# The Convolution Integral

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.

$\text{If:} \ \ \ \ \ \ f(t) \implies y(t) \\ \text{then:} \ f(t - t_{0}) \implies y(t - t_{0})$

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

$\text{If:} \ \ \ c_{1}f_{1}(t) \implies c_{1}y_{1}(t) \ \text{and} \ c_{2}f_{2}(t) \implies c_{2}y_{2}(t) \\ \text{then:} \ \ \ \ \ \ \ \ \ \ c_{1}f_{1}(t) + c_{2}f_{2}(t) \implies c_{1}y_{1}(t) + c_{2}y_{2}(t)$

## 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:

$\delta(t) = \begin{cases} \infty & \text{if } t = 0,\\ 0 & \text{if } t \neq 0\\ \end{cases}$

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

$f(0) = f(t) \cdot \delta(t)$

Equivilantly can be formulated as:

$f(0) = f(0) \cdot \delta(t)$

And with Time-invariance we note…

$f(1) = f(1) \cdot \delta(t - 1)$

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

$f_{n}(t) = \sum_{k = -\infty}^{\infty} f(k) \cdot \delta(t - k)$

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

$h_{k}(t) = \delta(t - k)$

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

$h_{k}(t) = h_{0}(t - k)$

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 $h_{k}(t)$.

$y(t) = \sum_{k = -\infty}^{\infty} f_{n}(k) \cdot h_{k}(t)$

or

$y(t) = \sum_{k = -\infty}^{\infty} f_{n}(k) \cdot h_{0}(t - k)$

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

$y(t) = f(t) * h(t) = \int_{-\infty}^{\infty} f(\tau) \cdot h(t - \tau)d\tau$

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