E12
Impulse Function and Response

The impulse can be represented as the derivative of a step

The figure in the upper left shows a ramp function.  As you go to the right T→0, and the function becomes a step. The figure on the left is the derivative of the ramp function - note that it has an area=1.  As T→0, the area remains one, but the width→0 and the height→∞.

Properties of the impulse

You should be able to explain all of these using the fact that the unit impulse is zero everywhere but when its argument is zero, and it has an area of one:

We will often use the "sifting" property of the impulse function (you should be able to explain all three forms).

Note:

The impulse response is the derivative of the unit step response.

If the input to a circuit (or any linear system) is a unit step, γ(t), the output is the unit step response, yγ(t).

If we delay the step, we simply delay the response:

If we scale the step (multiply by a constant), we simply scale the response

By linearity, if we apply the sum of two inputs, the output is simply the sum of the individual outputs:

Now if we take ΔT→0, the input is an impulse (the derivative of a step function), so the output is the impulse response (the derivative of the unit step response).

 

Calculating the impulse response.

We will calculate the impulse response as the derivative of the step response.  Two examples follow, along with a caveat.

Example 1:  The circuit shown

has a step response,

 

In this function γ(t) is the unit step - multiplying by γ(t) makes the step response equal to zero when time is less than zero.  To find the impulse response, simply differentiate (using chain rule)

 

Consider the second term which is of the form

Since the impulse function is zero everywhere except at t=0, we can rewrite it as

So the impulse response becomes

 

The second term is somewhat puzzling.  We take an impulse function (which has infinite height) and multiply it by zero.  What is the result?  It turns out that it doesn't matter.  The second term represents an impulse with an area of zero (i.e., it could have some finite height, but still has zero width).  An impulse of zero area has no affect on the system so

 

The step response and impulse response are shown below for a time constant of 0.5.

 

Example 2:  The circuit shown

has a step response,

 

So the impulse response is

 

Note that this time there is an impulse function as part of the impulse response. 

The step response and the impulse response are shown below.  Note that because there is a discontinuity in the step response, there is an impulse in the impulse response (the step response of the first example had no discontinuity).

Caveat:  Though it often is unimportant, the step function, γ(t), in

can be very important when calculating the impulse response.  It is tempting to make γ(t) implicit and write the step response of example 1 as

 

We then try to find the impulse response by differentiating:

 

In this case (and any case for which the step response doesn't include a discontinuity) we get the correct answer (for t>0).

However, for example 2, if we express the step response as

 

and try to find the impulse response by differentiating

 

then we get the wrong answer.

So... when calculating the impulse response as the derivative of the step response, remember to account for the fact that the step response is always multiplied by γ(t) (this is only important if there is a discontinuity in the step response - seen in example 2, but not example 1).