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Cheat Sheet

Signals & Systems For DummiesFrom Signals and Systems For Dummies by Mark Wickert

Signals and systems is an aspect of electrical engineering that applies mathematical concepts to the creation of 

product design, such as cell phones and automobile cruise control systems. Absorbing the core concepts of 

signals and systems requires a firm grasp on their properties and classifications; a solid knowledge of algebra,

trigonometry, complex arithmetic, calculus of one variable; and familiarity with linear constant coefficient (LCC)

differential equations.

Getting to Know the Mathematical Foundation of Signals and Systems

The study of signals and systems establishes a mathematical formalism for analyzing, modeling, and simulating electrical systems in the time, frequency,

and s- or z-domains. Signals exist naturally and are also created by people.

Some operate continuously (known as continuous-time signals); others are

active at specific instants of time (and are called discrete-time signals).

Signals pass through systems to be modified or enhanced in some way.

Systems that operate on signals are also categorized as continuous- or 

discrete-time.

Mathematics plays a central role in all facets of signals and systems.

Specifically, complex arithmetic, trigonometry, and geometry are mainstays of 

this dynamic and (ahem) electrifying field of work and study. This article

highlights the most applicable concepts from each of these areas of math for 

signals and systems work.

Complex arithmetic for signals and systems

Here are some of the most important complex arithmetic operations and

formulas that relate to signals and systems.

Trigonometry and Euler's formulas

This table presents the key formulas of trigonometry that apply to signals and

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systems:

Geometric series

 Among the most important geometry equations to know for signals and

systems are these three:

Recognizing Signal Properties and ClassificationsSignals — both continuous-time signals and their discrete-time counterparts

— are categorized according to certain properties, such as deterministic or 

random, periodic or aperiodic, power or energy, and even or odd. These traits

aren't mutually exclusive; signals can hold multiple classifications.

Here are some of the most important signal properties.

But wait! There's more. Signals can also be categorized as

exponential, sinusoidal, or a special sequence. The unit sample

sequence and the unit step sequence are special signals of interest

in discrete-time. All the continuous-time signal classifications have

discrete-time counterparts, except singularity functions, which

appear in continuous-time only.

Defining special signals that serve as building blocks for more complex

signals makes the creation of custom signal models to suit your needs more

systematic and convenient.

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Recognizing System Properties and Classifications

Part of learning about signals and systems is that systems are identified

according to certain properties they exhibit. Have a look at the core system

classifications:

Linearity: A linear combination of individually obtained outputs is equivalentto the output obtained by the system operating on the corresponding linear 

combination of inputs.

Time-invariant: The system properties don't change with time. A present

input produces the same response as it does in the future, less the time

shift factor between the present and future.

Memoryless: If the present system output depends only on the present

input, the system is memoryless.

Causal: The present system output depends at most on the present and

past inputs. Future inputs can't be used to produce the present output.

Stable: A system is bounded-input bound-output (BIBO) stable if allbounded inputs produce a bounded output.

This table presents core linear time invariant (LTI) system properties for both

continuous and discrete-time systems. Time-domain, frequency-domain, and

s/z -domain properties are identified for the categories basic input/output,

cascading, linear constant coefficient (LCC) differential and difference

equations, and BIBO stability:

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Signals and Systems: Working with TransformTheorems and Pairs

Both signals and systems can be analyzed in the time-, frequency-, and s-

and z-domains. Leaving the time-domain requires a transform and then an

inverse transform to return to the time-domain.

 As you work to and from the time domain, referencing tables of both transform

theorems and transform pairs can speed your progress and make the work

easier. Use this table of common pairs for the continuous-time Fourier 

transform, discrete-time Fourier transform, the Laplace transform, and the z -

transform as needed.

Working in the frequency domain means you are working with Fourier 

transform and discrete-time Fourier transform — in the s-domain.

Using Fourier transforms for continuous-time signals

Here is a short table of theorems and pairs for the continuous-time Fourier 

transform (FT), in both frequency variable

The forward and inverse transforms for these two notational schemes are

defined as:

. . . and here's the table:

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Applying Fourier transform to discrete-time signals

For discrete-time signals and systems the discrete-time Fourier transform(DTFT) takes you to the frequency domain. A short table of theorems and

pairs for the DTFT can make your work in this domain much more fun. The

discrete-time frequency variable is

The forward and inverse transforms are defined as:

. . . and here's the table:

Using the Laplace transform in the s-domain

For continuous-time signals and systems, the one-sided Laplace transform

(LT) helps to decipher signal and system behavior. It's also the best approach

for solving linear constant coefficient differential equations with nonzero initial

conditions. The one-sided LT is defined as:

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The inverse LT is typically found using partial fraction expansion along with LT

theorems and pairs. Here's a short table of LT theorems and pairs.

Letting the z-Transform help with signals and systemsanalysis

For discrete-time signals and systems, the z -transform (ZT) is the counterpart

to the Laplace transform. With the ZT you can characterize signals and

systems as well as solve linear constant coefficient difference equations. The

two-sided ZT is defined as:

The inverse ZT is typically found using partial fraction expansion and the use

of ZT theorems and pairs. Here is a short table of ZT theorems and pairs.

Exploring Signals and Systems: Core Concepts of Sampling Theory

Sampling theory links continuous and discrete-time signals and systems. For 

example, you can get a discrete-time signal from a continuous-time signal by

taking samples every T  seconds. This article points out some useful

relationships associated with sampling theory. Key concepts include the low-

pass sampling theorem, the frequency spectrum of a sampled continuous-time signal, reconstruction using an ideal lowpass filter, and the calculation of 

alias frequencies.

The table of properties begins with a block diagram of a discrete-time

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processing subsystem that produces continuous-time output y (t ) from

continuous-time input x (t ). This block diagram motivates the sampling theory

properties in the remainder of the table.

The process of converting continuous-time signal x (t ) to discrete-time signal

 x [n] requires sampling, which is implemented by the analog-to-digital

converter (ADC) block. The block with frequency response

represents a linear time invariant system with input x [n] and output y [n]. Thediscrete-time signal y [n] is returned to the continuous-time domain via a

digital-to-analog converter and a reconstruction filter.

Synthesizing Signals with the Fourier Series

Periodic signals can be synthesized as a linear combination of harmonically

related complex sinusoids. The theory of Fourier series provides the

mathematical tools for this synthesis by starting with the analysis formula,

which provides the Fourier coefficients X n corresponding to periodic signal x (t )

having period T 0.

Common periodic signals include the square wave, pulse train, and triangle

wave. This table shows the Fourier series analysis and synthesis formulas

and coefficient formulas for X n in terms of waveform parameters for the

provided waveform sketches:

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