line's contribution is \(|a_k|^2\). Plotting \(|a_k|^2\) against \(k\omega_0\) therefore shows how the signal's power is distributed across frequency, and that plot is the power spectrum.

Because the magnitude spectrum is even, the physical harmonic at \(k\omega_0\) carries \(|a_k|^2 + |a_{-k}|^2 = 2|a_k|^2\) — the same factor-of-two bookkeeping as before, and consistent with the amplitude-\(2|a_k|\) cosine having mean-square value \((2|a_k|)^2/2 = 2|a_k|^2\).

It also makes bandwidth definable. A periodic signal built from infinitely many harmonics has, strictly, infinite bandwidth; in practice the high harmonics carry so little power that discarding them changes nothing audible, visible or measurable. The essential bandwidth is the frequency below which some agreed fraction — commonly 95% or 99% — of the total power lies.

Take the symmetric square wave of Chapter 10, of amplitude \(\pm 1\), whose harmonic amplitudes are \(4/(\pi k)\) for odd \(k\). Its total power is 1 W into a \(1\,\Omega\) reference, and the power in the \(k\)-th harmonic is \(\tfrac12\big(4/\pi k\big)^2 = 8/(\pi^2k^2)\). Accumulating these gives a table that is worth remembering, because it explains why square waves survive being filtered so brutally.

Everything so far has been about signals. Now bring in a system, and the reason for all this machinery finally appears.

Chapter 7 established that an LTI system is completely described by its impulse response \(h(t)\), acting through the convolution integral \(y(t) = \int h(\tau)x(t-\tau)\,d\tau\). Feed it a single everlasting complex exponential \(x(t) = e^{j\omega t}\) and watch what the integral does.

The step that matters is trivial algebra: \(e^{j\omega(t-\tau)} = e^{j\omega t}e^{-j\omega\tau}\), and the factor \(e^{j\omega t}\) does not involve the integration variable, so it comes straight out. What is left inside is a definite integral over \(\tau\) whose value depends on \(\omega\) but not on \(t\). Call it \(H(j\omega)\).

The output is the input again, multiplied by a complex constant. In the language of linear algebra, \(e^{j\omega t}\) is an eigenfunction of every LTI system and \(H(j\omega)\) is the corresponding eigenvalue. No other family of signals does this: a step comes out of an \(RC\) network as an exponential, a triangle comes out rounded, but \(e^{j\omega t}\) comes out as \(e^{j\omega t}\), rescaled and rotated, and nothing else.

Look at the integral defining \(H(j\omega)\). It takes the impulse response and produces a function of frequency; it is, in fact, exactly the Fourier transform of \(h(t)\), which is why Chapter 13 can be read as the general theory of the operation we have just performed on one particular signal. For now, treat \(H(j\omega)\) as what it plainly is: a complete description of the system, one complex number per frequency, whose modulus is the gain the system applies at that frequency and whose argument is the phase shift.

Two practical remarks. First, if \(h(t)\) is real then \(H(-j\omega) = H^{*}(j\omega)\), so the frequency response of a real system obeys the same even-magnitude, odd-phase symmetry as the spectrum of a real signal. Second, the integral must converge for \(H(j\omega)\) to exist at all, and \(\int|h(\tau)|d\tau \lt \infty\) is exactly the BIBO stability condition of Chapter 8. Unstable systems have no frequency response, which is the mathematics agreeing with the obvious physics: you cannot measure the steady-state gain of a system that never reaches steady state.

We now have the two halves. A periodic input is a sum of complex exponentials with known weights; each exponential passes through the system multiplied by a known number. Linearity does the rest.

The output is a sum of the very same exponentials, so it is periodic with the same period \(T\), and its Fourier coefficients can be read off by inspection.

Three consequences follow immediately, and they are the reason engineers think in the frequency domain at all.

The first is that an LTI system creates no new frequencies. Whatever appears at the output was already present at the input; the system can amplify a harmonic, attenuate it, shift its phase, or kill it entirely by having \(H(jk\omega_0) = 0\), but it cannot put power at a frequency where \(a_k\) was zero. The set of frequencies is fixed by the input; only the weights are the system's business.

The second is that magnitudes multiply and phases add: \(|b_k| = |H(jk\omega_0)||a_k|\) and \(\angle b_k = \angle H(jk\omega_0) + \angle a_k\). The two spectral plots of the output are obtained from those of the input by a pointwise product and a pointwise sum — no integrals, no convolution, no differential equation.

The third is that the whole calculation is local in frequency. What happens to the seventh harmonic does not depend in any way on what happens to the third. That independence is what allows a system to be specified, designed and tested one frequency at a time, and it is precisely what a network analyser exploits when it sweeps a sinusoid across a band and records amplitude ratio and phase shift.

If a system multiplies each harmonic by \(H(jk\omega_0)\), then designing \(|H(j\omega)|\) to be large over one band of frequencies and small over another is a way of selecting which parts of a signal survive. A system built for that purpose is a filter, and the frequency band it passes is its passband, the band it rejects its stopband.

The four idealised shapes are the vocabulary of the subject. An ideal lowpass filter has \(|H| = 1\) for \(|\omega| \lt \omega_c\) and \(0\) beyond; an ideal highpass is its complement; an ideal bandpass passes \(\omega_1 \lt |\omega| \lt \omega_2\); an ideal band-stop removes exactly that band. Each is defined on both positive and negative frequencies, symmetrically, because a real filter acting on a real signal must produce a real output.

