Transform theory and applications for engineering & technology students

Abstract

This article presents a streamlined overview of Fourier, Laplace, and $\mathcal{Z}$ -transforms for a sophomore- or junior-level signals and systems course in electrical/computer engineering and engineering technology programs. Transform methods are introduced after foundational circuit analysis and serve as prerequisites to studies in communications, controls, power, and signal processing. Conventional textbooks define transforms formally and rely on tables of properties and transform pairs, often assuming prior mathematical preparation. However, this approach is challenging for engineering technology students, who may have completed only Calculus I and II yet are expected to meet ABET-driven learning outcomes requiring advanced mathematical competencies. To bridge the gap between formal theory and engineering applications, this article suggests an alternative instructional strategy that begins with an exposure to seven interconnected topics over one or two lectures, progressing from familiar to more complex material. These topics are revisited during the semester to reinforce understanding and contextualize applications. The sequence comprises: (i) logarithms and phasors; (ii) vector basis representation; (iii) polynomial functions; (iv) periodic functions and Fourier series; (v) Fourier transform for non-periodic functions; (vi) Laplace transform for functions not suited to Fourier analysis; and (vii) sampling and the $\mathcal{Z}$ -transform. By incrementally increasing conceptual complexity and leveraging visualization tools such as MATLAB, this approach demystifies transform theory and enhances students’ appreciation for the application of transforms in engineering contexts. The article offers a valuable resource for students, instructors, and textbook authors seeking a more incremental approach to transform theory instruction.

Introduction

Transform theory has had an invaluable impact in virtually all fields of science and engineering (E). For example, the Fourier series and the Fourier transform ( $\mathcal{F}T$ ) have enabled incredible advances in the analysis and design of communication systems and in signal/image processing. Similarly, Laplace and $\mathcal{Z}$ -transforms ( $\mathcal{L}T$ and $\mathcal{Z}T$ ) are the tools-of-choice in classical control system analysis and design techniques.

A typical second- or third-year electrical and computer E or engineering technology (ET) student taking a “Signals and Systems” or “Linear Systems” course may feel frustrated by the abstract and rapid introduction of various transforms in standard textbooks. E and ET programs often assume that prerequisite math courses have covered these topics in depth, but this may not be the case. Students are then expected to accept that certain problems become easier to solve by applying specific transforms. While examples and homework exercises may demonstrate this advantage, an air of mystery often remains, and much foundational understanding is often lost.

To illustrate the problem, consider the section of a typical control textbook that introduces the unilateral $\mathcal{L}T$ $F(s)$ of a function $f(t)$ as Golnaraghi and Kuo ¹ and Dorf and Bishop ²

$$F(s)={\int }_{0^{-}}^{\mathrm{\infty }}f(t)e^{-st}\text{d}t$$

and proceeds to cover properties and applications, such as linearity, the initial and final value theorems, and solutions of linear, time invariant, ordinary differential equations, including partial fraction expansions for computing the inverse $\mathcal{L}T$ .

It is normally assumed that the students have been adequately exposed to the $\mathcal{L}T$ in a preceding “Signals and Systems” or “Linear Systems” course. The problem is, however, that virtually all signals and systems textbooks of the last three decades^3–13 take a similar approach or at best, provide further insight using application areas scattered throughout the book. Exceptions exist, for example, ³ which includes a chapter on expansion theory and Fourier series. A drawback is that such a presentation uses an unnecessarily sophisticated mathematical language for the intended audience of second-semester sophomores or first-semester juniors. The material may therefore be inaccessible at that level without considerable additional lecture time or background from mathematics courses beyond calculus and differential equations. Furthermore, the educational challenge is exacerbated for ET students whose degree plans may require only Calculus II-level mathematics.

