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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Introductory course textbook on signals and systems with numerous examples and code snippets implemented in Python
Supported by code examples, Signals and Systems: Theory and Practical Explorations with Python is a textbook resource for a complete introductory course in systems and signals, enabling readers to run Python programs for convolution, discrete time Fourier transforms and series, sampling, and interpolation for a wide range of functions. Readers are guided step-by-step through basic differential equations, basic linear algebra, and calculus to ensure full comprehension of the exercises.
This book is supported by a companion website, hosting interactive material to draw functions, and run programs in Python; it is enriched with audiovisual material via linking to related videos. Links to resources that provide a deeper explanation about the important concepts in the book, such as the systems approach, complex numbers, harmony, the Euler equation, and Hilbert spaces, are also included.
Written by two highly qualified academics, topics covered include:
- Systems approach for modeling the natural and manmade systems and some application areas
- Representation of complex and real signals by basic functions, such as real and complex exponentials, unit step and unit impulse functions
- Properties of signals, such as symmetry, harmony, energy, power, continuity and discreteness
- Convolution and correlation operations for continuous time and discrete time signals and systems
- Representation of systems by impulse response, frequency response, transfer function, block diagram, differential and difference equations
- Properties of systems, such as linearity, time invariance, memory, invertibility, stability and causality
- Continuous time and discrete time Fourier analysis in Hilbert space and their extension to Laplaca transform and z-transform
- Filtering by Linear Time Invariant systems in time and frequency domains, covering low pass, high pass band pass and band reject filters
- Sampling theorems for continuous time and discrete time systems, covering A/D and D/A conversion, sampling and interpolation
Signals and Systems is an ideal textbook resource for a one semester introductory course on signals and systems for upper level undergraduate and graduate students in computer science, electrical engineering and data science. It is also a useful reference for professionals working in bioinformatics, robotics, remote sensing, and related fields.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
About the Companion Website
1 Introduction to Systems and Signals
1.1 Example Applications
1.2 Relationship Between Signals and Systems
1.3 Mathematical Representation of Signals and Systems
1.4 Operations on the Time Variable of Signals
1.5 Signals with Symmetry Properties
1.6 Complex Signals Represented by Complex Functions
1.7 Chapter Summary
Problems
Notes
2 Basic Building Blocks of Signals
2.1 LEGO Functions of Signals
2.2 King of the Functions: Exponential Function
2.3 Unit Impulse Function
2.4 Unit Step Function
2.5 Chapter Summary
Problems
3 Basic Building Blocks and Properties of Systems
3.1 Representation of Systems by Equations
3.2 Interconnection of Basic Systems: Series, Parallel, Hybrid, and Feedback Control Systems
3.3 Properties of Systems
3.4 Basic Building Blocks of Systems and Their Properties
3.5 Chapter Summary
Problems
4 Representation of Linear Time-Invariant Systems by Impulse Response and Convolution Operation
4.1 Representation of LTI Systems by Impulse Response
4.2 Properties of Impulse Response for LTI Systems
4.3 An Application of Convolution in Machine Learning
4.4 Chapter Summary
Problems
Note
5 Representation of LTI Systems by Differential and Difference Equations
5.1 Linear Constant-Coefficient Differential Equations
5.2 Representation of a Continuous Time LTI System by Differential Equations
5.3 Solving the Linear Constant Coefficient Differential Equations That Represent LTI Systems
5.4 Linear Constant Coefficient Difference Equations
5.5 Relationship Between the Impulse Response and Difference or Differential Equations
5.6 Block Diagram Representation of Differential Equations for LTI Systems
5.7 Chapter Summary
Problems
6 Fourier Series Representation of Continuous Time Periodic Signals
6.1 History
6.2 Mathematical Representation of Waves and Harmony
6.3 Dirichlet Conditions
6.4 Fourier Theorem
6.5 Frequency Domain and Hilbert Spaces
6.6 Response of a Linear Time-Invariant System to the Continuous Time Complex Exponential Input Signal
6.7 Convergence of the Fourier Series and Gibbs Phenomenon
6.8 Properties of Fourier Series for Continuous Time Functions
6.9 Trigonometric Fourier Series for Continuous Time Functions
6.10 Trigonometric Fourier Series for Continuous Time Even and Odd Functions
6.11 Chapter Summary
Problems
7 Fourier Series Representation of Discrete Time Periodic Signals
7.1 Fourier Series Theorem for Discrete Time Functions
7.2 Discrete Time Fourier Series Representation in Hilbert Space
7.3 Properties of Discrete Time Fourier Series
7.4 Discrete Time LTI Systems with Periodic Input and Output Pairs
7.5 Chapter Summary
Problems
8 Continuous Time Fourier Transform and Its Extension to Laplace Transform
8.1 Fourier Series Extension to Aperiodic Functions
8.2 Existence and Convergence of the Fourier Transforms: Dirichlet Conditions
8.3 Fourier Transforms
8.4 Comparison of Fourier Series and Fourier Transform
8.5 Frequency Content of Fourier Transform
8.6 Representation of LTI Systems in Frequency Domain by Frequency Response
8.7 Relationship Between the Fourier Series and Fourier Transform of Periodic Functions
8.8 Properties of Fourier Transform: For Continuous Time Signals and Systems
8.9 Laplace Transforms as an Extension of Continuous Time Fourier Transforms
8.10 Inverse of Laplace Transform
8.11 Continuous Time Linear Time-Invariant Systems in Laplace Domain
8.12 Chapter Summary
Problems
9 Discrete Time Fourier Transform and Its Extension to -Transforms
9.1 Fourier Series Extension to Discrete Time Aperiodic Functions
9.2 Dirichlet Conditions Are Relaxed for the Existence of Discrete Time Fourier Transform
9.3 Fourier Transform of Discrete Time Periodic Functions
9.4 Properties of Fourier Transforms for Discrete Time Signals and Systems
9.5 Discrete Time Linear Time-Invariant Systems in Frequency Domain
9.6 Representation of Discrete Time LTI Systems
9.7
z
-Transforms as an Extension of Discrete Time Fourier Transforms
9.8 Inverse of -Transform
9.9 Discrete Time Linear Time-Invariant Systems in -Domain
9.10 Chapter Summary
Problems
10 Linear Time-Invariant Systems as Filters
10.1 Filtering the Periodic Signals by Frequency Response
10.2 Filtering the Aperiodic Signals by Frequency Response
10.3 Frequency Ranges of Frequency Response
10.4 Filtering with LTI Systems
10.5 Ideal Filters for Discrete Time and Continuous Time LTI Systems
10.6 Discrete Time Real Filters
10.7 Continuous Time Real Filters
10.8 Chapter Summary
Problems
11 Continuous Time Sampling
11.1 Sampling
11.2 Properties of the Sampled Signal in Time and Frequency Domains
11.3 Reconstruction
11.4 Aliasing
11.5 Sampling Theorem
11.6 Sampling with Zero-Order Hold
11.7 Reconstruction with Zero-Order Hold
11.8 Sampling and Reconstruction with First-Order Hold
11.9 Chapter Summary
Problems
12 Discrete Time Sampling and Processing
12.1 Time Normalization
12.2 C/D Conversion:
12.3 D/C Conversion
12.4 Sampling the Discrete Time Signals
12.5 Reconstruction of Discrete Time Signal from Its Sampled Counterpart
12.6 Discrete Time Decimation and Interpolation
12.7 Chapter Summary
Problems
Bibliography
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Relationship between the unit step and unit impulse functions.