These shapes are idealisations, and Chapter 15 will show that none of them can be built: a magnitude response that is exactly zero over a band forces an impulse response that is non-zero for \(t \lt 0\), so an ideal filter is non-causal. That does not make them useless. They are the target a real design approximates, and they make the arithmetic of "which harmonics get through" completely transparent.

A filter you can build is the single \(RC\) lowpass network of Chapter 9, whose frequency response follows from the impedance divider or, equivalently, from applying the eigenfunction property to its differential equation:

Substituting \(x = e^{j\omega t}\), \(y = H e^{j\omega t}\) into the differential equation turns \(d/dt\) into multiplication by \(j\omega\) and reduces calculus to algebra — a foretaste of the whole of Part 5. Its magnitude and phase are

At \(\omega = \omega_c\) the magnitude is \(1/\sqrt{2} = 0.707\) and the phase is \(-45^\circ\). Since power goes as the square of magnitude, half the power is lost at that frequency — hence the names half-power frequency, \(-3\,\)dB frequency and cutoff, all for the same \(\omega_c\). Beyond cutoff the magnitude falls as \(\omega_c/\omega\), a decade of frequency costing a factor of ten: a gentle roll-off compared to the vertical wall of the ideal filter, but real.

The figure below shows the whole calculation in one picture for a square wave whose fundamental sits exactly at the cutoff. The input lines are multiplied, one at a time, by the height of the response curve above them.

Sometimes the object is not to reshape a signal but to move it — down a cable, through an amplifier, across a radio link — with its shape intact. Perfect transmission cannot mean "output equals input", because any physical channel imposes some delay and usually some gain. The honest definition allows both and forbids everything else.

What frequency response does that demand? Apply the definition harmonic by harmonic. If \(x\) has coefficients \(a_k\), then Chapter 11's shifting property gives \(y\) the coefficients \(K a_k e^{-jk\omega_0 t_d}\). Comparing with \(b_k = H(jk\omega_0)a_k\) and reading off at every frequency:

The phase requirement is the one that surprises people, so it is worth saying plainly why constant phase will not do. A phase shift of \(\theta\) at frequency \(\omega\) is a time shift of \(\theta/\omega\) seconds — the same angle at a higher frequency corresponds to a shorter delay. A system that shifts every harmonic by \(-90^\circ\) delays the fundamental by a quarter of a period, the third harmonic by a twelfth of a period, and the fifth by a twentieth. The harmonics arrive out of step, and a waveform assembled from misaligned harmonics is not the waveform you sent, even though every amplitude is untouched. Only phase proportional to frequency delays all harmonics equally, because then \(\theta/\omega = t_d\) is the same for all of them.

Two measures make the idea quantitative. The phase delay \(t_p(\omega) = -\angle H(j\omega)/\omega\) is the delay experienced by a single sinusoid of frequency \(\omega\). The group delay \(t_g(\omega) = -\,d\,\angle H(j\omega)/d\omega\) is the delay experienced by the envelope of a narrow band of frequencies centred on \(\omega\). For a distortionless channel both equal \(t_d\) at every frequency, since the phase is \(-\omega t_d\) and its slope is \(-t_d\).

For the \(RC\) lowpass they are not equal, and neither is constant. With \(\angle H = -\arctan(\omega/\omega_c)\),

Taking \(RC = 1\) ms, so \(\omega_c = 1000\) rad/s, gives the numbers below. At low frequency both delays tend to \(RC = 1\) ms; by the cutoff they have separated, and by \(5\omega_c\) the group delay has all but vanished while the phase delay is still a quarter of a millisecond. Different harmonics are delayed by different amounts — the channel disperses the signal — which is phase distortion made arithmetic.

The rule \(b_k = H(jk\omega_0)a_k\) has a contrapositive that is more useful in the laboratory than the rule itself. If a system produces output power at a frequency the input did not contain, that system is not LTI. There is no exception and no special case: the derivation used only linearity and time invariance, so a frequency that was not in the input can only come from the failure of one of those two properties.

The commonest culprit is a mild nonlinearity. Suppose an amplifier is very nearly linear but has a small cubic term, \(y(t) = x(t) + \varepsilon x^3(t)\), and drive it with a pure cosine \(x(t) = A\cos\omega_0 t\). Expanding the cube with \(\cos^3\theta = \tfrac34\cos\theta + \tfrac14\cos 3\theta\):

A frequency that was nowhere in the input has appeared at the output, its size growing as the cube of the drive level — which is why distortion in an amplifier worsens sharply as it is pushed hard. The standard figure of merit collects all the unwanted harmonics into one number.

Time variation is the other route to new frequencies, and it is not a defect but a tool. Multiplying a signal by a carrier, \(y(t) = x(t)\cos\omega_c t\), is a linear operation but a time-varying one, and Chapter 11's multiplication property showed it shifts every spectral line by \(\pm\omega_c\). Every radio transmitter ever built depends on that shift, and Chapter 28 will treat it properly. The lesson is that "no new frequencies" requires both halves of LTI, and that giving up either one buys a capability worth having.

Problems 1 to 3 exercise the reading of spectra; 4 to 6 push periodic signals through systems; 7 and 8 ask you to prove the results the chapter has been using. Sketch every spectrum before computing anything with it.