Analog and digital LTI systems — Adrian Darlington

Why LTI systems matter

An LTI system has two useful promises. Linearity lets us split an input into pieces, process each piece, and add the results. Time invariance says that moving an input in time moves its output by exactly the same amount. Together, these promises mean that one impulse response characterizes the entire system.

Linearity

T\{a x_1+b x_2\}=aT\{x_1\}+bT\{x_2\}

Scaling and addition pass through the system.

Time invariance

T\{x[n-k]\}=y[n-k]

A delayed input has the same output, delayed.

This is why filters, communication channels, and many physical circuits can be analyzed systematically. In the frequency domain, complex sinusoids are eigenfunctions of an LTI system: their frequency stays put while amplitude and phase change.

Building a signal from impulses

In discrete time, $\delta[n-k]$ is 1 only when $n=k$ . Multiply by $x[k]$ to recover one sample; sum the weighted impulses to reconstruct the whole sequence.

x[n]=\sum_{k=-\infty}^{\infty}x[k]\,\delta[n-k]

Continuous time has the parallel “sifting” identity. A Dirac impulse is not an ordinary finite-height spike: it is an idealized object whose integral picks out the signal value at one instant.

x(t)=\int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau

From impulse response to convolution

Let the response to an impulse at zero be $h[n]=T\{\delta[n]\}$ in discrete time, or $h(t)=T\{\delta(t)\}$ in continuous time. By time invariance, shifting the impulse shifts that response; by linearity, scaling the impulse scales it.

Discrete time

\begin{aligned}y[n]&=T\!\left\{\sum_k x[k]\delta[n-k]\right\}\\&=\sum_k x[k]T\{\delta[n-k]\}\\&=\sum_k x[k]h[n-k]\\&=(x*h)[n]\end{aligned}

Continuous time

\begin{aligned}y(t)&=T\!\left\{\int x(\tau)\delta(t-\tau)\,d\tau\right\}\\&=\int x(\tau)T\{\delta(t-\tau)\}\,d\tau\\&=\int x(\tau)h(t-\tau)\,d\tau\\&=(x*h)(t)\end{aligned}

The second line in each derivation uses linearity; the third uses time invariance. The sum runs over all integer $k$ , and the integral over all time. The result is convolution: add a shifted, scaled copy of the impulse response for every piece of the input.