Signals and Systems

Signals convey information. Systems transform signals.

Understanding of these will be developed by exploring their structure (syntax), and their interpretation (semantics). The relationship between the input and output is a declarative description of the system, while the process for transforming signals into outputs is an imperative description of the system. We'll model signals and systems mathematically, as functions. A signal maps a domain, such as time, or space, or light intensity, onto a range. A system works similarly, mapping signals from its domain onto output signals. The domain and range are both sets of signals, or signal spaces.

Signals.

Signals often carry information, as temporal or spatial patterns, which can be embodied in many different types of media.

Audio signals.

Sound can be represented as a function, $Sound:Time\; -> Pressure$, where $Pressure$ is the set of all possible air pressures, and $Time$ is a set expressing the duration of the signal.

One second of a voice signal, can be described $Voice:[0,1]\; -> Pressure$, and is often visualized as a waveform, which can be plotted as a function. Because computers cannot handle the full range of numbers implied by $[0,1]$, or the seemingly continuous nature of a waveform, what happens instead is that, roughly 8000 numbers (perhaps members of $Bin*$), per second are sampled into what is known as a discrete time signal, as they are defined only at discrete points in time.

A discrete, one second computer signal, can be thought of as a function that maps discrete time to 16bit integers: $ComputerVoice:DiscreteTime -> Ints16$.

• Their continuous counterparts work as you'd expect.

Hardware handles the task of turning the $ComputerVoice$ function into a $Sound$ function:

$HW: ((CompVoice:DiscTime -> Ints16) -> (Voice:[0,1] -> P))$.

The sound of an ideal 440Hz tone over an infinite $Real$ valued time-interval, can be modeled as $PureTone: Reals -> Reals$, where the function for converting time to pressure is $\forall t \in\; Reals,\; PureTone(t) = Psin(2 \pi \times 440t)$.

Images.

Grayscale images are represented by the function:

$Image: [0, 11] -> [0, B_{max}]$.

More generally:

$Image: VerticalSpace \times HorizontalSpace -> Intensity$, where $Intensity$ $= [black, white]$, is the intensity range measured in some scale.

Color pictures, are sometimes measured in terms of RGB values, and so a color picture is represented by the function:

$ColorImage: VerticalSpace \times HorizontalSpace -> Intensity$, with the values assigned by the function at any point $(x,y)$ is given by the triple $(r, g, b) \in Intensity$: $(r, g,b ) = ColorImage(x, y)$. Depending on the image, the spatial domain may be different, and different ranges of intensity, and the way color is assigned to points in the domain. For instance, a computer represents color images using colormap tables:

$Display: ColorMapIndexes -> Intensity$: $(r, g, b) = Display(x, y)$.

Since computers are finite, storing images requires discretizing domains and ranges, so that your computer may operate like so:

$ComputerImage: DiscreteVSpace \times DiscreteHSpace -> Ints8$, where: $DiscreteVSpace = \{1, 2, \ldots, 300\}$. $DiscreteHSpace = \{1, 2, \ldots, 300\}$. $Ints8 = \{0, 1, \ldots, 255\}$.

$ComputerImage$ can be said to store $200 \times 300$ pixels, with a pixel being a picture element, the value of which is $ComputerImage(row, column) \in Ints8$, where $row \in DiscreteVerticalSpace,\;and \;column\; \in DiscreteHorizontalSpace$.

A way that computers store images, represented by a function, is:

Video Signals

A video, is a sequence of images, with a display rate, typically frames per second, is a signal, the domain of which is discrete time, and can be represented as $FrameTimes = \{0, 1/30, 2/30, \ldots \}$, and the range of which is an $ImageSet$.

$Video: FrameTimes -> ImageSet$.

For analog videos:

$VideoFrame: DiscreteVSpace \times DiscreteHSpace -> Intensity$.

For any time $t \in FrameTimes$, the image $Video(t) \in ImageSet$, or, alternatively, we can say:

• $(r,g,b) = AltVideo(t,x,y)$

If these videos represent the same video, then $\forall t \in FrameTimes$, and also $\forall (x, y) \in DiscreteVSpace \times DiscreteHSpace$:

$(Video(t))(x,y) = AltVideo(t,x,y)$.

Signals Representing Physical Attributes

Changes in the attributes of physical objects are represented as functions of time and or space:

The position of an airplane: $Position: Time -> Reals^{3}$, where $\forall t \in Time$, where $PositionVelocity(t) = (x(t), y(t), z(t),v_x(t), v_y(t), v_z(t))$ gives the position and velocity at $t \in Time$.

On the other hand, a pendulum is represented by:

$\theta : Time \rightarrow [-\pi, \pi]$, where $\theta (t)$ is the angle at time $t$.

Or perhaps, a robot:

$(\theta_u, \theta_l) : Time \rightarrow [-\pi, \pi]^2$, where $\theta_u (t), \theta_l (t)$ are the angles made with the upper and lower arm at time $t$.

Sequences

Temporal or spatial information can be represented by functions of time or space variables, and as well, can be represented by sequences of symbols, occurring as a representation of data, or an event stream.

Binary Files

An N-bit file $b_1, b_2, \ldots, B_n$, where each $b_i \in Bin = \{0,1\}$, can be regarded as a funciton $File: \{1,2,\ldots, N\} -> Bin$, where the assignment $File(n) = b_n$ for every $n \in \{1, \ldots, N\; \}$. We can also take the range to be $EnglishWords$, with an N-word long English text being a function

$EnglishText: \{1,2,\ldots, N\} -> EnglishWords$.

Typically, data sequences are functions of the form $Data: Indices -> Symbols$, where $Indices \subset Nats$. An advantage of this is Data can be a discrete-time signal, however $Indices$ do not represent uniformly spaced instances of time. However, we can say that if $m,n \in Indices$, with $m < n$, then the $m^{th}$ symbol $Data(m)$ occurs before $Data(n)$, but just not the amount of time elapsed between them.

The other representation, event streams or traces, are formed from a log of significant events, and are functions of the form $EventStream: Indices -> EventSet$.

Discrete Signals And Sampling

Continuous time signals are considered so, because $Time$ is a continous domain of the form $[\alpha, \beta] \subset Reals$, and as well, $Image$ is a continuous 2-dimensional shape of the form $[a,b] \times [c,d] \subset Reals^2$