# The Delta Function

## Abstract

The delta function plays a central role in circuits and signals. It is very attractive to use for many reasons, including simplicity, flexibility, and constant spectrum. The delta function is a very localized pulse of very narrow pulse width (ideally zero) and very large pulse height (ideally infinite). While pulse width and height can be moved around, to some extent, they must be tied via the relation of unity pulse area! Typically we don’t need to play with zero and infinite pulse widths and heights; we can approximate a delta function by using a relatively narrow and high pulse, depending on the exact circuit at hand, but again subject to the unit area requirement. The delta function—ideally—also has a constant spectrum; that is the magnitude of its Fourier transform in frequency is 1 (and the phase zero). This means that any multiplication in the frequency domain yields the function being multiplied with; and this will be used to advantage when dealing with transfer functions of circuit. In this chapter we derive the pulse from various functions, such as from the unit step function, the negative exponential, and the sinc function. We illustrate the derivation process visually. Next we move to the sampling property of the delta function and wrap the chapter with the sampling property of the derivative of the delta function. The importance of the delta (impulse) function lies in system response which will be discussed in next chapter.