Fourier series

The cheapest way to make a tone with a microcontroller is to toggle a pin: high for half a period, low for the other half. The result is a square wave, and when Amara first drives the accessible pedestrian signal's speaker amplifier that way, the 880 Hz locator tone sounds harsh and buzzy rather than clean. The reason is that a square wave is not one frequency. It is a sum of sinusoids at 880 Hz, 2,640 Hz, 4,400 Hz and every other odd multiple of 880 Hz, and the higher ones are what the ear hears as buzz. The Fourier series makes that statement exact. It writes any reasonable periodic signal as a sum of harmonically related sinusoids, tells you how strong each one is, and, combined with lesson 2's eigenfunction property, tells you what an LTI system does to each one.

Harmonics

A periodic signal with fundamental period T₀ has fundamental frequency f₀ = 1/T₀ and angular frequency ω₀ = 2π/T₀. Its harmonics are the frequencies kω₀ for integer k: the first harmonic is the fundamental, the second is 2ω₀, and so on. Any sinusoid at a harmonic frequency completes a whole number of cycles in T₀, so a sum of such sinusoids repeats with period T₀ as well. Fourier's claim, made in the early nineteenth century and made rigorous later, is that the converse is true: almost any periodic signal can be built as such a sum.

The exponential Fourier series

The neatest form uses complex exponentials. A periodic signal x(t) with period T₀ is written as

The integral runs over any one period. The coefficient c_k is a complex number whose magnitude is the strength of the kth harmonic and whose angle is its phase. The k = 0 coefficient, c₀, is the average value of the signal over a period, its DC component. The formula for c_k works because complex exponentials at different harmonics are orthogonal: the integral over one period of e^(jkω₀t)e^(−jmω₀t) is zero unless k = m. Multiplying the series by e^(−jmω₀t) and integrating therefore picks out a single coefficient, much as the sifting property picks out a single value.

For a real signal, the coefficients come in conjugate pairs, c_{−k} = c_k*, so the positive and negative harmonics combine into real sinusoids. The equivalent trigonometric form is

so the amplitude of the real sinusoid at harmonic k is 2|c_k|. Some textbooks write the series with separate cosine and sine coefficients a_k and b_k instead; they carry the same information, with a_k = 2 Re{c_k} and b_k = −2 Im{c_k}.

Symmetry saves work

Symmetry in the waveform shows up as structure in the coefficients. An even signal has only cosine terms, so its coefficients are real. An odd signal has only sine terms, so its coefficients are purely imaginary, and its DC value is zero. A signal with half-wave symmetry, x(t + T₀/2) = −x(t), meaning the second half of each period is the first half turned upside down, has only odd harmonics. A square wave with zero average has half-wave symmetry, which is why it contains 880 Hz, 2,640 Hz and 4,400 Hz but no 1,760 Hz.

The line spectrum

Plotting |c_k| against frequency kf₀ gives the signal's line spectrum: a set of vertical lines at the harmonic frequencies, with heights equal to the harmonic strengths. Phase is plotted separately. The line spectrum is a second description of the same signal, in the frequency domain instead of the time domain, and many questions are easier there. How much of the buzz will a filter remove? Which harmonics fall inside the speaker's range? Will a harmonic land on a frequency that the noise detector is listening to? Each is answered by looking at the lines.

Pulse trains and duty cycle

Not every pin output has a 50 % duty cycle. A pulse train that is high for a time τ out of every period T₀ has duty cycle d = τ/T₀ and coefficients c_k = d·sin(πkd)/(πkd), measured from a pulse centered at t = 0. The DC value is c₀ = d, and the harmonic strengths follow an envelope that falls to zero wherever kd is a whole number. A 25 % duty cycle therefore has no fourth, eighth or twelfth harmonic but does have the second, sixth and tenth. As the pulses get narrower, the envelope widens and the harmonics stay strong to higher frequencies, which is why short, sharp pulses cause more radio interference than slow square waves. Lesson 7 shows that this envelope is the Fourier transform of a single pulse, sampled at the harmonic frequencies.

