Signals and Systems For Dummies
Overview
Getting mixed signals in your signals and systems course?
The concepts covered in a typical signals and systems course are often considered by engineering students to be some of the most difficult to master. Thankfully, Signals & Systems For Dummies is your intuitive guide to this tricky course, walking you step-by-step through some of the more complex theories and mathematical formulas in a way that is easy to understand.
From Laplace Transforms to Fourier Analyses, Signals & Systems For Dummies explains in plain English the difficult concepts that can trip you up. Perfect as a study aid or to complement your classroom texts, this friendly, hands-on guide makes it easy to figure out the fundamentals of signal and system analysis.
- Serves as a useful tool for electrical and computer engineering students looking to grasp signal and system analysis
- Provides helpful explanations of complex concepts and techniques related to signals and systems
- Includes worked-through examples of real-world applications using Python, an open-source software tool, as well as a custom function module written for the book
- Brings you up-to-speed on the concepts and formulas you need to know
Signals & Systems For Dummies is your ticket to scoring high in your introductory signals and systems course.
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About The Author
Mark Wickert, PhD, is a Professor of Electrical and Computer Engineering at the University of Colorado, Colorado Springs. He is a member of the IEEE and is doing real signals and systems problem solving as a consultant with local industry.
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CHEAT SHEET
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.
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A big wide world of properties is associated with signals and systems — plenty in the math alone! Here are ten unforgettable properties related to signals and systems work.
LTI system stability
Linear time-invariant (LTI) systems are bounded-input bounded-output (BIBO) stable if the region of convergence (ROC) in the s- and z-planes includes the
The s-plane applies to continuous-time systems, and the z-plane applies to discrete-time systems.
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Here are eleven common mistakes students make when trying to solve problems and how to avoid them. Slow down enough to think through solutions, and make sure your fundamental understanding of the core material is at least as good as your ability to work through detailed problems.
Miscalculating the folding frequency
In sampling theory, the alias frequencies fold over fs/ 2 (known as the folding frequency), where fs is the sampling frequency in hertz.
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Following are eleven signals and systems concepts that apply to the design of a signal processing system known as an audio graphic equalizer. When you listen to music on a portable music player or a computer, you can usually customize the sound— you can re-shape the frequency spectrum of the underlying music signal to suit your tastes using a set of ten tone controls.
For many people in the United States, AM radio — a legacy modulation scheme known as amplitude modulation (AM) — is the place to go for talk radio and sports. But AM is more than just commercial broadcasting. Aircraft use AM to communicate with the tower at an airport, and shortwave radio uses AM for international broadcasting.
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In reality, multiple radio stations operate in the same metro area, or market. When you tune in a signal at 750 kHz, another signal may be at 760 kHz. To find out whether the adjacent signal impacts the simple receiver design, assume that the interference is a single tone, Aicos(2πfit). The received signal is now of the form
with fi assumed to lie just outside the +/– 5-kHz channel bandwidth centered on fc.
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Options a and b are the fixed FIR and IIR notch filters, respectively. The simplicity of these filters is a major draw. But how well do they work? Characterizing the filters in the frequency domain is a good starting point for this assessment.
A sinusoidal signal of the form
won’t pass through these filters in steady state.
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The Fourier series is a powerful mathematical tool, and it applies to multiple branches of engineering and mathematics. The design of a frequency tripler is a good example of the Fourier series in action. For computer and electrical engineers, the Fourier series provides a way to represent any periodic signal as a sum of complex sinusoids via Euler’s formulas.
This case study focuses on the sled position control system, which moves the read/write laser assembly radially under the spinning CD. Check out the figure for a close-up of a CD/DVD electromechanical system.
Credit: Illustration by Mark Wickert, PhD
The first prototypes of the compact disc (CD) were independently developed by Sony and Philips around 1976, with the intended use of storing digital audio data.
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The real deal with a ten-band equalizer is that you can graphically visualize the spectral shaping you provide to the signal passing through the equalizer by just looking at the positions of slider gain controls, as shown.
Credit: Illustration by Mark Wickert, PhD
Here’s a collection of Python functions to display the exact frequency response.
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In the study of both continuous-time and discrete-time systems, the frequency response and its relationship to pole and zero locations in the s- and z-planes, respectively, are of fundamental importance.
This helpful video presents the mathematical link between poles and zero locations and the frequency response.
