LTI system and its significance

Linear time-invariant system

A system is LTI if it obeys:

• Linearity (superposition): $a x_1(t)+b x_2(t)\to a y_1(t)+b y_2(t)$.
• Time invariance: a time shift of the input gives the same shift of the output: $x(t-t_0)\to y(t-t_0)$.

Such a system is fully described by its impulse response $h(t)$; the output is $y(t)=x(t)*h(t)$ and $Y(f)=H(f)X(f)$.

Significance in communication engineering

1. Simple analysis: knowing only $h(t)$ or $H(f)$, the output for any input can be found by convolution or multiplication. This makes channel and filter analysis easy.
2. Frequency-domain design: sinusoids are eigenfunctions of LTI systems; a sinusoid passes with only a change of amplitude $\lvert H(f)\rvert$ and phase $\angle H(f)$. Filters, equalizers and channels are therefore designed using $H(f)$.
3. Model of channels: cables, free space (over short time) and most filters are well modelled as LTI, giving tools for distortion, bandwidth and delay calculations.
4. Distortionless transmission: the condition $H(f)=Ke^{-j2\pi ft_d}$ (flat amplitude, linear phase) is stated for LTI systems.
5. Noise analysis: for a random input, $S_y(f)=\lvert H(f)\rvert^2S_x(f)$, used to find output noise power of receivers.
6. Superposition of signals: in FDM many signals pass through the same channel without creating new frequencies; non-linear systems would create intermodulation.

Example: a telephone channel modelled as an LTI band-pass filter (300–3400 Hz) lets the designer predict exactly which speech frequencies pass and how much delay each suffers.