Chapter 6
Table 6.1 Summary of the continuous time Fourier series properties.
Table 6.2 Some popular continuous time periodic signals and their spectral c...
Chapter 7
Table 7.1 Summary of the properties of discrete time Fourier series.
Table 7.2 Some popular discrete time periodic signals and their spectral coe...
Chapter 8
Table 8.1 Basic properties of Fourier transform.
Table 8.2 Fourier transform pairs of popular continuous time functions.
Table 8.3 Properties of Laplace transform.
Table 8.4 Laplace transform pairs.
Chapter 9
Table 9.1 Properties of the discrete time Fourier transform.
Table 9.2 Fourier transform pairs of popular discrete time functions.
Table 9.3 Properties of -transform.
Table 9.4 -transform pairs for popular functions.
List of Illustrations
Chapter 1
Figure 1.1
Waterfall
by M.C. Escher
1
The puzzle on the left consists of the ...
Figure 1.2 Digital terrain model of a transportation area obtained from LIDA...
Figure 1.3 Visualizing anatomical regions during both the planning (a) and e...
Figure 1.4 Example of building detection using remote sensing applications.
3
Figure 1.5 Schematic representation of a system by a box, which consists of ...
Figure 1.6 Plot of the continuous time function, .
Figure 1.7 Plot of the discrete time sinusoidal function, . Note that the p...
Figure 1.8 Average monthly temperature in Ankara
Figure 1.9 The digital signal (black) is obtained by quantizing the range of...
Figure 1.10 The plot of a continuous time pulse signal, which is nonzero in ...
Figure 1.11 The plot of a discrete time pulse signal, which is nonzero in th...
Figure 1.12 Shift of a continuous time pulse signal, , given in Figure 1.10...
Figure 1.13 Time reversed versions of continuous time and discrete time puls...
Figure 1.14 Scaled continuous time and discrete time signals for .
Figure 1.15 The original discrete time signal is decimated by a factor of ....
Figure 1.16 Time-scaled continuous and discrete time pulse signals, given in...
Figure 1.17 The plot of shifted and squished continuous time signal, .
Figure 1.18 Plot of and .
Figure 1.19 Plot of and .
Figure 1.20 Plots of (a) , (b) , and (c) in Exercise 1.7.
Figure 1.21
Snakes
by the Dutch artist M.C. Escher, which depicts rotational...
Figure 1.22 A Penrose tiling with fivefold symmetrical two different rhombi.
Figure 1.23 Symmetric tiling Alhambra Palace.
6
Figure 1.24 A continuous time periodic signal with the fundamental period,
Figure 1.25 The continuous time cosine signal, , with amplitude and perio...
Figure 1.26 A discrete time periodic signal with fundamental period .
Figure 1.27 Plot of , for and .
Figure 1.28 An even signal has reflection symmetry about the vertical axis. ...
Figure 1.29 An odd signal has rotation symmetry about the origin. In other w...
Figure 1.30 Plot of parabolas for .
Figure 1.31 Plot of hyperbolas for .
Figure 1.32 An arbitrary continuous time signal, .
Figure 1.33 The odd and even parts of of Figure 1.32.
Figure 1.34 Plot of , and its even and odd parts.
Figure 1.35 Complex plane, where the real numbers are augmented by imaginary...
Figure 1.36 Plots of three complex numbers in Exercise 1.17.
Figure P1.1
Figure P1.2
Figure P1.3
Figure P1.4
Figure P1.5
Figure P1.6
Figure P1.9
Chapter 2
Figure 2.1 Continuous time exponential, , (a) for and (b) .
Figure 2.2 Discrete time real exponential function for (a) , (b) , (c) , ...
Figure 2.3 A painting, generated by Adobe Firefly using color harmony with f...
Figure 2.4 Periodic motion represented by a complex exponential function, in...
Figure 2.5 Periodic motion represented by a complex exponential function, in...
Figure 2.6 Plot of the superposition of the first two harmonics of .
Figure 2.7 Discrete time unit impulse function. We put a small dot at every ...
Figure 2.8 The plot of function. The width of this function is and the h...
Figure 2.9 Schematic representation of the unit impulse function in continuo...
Figure 2.10 Plot of
Figure 2.11 The plot of the discrete time unit step function, .
Figure 2.12 Addition of the unit impulse and its shifted version, . It is p...
Figure 2.13 We can also generate the unit impulse function by subtracting th...
Figure 2.14 A discrete time signal, which has nonzero values in the interval...
Figure 2.15 A function, which consists of two shifted impulse functions.
Figure 2.16 Plot of continuous time unit step function, . Note that this fu...
Figure 2.17 The well-behaved function approaches to the unit impulse funct...
Figure 2.18 The well-behaved function can be written in terms of two unit ...
Figure 2.19 We can slide the function all over the function , as we multi...
Figure 2.20 The continuous time unit impulse function is even.