This article addresses these shortcomings by suggesting a sequence of topics in a mathematically simple and relaxed language that helps the students develop a clearer grasp of transform theory. Although some of the topics may be found in the previously cited textbooks with varying degrees of depth, it is the sequence itself that is educationally effective. The seven sequential and highly interconnected topics listed below can be covered with examples in two standard 50-min lecture periods:

Logarithms and phasors: these introduce the notion of a transformation in already familiar settings such as alternating current circuit analysis.

Vectors in ${\mathrm{\Re }}^n$ : linear independence, span, and basis.

Extension to functions: the main example used is $n$ th-degree polynomials.

Periodic functions: by analogy with the previous items, the trigonometric Fourier series is introduced as a countably infinite sum of sinusoids (basis). The exponential form of the Fourier series is covered next as a prelude to the $\mathcal{F}T$ .

Non-periodic functions: the standard limiting argument is used to introduce the $\mathcal{F}T$ from the exponential form of the Fourier series. The notion that the $\mathcal{F}T$ is an uncountably infinite sum of sinusoids is stressed.

Functions that are not Fourier transformable: the function $f(t)=e^t$ is used to illustrate this topic. The integral convergence problem in the $\mathcal{F}T$ is resolved by introducing the complex variable $s=\sigma +j\omega$ and a region of convergence, thereby defining the bilateral $\mathcal{L}T$ . Causal signals and systems then lead to the unilateral $\mathcal{L}T$ .

The $\mathcal{L}T$ of a sampled signal: a typical signal $f(t)$ is sampled and its $\mathcal{L}T$ leads to the $\mathcal{Z}T$ .

The approach is for an intended audience consisting of E and ET students who have the standard background in circuit analysis, and vector manipulation in ${\mathrm{\Re }}^3$ as developed in mathematics, physics, or dynamics courses. The material in Topics 1–6 has been used effectively in the Junior-level “Linear Systems Analysis” (ELET 3301) and senior level “Control Systems” (ELET 4304) courses in ET at the University of Houston’s Technology Division of the Cullen College of Engineering, and in the senior level “Control Systems” (ETEC 4720) and graduate level “Analytical Methods in ET” (MSET 5040) in ET, College of Engineering at the University of North Texas (2014–2018). In ET, Topic 7 may be covered for juniors and seniors as an advanced application, but is more suitable for graduate students.

The remainder of this article is organized and presented as a typical lecture covering the suggested list of topics. In the interest of brevity, very few examples are included. Derivations and many of the more obvious remarks are also omitted. Instructors are encouraged to make a connection to senior-level courses in communications or controls, and to offer additional examples and math details if desired as part of their lecture materials. Finally, the expectation is that one returns and expands as needed on any of these subjects as the semester progresses.

The logarithm transform and phasors

The general objective is to explore how functions of time called signals, that are widely used in E, may be represented in various ways using an appropriate transformation. This is done to ease the application of certain mathematical operations in signal analysis and system design, and to uncover certain features of the signal that are not apparent in its original time-domain form.

Some of the most common transformations used in science and E are the $\mathcal{F}T$ , $\mathcal{L}T$ , and $\mathcal{Z}T$ . This lecture navigates through various mathematical tools and shows how these transformations have a unifying conceptual thread. Transform applications to solve E problems then become much easier to understand.

Consider first computing an expression such as

$$y=\frac{x_1^{1.25}}{x_2^{-0.25}}$$

for two real and positive numbers $x_1,x_2$ , where in this context, the exponential and division operations are deemed to be challenging. The logarithm transformation applied to both sides of the equation changes the domain and range from numbers to the natural logarithm of numbers, but more importantly, it transforms the operations into addition and subtraction, that is,

$$y=\frac{x_1^{1.25}}{x_2^{-0.25}}\mapsto \ln y=z=1.25\ln x_1-(-0.25)\ln x_2$$

which is easier to do. After adding the two quantities in the logarithm domain, a reverse operation is needed to convert the answer $z$ back to the original number domain. This operation is the natural logarithm inverse that converts $z$ to $y=e^z$ . It is important that any transformation of practical use be invertible to guarantee that uniqueness is not lost, so that in the above example, $z$ leads to one and only one possible answer $y$ without ambiguity.

