Properties and Convergence of Fourier Series

Chapter 10 supplied a complete recipe: given \(x(t)\), integrate to get \(a_k\); given \(a_k\), sum to get \(x(t)\). Followed literally, that recipe means a fresh integration by parts for every new waveform, which is both tedious and uninformative. Chapter 8 faced the same situation with convolution and answered it the same way — by establishing structural properties of the operation, so that new results follow from old ones by algebra rather than by calculus.

The properties below do two distinct jobs, and it is worth separating them. Some are labour-saving: a shifted, scaled or differentiated version of a signal whose coefficients you already know needs no new integral. Others are structural: they tell you what the spectrum of a signal must look like before you compute anything, and they are what allows an engineer to glance at a waveform and say "odd harmonics only, falling like \(1/k^2\)" without writing a line.

The second half of the chapter changes register. Everything so far has assumed that the infinite series in the synthesis equation converges to \(x(t)\). For a signal with jumps in it — a square wave, a sampled-and-held voltage, a switching waveform — that assumption deserves scrutiny, because each partial sum is a finite combination of smooth sinusoids and is therefore perfectly continuous. Something must happen in the limit, and what happens is more interesting than either "it works" or "it doesn't".

Linearity is immediate, and it is immediate because the analysis equation is an integral, which is itself linear. For any constants \(A\) and \(B\), the coefficients of \(Ax(t)+By(t)\) are \(Aa_k + Bb_k\). It is the property used most often and noticed least.

Time shifting is the one that repays study. Let \(y(t)=x(t-t_0)\) and compute its coefficients directly, substituting \(\tau = t-t_0\). The interval of integration shifts, but it is still an interval of one full period, so the notation \(\int_T\) absorbs it:

Both halves of that statement matter. The first says that magnitude alone cannot tell you when anything happened; all timing information in a signal lives in its phase. The second says that a pure delay is exactly a linear phase characteristic. Chapter 15 will invert this observation: a filter whose phase response is linear in \(\omega\) delays every harmonic by the same amount of time and therefore does not distort the waveform's shape, while a filter with non-linear phase moves the harmonics relative to one another and mangles it, even if every magnitude is left alone.

Time reversal follows the same substitution, this time \(\tau = -t\), which flips the limits and cancels the sign of \(d\tau\), leaving \(x(-t) \leftrightarrow a_{-k}\). Combined with the conjugation results of the next section, this is what turns statements about even and odd signals into statements about their spectra.

Time scaling is the property most often misremembered, because in the Fourier transform of Chapter 14 scaling stretches the spectrum, while here it does not. Let \(\alpha \gt 0\) and consider \(x(\alpha t)\). This is a different signal with a different period, \(T/\alpha\), and therefore a different fundamental frequency, \(\alpha\omega_0\). Substituting into the analysis equation with the new period returns the same numbers \(a_k\).

Conjugating the analysis equation gives \(\overline{a_k}\) as the coefficient of \(x^*(t)\) at index \(-k\), which is the compact statement \(x^*(t) \leftrightarrow a_{-k}^*\). Chapter 10 extracted the most important consequence — for real \(x\), \(a_{-k}=a_k^*\) — and the rest of the family follows by combining that with time reversal.

The chain of reasoning is short enough to reconstruct rather than memorise. If \(x\) is real, \(a_{-k}=a_k^*\). If \(x\) is additionally even, time reversal gives \(a_{-k}=a_k\); putting the two together forces \(a_k = a_k^*\), so \(a_k\) is real. If instead \(x\) is odd, time reversal gives \(a_{-k}=-a_k\), so \(a_k = -a_k^*\) and \(a_k\) is purely imaginary. Two facts, combined two ways, produce four results.

One more consequence is worth stating because it is used in the next chapter. Any real signal splits into its even and odd parts, \(x = \mathrm{Ev}\{x\} + \mathrm{Od}\{x\}\), as Chapter 4 established. Applying the table to each part separately shows that the even part carries exactly the real part of the spectrum and the odd part carries exactly the imaginary part:

What happens to the coefficients when two periodic signals of the same period are multiplied? Write both as series and multiply them out. A term of the product carries \(e^{j(l+m)\omega_0 t}\), so to collect the coefficient of the \(k\)th harmonic we must gather every pair with \(l+m=k\):

The companion property runs the other way. Ordinary convolution of two periodic signals diverges — the integral over an infinite range of a signal that never decays cannot converge — so the operation is redefined over a single period. Periodic convolution is

and its coefficients follow from swapping the order of integration and applying the time-shift property to the inner integral. The inner integral is the analysis integral of \(y(t-\tau)\), which by Section 11-2 equals \(T\,b_k e^{-jk\omega_0\tau}\):

Differentiate the synthesis equation term by term. Each \(e^{jk\omega_0 t}\) contributes a factor \(jk\omega_0\) and nothing else, because the exponential is its own derivative up to that constant:

These two rules turn into a technique. Differentiating a piecewise-linear waveform repeatedly reduces it, by Chapter 3's chain, to a train of steps and then to a train of impulses — and the coefficients of an impulse train are trivial, since sifting evaluates the analysis integral instantly. Compute those, then divide by \((jk\omega_0)^m\) to climb back up. Worked Example 2 does exactly this for the triangular wave.

