Working principle of PCM

Pulse Code Modulation (PCM) converts an analog signal into a sequence of binary code words by sampling, quantizing and encoding; the bits are then sent as pulses.

Transmitter
m(t)->[LPF]->[Sampler]->[Quantizer]->[Encoder]->line
     anti-    fs>=2W    L levels     n bits
     alias
Channel: --[Regenerative repeaters]-->
Receiver
-->[Regen]->[Decoder]->[Hold]->[Recon LPF]->m(t)

Transmitter

1. Anti-aliasing LPF limits the message to $W$ Hz.
2. Sampling at $f_s \ge 2W$ (flat-top, sample-and-hold), e.g. speech 3.4 kHz at 8 kHz.
3. Quantization rounds each sample to one of $L$ levels. For uniform steps:

$$\Delta = \frac{2m_p}{L}, \qquad N_q = \frac{\Delta^2}{12}$$

Non-uniform (µ/A-law) quantization is used for speech. 4. Encoding maps each level to an $n$-bit code word, $n = \log_2 L$. Bit rate and minimum bandwidth:

$$R_b = nf_s, \qquad B_T = \frac{R_b}{2} = nW\ (\text{at } f_s = 2W)$$

Channel

• Regenerative repeaters placed along the line re-time, re-shape and re-decide the pulses, so noise does not accumulate (major advantage of PCM).

Receiver

1. Regeneration of clean pulses.
2. Decoding: each code word is converted back to a quantized PAM sample.
3. Reconstruction LPF (cut-off $W$) recovers the analog signal, with only quantization noise.

Performance

$$\text{SQNR} = 1.5L^2 \;\Rightarrow\; \text{SQNR}_{dB} = 1.76 + 6.02n\ \text{dB}$$

(full-load sinusoid). Each extra bit adds 6 dB but increases bandwidth.

Example: telephony with $f_s = 8$ kHz, $n = 8$: $R_b = 64$ kbps, SQNR ≈ 49.9 dB.

Merits: high noise immunity, regenerative repeaters, easy encryption and TDM, uniform digital format. Demerits: larger bandwidth, need for synchronization, quantization noise.