Another example of transforming to a different domain where certain operations become much easier to perform is the use of phasors in electric circuit analysis. A series RLC circuit driven by a voltage source $v(t)$ shown in Figure 1 leads to the second-order, linear, ordinary differential equation (1) that governs the current $i(t)$

$$\frac{{\text{d}}^{2}i(t)}{\text{d}{t}^2}+\frac{R}{L}\frac{\text{d}i(t)}{\text{d}t}+\frac{1}{LC}i(t)=\frac{1}{L}\frac{\text{d}v(t)}{\text{d}t}$$

(1)

that can be solved given the input voltage $v(t)$ and a set of initial conditions $i(0)$ and $(\text{d}i/\text{d}t)(0)$ . When the excitation source is purely sinusoidal $v(t)=V_0\cos (\omega t+{\varphi }_{in})$ , the forced component of $i(t)$ is most conveniently found by transforming the circuit to its phasor-domain equivalent. The signals become phasors

$$\begin{array}{rl}v(t)& =R_e\{V_0{e}^{j(\omega t+{\varphi }_{\text{in}})}\}\leftrightarrow \overline{V}=V_0{e}^{j{\varphi }_{\text{in}}}\\ i(t)& =R_e\{I_0{e}^{j(\omega t+{\varphi }_{\text{out}})}\}\leftrightarrow \overline{I}=I_0{e}^{j{\varphi }_{\text{out}}}\end{array}$$

where $R_e$ denotes the operation of keeping the real part of the indicated complex function. The double arrow implies uniqueness or a one-to-one correspondence between the time/phasor pair $v(t)\leftrightarrow \overline{V}$ . It is useful to think of each circuit phasor as a vector in the complex plane having a magnitude, an angle, and rotating counterclockwise at a frequency of $\omega$ . For example, the phasor $\overline{V}$ is a vector of magnitude $V_0$ and angle ${\varphi }_{in}$ , rotating at the frequency of $\omega$ .

Figure 1.

A series RLC circuit.

Each circuit element is also transformed to its equivalent complex impedance

$$R⟶R;C⟶\frac{1}{j\omega C};L⟶j\omega L$$

Then, the single loop equation $\overline{V}=\overline{Z}\overline{I}$ , can be solved for the phasor current $\overline{I}=\overline{V}/\overline{Z}$ , where $\overline{Z}$ is the circuit complex impedance evaluated at the excitation frequency $\omega$ ,

$$\overline{Z}=R+j\omega L+\frac{1}{j\omega C}=|\overline{Z}|e^{j{\varphi }_z}$$

The result is

$$\overline{I}=\frac{\overline{V}}{\overline{Z}}=\frac{V_0{e}^{j{\varphi }_{\text{in}}}}{|\overline{Z}|e^{j{\varphi }_z}}=\frac{V_0}{|\overline{Z}|}{e}^{j({\varphi }_{\text{in}}-{\varphi }_z)}=I_0{e}^{j{\varphi }_{\text{out}}}$$

Similar to the logarithm example, a return to the original domain, in this case time, $\overline{I}\leftrightarrow i(t)$ , is easily done

$$i(t)=I_0\cos (\omega t+{\varphi }_{in}-{\varphi }_z)$$

In summary, a relatively challenging mathematics operation of solving the second-order differential equation (1) was effectively replaced by a much simpler one, in this case, algebraic manipulations of complex numbers.