Figure 2.21 A piecewise constant function, which does not have a compact ana...
Figure 2.22 A bounded piece-wise constant function.
Figure P2.1
Chapter 3
Figure 3.1 Black box representation of a system, where represents a functi...
Figure 3.2 Human visual system, represented by the
series connection
of two ...
Figure 3.3 Block diagram representation of the audio-visual system, as two p...
Figure 3.4 Block diagram representation of the audio-visual system, as a com...
Figure 3.5 Block diagram representation of a feedback control system.
Figure 3.6 An artificial neuron. It first takes the linear combination of th...
Figure 3.7 An artificial neural network, with one hidden layer, obtained by ...
Figure 3.8 When we cascade the system, with its inverse , the input of th...
Figure 3.9
Stable system
(a): When we put a glass billiard ball to the side o...
Figure 3.10 Linearity property. Given that and , a linear system represen...
Figure 3.11 Schematic representations of a scalar multiplier. Both represent...
Figure 3.12 An adder, which adds three inputs, to generate output
Figure 3.13 Schematic representation of a multiplier, which multiplies two i...
Figure 3.14 Schematic representation of an integrator.
Figure 3.15 Schematic representation of a differentiator.
Figure 3.16 Schematic representation of the unit delay operator.
Figure 3.17 Block diagram representation of the system, , represented by th...
Figure 3.18 Block diagram representation of the system,
Figure 3.19 Block diagram of the system in Exercise 3.19.
Figure 3.20 Cascaded representation of an incrementally linear system (given...
Figure 3.21 Block diagram of the system in Exercise 3.20.
Chapter 4
Figure 4.1 Representation of discrete time signals in terms of weighted summ...
Figure 4.2 If we observe the input and the corresponding output , can we ...
Figure 4.3 Impulse response is the response of an LTI system to a unit impul...
Figure 4.4 Relationship between the input and output signal of a discrete ti...
Figure 4.5 Relationship between the input and output signal of a continuous ...
Figure 4.6 Representation of LTI system by its impulse response, .
Figure 4.7 Associativity of convolution. Provided that and are impulse r...
Figure 4.8 Distributivity of convolution. Two block diagrams are equivalent....
Figure 4.9 When the impulse response of an LTI system is , the system acts ...
Figure 4.10 When the impulse response of an LTI system is , the system dela...
Figure 4.11 (a) Convolution of two unit step functions. Prior to integration...
Figure 4.12 While we convolute two exponential functions with negative decay...
Figure 4.13 Result of the convolution operation , in Exercise 4.6.
Figure 4.14 Output of a discrete time LTI system, represented by the impul...
Figure 4.15 When the impulse response of a discrete time LTI system is , th...
Figure 4.16 If the output of an LTI system represented by an impulse respons...
Figure 4.17 Block diagram for the LTI system represented by in Exercise 4....
Figure 4.18 Sample images from the MNIST (Modified National Institute of Sta...
Figure 4.19 The system that takes an input image (as a matrix), processe...
Figure 4.20 The optimal filter learned on the MNIST training set by minimi...
Chapter 5
Figure 5.1 Representation of a continuous time LTI systems by differential e...
Figure 5.2 For an LTI system represented by a linear constant-coefficient di...
Figure 5.3 Using the superposition property to find the particular solution ...
Figure 5.4 Representation of a discrete time LTI system with difference equa...
Figure 5.5 Solution of the difference equation of with initial condition,
Figure 5.6 Impulse response of an infinite impulse response (IIR) filter.
Figure 5.7 Impulse response of a finite impulse response (FIR) filter is the...
Figure 5.8 Schematic representation of an adder for two input signals.
Figure 5.9 Schematic representation of a scalar multiplier.
Figure 5.10 Schematic representation of unit delay operator.
Figure 5.11 Schematic representation of the unit advance operator.
Figure 5.12 Schematic representation of an integrator.
Figure 5.13 Schematic representation of a differentiator.
Figure 5.14 The output of the adder is , which is equal to If we integrat...
Figure 5.15 Realization of the differential equation, by a differentiator,...
Figure 5.16 Block diagram representation of the difference equation,
Figure 5.17 Block diagram representation of a differential equation in Exerc...
Figure 5.18 Block diagram representation of a differential equation in Exerc...
Chapter 6
Figure 6.1 Digital artwork showing colorful waves.
Figure 6.2 A simple rotation on a circle in a complex plane creates trigonom...
Figure 6.3 Plot of the periodic function, .
Figure 6.4 Magnitude vs. and phase vs. plots. The magnitudes are all...
Figure 6.5 The plot of , for and
Figure 6.6 A function with infinitely many discontinuities in a finite inter...
Figure 6.7 A two-dimensional Euclidean vector space, called two-tuples. A ve...
Figure 6.8 The plot of the spectral coefficients vs. in Exercise 6.6.
Figure 6.9 Plot of magnitude spectrum (a) and phase spectrum (b) of the Four...
Figure 6.10 A periodic function, called pulse train, which repeats itself at...
Figure 6.11 Plot of Fourier spectrum vs. , for and , for the signal ...
Figure 6.12 An LTI system, represented by , receives a complex exponential,...
Figure 6.13 Gibbs phenomenon. From left to right, we are approximating a squ...
Figure 6.14 A continuous time impulse train, has spectral coefficients, ,...
Figure 6.15 Square wave of width and period and its derivative.
Figure 6.16 Plot of with period .
Figure 6.17 Plot of trigonometric coefficients, and , for , with perio...
Figure 6.18 A periodic function, called square wave, which repeats itself at...
Chapter 7
Figure 7.1 The plot of . Its fundamental period is .
Figure 7.2 Magnitude and phase spectrum of . Both vs. and the vs. a...
Figure 7.3 Plot of the real and imaginary parts (top row) and the magnitude ...
Figure 7.4 Discrete time periodic square wave with a fundamental period , f...
Figure 7.5 Fourier series coefficients for the periodic square wave of Examp...
Figure 7.6 (a) Periodic signal and its representation as a sum of (b) the ...
Figure 7.7 Sequence that is consistent with the properties specified in th...
Figure 7.8 (a) The square-wave sequence in Example 7.12; (b) the sequence
Figure 7.9 An LTI system, where the periodic input output pairs are represen...
