The properties of LTI systems are important in two aspects:
\begin{enumerate}
\item time-domain behavior
\item frequency-domain behavior
\end{enumerate}

Lets first look at the time domain. Consider a signal $x: \Z \to \R$.

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How do we split the signal into its components? We can write the signal as a sum of scaled and shifted versions of the most fundamental signal, the Kronecker delta. Therefore, we can write the signal as $x(n) = \sum_k x(k)\delta(n-k)$. The \emph{Kronecker Delta} function, also known as \emph{Discrete-Time Impulse Signal} (aka Unit Impulse), is defined by

$$\delta(n) = \begin{cases}
1 & n=0 \\
0 & \mbox{otherwise} \\
\end{cases}
$$

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In the case of the signal below, we have $x(n) = x(-1)\delta(n+1) + x(0)\delta(n) + x(1)\delta(n-1) + x(2)\delta(n-2)$.

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If $\delta(n) \to \fbox{F} \to f(n)$, then $f(n)$ is the \emph{impulse response} of the LTI system $F$. This is important because for any LTI system in discrete-time, if the response to the unit impulse ($\delta(n)$) is known, then the output can be determined.

\begin{align*}
x(n)& = x(-1)\delta(n+1) + x(0)\delta(n) + x(1)\delta(n-1) + x(2)\delta(n-2) \\
y(n)& = x(-1)f(n+1) + x(0)f(n) + x(1)f(n-1) + x(2)f(n-2) 
\end{align*}

More generally, any input signal can be represented as a linear combination of scaled and shifted impulses.
\begin{align*}
x(n) &= \sum \limits_{m\in\Z} x(m)\delta(n-m) \\
y(n) &= \sum \limits_{m\in\Z} x(m)f(n-m) \\
\end{align*}

%%% CONVOLUTION SUM

The output signal $y$ can be written as a sum, also known as the convolution sum:
$$y(n) = \sum \limits_{m\in\Z} x(m)f(n-m) = (x * f)(n)$$.

Convolution has the property of commutativity: Let $m = n-k$:   % k = n-m
$$y(n) = \sum \limits_{k\in\Z} f(m)x(n-k) = (f*x)(n)$$.

Hence, $(x*f)(n) = (f*x)(n)$:

\begin{align*}
x \to & \fbox{f} \to y = x * f \\
f \to & \fbox{x} \to y = f * x
\end{align*}

Any function can be thought of as a signal, and any signal may be thought of as a response---but the input signal and the impulse response must share the same domain.


{\bf Continuous-time}


We also have a continuous-time impulse, the Dirac Delta, which allows us to characterize continuous-time signals. This function can be characterized by integration: $ \int_{-\infty}^{\infty} \delta(\tau)d\tau = 1 $

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Hence, the input signal can be written as scaled and shifted versions of the Dirac delta:

$$ x(t) = \infint x(\tau)\delta(t-\tau) $$

Which leads us to the convolution integral. The corresponding convolution formula in continuous-time is given by

$$ y(t) = \infint x(\tau) f(t-\tau) d\tau $$


Time-Domain Behavior