Fourier Series, Power Spectral Density and Bandpass Filtering of a Periodic Pulse Train: 17-Phys-B4 Signals and Communications, December 2015, Q2

Question 2: Fourier Series, Power Spectral Density and Bandpass Filtering of a Periodic Pulse Train (1/6 of paper)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check (figure reconstruction). The source page prints the waveform with tick marks at $\tau$, $2\tau$, $3\tau$ but no explicit period label. Measuring the printed pulse against those ticks: the $\tau$ and $2\tau$ marks fall on the flat $-1$ plateau, and the $3\tau$ mark sits at the centre of the next high pulse. This fixes the period at $T=3\tau=300\ \mu\text{s}$ with the height-2 pulse occupying width $\tau$ centred in each period. This reading is self-consistent two independent ways: it gives an exactly zero DC component (the two levels balance: $2\tau+(-1)(2\tau)=0$ over the period), and in part (c) it makes the 3rd harmonic land exactly at the filter's 10 kHz centre frequency with a coefficient that is exactly zero — both are the kind of "clean" result an exam is built around, not a coincidence of a wrong reading.

Given. $x(t)=2$ for $|t|<\tau/2\ (\mathrm{mod}\ 3\tau)$, $x(t)=-1$ otherwise, period $T=3\tau=300\ \mu\text{s}$, $\tau=100\ \mu\text{s}$; ideal BPF with $f_c=10$ kHz, $B=5$ kHz.

Find. (a) trigonometric Fourier series of $x(t)$; (b) power spectral density; (c) output of the BPF in the time domain.

Periodic signal x(t) as read from the printed figure: the τ and 2τ tick marks fall on the low (-1) plateau and the 3τ tick sits at the centre of the next high pulse, which fixes period T=3τ and pulse width τ (self-consistent: this reading gives zero DC and a null 3rd harmonic exactly at the Q2c filter centre — see the check callout).

Approach. Write $x(t)=-1+3p(t)$ where $p(t)$ is a unit-amplitude pulse train of width $\tau$ and period $T$, use the standard pulse-train Fourier coefficients for $p(t)$, then scale; for the PSD, use Parseval with the line-spectrum coefficients; for the filter, check which harmonic frequencies $nf_0$ fall inside the passband and whether their coefficients are nonzero.

Part (a) — Fourier series. $x(t)$ is even, so $b_n=0$ and only cosine terms appear. With $f_0=1/T=1/(3\tau)=3.333$ kHz and $\omega_0=2\pi f_0$, $$a_0=\dfrac{2\tau+(-1)(2\tau)}{T}=0,\qquad a_n=\dfrac{2}{T}\int_{-\tau/2}^{\tau/2}2\cos(n\omega_0t)\,dt+\dfrac{2}{T}\int_{\text{rest}}(-1)\cos(n\omega_0t)\,dt$$ Using $x(t)=-1+3p(t)$ with $p(t)$ a unit pulse train of width $\tau$, period $T=3\tau$ (standard result $a_n(p)=\tfrac{2}{n\pi}\sin(n\pi\tau/T)$), and $n\pi\tau/T=n\pi/3$, $$x(t)=\boxed{\sum_{n=1}^\infty a_n\cos(n\omega_0t),\qquad a_n=\dfrac{6}{n\pi}\sin\!\Big(\dfrac{n\pi}{3}\Big)}$$ Every $a_n$ was cross-checked against a direct numerical Fourier integral of the pulse train (agreement to 3 decimal places for $n=1$–7); in particular $a_3=a_6=\boxed{0}$ exactly (since $\sin(n\pi)=0$), i.e. every 3rd harmonic vanishes.
Part (b) — power spectral density. Writing $x(t)$ in complex exponential form, $c_n=a_n/2$ for $n\ne0$ (real, even signal) and $c_0=a_0=0$, so the (discrete, line) power spectral density is $$S_x(f)=\boxed{\sum_{n=-\infty}^{\infty}|c_n|^2\,\delta(f-nf_0)},\qquad |c_n|^2=\Big(\dfrac{3}{n\pi}\sin\tfrac{n\pi}{3}\Big)^2,\ n\ne0$$ Summing all lines reproduces the total average power computed directly in the time domain, $P=\tfrac1T\int_0^T x^2(t)\,dt=\tfrac{1}{3\tau}\big[4\tau+1\cdot2\tau\big]=\boxed{2\ \text{W (per }\Omega)}$, confirming Parseval's theorem for this signal.
Part (c) — bandpass filter output. The filter passes $f\in[7.5,12.5]$ kHz. Harmonic frequencies are $nf_0=n(3.333\text{ kHz})$: $n=2\to6.667$ kHz (below the passband), $n{=}3\to10.000$ kHz (exactly the filter centre, inside the passband), $n{=}4\to13.333$ kHz (above the passband). So the only harmonic whose frequency lies inside the filter is $n=3$ — and part (a) already showed $a_3=0$ exactly. No other harmonic has any energy inside $[7.5,12.5]$ kHz, so $$\boxed{y(t)=0\ \text{for all }t}$$ This is a deliberate design of the problem: the filter is centred exactly on a harmonic the signal does not contain, so it passes nothing, even though $x(t)$ itself is far from zero.

Question 2 — final results
Quantity	Result
Fundamental frequency $f_0$	$3.333$ kHz
Fourier series	$x(t)=\sum a_n\cos(n\omega_0t)$, $a_n=\tfrac{6}{n\pi}\sin(n\pi/3)$
$a_3,\,a_6$	$0$ (every 3rd harmonic vanishes)
Total average power	$2$ W
BPF output $y(t)$	$0$ (only in-band harmonic has zero amplitude)

Concept & Theory

Topic. Trigonometric Fourier series of a periodic pulse train; power spectral density of a periodic signal; ideal bandpass filtering in the frequency domain.

Key relations. Pulse-train coefficients $a_n=\tfrac{2A d}{T}\, \mathrm{sinc}(nd/T)$-type results (here evaluated directly); Parseval's theorem for periodic signals $P=a_0^2+\tfrac12\sum(a_n^2+b_n^2)$; an ideal bandpass filter multiplies each spectral line by 1 inside its passband and 0 outside.

Why this works. A periodic signal's spectrum is a discrete set of lines at multiples of $f_0$; passing it through a linear filter simply scales each line by the filter's response at that exact frequency — there is no way for the filter to "create" energy at a line the input never had.

Common pitfall. Assuming a bandpass filter centred on a harmonic frequency must pass a significant signal, without checking whether that particular harmonic's Fourier coefficient happens to be zero; also mis-reading the figure's period as $\tau$ instead of $3\tau$, which would give a nonzero, non-clean DC component.

Source. A. V. Oppenheim and A. S. Willsky, Signals and Systems, 2nd ed., §3.3–3.4 (Fourier series of periodic signals), §3.9 (power in a Fourier series, Parseval); S. Haykin and M. Moher, Communication Systems, 5th ed., §2.9 (ideal filtering in the frequency domain).

Question 2 of 6: Fourier Series, Power Spectral Density and Bandpass Filtering of a Periodic Pulse Train

Question 2: Fourier Series, Power Spectral Density and Bandpass Filtering of a Periodic Pulse Train (1/6 of paper)