Figure 7.10 A discrete time LTI system, represented by the impulse response,...
Chapter 8
Figure 8.1 (a) Given an aperiodic function , which is nonzero in a finite i...
Figure 8.2 (a) The discrete spectral coefficients, of a periodic function,...
Figure 8.3 Sketch of the signal, .
Figure 8.4 Fourier transform of the signal, , as depicted in Figure 8.3.
Figure 8.5 (a) The rectangular pulse signal of the example and (b) its Fouri...
Figure 8.6 Fourier transform pair of Exercise 8.5: (a) Fourier transform, ...
Figure 8.7 The exponential function is obtained at the output of an LTI syst...
Figure 8.8 Magnitude and phase spectrum of the frequency response,
Figure 8.9 Comparison of the spectral coefficients and Fourier transform of ...
Figure 8.10 Impulse train in time domain (a), its spectral coefficients (b),...
Figure 8.11 Decomposing a signal into the linear combination of two simpler ...
Figure 8.12 An integrator.
Figure 8.13 (a) A signal for which the Fourier transform is to be evaluate...
Figure 8.14 A sample block diagram representation in time domain. Note that ...
Figure 8.15 The block diagram representation of Figure 8.14 in frequency dom...
Figure 8.16 A bandlimited signal, for .
Figure 8.17 Fourier transform, , of .
Figure 8.18 Amplitude modulation: The signal is shifted to high frequencie...
Figure 8.19 Duality property which shows the relationships between the analy...
Figure 8.20 Region of convergence for the Laplace transform of .
Figure 8.21 Region of convergence for the Laplace transform of the unit step...
Figure 8.22 Region of convergence for the Laplace transform of .
Figure 8.23 Two-sided function .
Figure 8.24 Region of convergence for the Laplace transform of .
Figure 8.25 Region of convergence of the transfer function for ROC .
Figure P8.8a Continuous time periodic signal of Problem 8.8.b.
Figure P8.8b Continuous time signal of Problem 8.8.c.
Figure P8.9 Continuous time signal of Problem 8.9.
Figure P8.17a Magnitude and phase plots of , in problem 8.17.
Figure P8.17b Frequency domain signal of Problem 8.17b
Figure P8.20 Frequency domain signal of Problem 8.20.
Figure P8.22 Continuous time signal of Problem 8.22b.
Figure P8.29 Block diagram representation of an LTI system in Problem 8.29....
Chapter 9
Figure 9.1 (a) A finite duration signal . (b) It is repeated at every funda...
Figure 9.2 Magnitude and phase plots of for
Figure 9.3 Magnitude and phase plots of for
Figure 9.4 (a) Signal and (b) its Fourier transform .
Figure 9.5 (a) Rectangular pulse for and (b) its Fourier transform.
Figure 9.6 Fourier transform of a discrete impulse function, is constant. ...
Figure 9.7 Impulse train preserves its analytic form in both time and freque...
Figure 9.8 Impulse train, shifted by
Figure 9.9 Magnitude and phase spectrum of the complex discrete time domain ...
Figure 9.10 Magnitude and phase spectra of , in Exercise 9.8.
Figure 9.11 Fourier transform of .
Figure 9.12 Fourier transform of
Figure 9.13 Fourier transform of
Figure 9.14 Fourier series coefficients and Fourier transform of the periodi...
Figure 9.15 If we replace by in the discrete time Fourier transform of
Figure 9.16 A discrete time linear time-invariant system with two parallel i...
Figure 9.17 A block diagram representation of discrete time LTI systems, wit...
Figure 9.18 Magnitude and phase spectrum of the frequency response of a firs...
Figure 9.19 Impulse response of a first-order difference equation for .
Figure 9.20 Plot of the unit step response for
Figure 9.21 Block diagram representation of a feedback control system, repre...
Figure 9.22 ROC for the -transform of finite duration signals.
Figure 9.23 ROC for the -transform of right-sided signals.
Figure 9.24 ROC for the -transform of left-sided signals.
Figure 9.25 ROC for the -transform of two-sided signals.
Figure 9.26 ROC for the -transform of .
Figure 9.27 ROC for the -transform of .
Figure 9.28 ROC for the -transform of .
Figure 9.29 ROC for .
Figure 9.30 ROC for .
Figure 9.31 ROC for .
Chapter 10
Figure 10.1 When the input of a discrete time LTI system is the superpositio...
Figure 10.2 When the input of a continuous time LTI system is the superposit...
Figure 10.3 Frequency response of continuous time ideal low-pass filter with...
Figure 10.4 Frequency response of discrete time ideal low-pass filter with c...
Figure 10.5 Frequency response of continuous time ideal high-pass filter wit...
Figure 10.6 Frequency response of discrete time ideal high-pass filter with ...
Figure 10.7 Frequency response of continuous time ideal band-pass filter bet...
Figure 10.8 Frequency response of discrete time ideal band-pass filter betwe...
Figure 10.9 Frequency response of continuous time ideal band-reject filter, ...
Figure 10.10 Frequency response of discrete time ideal band-reject filter cu...
Figure 10.11 One full period of the Fourier transform of an input, for a dis...
Figure 10.12 Impulse response of a discrete time ideal low-pass filter.
Figure 10.13 The frequency domain representation of the output signal, when ...
Figure 10.14 (a) Frequency response of a continuous time low-pass filter w...
Figure 10.15 (a) Magnitude and (b) phase spectrum of a low-pass filter, repr...
Figure 10.16 The input signal is defined as the addition of two signals: (a)...
Figure 10.17 The plot of the output signal when the input and impulse resp...
Figure 10.18 One full period of the magnitude and phase plots of the frequen...
Figure 10.19 Magnitude and phase plot of the frequency response,
Figure 10.20 The frequency response of a band-stop filter.
Figure 10.21 The frequency response of a notch filter.
Figure 10.22 Magnitude and phase plots of the frequency response, of the f...
Figure 10.23 Block diagram representation of a first-order differential equa...
Figure 10.24 Magnitude and the phase plot of the frequency response of the f...
Figure 10.25 The impulse response and unit step response of a low-pass filte...
Figure 10.26 We take the derivative of the input signal by a differentiator....
