0.4 Signal processing in processing: convolution and filtering

Systems

For our purposes, a system is any processing element that, given as input a sequenceof samples $x(n)$ , produces as output a sequence of samples $y(n)$ . If the samples are coming from a temporal series we talk about discrete-time systems . In this module we will not be concerned with continuous-time processing, eventhough the principles here described can be generalized to functions of continuous variable. Instead, the sequence ofnumber can come from the sampling of an image, and in this case it will be appropriate to talk of discrete-space systems and use two indeces $m$ and $n$ if sampling is done by a rectangular grid of rows and columns.

In this module we are only dealing with linear systems , thus meaning that the following principle holds:

Superposition principle

If $y_1$ and $y_2$ are the responses to the input sequences $x_1$ and $x_2$ then the input $a_1{x}_1+a_2{x}_2$ produces the response $a_1{y}_1+a_2{y}_2$

Another important concept is time (and space) invariance.

Time invariance

A system is time-invariant if a time shift of $D$ samples in the input results in the same time shift in the output, i.e., $x(n-D)$ produces $y(n-D)$ .

A series connection of linear time-invariant (LTI) blocks is itself a linear and time-invariant system, and the order of blocks can be changedwithout affecting the input-output behavior.

LTI systems can be thoroughly described by the response they give to a unit-magnitude impulse.

The impulse in discrete time (space)

is the signal $\delta$ with value $1$ at the instant zero (in the point with coordinates $$

0 0

), and $0$ in any other instant (point).

Impulse response and convolution

We call $h$ the output signal of a LTI system whose input is just animpulse. Such output signal is called impulse response . Since any discrete-time (-space) signal can be thought of as a weighted sum of translated impulses, each samplethat shows up to the input activates an impulse response whose amplitude is determined by the value ofthe sample itself. Moreover, since the impulse responses are activated at a distance of one sampling step from each other andare extended over several samples, the effect of each input sample is distributed over time, on a number of contiguoussamples of the output signal. Being the system linear and time-invariant, the successive impulse responses sum theireffects. In other words, the system has memory of the past samples, previously given as input to the system, and it usessuch memory to influence the present.

To have a physical analogy, we can think of regular strokes of a snare drum. The response to each stroke is distributed intime and overlaps with the responses to the following strokes.