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Continuous-time signals and systems never take a break. When a circuit is wired up, a signal is there for the taking, and the system begins working — and doesn’t stop. Keep in mind that the term signal is used here loosely; any one specific signal may come and go, but a signal is always present at each and every time instant imaginable in a continuous-time system.
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Visualize the big picture of the AM radio transmitter, receiver, and interfering signals with a system block diagram. Each block in the diagram has an underlying mathematical model.
Start with the AM signal model
The signal model for an AM signal is
where Ac is the carrier amplitude, fc is the carrier frequency, m(t) is the message signal, and the modulation index is
The modulation index controls how much of the signal is made up of the message and how much is pure carrier.
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The system block diagram for the sled (coarse laser position) control system on a CD/DVD drive is composed of several subsystems. Start with a continuous-time linear time-invariant (LTI) system model of the drive motor and mechanical attributes of the sled drive train (which includes screw gear and sled track).
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After the system has been linearized, a system block diagram utilizing Laplace transform (LT) techniques for feedback control of the vehicle velocity can be constructed. The differential equation can now be taken to the s-domain by taking the Laplace transform (LT) of both sides.
Taking the LT of all time domain quantities produces corresponding s-domain quantities.
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The block diagram of a system that uses a digital filter to remove an SNOI and retain the SOI is shown. This system assumes that the signals originate in continuous time and, after filtering, are returned to the continuous-time domain.
Credit: Illustration by Mark Wickert, PhD
The analog-to-digital converter (ADC) and digital-to-analog converter (DAC) interfaces represent how a real-time system is configured; but later, when simulating this system, you’ll use wav files to process prerecorded speech (the dashed lines).
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A basic digital spectrum analyzer is shown in the figure. The default window, w[n], is a constant of one over the capture interval of Nr samples. The FFT works with a finite length discrete-time signal. The window function w[n] is a design parameter that you may consider changing later in the process. The antialising filter ensures that signals greater than fs/2 don’t enter the ADC.
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A signal is classified as deterministic if it’s a completely specified function of time. A good example of a deterministic signal is a signal composed of a single sinusoid, such as
with the signal parameters being:
A is the amplitude, f0 is the frequency (oscillation rate) in cycles per second (or hertz), and
is the phase in radians.
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The various blocks of a digital communication system are a hybrid of discrete-time signal generation and filtering, continuous-time signal processing at baseband frequencies, and continuous-time radio frequency (RF) up and down conversion.
Start with the signal
A digital communications signal at baseband takes the form
where ak is a bit sequence that’s been translated from binary 0/1 values to +/– 1 values, p(t) is a pulse shape, and A is an amplitude scale factor.
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Discrete-time signals and systems march along to the tick of a clock. Mathematical modeling of discrete-time signals and systems shows that activity occurs with whole number (integer) spacing, but signals in the real world operate according to periods of time, or the update rate also known as the sampling rate.
Signals are sometimes classified by their symmetry along the time axis relative to the origin, t = 0. Even signals fold about t = 0, and odd signals fold about t = 0 but with a sign change. Simply put,
To check the even and odd signal classification, use the Python rect() and tri() pulse functions to generate six aperiodic signals.
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Before getting into the closed-loop system function of the CD/DVD case study, consider a few attributes of the open-loop system function by writing it out, leaving Ka as the only variable not defined:
You can find the pole-zero plot by using PyLab and custom function splane(b,a) found at ssd.py. This function returns the system function numerator and denominator polynomial coefficients as ndarrays b and a.
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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.
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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).
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A peaking filter for an audio graphic equalizer provides gain or loss (attenuation) at a specific center frequency fc. A peaking filter has unity frequency response magnitude, or 0 dB gain, at frequencies far removed from the center frequency.
At the center frequency fc, the frequency response magnitude in dB is GdB, which is continuously adjustable over a range of, say, +/– 12 dB.
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To successfully apply the various signals and systems concepts as part of practical engineering scenarios, you need to know what analysis tools are available. The figure shows the mathematical relationships between time, frequency, and the s- and z-domains. This portrayal offers perspective on a well-rounded study of signals and systems and reveals that you can establish relationships between domains in more than one way.
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Computer tools play a big part in modern signals and systems analysis and design. LCC differential and difference equations are a fundamental part of simple and highly complex systems. Fortunately, current software tools make it possible to work across domains with these LCC equations without too much pain.
