# Introduction to Signals and Systems

## Types of Signals

### Properties of the Unit Impulse

$f[n]\delta[n] = f[0]\delta[n]$

$f[t]\delta[n-N] = f[N]\delta[n-N]$

$f(t)\delta(t) = f(0)\delta(t)$

$f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)$

$\delta(at) = \frac{1}{|a|}\delta(t)$

## Signal transformations

Signals can be transformed by modifying the variable.

$x(t - \tau)$: Shift a signal right by $\tau$ steps.

$x(-t)$: Rotate a signal about the $t=0$

$x(kt)$: Stretch a signal by a factor of $k$

These operations can be combined to give more complex transformations. For example, $y(t) = x(\tau - t) = x(-(t-\tau))$ flips $x$ and shifts it right by $\tau$ timesteps. This is equivalent to shifting $x$ left by $\tau$ timesteps and then flipping it.

## Convolution

While written in discrete time, these properties apply in continuous time as well.

$(x*\delta)[n] = x[n]$

$x[n]*\delta[n-N]=x[n-N]$

$(x*h)[n] = (h*x)[n]$

$x * (h_1 + h_2) = x*h_1 + x*h_2$

$x * (h_1 * h_2) = (x * h_1) * h_2$

## Systems and their properties

Notice: The above conditions on linearity require that $x(0) = 0$ because if $a = 0$, then we need $y(0) = 0$ for scaling to be satisfied

## Exponential Signals

Exponential signals are important because they can succinctly represent complicated signals using complex numbers. This makes analyzing them much easier.

$x(t) = e^{st}, x[n] = z^n (s, z \in \mathbb{C})$

Last updated