There is also a structural reading, and it is one of the most useful pieces of intuition in the subject. Differentiation multiplies \(a_k\) by \(k\); so if \(dx/dt\) is to have coefficients that stay bounded, \(a_k\) must fall off at least as fast as \(1/k\). Running the argument in reverse relates the smoothness of the waveform to the decay of its spectrum.

Chapter 2 defined the average power of a periodic signal as \(P = \frac{1}{T}\int_T |x(t)|^2 dt\). The question Parseval answers is whether that power can be attributed to individual harmonics, and the answer — that it can, with no cross-terms whatsoever — is a direct consequence of orthogonality.

Write \(|x|^2 = x\,x^*\), substitute the series for each factor, and integrate:

The bracket is \(1\) when \(k=m\) and \(0\) otherwise, so the double sum collapses onto its diagonal.

For a real signal the \(\pm k\) lines have equal magnitude, so they can be merged: \(P = a_0^2 + 2\sum_{k\ge1}|a_k|^2\), which in amplitude–phase notation reads \(P = C_0^2 + \tfrac12\sum_{k\ge1}C_k^2\). The factor of one half is the familiar statement that a sinusoid of peak amplitude \(C\) has mean-square value \(C^2/2\); Parseval simply says that these add.

The practical use is to decide how many harmonics matter. A specification such as "total harmonic distortion below 1%" is a statement about the tail of this sum, and truncating a series is only defensible once you know what fraction of the power the discarded terms carried. Worked Example 3 works that out for the square wave, with a result that surprises most people the first time.

Everything up to this point assumed that the synthesis equation reproduces \(x(t)\). Fourier asserted it for all signals; that assertion is false, and identifying the correct hypotheses occupied mathematicians for most of the nineteenth century. Dirichlet's conditions are sufficient — not necessary — and they are satisfied by every signal an engineer is likely to meet.

Condition (1) is what makes the analysis integral exist at all: if \(\int_T|x|\,dt\) is infinite then \(a_k\) itself is undefined and there is nothing to discuss. Conditions (2) and (3) exclude signals that oscillate or jump infinitely often within a finite time. Each condition has a standard signal that violates it and only it.

None of these three is a signal any instrument could produce, and that is the honest summary of the conditions: they are a fence around pathologies, not a constraint on engineering. The clause that does bite in practice is the second half of the theorem — the behaviour at a jump. At a discontinuity the series does not converge to either one-sided value but to their average, and it does so for a reason that is easy to see: the series has no way of distinguishing the two sides, since every one of its terms is continuous. Confronted with a choice between \(x(t^-)\) and \(x(t^+)\), it splits the difference.

This is why Chapter 10's sawtooth series returned \(T/2\) at \(t=0\) while the signal was defined to be \(0\) there, and why the square-wave series returns \(0\) at every switching instant. In both cases the series is right and the point value is irrelevant — as Chapter 3 observed, changing a signal at isolated instants changes no integral, and integrals are all the coefficients ever saw.

There is a second convergence theorem, with a weaker hypothesis and a weaker conclusion, and for engineering purposes it is the more important of the two. It asks only that the signal have finite energy in a period, \(\int_T |x|^2dt \lt \infty\), and it concludes not that the series matches \(x\) at each instant, but that the energy of the error goes to zero.

To state it, define the \(N\)th partial sum and its error. In fact it pays to be more ambitious and ask which coefficients \(c_k\) give the best \(N\)-term harmonic approximation, rather than assuming the Fourier ones are optimal. Let \(\hat{x}_N(t)=\sum_{|k|\le N} c_k e^{jk\omega_0 t}\) and expand the mean-square error, using orthogonality at every step:

Now complete the square on the last two sums, adding and subtracting \(\sum|a_k|^2\):

The last sum is a sum of squares and is the only part that depends on our choice. It is smallest — zero — precisely when \(c_k = a_k\) for every \(k\).

So the error energy vanishes. What does not vanish is the maximum of the error, and the discrepancy between those two statements is the Gibbs phenomenon.

Look again at the square-wave partial sums of Chapter 10. Near each jump the partial sum overshoots, and adding harmonics does not remove the overshoot; it only squeezes it into a narrower interval on either side of the discontinuity. The peak height converges to a fixed number, and that number is not 1.

The location of the peak can be found exactly. Differentiating the partial sum through harmonic \(2M-1\) and using the identity \(\sum_{m=1}^{M}\cos(2m-1)\theta = \sin(2M\theta)/(2\sin\theta)\) shows that the derivative first vanishes when \(2M\omega_0 t = \pi\), that is at \(t = T/(4M)\). Evaluating the sum there and letting \(M\to\infty\) turns it into a Riemann sum for an integral:

Problems 1 to 4 should be solved with the properties alone — if you find yourself setting up an analysis integral, look again. Problems 5 to 8 concern power and convergence.