LCC differential and difference equations are completely characterized by the {ak} and {bk} coefficient sets.
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Here is a detailed analytical solution to a convolution integral problem, followed by detailed numerical verification, using PyLab from the IPython interactive shell (the QT version in particular). The intent of the numerical solution is to demonstrate how computer tools can verify analytical solutions to convolution problems.
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Mastering convolution integrals and sums comes through practice. Here are detailed analytical solutions to one convolution integral and two convolution sum problems, each followed by detailed numerical verifications, using PyLab from the IPython interactive shell (the QT version in particular).
Continuous-time convolution
Here is a convolution integral example employing semi-infinite extent signals.
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Here’s a system-level look at the signals and systems model of a karaoke machine — an audio playback system with a powerful speaker that allows a person to sing over recorded music. A multimedia interface includes a TV to display and update lyrics as the music progresses.
From a high-level signals and systems viewpoint, a particular design attribute of this system is that it contains a sensor, a microphone, and two audio transducers (the left and right channel speakers).
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With the simplification to the open-loop system function in place, you can dive in and find the closed-loop system function of the CD/DVD case study, with full variable substitution. Here’s the closed-loop system function H(s) with the simplified open-loop system function applied:
With Ka = 50, you have a natural frequency of 15.
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Increasing the data record length, denoted Nr in this case study, improves spectral resolution, but the decay rate of the spectral leakage still needs to be addressed. Many window choices are available in common signal processing tool sets. The SciPy signal package is no exception. To get to the point, the figure provides plots of for some popular windows.
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A type of signal classification you need to be able to determine is periodic versus aperiodic. A signal is periodic if x(t) = x(t + T0), where T0, the period, is the largest value satisfying the equality. If a signal isn’t periodic, it’s aperiodic.
When checking for periodicity, you’re checking in a graphical sense to see whether you can copy a period from the center of the waveform, shift it left or right by an integer multiple of T0, and if it perfectly matches the signal T0 seconds away.
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To classify a signal x(t) according to its power and energy properties, you need to determine whether the energy is finite or infinite and whether the power is zero, finite, or infinite. The measurement unit for power and energy are watts (W) and joules (J).
In circuit theory, watts delivered to a resistor of R ohms is represented as
where V is voltage in volts and I is current in amps.
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You’re given the task of designing an analog (continuous-time) filter to meet the amplitude response specifications shown. You also need to find the filter step response, determine the value of the peak overshoot, and time where the peak overshoot occurs.
The objective of the filter design is for the frequency response magnitude in dB (20log10|H(f)|) to pass through the unshaded region of the figure as frequency increases.
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The zero-order-hold (ZOH), which is inherent in many digital-to-analog converters (DACs), holds the analog output constant between samples. The action of the ZOH introduces frequency droop, a roll off of the effective DAC frequency response on the frequency interval zero to one-half the sampling rate fs, in reconstructing y(t) from y[n].
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Signals — 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.
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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 equivalent to the output obtained by the system operating on the corresponding linear combination of inputs.
Signals, both continuous and discrete, have attributes that allow them to be classified into different types. Three broad categories of signal classification are periodic, aperiodic, and random.
Periodic signals
Signals that repeat over and over are said to be periodic. In mathematical terms, a signal is periodic if
x(t + T) = x(t) (continuous-time)
x[n + N] = x[n] (discrete-time)
The smallest T or N for which the equality holds is the signal period.
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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.
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You may not fully appreciate the mathematics of signals and systems on your first encounter. To enhance your understanding, you can use computer animation to bring to life some of the more difficult concepts and hopefully make a lasting impression.
To that end, a collection of five graphical user interface (GUI) apps are available for download.
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You probably have some level of familiarity with consumer electronics, such as MP3 music players, smartphones, and tablet devices, and realize that these products rely on signals and systems. But you may take for granted the cruise control in your car.
Here, the signals and systems framework in three familiar devices are shown at the block diagram level — a system diagram that identifies the significant components inside rectangular boxes, interconnected with arrows that show the direction of signal flow.
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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.
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The block diagram is a hybrid of discrete-time and continuous-time signals and systems. When studying the performance of a digital communications scheme, working with the complex baseband, also known as complex envelope signals, is convenient.
