2.2. Convolution

\(\newcommand{\op}[1]{{\mathsf #1}}\)

A linear shift invariant system can be described as convolution of the input signal. The kernel used in the convolution is the impulse response of the system.

A (continuous time) Shift Invariant Linear System is characterized with its impulse response.

A proof for this fact is easiest for discrete time signals. The proof for discrete time signals is left as an exerise for the reader. Here we consider continuous time signals.

Let \(x\) be the input signal to a linear system \(\op L\) and let the output be \(y=\op L x\). We can write \(x\) as an integration (summation) of shifted pulses:

\[x(t) = \int_{-\infty}^{\infty} x(u)\delta(u-t)du\]

Because \(\delta(x)=\delta(-x)\) we can also write:

\[x(t) = \int_{-\infty}^{\infty} x(u)\delta(t-u)du = \int_{-\infty}^{\infty} x(u)\delta_u(t)du\]

where \(\delta_u(t)\) is the function \(\delta\) shifted to the left over \(u\). Now look at \(\op L x\). Because of the linearity of \(\op L\) we may write:

\[(\op L x)(t) = \int_{-\infty}^{\infty} x(u)(\op L\delta_u)(t)du\]

Shift invariance of the operator implies that \((\op L\delta_u)=(\op L \delta)_u\), i.e. first shifting and then applying the operator is the same as first applying the operator and then shift.

Obviously \(\op L \delta\) is the pulse response of the linear system, let’s call it the function \(h\), then we get:

\[(\op L x)(t) = y(t) = \int_{-\infty}^{\infty} x(u)h(t-u)du\]

or equivalently:

\[y = x \ast h\]

the output of a shift invariant system is given by the convolution of the input signal with the impulse response function of the system. In the signal processing literature it is common to write:

\[y(t) = x(t) \ast h(t)\]

Although this is a bit sloppy notation (for a mathematician this looks like an expression involving real numbers not functions) it is used a lot and even in some cases it helps to make clear what the functions involved are.

Consider the case of discrete time signals. Let \(x[n]\) be the input signal to a linear LTI system that is characterized with its impulse response \(h[n]\). The output signal then is given by:

\[y[n] = x[n] \ast h[n]\]

So although mathematically quite sloppy this notation allows clear distinction between coninuous time and discrete time systems.

A (discrete time) Shift Invariant Linear System is characterized with its impulse response.