signals & systems for dummies cheat sheet - for dummies
Post on 07-Jan-2016
534 Views
Preview:
DESCRIPTION
TRANSCRIPT
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 1/8
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
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 2/8
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.
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 3/8
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:
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 4/8
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:
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 5/8
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:
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 6/8
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
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 7/8
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:
7/17/2019 Signals & Systems for Dummies Cheat Sheet - For Dummies
http://slidepdf.com/reader/full/signals-systems-for-dummies-cheat-sheet-for-dummies 8/8
Copyright © 2015 & Trademark by John Wiley & Sons, Inc. All rights reserved.
About usAbout For Dummies
Contact Us
Privacy Policy
Terms and Conditions
Advertise with Us
Manage Your E-Mail
Newsletters
Help
More From DummiesMobile Site
My Account
Dummies B2B Services
Online Test Banks
eLearning
Mobile Apps
Dummies Store
Sweepstakes
Topics A-ZSmall Business & Careers
Computers & Software
Consumer Electronics
Crafts & Hobbies
Education & Languages
Food & Drink
Games
Health & Fitness
Home & Garden
Internet & Social Media
Music & Creative Arts
Personal Finance
Pets
Photography & Video
Relationships & Family
Religion & Spirituality
Sports & Outdoors
top related