Pulse Code Modulation (PCM)
Digital Communication System Fundamentals
1. The Bit Rate Calculation
In PCM, the Data Rate (\(R_b\)) is the speed at which digitized information is generated. It is the product of the sampling frequency and the number of bits per sample.
\[ R_b = n \times f_s \]
Where:
- \(n\): Bits per sample (\(n = \log_2 L\), where \(L\) is the number of levels).
- \(f_s\): Sampling frequency (must satisfy \(f_s \ge 2f_m\)).
2. Minimum Transmission Bandwidth
The theoretical minimum bandwidth required to transmit a binary PCM signal without Inter-Symbol Interference (ISI) is the Nyquist Bandwidth.
\[ B_{min} = \frac{R_b}{2} \]
For a practical baseband system using pulses, the bandwidth is often approximated as \(B \approx R_b\) to account for filter roll-off and guard bands.
3. Practical Applications of PCM
PCM is the backbone of almost all modern digital audio and telecommunication standards.
Digital Telephony
Used in PSTN and VoIP (G.711 standard) to digitize voice calls at 64 kbps.
Audio Storage (CDs)
Compact Discs use PCM with a sampling rate of 44.1 kHz and 16-bit resolution.
Space Telemetry
Used to send sensor data from satellites to ground stations due to its high noise immunity.
Digital Video
Early digital video formats used PCM for uncompressed audio tracks to maintain high fidelity.
Example: CD-Quality Audio
For a standard high-fidelity audio signal sampled at 44.1 kHz with 16-bit depth:
\[ R_b = 16 \times 44,100 = 705,600 \text{ bps} \]
The minimum bandwidth required for this single channel is:
\[ B = \frac{705.6 \text{ kbps}}{2} = 352.8 \text{ kHz} \]