# Introduction to Signals and Systems

## Types of Signals

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#### Definition 1

A signal is a function of one or more variables
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#### Definition 2

A continuous signal $$x(t)$$ maps $$\mathbb{R} \rightarrow \mathbb{R}$$
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#### Definition 3

A discrete signal $$x\[n]$$ maps $$\mathbb{Z} \rightarrow \mathbb{R}$$
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### Properties of the Unit Impulse

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#### Definition 4

The unit impulse in discrete time is defined as

$$\delta\[n] = \begin{cases} 1 & \text{if } n = 0\ 0 & \text{else } \end{cases}$$
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* $$f\[n]\delta\[n] = f\[0]\delta\[n]$$
* $$f\[t]\delta\[n-N] = f\[N]\delta\[n-N]$$

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#### Definition 5

The unit impulse in continuous time is the dirac delta function

$$\delta(t)=lim\_{\Delta\rightarrow 0}\delta\_{\Delta}(t) \qquad \delta\_{\Delta}(t)=\begin{cases} \frac{1}{\Delta}, & \text{if } 0 \leq t < \Delta \ 0 & \text{else.} \end{cases}$$
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* $$f(t)\delta(t) = f(0)\delta(t)$$
* $$f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)$$
* $$\delta(at) = \frac{1}{|a|}\delta(t)$$

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#### Definition 6

The unit step is defined as

$$u\[n] = \begin{cases} 1 & \text{if } n \geq 0\ 0 & \text{else.} \end{cases}$$
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## 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

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#### Definition 7

The convolution of two signals in discrete time

$$(x\*h)\[n] = \sum\_{k=-\infty}^{\infty}{x\[k]h\[n-k]}$$
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#### Definition 8

The convolution of two signals in continuous time

$$(x\*h)(t) = \int\_{-\infty}^{\infty}{x(\tau)h(t-\tau)d\tau}$$
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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

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#### Definition 9

A system is a process by which input signals are transformed to output signals
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#### Definition 10

A memoryless system has output which is only determined by the input's present value
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#### Definition 11

A causal system has output which only depends on input at present or past times
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#### Definition 12

A stable system produces bounded output when given a bounded input. By extension, this means an unstable system is when $$\exists$$a bounded input that makes the output unbounded.
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#### Definition 13

A system is time-invariant if the original input $$x(t)$$ is transformed to $$y(t)$$, then $$x(t-\tau)$$ is transformed to $$y(t-\tau)$$
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#### Definition 14

A system $$f(x)$$ is linear if and only if for the signals $$y\_1(t) = f(x\_1(t)), y\_2(t) = f(x\_2(t))$$, then scaling ($$f(a x\_1(t)) = a y(t)$$ and superposition ($$f(x\_1(t) + x\_2(t)) = y\_1(t) + y\_2(t)$$) hold.
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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

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#### Definition 15

The impulse response of a system $$f\[x]$$ is $$h\[n] = f\[\delta\[n]]$$, which is how it response to an impulse input.
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#### Definition 16

A system has a Finite Impulse Response (FIR) if $$h\[n]$$decays to zero in a finite amount of time
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#### Definition 17

A system has an Infinite Impulse Response (IIR) if $$h\[n]$$does not decay to zero in a finite amount of time
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## 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})$$

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#### Definition 18

The frequency response of a system is how a system responds to a purely oscillatory signal
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