Vector representation

Consider the familiar set of vectors used to label the perpendicular axis of the $(x,y)$ -plane

$${\overline{u}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right];{\overline{u}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

(2)

It turns out this is a very convenient set of vectors because any other vector $\overline{v}\in {\mathrm{\Re }}^2$ is uniquely represented as a linear combination of $({\overline{u}}_1,{\overline{u}}_2)$ , meaning there exist unique scalars $(a_1,a_2)$ such that

$$\overline{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=a_1{\overline{u}}_1+a_2{\overline{u}}_2=a_1\left[\begin{array}{c}1\\ 0\end{array}\right]+a_2\left[\begin{array}{c}0\\ 1\end{array}\right]$$

(3)

Clearly, $(a_1,a_2)=(v_1,v_2)$ are just the components of $\overline{v}$ and no calculations were needed at all to find them. The reason $(a_1,a_2)$ are so easily found is that the chosen vectors $(u_1,u_2)$ in addition to having unit length, have an inner angle of $90\circ$ and the set forms an orthonormal coordinate system.

Orthogonality of $({\overline{u}}_1,{\overline{u}}_2)$ can be easily seen geometrically, but also algebraically from the inner product operation between two vectors $\overline{v},\overline{w}$ defined by

$${\overline{v}}^{\mathrm{\top }}\overline{w}=v_1{w}_1+v_2{w}_2=|\overline{v}||\overline{w}|\cos (\theta )$$

where $(\cdot {)}^{\mathrm{\top }}$ is the transpose operation, and $\theta$ is the inner angle between $\overline{v}$ and $\overline{w}$ . Hence, ${\overline{u}}_1^{\mathrm{\top }}{\overline{u}}_2=0\Rightarrow \theta =90\circ$ .

The orthonormality property is very relevant to transform theory, as shown next. Consider the following calculation: multiply equation (3) from the left by ${\overline{u}}_1^{\mathrm{\top }}$ and find that the coefficient $a_1$ is easily computed to be

$${\overline{u}}_1^{\mathrm{\top }}\overline{v}=a_1{\overline{u}}_1^{\mathrm{\top }}{\overline{u}}_1+a_2{\overline{u}}_1^{\mathrm{\top }}{\overline{u}}_2\Rightarrow a_1=v_1$$

(4)

Similarly, repeating these operations with ${\overline{u}}_2$ yields $a_2=v_2$ . The vector $\overline{v}$ is then said to be expressed with respect to the set $({\overline{u}}_1,{\overline{u}}_2)$ . The extension to the $n$ -dimensional space ${\mathrm{\Re }}^n$ , where each vector has $n$ components, is straightforward.

As an exercise, consider expressing $\overline{v}=[-0.5\pi {]}^{\mathrm{\top }}$ with respect to the unit vectors but also with respect to a second set of arbitrarily chosen vectors, such as

$${\overline{w}}_1=\left[\begin{array}{c}1\\ 2\end{array}\right];{\overline{w}}_2=\left[\begin{array}{c}3\\ 4\end{array}\right]$$

which are not special in any obvious way, that is, they do not have unit length and their inner angle is $\theta =10.3\circ$ . The result is $a_1=0.5(2+3\pi )$ and $a_2=-0.5(1+\pi )$ . The example highlights the convenience of using the orthonormal set $({\overline{u}}_1,{\overline{u}}_2)$ instead. This fact will again be exploited in the context of Fourier series representations of periodic functions.

How about other possible representations of $\overline{v}$ ? Consider the set of vectors $({\overline{u}}_1,{\overline{u}}_2,{\overline{w}}_3)$ , where ${\overline{w}}_3$ is arbitrary. It turns out that there is an infinite number of possible representations for $\overline{v}$ , so uniqueness is lost. What makes one representation better than another? The answers lie in the observation that because ${\overline{w}}_3$ itself can be written as a linear combination of $({\overline{u}}_1,{\overline{u}}_2)$ , in some sense the set $({\overline{u}}_1,{\overline{u}}_2,{\overline{w}}_3)$ is redundant and provides no more flexibility or completeness than $({\overline{u}}_1,{\overline{u}}_2)$ to represent a vector $\overline{v}\in {\mathrm{\Re }}^2$ . These questions are formally addressed in a linear algebra course under the concepts of linear independence, span, and basis. For our purposes, it suffices to say that the orthonormal set $({\overline{u}}_1,{\overline{u}}_2)$ is a basis for ${\mathrm{\Re }}^2$ and contains just enough flexibility (or completeness) to allow us to write any arbitrary vector $v\in {\mathrm{\Re }}^2$ as a unique linear combination of $({\overline{u}}_1,{\overline{u}}_2)$ .