Figure 10.27 Magnitude and the phase plots of the frequency response,
Chapter 11
Figure 11.1 Given finitely many discrete points, we can define infinitely ma...
Figure 11.2 Block diagram representation of sampling process. A continuous t...
Figure 11.3 Continuous time signal is multiplied by an impulse train to ...
Figure 11.4 A band-limited signal ranges , in time domain. However, since...
Figure 11.5 Impulse train in time domain (left) and its Fourier transform:
Figure 11.6 Sampled signal in time and frequency domains.
Figure 11.7 Plots of , , and in Exercise 11.1.
Figure 11.8 Plot of the magnitude and phase of in Exercise 11.1.
Figure 11.9 Reconstruction filter , in frequency domain.
Figure 11.10 Reconstructed signal and its inverse Fourier transform.
Figure 11.11 Sampling a continuous time signal and reconstruction of the s...
Figure 11.12 Reconstructed signal is obtained by filtering the sampled sig...
Figure 11.13 Fourier transform of the sampled signal and the ideal low-pas...
Figure 11.14
Aliasing:
If , then, ’s in the sampled signal overlap with ...
Figure 11.15 Cosine function with period in time domain and its Fourier tr...
Figure 11.16 Impulse train sampling with period . (a) Original signal and...
Figure 11.17 Impulse train sampling with sampling frequency, . (a) Original...
Figure 11.18
Sampling with zero-order hold
: Generation of a piecewise constan...
Figure 11.19 An LTI system represented by the impulse response .
Figure 11.20 The impulse response function is the shifted version of . In...
Figure 11.21 Block diagram representation of sampling with zero-order hold....
Figure 11.22 The reconstruction filter receives the zero-order hold sample...
Figure 11.23 Magnitude and phase plots of the reconstruction filter, , for ...
Figure 11.24 (a) Reconstructed signal in time domain from impulse train samp...
Figure P11.7a
Figure P11.7b
Figure P11.8a
Figure P11.8b
Figure P11.10
Figure P11.11a
Figure P11.11b
Figure P11.12
Figure P11.13a
Figure P11.13b
Figure P11.13c
Chapter 12
Figure 12.1 Block diagram representation of a general digital signal process...
Figure 12.2 Converting a continuous time weighted impulse function of the sa...
Figure 12.3 C/D conversion: Conversion of a continuous time signal into a di...
Figure 12.4 Comparison of the continuous time sampled signal and its discr...
Figure 12.5 D/C conversion: Recovering a continuous time signal from its dis...
Figure 12.6 A continuous LTI system with a differentiation operator.
Figure 12.7 Magnitude and phase plots of the continuous time band-limited di...
Figure 12.8 Discrete time counterpart of a band limited continuous time diff...
Figure 12.9 An LTI system represented by and its discrete counterpart , f...
Figure 12.10 Discrete time counterpart is time normalized version of the c...
Figure 12.11 The magnitude and phase plots of the frequency response of the ...
Figure 12.12 The magnitude and phase plots of frequency responses of the dis...
Figure 12.13 Impulse train sampling of a discrete signal .
Figure 12.14 Fourier transform of discrete time impulse train and Fourie...
Figure 12.15 Time and frequency domain representations of the discrete time ...
Figure 12.16 Reconstruction filter for discrete time sampling.
Figure 12.17 Frequency response, , of the reconstruction filter for discret...
Figure 12.18 When a discrete time signal is down-sampled, we only keep the...
Figure 12.19 Fourier transform of the decimated signal .
Figure P1
Figure P2
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
About the Companion Website
Begin Reading
Bibliography
Index
End User License Agreement
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Signals and Systems
Theory and Practical Explorations with Python
Fatoş Tunay Yarman Vural
Middle East Technical University
Ankara, Turkey
Emre Akbaş
Middle East Technical University
Ankara, Turkey
This edition first published 2025© 2025 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data:
Names: Yarman Vural, Fatoş Tunay, author. | Akbaş, Emre, author.
Title: Signals and systems : theory and practical explorations with Python / Fatoş Tunay Yarman Vural, Emre Akbaş.
Description: Hoboken, NJ : Wiley, 2025. | Includes bibliographical references and index.
Identifiers: LCCN 2024017990 (print) | LCCN 2024017991 (ebook) | ISBN 9781394215751 (hardback) | ISBN 9781394215768 (adobe pdf) | ISBN 9781394215775 (epub)
Subjects: LCSH: Signal theory (Telecommunication) | System analysis. | Python (Computer program language)
Classification: LCC TK5102.5 .Y37 2025 (print) | LCC TK5102.5 (ebook) | DDC 621.382/23 – dc23/eng/20240515
LC record available at https://lccn.loc.gov/2024017990
LC ebook record available at https://lccn.loc.gov/2024017991
Cover Design: WileyCover Image: © MR.Cole_Photographer/Getty Images
Dedicated to those,
who endeavor with love and wisdom
to better the world!
About the Authors
Fatoş Tunay Yarman Vural received her B.S. and M.S. degrees from the Technical University of Istanbul and Bogazici University, Turkey. She received her Ph.D. degree from Princeton University in 1981. She was an Assistant Professor at Drexel University and a research fellow at the Massachusetts Institute of Technology in the United States between 1985 and 1987. She joined the Department of Computer Engineering, Middle East Technical University as a faculty member in 1992. In 1996, she became the Department Chairperson until 2000. Between 2000 and 2008, she served as the Assistant President responsible for research and industrial relationships at Middle East Technical University. Dr. Yarman Vural organized dozens of workshops, national and international meetings, and conferences during her academic career.
Her research areas include neuroinformatics, machine learning, pattern recognition, computer vision, and image processing, where she has published more than 200 papers. She accomplished about 100 national and international research and industrial projects. She founded research centers at Middle East Technical University related to modeling, simulation, ethics, science and society, and e-government. She has founded various programs in Turkey for faculty development, research project development, science and society relations, distance education, and postdoctoral studies. She was the Turkish representative for research in NATO between 2004 and 2007. Dr. Yarman Vural is a member of IEEE Computer Society. She is also a member of Turkish Informatics Association, Turkish Intelligence Association, and Turkish Informatics Foundation.
