The impulse function, also called the Dirac delta function, is commonly used in statistical signal processing, and on the CSP Blog (examples: representations and transforms). I think we’re a bit casual about this usage, and perhaps none of us understand impulses as well as we might.
Enter C. Candan and The Literature [R155].
Candan explains what a generalized function is, and focuses on the impulse and applications of the impulse that are useful to engineers specializing in statistical signal processing. I encourage you to go get the paper and read the whole thing, but I’m going to entice you further by discussing a few highlights.
Candan starts out by defining the linear functional, which is a function of a function. So a functional takes an entire function as input and returns a number as output if we express the idea in systems-oriented language.
The next step is to define a kind of equality using functionals: generalized equality. Two functions are equal in this generalized sense if their two functionals produce the same result when applied to all test functions in some broad class of interest.
Then Candan starts to focus on a particular set of functionals that will lead to the impulse. For example, functionals defined by a rectangle.
Then the idea is to sneak up on the delta function by looking at a sequence of functions (the and that define the functional, not the test functions ). For example, consider the sequence of unit-area rectangles shown in Figure 3 (Candan’s Figure 1). (See also SPTK Signal Representations Figure 3 and Eq. (23).) The limit of the sequence of rectangles doesn’t exist in the ordinary sense, since the limit function at is unbounded. But Candan uses that sequence to define the delta function in terms of generalized equality. A similar sequence of sinc() functions, illustrated in Figure 2, also leads to the delta function.
This development leads to the various properties of the impulse function that we use in signal processing, but expressed with a notation that reminds the reader about the connection to generalized equality. Candan’s table of these familiar properties is reproduced in Figure 5.
The payoff of this unusually careful treatment of the impulse function comes when Candan looks at some familiar examples from probability theory and communication-signal modeling. In Candan’s second example, he wants to find the probability density function for a random variable that is the square of a second random variable. The sequence of equations for this example is shown in Figure 6. If you’ve ever read a probability-theory book with this example (The Literature [R149]), or tried it yourself, you’ll realize this is an easy way to solve the problem compared to conventional solutions.
The (a) equality follows from elementary probability theory–you integrate the joint density over one variable to obtain the density for the other. The (b) equality follows from the definition of conditional probability densities, and the (c) equality arises because if the random variable is known to take the value , then the value of the random variable is determined, so that the conditional density is an impulse centered at . Continuing on to (d), we use the fact that the impulse is an even function. To obtain equality (e), we use the Advanced Scaling property in Figure 5 together with . Finally, (f) follows from the Sifting property.
Example 4, in Figure 7, shows how to use the generalized equality and impulse properties to show that the Fourier transform of a constant is equal to (in the generalized sense) an impulse function.
I use impulses in a variety of Signal Processing ToolKit posts, including Representations, Linear Time-Invariant Systems, Ideal Filters, and Convolution. They also come up naturally in the study of the structure of spectral moments and cumulants (the cyclic polyspectra).