# The Unit Impulse Function

Contents

## Time Domain Description

One of the more useful functions in the study of linear systems is the "unit impulse function."   An ideal impulse function is a function that is zero everywhere but at the origin, where it is infinitely high.  However, the area of the impulse is finite.  This is, at first hard to visualize but we can do so by using the graphs shown below.

Consider first the ramp function shown in the upper left.  It is zero for t<0 and one for t>T, and goes linearly from 0 to 1 as time goes from 0 to T.  If we let T→0, we get a unit step function, γ(t) (upper right).  If we take the derivative of our ramp function (lower left), we get a rectangular pulse with height 1/T (the slope of the line) and width T.  This rectangular pulse has area (height·width) of one.  If we take the limit as T→0, we get a pulse of infinite height, zero width, but still with an area of one; this is the unit impulse and we represent it by δ(t).  Since we can't show the height of the impulse on our graph, we use the vertical axis to show the area.  The unit impulse has area=1, so that is the shown height.
Note: this derivation of an impulse function is not unique.  The important result is that the function has zero width and an area of one.

## Laplace Domain Description

The relationship between step function and impulse function is even more obvious in the Laplace Domain.  The definitions for both are given below.

 Step Function Impulse Function

Clearly from these definitions we have Δ(s)=s·Γ(s). Since multiplication by "s" in the Laplace Domain is equivalent to differentiation in time this tells us that the unit impulse function is simply the derivative of the unit step function.

##### Key Concept: The Impulse Function

The unit impulse function has zero width, infinite height and an integral (area) of one.  We plot it as an arrow with the height of the arrow showing the area of the impulse.

To show a scaled input on a graph, its area is shown on the vertical axis.  In the diagram below the area of the impulse function is "A."  For a scaled impulse (i.e., A·δ(t)), the multiplier in front of the impulse is the area.

##### Aside: Other definitions of impulse

Note, there are other, equally valid, definitions of an impulse.  The only important result is that the function has width approaching zero, height approaching infinity and an area of one.  For example, consider a Gaussian curve.

It is well known that the area under this graph is always one one (ref).  The graph below shows the function for several values of  σ.

Clearly as σ→0, f(0)→∞, and the width→0, but the area under the curve remains one.  This definition of the impulse is often used in statistics.

## Properties of the Unit Impulse

### The integral of the unit impulse

The integral of the impulse is one.  So if we consider the integral (with b>a)

In other words, if the integral includes the origin (where the impulse lies), the integral is one.  If it doesn't include the origin, the integral is zero.

Likewise, and by similar reasoning, if the impulse is not at the origin (and b>a)

### The sifting property of the impulse

Let us now evaluate the integral of a function multiplied by an impulse at the origin.

We can simplify this integral by noting that because the impulse is zero everywhere except when t=0 we can replace δ(t)·f(t) by δ(t)·f(0).

We can now pull f(0) out of the integral (because it is a constant).  So

More generally, and by the same reasoning, we can write (with b>a)

Likewise because δ(t-T) is zero except at t=T we can show (if b>a)

This is called the "sifting" property because the impulse function δ(t-T) sifts through the function f(t) and pulls out the value f(T).

##### Key Concept: Sifting Property of the Impulse

If b>a, then

Said another way, we replace the value of "t" in the function f(t) by the value of "t" that makes the argument of the impulse equal to 0 (in this case, t=T).

##### Example: Another integral problem

Assume a<b, and evaluate the integral

Solution:
We now that the impulse is zero except at t=0 so

and

Again, we replace the value of "t" in the function that makes the argument of the impulse equal to 0 (in this case, t=0).

### Convolution with an impulse

Another important property of the impulse is that convolution of a function with a shifted impulse yields a shifted version of that function.

We prove this by using the definition of convolution (first line, below). By the sifting property, we replace "λ" in the function by the value of "λ" that makes the argument of the impulse equal to zero (in this case λ=T) and we get the desired result (the third line).

##### Key Concept: Convolution with a shifted impulse

Convolution of a function with a shifted impulse yields a shifted version of that function.

##### Aside:An alternate proof

We can also do the proof using the alternate form of the impulse (recall that the convolution is commutative).

References