0.3 The communications channel

The effects of signal propagation are usually modeled as:

where

$H\left(f\right)$ : linear filtering due to multipath propagation
$w\left(t\right)$ : additive noise/interference.

Noise/interference sources include:

• electronic circuitry (“thermal/shot noise” or “quantization noise”); usually broadband,
• other comm systems (“multi-access interference” or “co-channel interference”); broadband or narrowband.

SNR can be improved with appropriate filtering at receiver:

• signal-to-noise ratio (SNR) $=\frac{\text{signal}\text{energy}}{\text{noise}\text{energy}}=\frac{{\mathcal{E}}_s}{{\mathcal{E}}_n}$
• roughly speaking, average SNR can be improved by filtering out the frequencies dominated by noise:

Filtering due to multipath propagation

The signal may propagate along paths with different lengths:

Since different path lengths imply different path delays:

which can be written as $x\left(t\right)=s\left(t\right)*h\left(t\right)$ for

The result is an $\left|H\right(f\left)\right|$ that varies with f , implying frequency-dependent signal attenuation:

t_max = 1; Ts = 1/1000;c1 = 1; tau1 = 0.02; c2 = -0.5; tau2 = 0.08;c3 = 0.2; tau3 = 0.10; h = zeros(1,t_max/Ts);h(tau1/Ts) = c1/Ts; h(tau2/Ts) = c2/Ts;h(tau3/Ts) = c3/Ts; plottf(h,Ts);