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Pulse Code Modulation (PCM)


Pulse Code Modulation (PCM)

Pulse Code Modulation (PCM) is a digital representation of an analog signal. It is the standard method used for converting analog audio, video, and other signals into a digital format for transmission, processing, or storage.

Block Diagram Quantization Equations Applications

Block Diagram

The PCM process can be broken down into two main parts: the transmitter (analog-to-digital) and the receiver (digital-to-analog).

┌──────────────┐ │ Analog │ │ Message Signal│ └───────┬──────┘ │ ┌─────▼─────┐ │ LPF │ └─────┬─────┘ │ ┌─────▼─────┐ │ Sampler │ └─────┬─────┘ │ ┌─────▼─────┐ │ Quantizer │ └─────┬─────┘ │ ┌─────▼─────┐ │ Encoder │ └─────┬─────┘ │ PCM Output to Channel │ ┌─────────────────▼───────────────────┐ │ CHANNEL │ │ ┌─────────────────────────────┐ │ │ │ Regenerative Repeater │ │ │ └──────────────┬──────────────┘ │ │ ... │ │ ┌──────────────▼──────────────┐ │ │ │ Regenerative Repeater │ │ │ └─────────────────────────────┘ │ └─────────────────┬───────────────────┘ │ Channel Output │ ┌──────▼──────┐ │ Regeneration │ │ Circuit │ └──────┬──────┘ │ ┌──────▼──────┐ │ Decoder │ └──────┬──────┘ │ ┌──────▼──────┐ │Reconstruction│ │ Filter │ └──────┬──────┘ │ ┌──────▼──────┐ │ Destination │ └──────────────┘

Fig: Block Diagram of a PCM System

Transmitter (Analog-to-Digital Conversion)

  1. Low-Pass Filtering: Removes high-frequency components that could cause aliasing.
  2. Sampling: Converts continuous-time signal into discrete-time signal at rate \(f_s\).
  3. Quantization: Approximates sample amplitude to discrete levels.
  4. Encoding: Assigns unique binary codewords to levels.

Receiver (Digital-to-Analog Conversion)

  1. Regeneration: Prevents noise accumulation using repeaters.
  2. Decoding: Converts binary back to amplitude levels (staircase signal).
  3. Reconstruction Filtering: Smooths the signal back to analog form.

Interactive PCM Simulator

Test various sampling frequencies, quantization levels, and bit rates to visualize how signal quality changes in real-time.

Launch Simulator Tool

The Quantization Process

Quantization maps continuous-amplitude samples to a finite set of discrete levels. This is the main source of error in PCM systems.

Quantization Error (Noise)

The error \(e_q\) ranges from \(-\Delta/2 \le e_q \le \Delta/2\), where \(\Delta\) is the step size.

$$ \Delta = \frac{x_{\text{max}} - x_{\text{min}}}{L} = \frac{x_{\text{max}} - x_{\text{min}}}{2^n} $$

Types of Uniform Quantizers

  • Mid-Tread Quantizer: Origin lies on a tread (zero level exists).
    • Output ↑ 3Δ | ┌─────── | │ 2Δ | ┌──────┘ | │ Δ | ┌──────┘ | │ 0 ─┼─┼───────────→ Input | ┌─────┘ -Δ | │ | ┌────┘ -2Δ | │ └┘ -3Δ |
  • Mid-Rise Quantizer: Origin lies on a riser (no zero level).

Noise and Signal Quality (SQNR)

The quality is measured by the Signal-to-Quantization Noise Ratio (SQNR).

$$ \sigma_e^2 = \frac{\Delta^2}{12} \quad \text{and} \quad \text{SQNR} = \frac{P_s}{\sigma_e^2} $$

The 6dB Rule: For every additional bit, SQNR improves by ~6 dB.

$$ (\text{SQNR})_{\text{dB}} \approx 1.76 + 6.02n $$

Bit Rate (Rb)

$$ R_b = n \times f_s $$

Solved Example: TDM & Bandwidth

Question: Six signals are multiplexed using TDM, and the number of quantization levels is 256. Message frequency \(f_m = 5 \text{ KHz}\). Find Transmission Bandwidth.

Given: - N = 6 (Signals) - fm = 5 KHz - L = 256 levels → n = log₂(256) = 8 bits - fs = 2 × fm = 10 KHz Calculation: Bit Rate (Rb) = N × n × fs Rb = 6 × 8 × 10 = 480 kbps Bandwidth (B) = Rb / 2 B = 480 / 2 = 240 KHz

Deep Dive: Sampling Technique

Sampling is the bridge between analog and digital. It is essentially a switching approach. To prevent aliasing, we follow the Nyquist Criterion:

Sampling Illustration
The sampling frequency should be at least twice the message signal’s frequency.

Computers only comprehend binary ('0' and '1'). Quantization gives these samples meaning. For a 4-bit quantizer, we map signal values to 16 distinct levels (0000 to 1111).

Applications of PCM

  • Digital Telephony: Standard for PSTN and VoIP calls.
  • Digital Audio: Audio CDs (16-bit PCM @ 44.1 kHz), WAV, and AIFF files.
  • Space Communication: Robust against noise in deep space telemetry.
  • ISDN: Integrated services over digital telephone lines.

MATLAB Visualizations

Get MATLAB Code for these simulations

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