Power and Parseval's relation

The average power of a periodic signal can be computed in either domain. Parseval's relation says

Each harmonic contributes its own power, and the powers add because the harmonics are orthogonal. For the square wave, the time-domain power is simply A². The fundamental alone contributes 2|c₁|² = 8A²/π² ≈ 0.81A², so 81 % of the square wave's power is in the 880 Hz tone that Amara wants and 19 % is in the harmonics that make the buzz. The ratio of harmonic content to fundamental is called the total harmonic distortion, or THD:

For a square wave the THD is about 48 %, which is why it sounds so different from a sine.

Convergence and the Gibbs phenomenon

A Fourier series converges to the signal at every point where the signal is continuous, provided the signal satisfies mild conditions, known as the Dirichlet conditions, that every practical waveform meets: finite energy per period and finitely many jumps and wiggles. At a jump, the series converges to the midpoint of the jump. Near a jump, however, a partial sum with N harmonics always overshoots by about 9 % of the jump size, and adding more harmonics squeezes the overshoot closer to the edge without making it smaller. This is the Gibbs phenomenon. It appears whenever a signal with sharp edges is passed through a filter that cuts off its high harmonics abruptly, which is one reason filter designers sometimes prefer gentle cut-offs.

LTI systems and periodic inputs

Now combine the Fourier series with the eigenfunction property. Each harmonic e^(jkω₀t) passes through an LTI system as H(jkω₀)e^(jkω₀t), where H(jω) is the system's frequency response. By linearity, the response to the whole series is

So the output is periodic with the same period, and its Fourier coefficients are the input coefficients multiplied by the frequency response at each harmonic. A filter cannot create new harmonics; it can only make existing ones stronger or weaker and shift their phases. For the RC low-pass filter, lesson 8 shows that H(jω) = 1/(1 + jωRC). Amara adds a two-stage RC filter between the pin and the amplifier to soften the buzz.

import math

def square_partial(t, N, T0=1.0):
    w0 = 2 * math.pi / T0
    return sum(4 / (math.pi * k) * math.sin(k * w0 * t) for k in range(1, N + 1, 2))

for N in (5, 19, 99):
    peak = max(square_partial(i / 20000, N) for i in range(2000))   # first tenth of a period
    print(f"harmonics up to {N:3d}: peak = {peak:.4f} (overshoot {100*(peak-1)/2:.1f} % of the jump)")

P_total = 1.0                                  # square wave of amplitude 1
for N in (1, 3, 9, 49):
    P = sum(0.5 * (4 / (math.pi * k)) ** 2 for k in range(1, N + 1, 2))
    print(f"power in harmonics up to {N:2d}: {100*P/P_total:.1f} %")

f0, RC, stages = 880.0, 0.36e-3, 2
def amp(k):
    H = 1 / math.sqrt(1 + (2 * math.pi * k * f0 * RC) ** 2)
    return 4 / (math.pi * k) * H ** stages
harm = math.sqrt(sum(amp(k) ** 2 for k in range(3, 400, 2)))
print(f"THD of raw square wave: {100*math.sqrt(1 - 8/math.pi**2)/math.sqrt(8/math.pi**2):.1f} %")
print(f"THD after {stages} RC stages: {100*harm/amp(1):.1f} %")

harmonics up to   5: peak = 1.1884 (overshoot 9.4 % of the jump)
harmonics up to  19: peak = 1.1798 (overshoot 9.0 % of the jump)
harmonics up to  99: peak = 1.1790 (overshoot 9.0 % of the jump)
power in harmonics up to  1: 81.1 %
power in harmonics up to  3: 90.1 %
power in harmonics up to  9: 96.0 %
power in harmonics up to 49: 99.2 %
THD of raw square wave: 48.3 %
THD after 2 RC stages: 4.6 %

The overshoot stays near 9 % of the jump however many harmonics are added. Half of the 2 units of jump is 1, so a peak of about 1.18 means 0.18 above the top, or 9 % of the full jump. The power table shows the fundamental alone carrying about 81 % of the power and the first five odd harmonics about 96 %. The last two lines show why Amara's two RC stages work: the THD falls from about 48 % to under 5 %, close to a clean tone, because each stage hits the higher harmonics harder than the fundamental.