Impulse Response

[[tab Introduction]]

• What does it mean?
• What is its significance for LTI Systems?

Example of Practical LTI System

The impulse response completely characterizes an LTI System. We can find the output $H(x[n])$ for any $x[n]$ if $h[n]$ is known.

Example

Any input signal $x[n]$ can be written as:(1)

\begin{align} x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k] \end{align}

where $x[k]$ is the value taken by the signal $x[n]$ at $n=k$

(2)

\begin{align} y[n]=H(x[n])=\sum_{k=-\infty}^{\infty}x[k]h[n-k] \end{align}

The above equation is known as the Convolution Sum and is written as:

(3)

\begin{equation} y[n]=x[n]*h[n] \end{equation}

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[[tab Computing Impulse Response]]

Let the output of a DT System $y[n]$ be related to its input $x[n]$ according to:

(4)

\begin{align} y[n]-0.5y[n-1]=2x[n] \nonumber \end{align}

with initial conditions $y[-1]=0$.Compute the impulse response of the system $h[n]$.

**Solution**

Solution: $x[n]$ is the impulse function and so $x[n]=\delta[n]$

(5)

\begin{align} x[0]=1, x[n]=0 \hspace{0.2cm} \text{for} \hspace{0.2cm} n \neq 0 \nonumber \end{align}

(6)

\begin{align} y[0]-0.5y[n-1]=2x[0]=2*1=2 \nonumber \end{align}

(7)

\begin{align} y[1]-0.5y[0]=2x[0]=2*x[1]=0 \nonumber \end{align}

(8)

\begin{align} y[2]-0.5y[1]=2x[2]=0 \nonumber \end{align}

So,

(9)

\begin{align} y[n]-0.5y[n-1]=2x[n]=0 \nonumber \end{align}

(10)

\begin{align} y[n]=0.5y[n-1] \nonumber \end{align}

(11)

\begin{align} y[n]=2\bigg(\frac{1}{2}\bigg)^n n\geq 0 \nonumber \end{align}

(12)

\begin{align} h[n]= \begin{cases} \bigg(\frac{1}{2}\bigg)^n & n\geq 0\\ \nonumber 0 & n<0 \nonumber \end{cases} \end{align}

Remark:

1. Impulse response is a signal.
2. It is the response of the system to an input which is the impulse signal.
3. Find $y[n]$ when $x[n]=\delta[n]$

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