Question 4 of 7: Modulator outputs and demodulators for a square-wave message (20 marks)

Question 4 — Modulator outputs and demodulators for a square-wave message (20 marks)

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 $m(t)$ is a zero-mean square wave of unit amplitude, so $m(t) = \pm1$ V with period $T_m$ and $|m|_{\max} = 1$. Carrier $c(t) = A_c\cos\omega_c t$ with $\omega_c \gg 2\pi/T_m$ (typical values assumed where the question leaves a parameter open: unit carrier amplitude, a carrier many times the message rate, and for FM a peak deviation $\Delta f$ of a few message-rate multiples).

Find. For each of the four schemes, a labelled sketch of the modulator output and a block diagram of the matching demodulator.

Approach. Write each modulated signal as an explicit function of the two message levels $m = \pm1$, read the envelope and phase behaviour off that expression, and then choose the demodulator that inverts exactly the property the modulator used to carry the information — envelope for AM, coherent multiplication for suppressed-carrier schemes, and frequency-to-amplitude conversion for FM.

(a) AM with modulation index $a = 0.5$

The modulated wave is $s(t) = A_c\bigl[1 + a\,m(t)\bigr]\cos\omega_c t$. With $a = 0.5$ and $m = \pm1$ the envelope alternates between two strictly positive levels:

Because $a \lt 1$, the envelope never reaches zero and never inverts, so no phase reversal occurs — this is exactly the condition under which envelope detection is valid.

The demodulator is therefore the simplest of the four: a diode envelope detector recovers $A_c[1 + a\,m(t)]$, a d.c. block removes the constant $A_c$, and a baseband low-pass filter removes the carrier ripple, leaving a scaled replica of the square wave.

(b) DSB with suppressed carrier

Here $s(t) = A_c\,m(t)\cos\omega_c t$. The envelope $|m(t)|A_c$ is constant at $A_c$, but the carrier phase jumps by $180^\circ$ at every message transition, because $m$ changes sign. The transmitted power is entirely in the sidebands, which is the efficiency advantage of DSB-SC; the price is that an envelope detector would recover $|m(t)|$, a constant, and lose the information completely.

Coherent (synchronous) detection is mandatory. Multiplying by a locally generated $2\cos\omega_c t$ of the correct frequency and phase gives $A_c m(t)\bigl[1 + \cos 2\omega_c t\bigr]$, and a low-pass filter removes the double-frequency term. The local carrier is obtained from the received signal itself by a squaring loop or, more usually, a Costas loop; a phase error $\phi$ scales the output by $\cos\phi$, so lock quality directly sets the recovered amplitude.

(c) Frequency modulation

The instantaneous frequency is $f_i(t) = f_c + k_f m(t)$, so with a two-level message the carrier simply hops between $f_c + \Delta f$ and $f_c - \Delta f$, where $\Delta f = k_f$ for unit-amplitude $m$ — this is binary FSK. The envelope is rigidly constant, which is why FM is immune to amplitude disturbances, and the phase is continuous through each transition provided the modulator integrates $m(t)$ (as a true FM modulator does). Taking a typical deviation ratio, Carson’s rule gives a transmission bandwidth $B \approx 2(\Delta f + f_m)$.

The classical demodulator converts frequency deviation into amplitude deviation and then detects the envelope: a hard limiter first strips any amplitude disturbance, a differentiator (or slope circuit) gives an output proportional to $\omega_i$, an envelope detector recovers that amplitude, and a low-pass filter cleans up the baseband. A phase-locked loop is the modern equivalent, with the loop-filter voltage serving directly as the demodulated output.

(d) Phase modulation with $k_p = \pi/2$ rad/V

Here $s(t) = A_c\cos\bigl[\omega_c t + k_p m(t)\bigr]$ with $k_p m = \pm\pi/2$. Expanding,

because $\cos(\pm\pi/2) = 0$ and $\sin(\pm\pi/2) = \pm1$. The carrier term vanishes completely. With this particular modulator constant, phase modulation of a $\pm1$ square wave degenerates into a DSB-SC signal on a quadrature carrier — that is, into binary phase-shift keying. The envelope is constant and the phase reverses by $180^\circ$ at every transition, so the waveform is indistinguishable in form from part (b).

The demodulator is consequently the coherent detector again, but referenced to the quadrature carrier $-2\sin\omega_c t$: multiplying and low-pass filtering returns $A_c m(t)$. In the general PM case the low-pass filter output is $\tfrac{1}{2}A_c\sin(k_p m)$ and an $\arcsin(\cdot)/k_p$ stage is needed to linearise it, but at $k_p = \pi/2$ with a two-level message that stage collapses into a simple sign (bit) decision.