# Mathematics Meets Signal Processing: Exploring the Convolution Integral

**Table of Contents**

### Introduction

Since signals are sets of data or information and systems process said data, we are interested in the analysis of systems. When we deal with a special type of system that contains the properties of **linearity** and **time-invariance**, we are able to construct methods of analysis that are extremely useful for Linear Time-invariant (LTI) systems. Fourier analysis, which will be a seperate blog post, and the convolution integral are examples of exploiting system properties to decompose inputs into basic signals which are easy to work with analytically. Let’s have a little refresher first with these two properties.

## Linearity and Time Invariance

**Time-invariance** is the property of a system that when an input is shifted in time, then it’s subsequent output is shifted by the same amount of time.

**Linearity** implies set of independent outputs can be superimposed into one output.

## Convolution

Before we begin convolution, we must represent any arbitrary signal as a summation over a set of infinitely many weighted impulses \( \delta(x) \) . Recall:

Since for all values \(t \neq 0, f(t) = 0\) due to multiplication by \(\delta(t)\) …

Equivilantly can be formulated as:

And with Time-invariance we note…

Therefore we can construct any arbitrary \(f(t)\) as \(f_{n}(t)\) using this idea.

We denote the system response *beginning* at k (unit impulse response) at time to a impulse signal delayed by k as:

Due to our good old friend Time-invariance, we can rewrite the unit impulse system response *beginning* at 0 delayed by k as:

Now if we input this newly constructed arbitrary function into a LTI system we note the response of the linear combination of these inputs \( f_{n}(t)\) is a linear combination of each weighted impulse response \(h_{k}(t)\) .

or

Now with a little big of calculus and replacing our notation and variable k with \(\tau\) .

Intuitively, this integral could be thought of as some infinitesimally thin sample of f(t) we denote as \(d\tau\) input into our system where we obtain a infinitesimely thin system response \(h(t - d\tau)\) repeated over infinity to produce a summation of infinitely thin responses to the system. I highly recommend this video to give a good visual of what is going on.

## Conclusion

Now the real incredible part is now that we are capable of characterizing the entire system simply by the transmittion of some instantaneous excitation into the system!

## References

Lecture 4, Convolution by *Alan V. Oppenheim*. MIT RES.6.007 Signals and Systems, Spring 2011