AEiE0604 Data communication and information theory¶
Analog-to-digital communication process¶
Typical steps:
- Sampling.
- Quantization.
- Encoding.
- Line coding or modulation for transmission.
- Error control as needed.
Source coding¶
- Source coding removes redundancy to represent information efficiently.
- It is different from channel coding, which adds controlled redundancy for error protection.
Pulse modulation and PCM¶
| Technique | Core idea |
|---|---|
| PAM | Vary pulse amplitude |
| PWM | Vary pulse width |
| PPM | Vary pulse position |
| PCM | Sample, quantize, and binary-encode |
PCM essentials:
- Sampling converts continuous time to discrete time.
- Quantization converts continuous amplitude to discrete amplitude levels.
- Encoding maps each quantized level to bits.
Sampling theorem cue¶
- For faithful reconstruction of a bandlimited signal of highest frequency \(f_m\), sample at:
\[
f_s \ge 2f_m.
\]
- The rate \(2f_m\) is the Nyquist rate.
Quantization types and quantization noise¶
| Type | Recognition cue |
|---|---|
| Uniform quantization | Equal step size |
| Nonuniform quantization | Unequal step size, often better for wide dynamic range speech |
- Quantization error is the difference between actual sample amplitude and quantized level.
- Quantization noise power decreases when step size decreases.
- More bits per sample usually improve SQNR but increase bit rate.
Shannon-Hartley capacity theorem¶
\[
C = B\log_2(1+\text{SNR})
\]
where \(C\) is channel capacity in bit/s, \(B\) is bandwidth in Hz, and SNR is linear, not dB.
Trap:
- Do not substitute SNR in dB directly into the logarithm formula.
Multiplexing¶
| Method | Principle |
|---|---|
| FDM | Different frequency bands |
| TDM | Different time slots |
| WDM | Different optical wavelengths |
| CDM | Different spreading codes |
Random signals, white noise, thermal noise¶
- A random process is a family of random variables indexed by time.
- White noise has flat power spectral density over the band of interest in the idealized model.
- Thermal noise power over bandwidth \(B\) is:
\[
N=kTB
\]
where \(k\) is Boltzmann constant and \(T\) is absolute temperature.
Information measure¶
- Self-information of an event with probability \(p\) is:
\[
I = -\log_2 p
\]
- Entropy is average information per symbol.
- Rare events carry more information than common events.
Line codes and pulse shaping¶
| Line code cue | Meaning |
|---|---|
| NRZ | No return to zero within bit interval |
| RZ | Returns to zero within bit interval |
| Manchester | Transition encodes clock and data |
| Bipolar / AMI | Alternating polarity for ones |
- Pulse shaping limits bandwidth and controls intersymbol interference.
- Nyquist pulse-shaping ideas aim to reduce ISI at sampling instants.
Error control coding techniques¶
- Error detection codes detect corruption.
- Error correction codes can recover some errors without retransmission.
- Block codes, parity, Hamming, and convolutional ideas are standard families.
Information-and-data examples¶
- If a baseband signal has highest frequency 4 kHz, minimum sampling rate is 8 kHz.
- If bandwidth is doubled while SNR stays same, Shannon capacity doubles proportionally to \(B\).
- If coding removes redundancy for compression, it is source coding, not channel coding.
AEiE0604 revision box¶
- Sampling theorem: \(f_s\ge2f_m\).
- PCM = sampling + quantization + encoding.
- Capacity: \(C=B\log_2(1+\text{SNR})\) with linear SNR.
- White noise is idealized flat-PSD noise.
- Source coding removes redundancy; channel coding adds controlled redundancy.