Physical Layer: Channel Capacity, Nyquist-Shannon Bounds & Switching

Every physical transmission medium acts as a band-pass filter that attenuates high-frequency Fourier components of transmitted square waveforms. As pulses travel along copper or fiber, high frequencies decay rapidly, rounding the sharp pulse transitions and leading to Intersymbol Interference (ISI).

To transmit digital symbols without overlapping each other at the receiver, the symbol transmission rate (baud rate) cannot exceed twice the available analog bandwidth B (in Hertz). This fundamental physical constraint leads directly to the classical Nyquist and Shannon channel bounds.

In 1928, Harry Nyquist proved that for an ideal noiseless channel of bandwidth B Hertz, the maximum symbol transmission rate without ISI is exactly 2B baud (symbols per second).

If each symbol encodes M distinct discrete voltage levels, each symbol carries log2(M) bits of information.

Nyquist Maximum Data Rate Formula:

$C = 2 × B × \log_2(M) \quad bits per second (bps)$

Implication: On a noiseless channel, data rate can be arbitrarily increased simply by increasing the number of signaling levels M. However, in the real world, thermal noise limits how closely signaling levels can be spaced before the receiver confuses adjacent levels.

In 1948, Claude Shannon published the mathematical foundation of information theory, proving that random thermal noise (Additive White Gaussian Noise, AWGN) imposes an insurmountable upper limit on the error-free information capacity of any physical channel.

Shannon-Hartley Capacity Formula:

$C = B × \log_2(1 + SNR) \quad bits per second (bps)$

Where:

• B is the channel bandwidth in Hertz (Hz).

• SNR is the Signal-to-Noise Ratio (power ratio S / N) in linear units, NOT decibels.

Converting SNR from Decibels to Linear:

$SNR_{dB} = 10 \log_{10}(SNR) \implies SNR = 10^{\frac{SNR_{dB}}{10}}$

Key Principle: No matter how many signal levels M you engineer, you CANNOT exceed Shannon capacity C on a channel with bandwidth B and signal-to-noise ratio SNR without introducing errors!