Signal

A signal is a way of carrying information from one place to another.

The Domain of a signal is almost always time. The Codomain is usually some kind of measure of amplitude, whether it be decibels, voltage, etc. An exception to basic the rule of thumb are inputs involving both space and time, such as sensor data from a satellite or a waymo or something, where the domain is spatiotemporal rather than just temporal, consisting of coordinates and time, and the codomain is some kind of multidimensional representation of color, usually RGBA, rather than a single scalar. Anything involving CUDA, is really signal processing. Anything involving tacit programming, is signal processsing.

Levels of Discretization

Analog Signal

Analog Signal

An analog signal is a Continuous wave. It is typically denoted as $x(t)$, where $x$ is the function of the wave and $t$ is time.

An example of an analog signal is sound; imagine a guitar string vibrating at an 440 Hz A. Not only is the string a continuous wave, but the compression waves it imparts onto the air, the sound, are also continuous waves.

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Discrete-Time Signal

Discrete-Time Signal

A discrete-time signal is the result of Sampling a subset of the domain of an Analog Signal, creating a new signal where the Domain is discrete, while the Codomain remains Continuous. It is typically denoted as $x[n]$, where $x$ is the function of the signal, and $n$ is the index.

An example of a discrete-time signal is within the process of digitally recording a guitar. While a microphone can hear any pitch, it can not convey every instant of a sound to a computer. Therefore, a broker responsible for converting the analog signal of the microphone into something the computer can understand is required. A Analog-to-Digital Converter decides on a certain sampling rate at which to note what the microphone hears at that moment so that it can eventually send a finite amount of information to the computer. At a certain stage within the converter, the signal is represented with discrete time, but the codomain is still continuous.

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Digital Signal

Digital Signal

A digital signal is the result of Quantizing the values of the codomain of a Discrete-Time Signal, creating a new signal where both the Domain and Codomain are discrete. It is typically can be denoted as $x[n]$, where $x$ is the function of the signal, and $n$ is the index. It can also be denoted as $\hat{x}[n]$ or $x_q[n]$ to distinguish it from its discrete-time counterpart, and to show that it has been quantized.

An example of a digital signal would be the final steps of the Analog-to-Digital Converter before it sends the final digital signal to the computer; it has to represent that signal in binary. Any means of representing all continuous values with finite digits of precision will lead to loss, so regardless of if you choose IEEE 754 Doubles, or some kind of integer, you will be performing some level Quantization to the signal. The result is a signal with both discrete time and amplitude.

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Signal Transformations

Signals can be scaled, shifted, and flipped. They are subject to all of the typical Linear Transformations that maintain the validity of the Function. That is, not rotations, as they can break the vertical line test.

Signal Parity

Any signal can be uniquely decomposed into the sum of an Even Signal and an Odd Signal, by the Even-Odd Decomposition Property.

Even Signal

An even signal is a signal whose underlying function is an Even Function

Odd Signal

An odd signal is a signal whose underlying function is an Odd Function

Special Signals

Unit Sample

Unit Sample

The Kronecker Delta Function as a Discrete-Time Signal, denoted $\delta [n]$. Serves as a primitive building-block signal from which more complex discrete-time signals can be constructed.

$\delta [n]=\{\begin{array}{ll}1,& n=0\\ 0,& \text{otherwise}\end{array}$

Impulse Representation

Impulse

In Discrete-Time land, an Impulse $\delta [n]$ is a Signal that is the Unit Sample

$$\begin{array}{r}\delta [n]=\{\begin{array}{ll}1,& n=0\\ 0,& n\ne 0\end{array}\\ \\ \delta [n]=[1,\vec{0}]\end{array}$$

Note that for Convolutions, this unit impulse is like the Identity

$$x[n]\ast \delta [n]=x[n]$$

Similar to the Step Function, you can scale or shift the delta function. Furthermore, you can compose these scaled and shifted delta functions to create ANY given signal.

For example, $x[n]$ is a signal that goes: 1, 2, 3.

$$x[n]=1\cdot \delta [n]+2\cdot \delta [n-1]+3\cdot \delta [n-2]$$Link to original

Sifting Property

Sifting Property

You can also use the delta function to “sift” out the output of a signal at a given $n_0$

$$x[n_0]=\sum _{n=-\infty }^{\infty }x[n]\delta [n-n_0]$$

This can be used for either constructing or deconstructing a signal.

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Relationship to Unit Step

$\delta [n]=u[n]-u[n-1]$

Categorical Structure

From Category Theory point of view, $\delta [n]$ is the Identity Morphism for Convolution of a signal: $x[n]\ast \delta [n]=x[n]$ for any signal $x[n]$.

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Unit Step

Unit Step

The Heaviside Step Function as a Discrete-Time Signal, denoted $u[n]$.

$u[n]=\{\begin{array}{ll}1,& n\ge 0\\ 0,& n<0\end{array}$

Relationship to Unit Sample

$u[n]={\sum }_{k=-\infty }^n\delta [k]$

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Periodicity

A signal is periodic if it repeats its pattern exactly over and over again at regular, predictable intervals.

A discrete signal $x[n]$ is periodic iff $\exists N\in {\mathbb{Z}}^{+},s.t.x[n+N]=x[n],\forall n\in \mathbb{Z}$

This is important because a function in continuous-time can be periodic, but if that period is not an integer, then it is not neccessarily periodic in discrete-time! Alternatively, if a function’s period is an integer, it will be period in both continuous, and discrete-time (assuming step = 1).