So, what is this sifting property Mimi keeps mentioning?

In case you were wondering, and you probably were not, the sifting property Mimi referenced in class was mentioned (only briefly in my class) in ECE 202.
Formally, it is a property obeyed by the Dirac delta function as such:
$ \int_{-\infty}^{\infty}f(\tau)\delta(t-\tau)d\tau=f(t) $

Student 1

Ok, that's all well and good, but why?

Fifth year senior

What this does is it allows us to pick, or 'sift' out, hence the name, a particular value of the function in the integral at an exact instant in time.

Student 1

Doesn't the function do that by itself outside of the integral anyways?

Fifth year senior

Good, question, I'm glad I paid you to ask it. One of its uses is in helping develop and understand the idea of convolution.

Student 1:

...

Fifth year senior

Ok, so we know that a LTI system can be completely described by it's unit impulse response, but why is that? The sifting property shows us mathematically that any function, say a C.T. or D.T. signal, can be described as the sum of an infinite (or finite for most D.T. cases) number of shifted and scaled impulses. Knowing this, is makes sense that if we know what the system will do to a single impulse, and we know that the signal can be broken up into single impulses, we can then apply the 'effect' of the system to each individual impulse of the signal, sum them, and find the resulting output.

Student 1

So all we need to know to find the output of a LTI system is its input and its response to an impulse function?

Fifth year senior

Sounds simple, doesn't it. I hope you like algebra...

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva