# Introduction to Signals and Systems

- $f[n]\delta[n] = f[0]\delta[n]$
- $f[t]\delta[n-N] = f[N]\delta[n-N]$

- $f(t)\delta(t) = f(0)\delta(t)$
- $f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)$
- $\delta(at) = \frac{1}{|a|}\delta(t)$

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

- $(x*\delta)[n] = x[n]$
- $x[n]*\delta[n-N]=x[n-N]$
- $(x*h)[n] = (h*x)[n]$
- $x * (h_1 + h_2) = x*h_1 + x*h_2$
- $x * (h_1 * h_2) = (x * h_1) * h_2$

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 satisfiedExponential 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 modified 6mo ago