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Impulse Response of a System

A discrete signal sn can be written as a linear combination of delayed impulses.

sn=k=0k=N1skδnk

sn=s0δn+s1δn1++sN1δn(N1)

If the system is linear we can compute the response of the system to the signal sn if we know the impulse response of the system to each impulse in the signal.

For a time invariant system, this is easy to do because the characteristics of its impulse response does not change with time: The system as only one impulse response.

For a time variant system, however, this is not so easy.

Time Variant System

A time variant system's impulse response changes with time.

       δnkhkn  or  T[δnk]=hkn

That is, the system produces the output hkn in response to the impulse input δnk occurring at time k. The characteristics of the system's impulse response changes as a function of time: The system can potentially have many impulse responses (at least two.)

Time Invariant System

A time invariant system's impulse response does not change with time.

       δnkhnk  or  T[δnk]=hnk

Namely, the system produces the output hnk in response to the impulse input δnk occurring at time k. The characteristics of the system's impulse response does not change as a function of time, but the response is shifted in time by k units of time.

Response of a Linear System

       x1ny1n  or  T[x1n]=y1n

       x2ny2n  or  T[x2n]=y2n

       x1n+x2ny1n+y2n  or  T[x1n+x2n]=y1n+y2n

The response of the system to the sum of individual inputs is identical to the sum of system's response to each individual input.

We can represent a discrete signal sn as a linear combination of delayed impulses.

sn=k=0k=N1skδnk

We can then compute the response of the system to each individual component of the linear combination and add up all the responses.

T[sn]=k=0k=N1T[skδnk]=k=0k=N1skT[δnk]

The response of the system to each individual component of the linear combination will be the product of the value of the signal sn at time k, sk, and the response of the system to the impulse occurring at time k, T[δnk].

sn=s0δn+s1δn1++sN1δn(N1)

T[sn]=s0T[δn]+s1T[δn1]++sN1T[δn(N1)]

Then the response of a time variant system is given by:

yn=T[sn]=k=0k=N1skhkn

yn=T[sn]=s0h0n+s1h1n1++sN1hN1n

And the response of a time invariant system is given by:

yn=T[sn]=k=0k=N1skhnk

yn=T[sn]=s0hn+s1hn1++sN1hn(N1)

An Example

We compute the response yn of a linear system system to the input signal sn.

Input Signal

Given the signal

       sn={2,3,5,7}

We can write it in terms of delayed impulses as

       sn=s0δn+s1δn1+s2δn2+s3δn3
       sn=2δn+3δn1+5δn2+7δn3

The input signal consists of 4 impulses.

If we know the impulse response of the system for each impulse, we can compute its response yn to the input signal sn.

Time Variant System

Since a time variant system can potentially have more than one impulse response, we write its response yn as:

       yn=s0h0n+s1h1n+s2h2n+s3h3n
       yn=2h0n+3h1n+5h2n+7h3n

Notice the subscripts. They account for the fact that the system can respond differently to each impulse in the input.

Time Invariant System

Since a time invariant system has only one impulse response, we write its response yn as:

       yn=s0hn+s1hn1+s2hn2+s3hn3
       yn=2hn+3hn1+5hn2+7hn3

Notice the time delays. They account for the fact that the system responds identically to each impulse in the input but the response is shifted in time by the same amount as the corresponding impulse is shifted in time.

[This content was created with ML, available from the App Store.]