Question 4 of 7: Modulator Waveforms and Demodulators for a Square-Wave Message

Question 4: Modulator Waveforms and Demodulators for a Square-Wave Message (1/5 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.

Given. Message $m(t)$ = zero-mean square wave, unit amplitude ($m(t)=\pm1$); carrier $\cos(2\pi f_ct)$ of assumed amplitude $A_c$ and frequency $f_c$ (typical values, not specified by the question).

Find. For each of AM ($a=0.5$), DSB-SC, FM, and PM ($k_p=\pi/2$ rad/V): the modulator output waveform, sketched, and a demodulator block diagram.

Approach. A square-wave message only ever takes the two values $\pm1$, so each modulator output reduces to a simple two-state keying pattern: AM keys the envelope between two nonzero levels, DSB-SC keys the carrier phase by 180°, FM keys the instantaneous frequency between two values, and PM keys the phase directly by $k_p\Delta m$. Each demodulator follows directly from the corresponding CW-modulation theory (envelope detection, coherent/synchronous detection, discrimination).

1. Part (a) — AM, a = 0.5. $s(t)=A_c[1+a\,m(t)]\cos(2\pi f_ct)$ with $m(t)=\pm1$ collapses to two possible envelope levels: $$s(t)=\boxed{A_c(1\pm0.5)\cos(2\pi f_ct)=\{1.5A_c,\ 0.5A_c\}\cdot\cos(2\pi f_ct)}$$ i.e. the carrier envelope switches between $1.5A_c$ and $0.5A_c$ every time $m(t)$ transitions — a two-level "envelope-shift-keyed" carrier that never crosses zero (since $a=0.5<1$, no overmodulation). It is demodulated exactly like ordinary AM: an envelope (diode + RC) detector recovers the two-level envelope directly, and a DC-blocking stage removes the $A_c$ offset to leave $m(t)$.

2. Part (b) — DSB-SC. $s(t)=A_c\,m(t)\cos(2\pi f_ct)$ with $m(t)=\pm1$ is $$s(t)=\boxed{\pm A_c\cos(2\pi f_ct)}$$ a constant-envelope carrier whose PHASE flips by 180° at every message transition (no envelope break is visible — only a phase discontinuity, the classic signature that rules out envelope detection and forces coherent demodulation). A multiplier against a LOCAL, phase-synchronous copy of the carrier followed by an LPF recovers $m(t)$: $s(t)\cos(2\pi f_ct)=A_c m(t)\cos^2(2\pi f_ct)=\tfrac{A_c}{2}m(t)+\tfrac{A_c}{2}m(t)\cos(4\pi f_ct)$, and the LPF removes the $2f_c$ term.

3. Part (c) — frequency modulation. FM's instantaneous frequency is $f_i(t)=f_c+k_fm(t)$; with $m(t)=\pm1$ this is a two-level frequency-shift-keyed carrier (continuous phase, since the FM modulator integrates frequency into phase across the transition): $$s(t)=\boxed{A_c\cos\!\Big(2\pi f_ct+2\pi k_f\!\int_0^t m(t')\,dt'\Big),\quad f_i\in\{f_c+k_f,\ f_c-k_f\}}$$ Demodulation uses an FM discriminator: a limiter (removes any envelope/amplitude noise, since FM carries no information there), a differentiator (converts frequency deviation to a proportional AMPLITUDE variation, $d/dt$ of a $\cos(\cdot)$ brings down the instantaneous frequency as a multiplying factor), and an envelope detector to read that amplitude back out as $m(t)$.

4. Part (d) — phase modulation, kp = π/2 rad/V. PM's instantaneous phase is $\theta_i(t)=2\pi f_ct+k_p\,m(t)$; a transition of $m(t)$ from $-1$ to $+1$ (a swing of $\Delta m=2$) jumps the phase by $$\Delta\theta=k_p\Delta m=\dfrac{\pi}{2}\times2=\boxed{\pi\ \text{rad (a full }180^\circ\text{ inversion, identical in shape to the DSB-SC case)}}$$ so with this particular $k_p$, PM of a $\pm1$ square wave reproduces exactly the same $\pm A_c\cos(2\pi f_ct)$ waveform as part (b) — a coincidence of this specific $k_p=\pi/2$ value, not a general PM property. Demodulation is therefore also a coherent phase detector: multiply by a local $\sin(2\pi f_ct)$ (in quadrature, so the detector output is proportional to $\sin(\Delta\theta_i)\approx\theta_i$ for small deviations, or directly resolves the $\pm1$ keying here) and low-pass filter.