Emre Akbaş is an Associate Professor at the Department of Computer Engineering, Middle East Technical University (METU). Prior to joining METU, he was a postdoctoral research associate at the Department of Psychological and Brain Sciences, University of California, Santa Barbara. He received his PhD degree from the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign in 2011. His BS and MS degrees are both from the Department of Computer Engineering, METU. Dr. Akbas’s research received the Beckman Institute’s Cognitive Science AI Award, “METU thesis of the year” award (three times), the Parlar Foundation Research Incentive Award, and the Young Scientist Award of Science Academy, Turkey. His research interests are in computer vision and deep learning with a focus on object detection and human pose estimation.
Preface
Through the course of civilization, humankind has been driven by a great curiosity to grasp the relationships between the signals observed in nature and the systems that orchestrate them. From the rhythmic motion of celestial bodies to the intricate processes of life, we humans develop tools and methodologies that both illuminate and empower us in our interactions within the universe of which we are an integral part.
This book presents the fundamental principles and methodologies for modeling natural phenomena and creating human-made systems by blending theory and practical exploration. The chapters provide insights into systems and their observable signals, going beyond mere abstract formulations. The readers are invited to appreciate studying rigorous mathematical concepts such as symmetry while they walk around the beautiful ceramic tile decorations of Alhambra Palace and to be inspired by the impossible tiles of Roger Penrose. They enjoy the musical harmony while they grasp the harmonically related complex exponential functions. They find themselves in the infinite dimensions of Hilbert spaces while they study the abstract concepts of Fourier series and transforms. They appreciate the aesthetics of formalism of intricate systems, which can be represented as an ensemble of simple systems interconnected by signals, revealing the inherent wholeness and implicate order.
This approach is reflected in the following topics covered in this book:
Chapter 1
introduces the holistic approach of the book to modeling natural and human-made systems and their manifestations as signals. Mathematical foundations of signals and systems, together with their relationships, are provided.
Chapter 2
explains the Lego functions, namely trigonometric functions, exponential functions, unit steps, and unit impulse functions. Properties of these basic functions, such as symmetry, harmony, continuity, and discreteness, are investigated. In the rest of the chapters, we assemble the Lego functions to construct more elaborate signals, which are observed both in nature and in human-made machinery.
Chapter 3
explores the crucial properties of continuous time and discrete time systems, such as linearity, time invariance, stability, invertibility, and memory. These properties enable us to simplify the design and analysis of elaborate systems. Specifically, we model and implement systems with well-defined methodologies, when a system is linear and time-invariant (LTI). We also define some building blocks of LTI systems, such as adders, integrators, differentiators, delay operators, and scalar multipliers.
Chapter 4
defines a unique function, called the impulse response, which represents linear time-invariant systems. We describe an essential operation, called convolution, which relates the input–output pair of an LTI system through the impulse response.
Chapter 5
models continuous time and discrete time LTI systems by their dynamic nature using differential and difference equations. We study the solution to the differential and difference equations to investigate the relative rate of change between the input and output signals of an LTI system.
Chapters 6
and
7
introduce continuous time and discrete time Fourier series representations in Hilbert spaces, respectively. A periodic function is represented as a vector in this infinite-dimensional space, where the spectral coefficients correspond to the coordinates of the function.
Chapters 8
and
9
introduce continuous- and discrete time Fourier transforms for an aperiodic function, which satisfy a set of conditions, called Dirichlet conditions. We also extend the Fourier transform to the Laplace transform for continuous functions and to the z-transform for discrete functions, respectively.
Chapter 10
explains how signals are reshaped by linear time-invariant filters in time and frequency domains, covering low-pass, high-pass, band-pass, and band-reject filters.
Chapters 11
and
12
explain the pioneering sampling theorem of Claude Shannon for continuous time and discrete time signals and systems, which bridges the continuous world to the discrete world, opening the door to the digital era.
The chapters of the book are crafted by a rigorous formalism with a wide range of practical exercises and problems.
The theoretical content of the book is enriched by Python code snippets, providing practical implementation of key concepts. For example, elementary signal operations for time scale, time reverse, and translation are implemented, where the students plug and play a wide range of parameters to observe the operations on the functions. Also, students can run the Python codes for convolution, Fourier transforms and series, sampling, and interpolation for a wide range of functions with different sets of parameters. The companion website of the book enables the students to interact with systems and signals by changing the parameters of functions, providing a rich educational experience.
Some of the concepts behind the classic formulations are explained and interpreted with their historical progresses. For example, the origin and meaning of Euler’s number, the rationale behind complex numbers, the impact of complex exponentials as a building block of many complicated functions, the meaning of convolution operations, the concept of harmony, and the impact of the sampling theorem on the development of digital era are explained in a simple nontechnical language along with a mathematical formalism.
The book is designed as a one-semester (14 weeks with 3 lecture hours per week) undergraduate course material for science and engineering students, specifically computer science and engineering, electrical and electronics engineering, physics, informatics, and applied mathematics. Each chapter can be covered in 3 lecture hours per week, excluding Chapters 8 and 9, which can be split into two weeks each. For shorter semesters, the instructors can skip the mathematical preliminaries of Chapter 1. Laplace and z-transforms of Chapters 8 and 9 and Chapter 10 in full can also be omitted.
The students are expected to gain the basic knowledge and skills, which will be very useful for their future education and research in their undergraduate and graduate studies in the field of communication, computer vision, machine learning, signal, and image and video processing, among others.
Bridging the gap between theory and practice, the book also serves to the professionals, working in the above disciplines and a wide range of multidisciplinary fields, including bioinformatics, robotics, neuroscience, remote sensing, aeronautics, seismology, biomedical engineering, process control, astrophysics, cosmology, energy, and mechatronics.
Ironically, we owe this book to COVID-19 pandemic, where we ceased the class lectures and switched to online education. At that time, we urgently needed some course materials for our students to conduct the online education. We rapidly converted our handwritten notes, accumulated over years of experience, into digital files, adding a few animations and videos along the way. On the way, we crafted numerous examples and figures with the invaluable assistance of our students and colleagues. After five years of dedicated effort, we completed this book, which is now available in both PDF format with a hardcover option and for e-book readers. Additionally, the book is complemented by a companion website offering interactive materials, where readers can visualize functions, view videos, and execute Python programs. This is a brief history of our journey on the production of this book.
