8. Pulse Modulation: The Bridge to Digital // Analog Communications // bhaswanth

We end with a brief introduction to pulse modulation, the conceptual stepping stone from continuous analog to discrete digital communications. Chapter 12 will pick up the digital side in detail; here we plant the seeds.

8.1 Sample-and-hold: the act of sampling

Sample the message $m(t)$ at uniform intervals $T_s = 1/f_s$ . The output is a sequence of discrete samples $m(nT_s)$ . To represent these samples physically, we typically use a sample-and-hold circuit: a switch and a capacitor. At sample times, the switch closes, the cap charges to the current value of $m(t)$ , the switch opens, the cap holds the value until the next sample.

This is the front-end of every ADC and the implicit first step of every digital communication system. The sampling theorem (Chapter 3) tells us that as long as $f_s > 2W$ , where $W$ is the bandwidth of $m(t)$ , we lose nothing by sampling.

8.2 Pulse Amplitude Modulation (PAM)

The transmitted signal is a pulse train where each pulse's height equals the message at the sampling instant:

$s_\text{PAM}(t) = \sum_n m(nT_s)\,p(t - nT_s)$

where $p(t)$ is some pulse shape (rectangular, Gaussian, sinc-windowed). PAM is conceptually transparent: the message lives in the pulse heights.

PAM is the implicit first step of every PCM digital telephony system: sample, hold, and quantize. It also appears as a signaling scheme in some short-range digital systems (PAM-4 in modern Ethernet uses four-level pulses to send two bits per symbol, doubling spectral efficiency).

8.3 Pulse Width Modulation (PWM)

Each pulse has a fixed amplitude and a width proportional to the message:

plaintext

m=1  ████ ████ ████ ████
m=0.5 ██   ██   ██   ██  
m=0  •    •    •    •   
       T_s    T_s    T_s

PWM is everywhere in modern electronics:

Class-D audio amplifiers. The audio is PWM-modulated at hundreds of kHz, then a low-pass filter (LC) at the output integrates the PWM and reconstructs the audio. The transistors only switch fully on or fully off, so they dissipate almost no power; efficiencies of 90% are routine. Every modern smartphone speaker driver, every Bluetooth speaker, every car audio amplifier built since 2010 is class-D.
DC motor speed control. PWM the motor's drive voltage; the motor's mechanical inertia averages the pulses to an effective DC level proportional to the duty cycle.
LED dimming. PWM the LED at hundreds of Hz; your eye averages and perceives proportional brightness. (Watch out: cheap LED drivers PWM at frequencies low enough to flicker visibly, especially when seen out of the corner of your eye, where temporal resolution is higher than central vision.)
Servo control. Hobby servos take a pulse 1 to 2 ms wide every 20 ms; pulse width sets shaft angle.

8.4 Pulse Position Modulation (PPM)

Constant pulse amplitude and width; the timing of each pulse encodes the message. The pulse shifts left or right of its nominal slot in proportion to the message.

PPM is more noise-resistant than PAM because amplitude noise does not affect timing as much (provided the receiver can identify pulse edges accurately). Used in some optical fiber systems, infrared remote controls (the IR remote sending a PPM-coded command to your TV is an everyday example), and deep-space probe communications.

8.5 Pulse Code Modulation (PCM)

PCM is the next conceptual step: each sampled value is digitally encoded (quantized into bits) before transmission. PCM is the foundation of all digital telephony and digital audio. At 8 kHz sampling and 8 bits per sample (with $\mu$ -law companding for dynamic range), PCM gives 64 kbps per voice channel, the rate of every digital telephone channel since the 1960s. CD audio is PCM at 44.1 kHz / 16 bits per channel.

PCM is where analog ends and digital begins. We will pick it up in Chapter 12 (digital communications) and again in Chapter 24 (DSP and side channels), where the same digital streams are analyzed for what they leak about cryptographic operations.

8.6 Time-division multiplexing (TDM)

Pulse modulation enables a sharing scheme orthogonal to FDM: take many sampled message signals, interleave their samples in time, and transmit the resulting interleaved pulse stream. Different streams occupy different time slots rather than different frequencies. Classic example is the T1 line in North American telephony: 24 voice channels each sampled at 8 kHz with 8 bits, multiplexed into a 1.544 Mbps stream with 24 voice samples per frame at 8000 frames per second.

TDM and FDM are complementary; many real systems use both at once. A mobile phone uses FDM at the air interface (different cellular bands), TDM within a given band (GSM time slots), and CDMA (a third multiplexing scheme based on spread-spectrum codes) inside each slot in some systems. The multiplexing layer in a real protocol stack is usually a hybrid.