Fourier series

The notion of a basis vector set from the previous section has been extended to spaces containing functions of the scalar $t$ . For example, using the set $(1,t,t^2,t^3)$ , we can generate any cubic polynomial $f(t)$ of the form

$$f(t)=a_0+a_{1}t+a_2{t}^2+a_3{t}^3$$

so that $f(t)$ is uniquely identified by the set of coefficients $(a_0,a_1,a_2,a_3)$ . We could use a different set of functions, for example, $(3.5,2t-1,\pi t^2,7t^3-4t^2+5t+2)$ , which offers no clear advantage over the convenience of the set $(1,t,t^2,t^3)$ .

Next, consider the space of functions that are periodic in time with period $T$ , that is, $f(t+T)=f(t)$ , so the fundamental shape of the function repeats itself every $T$ seconds. For example, $f(t)=\sin (4.2t)$ is periodic with period $T=2\pi /4.2$  s. Periodicity is quite natural in the sciences and in E. In some applications, one can also create a periodic function by extension, for instance, $f(t)=e^t$ is not periodic; but if we stipulate that $f(t)$ repeats itself every $T=1.92$  s, the result is the shape of $f(t)=e^t$ repeating every $T$ seconds. Other common signals created by extension are a square wave (train of pulses) and a triangular wave or saw-tooth.

Is there an equivalent notion of basis for such a function space? The French mathematician and physicist Jean-Baptiste Joseph Fourier (1768–1830) used earlier work by other giants such as Newton, Euler, and Bernoulli in solving problems related to heat transfer, and managed to introduce the countably infinite set of functions $(\sin (n{\omega }_{0}t),\cos (m{\omega }_{0}t)$ , where ${\omega }_0=2\pi /T$ and $n,m$ are integers $0,1,2,\dots$ (hence the term countably infinite), as a very convenient way to represent a periodic function $f(t)$ . Here, ${\omega }_0$ is called the fundamental frequency, and the $n$ th (or $m$ th) frequency $n{\omega }_0$ ( $m{\omega }_0$ ) is a harmonic frequency, that is, an integer multiple of the fundamental. In so doing, he provided solutions to heat transfer problems described by a partial differential equation (heat equation), and his method has been extended to a broad variety of problems in acoustics, vibrations, signal and image processing, communications, solutions of differential equations, and many more!

The history of the Fourier series is quite interesting to read as it involves many well-known mathematics figures of the 18th and 19th centuries. It took many years for the community to finally accept the Fourier series as a viable analysis approach, even if it contained several open technical questions, including convergence and what exactly happens at a point of discontinuity of a function such as a pulse train. The whole subject is fascinating, and you are highly encouraged to read more about it, even from a historical perspective.

We will assume in this presentation that $f(t)$ is a periodic and practical signal (also referred to as “nice”) satisfying some technicalities known as the Dirichlet conditions, meaning the function admits a formal Fourier series representation. The Fourier series expansion for such $f(t)$ with period $T$ , and fundamental frequency ${\omega }_0=2\pi /T$ is

$$f(t)=a_0+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t))$$

(5)

where the sinusoidal functions’ amplitudes are tuned to the unique values $(a_n,b_n);n=1,2,\dots$ given by the expressions

$$\begin{array}{rl}& a_n=\frac{2}{T}{\int }_{T}f(t)\cos (n{\omega }_{0}t)\text{d}t\\ & b_n=\frac{2}{T}{\int }_0^{T}f(t)\sin (n{\omega }_{0}t)\text{d}t\end{array}$$

The coefficient $a_0$ is called the direct current (DC) component in electrical applications, or the average value given by

$$a_0=\frac{1}{T}{\int }_{T}f(t)\text{d}t$$

and the above integrals may be done over any arbitrary full period. Also, the sinusoidal functions are each at very specific frequencies, namely, integer multiples of the fundamental ${\omega }_n=n{\omega }_0$ .