Now, we invite you to join us on this journey of exploring and designing systems and their expressions through signals, with the interplay of theory, practice, aesthetics, and curiosity.
AnkaraMay 2024
Fatoş Tunay Yarman Vural
Emre Akbaş
Acknowledgments
Our journey in crafting this book was a dynamic process, commencing with our handwritten notes, which we diligently transcribed into Markdown and then converted into HTML pages. Subsequently, we transitioned to interactive Jupyter notebook files before ultimately transforming the content into LaTeX format alongside the creation of a companion website. This back-and-forth iterative process involved continuous refinement, with both new content additions and existing content revisions. We would not be able to overcome all the challenges without the assistance of our students. We extend our sincere gratitude to the students who contributed to the creation of this textbook. Their dedication to excellence is reflected in the high-quality figures, meticulous typesetting, and adept problem-solving skills they exhibited throughout the project. This book would not have been realized in a short amount of time without their support.
We would like to thank our students Anıl Eren Göçer, Baran Yancı, Can Ufuk Ertenli, Kıvanç Tezören, Robin Koç, and Zafer Bora Yılmazer for their general assistance with technical issues, typesetting, and creation of figures. Their dedication and enthusiastic participation were invaluable.
We have a long list of students to thank for Matplotlib figures and TIKZ drawings: Emin Sak, Hakan Gürsoy, Meriç Buğra Haliloğlu, Kenan Kartal, Deniz Eren Yılmaz, Arda Çavuşoğlu, Adil Ahmadli, Buket Zeren, Görkem Yiğit, Çağla B. Çam, and Tuğba Tümer. We thank Furkan Genç and Tahira Kazimi for creating the initial typesets of LaTeX tables.
Special acknowledgment is due to Güneş Sucu, Can Ufuk Ertenli, and Çağlar Seylan for their assistance in the initial transcribing of handwritten notes to Markdown format, and to Ulaş Aydın, Alpay Özkan, Ali Doğan, and Ömer Köse for their invaluable aid in resolving typesetting challenges across various Markdown and LaTeX setups. We thank Oğuz Gödelek and Yavuz Durmazkeser for their assistance with Python programming problems.
Our students’ support extended to reviewing the content for typos, grammatical errors, and mathematical errors, for which we thank Gökçen Gökçeoğlu, Fırat Ağış, Elif Sena Kuru, Kıvanç Filizci, Defne Ekin, and Syed Ebad Hyder. We would like to extend our gratitude to all of the students who played a part in creating this book. While we have endeavored to acknowledge each of them by name, it is possible that we may have inadvertently overlooked some. We wish to emphasize that any omissions were entirely unintentional, and we express our sincere gratitude for the support and assistance of all who have contributed.
We are grateful to Göktürk Üçoluk for the icons that we use to denote interactive content (video or code). He designed these icons and generously donated them to our book.
We appreciate the supportive environment fostered by our colleagues in the Department of Computer Engineering at Middle East Technical University, particularly Sinan Kalkan and Göktürk Üçoluk, who offered invaluable guidance and shared their experiences in book writing.
We also acknowledge the invaluable assistance provided by Becky Cowan, Sandra Grayson, Sundaramoorthy Balasubramani, and Nandhini Karuppiah from Wiley, whose expertise and guidance were essential in shaping the content.
Additionally, we thank our PhD advisors, Bradley William Dickinson and Narendra Ahuja, for their mentorship and influence on our development as scientists and authors.
Finally, we extend our heartfelt gratitude to our beloved family members, Huseyin Vural, Eren Karaca Akbaş, Derviş Can Vural, Ekin Akbaş, Tolga Yarman, Ayşe Yarman Öztekin, Binboğa Sıddık Yarman, and Faruk Ağa Yarman. Their unwavering support and understanding throughout the book-writing journey have been invaluable.
This textbook stands as a testament to the collective efforts and contributions of all those involved.
Fatoş Tunay Yarman VuralEmre Akbaş
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/signalsandsystems
This website offers interactive materials, where readers can visualize functions, view videos, and execute Python programs.
1Introduction to Systems and Signals
“There is nothing more practical than a good theory!”
Vladimir Vapnik
This book is about the mathematical representation of systems and signals.
Let us start by describing the meaning of the words systems and signals.
The origin of the word systems dates back to 15th century, when it was used as a Latin word systema, which means the entire universe. Since then, this very wide meaning has narrowed to a set of connected items or devices that operate together. In the context of this book, a system can be defined as a unified collection of interrelated and interdependent parts. And in many cases, it is more than the summation of its parts.
The aforementioned definition is quite flexible and may cover both natural or human-made systems. It can be as large as a planet, a star, or a galaxy, or as small as a single cell, a molecule, or a microchip.
In this book, we shall use the systems approach to model, analyze and investigate the natural systems, and design and implement human-made systems.
Motivating Question: What is the systems approach?
The systems approach is a holistic paradigm to mathematically represent a system. Holism is the philosophy, which accepts a system as a whole, not only as a collection of its parts. It is the opposite of the reductionist paradigm, which assumes that a complex system can be represented by its simpler components. For example, in a reductionist paradigm, a puzzle can be represented by the collection of its pieces, which come in a box. However, when we turn the box of puzzle over a table, we see all the pieces, but we cannot perceive the theme of it. On the other hand, in the systems approach, we need to do the puzzle and look at the ordered puzzle to see that it consists of a picture (Figure 1.1).
In order to model a system using the holistic paradigm, we not only represent the attributes of its multiple components, but also formulate their inter-relationships, considering the objective of the entire system. This approach implicitly models the synergy created by a system.
Learn more about the systems approach @ https://384book.net/v0101
Figure 1.1Waterfall by M.C. Escher1 The puzzle on the left consists of the pieces of the entire lithograph, but have no meaning. In order to observe the falling water of the watermill, we need to solve the puzzle.
Source: The M.C. Escher Company B.V/https://mcescher.com/gallery/impossible-constructions/#iLightbox[galleryimage1]/5/last accessed March 09, 2024.
