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