Impulse Response

[[tab Introduction]]

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

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.
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 $n=k$

\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:

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


[[tab Computing Impulse Response]]

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

\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 $h[n]$.



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