Fourier series expansions are conceptually quite similar to the vector expansions of the previous section. It turns out that much like the orthogonal set (2) is a basis for ${\mathrm{\Re }}^2$ , the set of sinusoidal functions $(\sin (n\omega t),\cos (m\omega t)),n,m=1,2,\dots$ is a basis for the space of periodic functions. Moreover, the calculation that led to equation (4) exploited the orthogonality of unit vectors $({\overline{u}}_1,{\overline{u}}_2)$ . It can be shown that for $n,m\ne 0$ the functions $(\sin (n\omega t),\cos (m\omega t))$ are also orthogonal in the sense that they satisfy the following definite integrals called the orthogonality conditions:

$$\begin{array}{rl}& {\int }_T\sin (n{\omega }_{0}t)dt={\int }_T\cos (n{\omega }_{0}t)dt=0\\ & {\int }_T\sin (n{\omega }_{0}t)\cos (m{\omega }_{0}t)dt=0\\ & {\int }_T\sin (n{\omega }_{0}t)\sin (m{\omega }_{0}t)dt=\frac{T}{2}{\delta }_{nm}\\ & {\int }_T\cos (n{\omega }_{0}t)\cos (m{\omega }_{0}t)dt=\frac{T}{2}{\delta }_{nm}\end{array}$$

where ${\delta }_{nm}=1,n=m$ and ${\delta }_{nm}=0,n\ne m$ .

Returning to the steps leading to equation (4), the expressions for the Fourier coefficients can be derived in a similar fashion: multiply both sides of equation (5) by $\cos (m{\omega }_{0}t)$ , integrate over $(0,T)$ , and use the above orthogonality conditions to find $a_n$ . Repeat using $\sin (m{\omega }_{0}t)$ to get the coefficients $b_n$ .

We conclude this section with the complex form representation of the Fourier series. Using the identities

$$\begin{array}{rl}\cos (n{\omega }_{0}t)& =\frac{e^{jn{\omega }_{0}t}+e^{-jn{\omega }_{0}t}}{2}\\ \sin (n{\omega }_{0}t)& =\frac{e^{jn{\omega }_{0}t}-e^{-jn{\omega }_{0}t}}{2j}\end{array}$$

and after a little algebra, one finds that

$$f(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{jn{\omega }_{0}t}$$

(6)

where

$$c_n=\frac{1}{T}{\int }_{-T/2}^{T/2}f(t)e^{-jn{\omega }_{0}t}\text{d}t=\frac{1}{2}(a_n-j{b}_n)$$

are the complex Fourier coefficients. One can verify that $c_0=a_0$ , $|c_n|=(1/2)\sqrt{a_n^2+b_n^2}$ , and $c_{-n}=c_n*$ (where $(\cdot )*$ means complex conjugate) for real functions $f(t)$ . Tables exist that list Fourier series expansions for a large number of signals of interest in science and E.

An application of Fourier series

This example illustrates the opportunity to connect the Fourier series with phasors. First, the differential operator $p=\text{d}/\text{d}t$ is introduced as a convenient notation change so that a capacitor’s voltage to current relation $i(t)=C(\text{d}v(t)/\text{d}t)$ , and an inductor’s $v(t)=L(\text{d}i(t)/\text{d}t)$ are replaced, respectively, by

$$i(t)=Cpv(t)\text{and}v(t)=Lpv(t)$$

which identify the capacitor’s operational impedance as $Z_c(p)=1/Cp$ and the inductor’s as $Z_L(p)=Lp$ .

