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Textbook by: Carlos E. Davila. E-mail the author

# The Unit Impulse Function

Module by: Carlos E. Davila. E-mail the author

## The Unit Impulse Function

The unit impulse is very useful in the analysis of signals, linear systems, and sampling. Consider the plot of a rectangular pulse in Figure 1. Note the height of the pulse is 1/τ1/τ and the width of the pulse is ττ. So we can write

- x p ( t ) d t = 1 - x p ( t ) d t = 1
(1)

As we let ττ get small, then the width of the pulse gets successively narrower and its height gets progressively higher. In the limit as ττ approaches zero, we have a pulse which has infinite height, and zero width, yet its area is still one. We define the unit impulse function as

δ ( t ) lim τ 0 x p ( t ) δ ( t ) lim τ 0 x p ( t )
(2)

The area under δ(t)δ(t) is one, and so we can write

- δ ( t - τ ) d t = 1 - δ ( t - τ ) d t = 1
(3)

If we multiply the unit impulse by a constant, KK, its area is now equal to that constant, i.e.

- K δ ( t - τ ) d t = K - K δ ( t - τ ) d t = K
(4)

The area of the unit impulse is usually indicated by the number shown next to the arrow as seen in Figure 2.

Suppose we multiply the signal x(t)x(t) with a time-shifted unit impulse, δ(t-τ)δ(t-τ). The product is a unit impulse, having an area of x(τ)x(τ). This is illustrated in Figure 3.

In other words,

- x ( t ) δ ( t - τ ) d t = x ( τ ) - x ( t ) δ ( t - τ ) d t = x ( τ )
(5)

Equation Equation 5 is called the sifting property of the unit impulse. As we will see, the sifting property of the unit impulse will be very useful.

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