Understanding the Zero-State Response
Understanding the Zero-State Response Swapnil Sunil Jain 15 January, 2007
Now, let a LTI system with input and output be described by the following differential equation
and the initial conditions: .
To find the zero-state response, we first set all the initial conditions to zero i.e.
Then we note that x(t) can be written in the following way (due to the shifting property of ):
Now, we define a new function h(t) (known as the impulse response of the LTI system ) the following way11Simply put, the impulse response of a system is the output we get when we drive the input by the impulse function
Then, due to the time-invariance property of the LTI system , it is true that, for some constant
and due to the linearity property of the LTI system , it is also follows that, for some constant ,
Then, it follows from (2), (4) and (5) that22This result makes sense if we think of the integral as an infinite sum and treat x(u) as some constant with respect to time t.
Now, we define the output in (6) as the zero-state response of x(t) i.e.
and we also define a new binary operation ’*’, called the convolution, as
Using this notation,
Thus, in order to find for a given input , we need to find . In order to do this, we first define a function that satisfies the following equality:
Then the following also holds:
Adding all the above equations together we get
or equivalently,33One could have also easily come up with the following result by simply operating b(D) on both sides of (8)
Since the above equation has the same form as (1) with , it follows that (since h(t), by definition, is the output when the input is the impulse function). Hence,
where is given by
Thus, in order to find , we need to find first. We will do this with the help of an example. Given,
with initial conditions . In order to get on the R.H.S, must be plus some ordinary function i.e.
For , we have
since for . Furthermore, we now get new initial conditions at . Thus,
In general, for , the initial condition involving the n-1 derivative would be equal to 1 and all the other initial conditions would be 0. Thus, if
then for ,
In summary, we found out that the zero-state response is given by
where is the impulse response of the system and is given by
where is the solution of the differential equation
with initial conditions .
|Title||Understanding the Zero-State Response|
|Date of creation||2013-03-11 19:27:00|
|Last modified on||2013-03-11 19:27:00|
|Last modified by||(0)|