The IQ signal xc(t) has associated complex envelope,
To get this signal, rely on the extended version of the phasor addition formula:
Here, the complex envelope,
is given by
Note that you can get the original signal xc(t) by multiplying (spinning) the complex envelope by the complex sinusoid
Then taking the real part:
The block diagram performs this operation in the transmitter when the sine and cosine multiplier outputs are summed together.
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As a simple experiment, this case study generates three tightly packed SRC shaped BPSK signals. The signal of interest is centered at f = 0, while the cochannel signals — those that are nearest neighbors — are at offsets of +/– 1.5 Rb Hz. The power level of the adjacent signals is 0 dB relative to the desired signal.
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Simulate Options a, b, and c by using a speech file of a male voice reading phrases. The file enters the simulation as a wav file:
In [<b>685</b>]: fs,s = ssd.from_wav('OSR_us_000_0030_8k.wav')
You can generate the combination SOI + SNOI by using a custom function that returns r given the speech input s, the signal-to-interference ratio (SIR) in dB — μ = 0.
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To finish off this case study, simulate the system in Python. To give you a feel for sinusoidal spectrum analysis and window selection, here’s a Python simulation that utilizes the test signal:
Assume that the sampling rate is 10 kHz, which is greater than twice the highest frequency of 3,000 Hz. The first challenge is resolving the two equal amplitude sinusoids at 1,000 and 1,100 Hz (f = 100 Hz).
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You can simulate the complete CD/DVD drive control system by using a step function input for r(t), which is the commanding signal for the sled positioner. The output should be scaled in millimeters for a voltage signal at the input. The convolution theorem states that y(t) = r(t) x h(t); h(t) is the impulse response of the closed-loop system function H(s).
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In this cruise control case study, you can use Pylab and the SciPy function lsim() to evaluate the cruise control performance with a 4 percent grade. Note that a grade of 4 percent means rise over run of 4/100, which is equivalent to
Set T = 10 s, vmax = 120 mph and v0 = 75 mph. Units conversion is handled inside cruise_control().
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This case study begins with model construction, starting from basic physics. Special consideration is given to incorporate wind resistance and include as a system disturbance, the onset of a hill. The laws of motion say that given a vehicle mass m and engine supplied force cw(t), where c is proportionality constant and 0 ≤ w(t) ≤ 1 represents the engine throttle, Ftot(t) = mv(t) = cw(t), t ≥ 0.
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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 Xncorresponding to periodic signal x(t) having period T0.
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You can test the AM receiver by using an actual voice message. The value of this test lies in actually listening to the speech before and after it goes through the communications link. Your ears hear differences that aren’t always easy to discern from graphical displays alone.
Also, because the voice message occupies a wider spectral bandwidth than the single sinusoid, the nonlinear signal processing of the envelope detector is more realistically characterized in the presence of strong interference.
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Computer and electrical engineers work through a process that allows them to test, or model, potential solutions to find out whether the idea is likely to work in the real world. For products that rely on signal processing, engineers use signals and system modeling and analysis to reveal what’s possible.
When you’re trying to quickly prove a solution approach, you’ll often turn to behavioral level modeling of certain elements of the overall system to avoid low-level implementation details.
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The ZT doesn’t converge for all sequences. When it does converge, it’s only over a region of the z-plane. The values in the z-plane for which the ZT converges are known as the region of convergence (ROC).
Convergence of the ZT requires that
The right side of this equation shows that x[n]r-n is absolutely summable (the sum of all terms |x[n]r–n| is less than infinity).
The z-transform (ZT) is a generalization of the discrete-time Fourier transform (DTFT) for discrete-time signals, but the ZT applies to a broader class of signals than the DTFT. The two-sided or bilateral z-transform (ZT) of sequence x[n] is defined as
The ZT operator transforms the sequence x[n] to X(z), a function of the continuous complex variable z.
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The great attributes of discrete-time signals and systems rely on the ability to interface with the continuous-time domain. Analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) are the electronic subsystems that convert signals between continuous-time and discrete-time signal forms. This figure shows how to implement the interface of these two subsystems.
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Working across domains is a fact of life as a computer and electronic engineer. Solving real computer and electrical engineering tasks requires you to assimilate the vast array of signals and systems concepts and techniques and apply them in a smart and efficient way. Here’s an example problem that shows how analysis and modeling across the time, frequency, and s- and z-domains actually works.
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