The origin of the word signal is even older than that of systems, dating back to 13th. century. It comes from the Latin word signale, which means anything that serves to indicate or communicate information.
When we observe a signal, we assume that there is a source system, which generates the signal. Thus, signals can be considered as partial information about the systems. In most cases, systems can be modeled and represented by a collection of subsystems. The interrelations among the subsystems of a system can be modeled by the received input signal(s) and the generated output signal(s), i.e., signals, of each subsystem.
In summary, the response of a system to a specific set of input signals provides information about the properties of systems. Signals describe the interrelations among the parts of a system. Loosely speaking, signals are the measurements of our varying observations about a system and/or its parts.
1.1 Example Applications
Models for representing signals and systems are widely used in electrical engineering and computer science for filter design, control, communications, computer vision, machine learning, speech, image, and video processing. The formalism of signals and systems is also used in a wide range of multidisciplinary areas, including bioinformatics, robotics, neuroscience, remote sensing, aeronautics, seismology, biomedical engineering, chemical process control, energy and mechatronics, astronomy, and cosmology.
Let us give some examples, where the methodologies of the systems approach are intensively utilized, in the modeling, design, and implementation stages of natural and human-made systems. Most of these models are generated by using the signals measured at the input and/or output of the systems.
Figure 1.2 Digital terrain model of a transportation area obtained from LIDAR scanning.
Source: black_mts/Adobe Stock.
1.1.1 Three-Dimensional World Models by LIDAR Signals
Light detection and ranging (LIDAR) signals are generated by a source that emits laser beams. These signals bounce off the surrounding objects and return to a sensor. Systems approach is, then, used to create a three-dimensional representation of the physical environment by measuring the elapsed time for each laser pulse to return to the sensor (Figure 1.2).
Learn more about the LIDAR example @ https://384book.net/v0102
1.1.2 Modeling the Brain Networks from the Brain Signals
Functional magnetic resonance imaging (fMRI) technique records the brain signals, which indirectly measure the activities in the anatomical regions. It is possible to model and analyze the cognitive processes, such as vision, speech, and memory of the human brain from the fMRI signals.
Representing brain activities by networks is crucial to understand various cognitive states. It is possible to extract brain networks from the fMRI recorded while the subjects perform a predefined cognitive task. Figure 1.3 shows two brain networks for planning and execution phases, while the subject solves a complex problem. The suggested computational network model can successfully discriminate the planning and execution phases of complex problem-solving process with more than 90% accuracy, when the estimated dynamic networks, extracted from the fMRI data, are classified by a machine learning algorithm.
Figure 1.3 Visualizing anatomical regions during both the planning (a) and execution (b) phases while the selected subject solves a complex problem.2
Source: With permission of IEEE.
Figure 1.4 Example of building detection using remote sensing applications.3
Source: With permission of IEEE.
An example: speech synthesis from neural decoding of spoken sentences @ https://384book.net/v0103
1.1.3 Detecting the Buildings from the Remote-Sensed Satellite Images
Remote sensing images are recorded by measuring the signals of several electromagnetic waveforms reflected from the earth’s surface. These signals are used to extract various information, such as measuring environmental pollution or climate change, the growth rate of cities or green areas, etc.
One important application of remote sensing is to detect the buildings in municipalities. For this purpose, a multidimensional signal measured from the earth’s surface is modeled to filter the buildings in the remotely sensed data, as shown in Figure 1.4.
1.1.4 Noise Reduction in Old Records
Due to the technological limitations of their time, the old gramophone recordings are mostly noisy. These recordings can be cleaned by using methodologies of signal processing. Additive noise is partially eliminated by estimating a mathematical model for the noise and subtracting it from the corrupted signal. An example of noise reduction can be found in the companion website of the book.
Noise reduction on “O’sole mio” @ https://384book.net/v0104
1.2 Relationship Between Signals and Systems
The aforementioned brief descriptions and examples of signals and systems show that there is a remarkable relationship between the signals and the underlying system, which generates the signal. Philosophically, one may consider the signals as the manifestation of systems. We, humans, can perceive the physical world through these manifestations. Heraclitus of Ephesus summarizes this view by his famous saying:
which translates to English as:
All flows!
Almost 2500 years ago, Heraclitus claimed that everything changes. Since then, as we study the nature, we discover some invariant laws, which lie behind the changes. Although we can only perceive the world of flux, these invariant laws govern our changing observations. In other words, we can only perceive variances, generated by the invariant laws, which govern the natural systems. Our aim is to find these invariant laws, manifested through our varying observations.
To analyze and understand a natural system or design and implement a human-made system, we need rigorous mathematical representations of signals, which correspond to our varying observations. Based on these observations, we can model the invariant rules of a system, which administers a set of prescribed tasks.
Motivating Question: How can we analyze and understand the laws that govern the natural systems? How can we design a human-made system to achieve a specific goal?
The answers to these questions require mathematical representation of systems and/or their subsystems. To follow the holistic approach, we also need mathematical representation of signals, which describe interrelations among the subsystems and the interaction between a system with its environment.
1.3 Mathematical Representation of Signals and Systems
There are many ways to formally represent signals and systems. In the context of this book,
signals are represented by
functions
,
systems are represented by
equations and/or algorithms
.
Loosely speaking, a system receives an input signal, represented by a function , and generates an output signal, represented by a function , for this particular input. The relationship between the input and output signals provides us with system equation or algorithm (Figure 1.5).
Throughout this book, we shall study the signals by representing and manipulating them with well-known mathematical objects, namely functions. We shall, also, study the systems by establishing the relationship between the input and output signals using a class of equations, namely the differential and integral equations. We pay special attention to linear systems, not only because of their mathematical tractability, but also because they open the door to analyze and design a wide range of nonlinear systems.
Figure 1.5 Schematic representation of a system by a box, which consists of an equation or an algorithm. A system receives an input signal, and generates an output signal, . The equation or algorithm relates the input signal to the output signal.
In the rest of this chapter, we shall provide a brief overview of functions to represent and manipulate signals. We shall, also, study a very interesting property of functions, called symmetry.
1.3.1 Signals Represented by Functions
Let us start by recalling the definition of a function. A function associates the elements in a domain set and the elements in a range set. Formally, a function,