To illustrate the approach, consider a series RC circuit driven by a voltage source $v_s(t)$ shown in Figure 2 driven by the square wave voltage source of Figure 3 with a Fourier series

$$v_s(t)=\frac{4}{\pi }(\sin (t)+\frac{1}{3}\sin (3t)+\frac{1}{5}\sin (5t)+\cdots )$$

Figure 2.

A series RC circuit.

Figure 3.

Square wave voltage source driving a series RC circuit.

By voltage division and setting $\mathrm{RC}=1$ , we can write

$$v_c(t)=G(p)v_s(t)=\frac{\frac{1}{Cp}}{\frac{1}{Cp}+R}{v}_s(t)=\frac{1}{p+1}{v}_s(t)$$

(7)

where $G(p)$ is the operational transfer function (OTF) of the RC circuit. Using superposition, the steady-state response to the $n$ th-harmonic input

$$v_{sn}(t)=A_n\sin ({\omega }_{n}t)=A_n\cos ({\omega }_{n}t-\frac{\pi }{2})$$

is simply the corresponding $n$ th-harmonic of the Fourier series representation of $v_c(t)$ obtained from equation (7) by replacing time-domain functions by their phasor representations and $p\to j{\omega }_n$ ,

$$V_{cn}=\frac{1}{1+j{\omega }_n}{V}_{sn}=\frac{1}{1+jn{\omega }_0}{A}_n{e}^{-j\frac{\pi }{2}}$$

where ${\omega }_0=2\pi /T=1$ . Computing the first few terms, $v_c(t)$ may be written as follows:

$$\begin{array}{rl}{v}_c(t)& =0.9\cos (t-2.36)+0.1342\cos (3t-2.82)+\\ & +0.05\cos (5t-2.94)+0.026\cos (7t-3.00)\cdots \end{array}$$

illustrating the low-pass filter behavior of this circuit. Other RLC circuit examples are used at this point to gain practice in developing the OTF $G(p)$ using nodal, mesh, or Thevenin equivalents and the Symbolic Toolbox of MATLAB (see Appendix A, note 1).

In general, high-frequency components (large $n$ and $m$ ) are needed to better approximate sharp edges such as a corner; also, the magnitude of the coefficients decreases as $n,m$ increase. This observation is true for noise, which is characterized by high-frequency components. Hence, in audio applications where a signal may be corrupted by noise, its Fourier series identifies those frequency components (usually the first few) that are absolutely needed to represent the signal itself. A filter can then be designed to attenuate the higher frequency components and thus improve the audio quality.

The $\mathcal{F}T$

Fourier series permit the representation of periodic functions as a countably infinite sum of sinusoids, each of a certain amplitude and discrete frequency called harmonics. In E applications, sometimes we are interested in functions that do not have a Fourier series representation. This means the function does not belong to the function space generated by the sine and cosine functions in the previous section. One such function is a single pulse shown in Figure 4. The function $p_2(t)$ cannot be built by summing discrete frequency components. The $\mathcal{F}T$ addresses this situation by expanding the function set from discrete frequency components $\{\cos (n{\omega }_{0}t)\}$ to a continuum of frequency components $\{\cos (\omega t)\}$ .

Figure 4.

Unit pulse signal.

For the purpose of this presentation, it suffices to state that for functions of time that satisfy certain technical requirements, there is a unique representation of $f(t)$ by a continuous function $F(\omega )$ in the frequency domain given by the relations

$$F(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-j\omega t}\text{d}t\leftrightarrow f(t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(\omega )e^{j\omega t}\text{d}\omega$$

(8)

called the $\mathcal{F}T$ pair, where once again, the double arrow implies a one-to-one correspondence or uniqueness of the pair $f(t)\leftrightarrow F(\omega )$ . One can rely on tables listing many signals of interest in science and E and their $\mathcal{F}T$ .

Upon examining the above relation and with reference to the complex representation of the Fourier series in equation (6), the following observations can be made:

The discrete frequency $n{\omega }_0$ became a continuous variable $\omega$ .

The summation done for countably infinite harmonic frequency components became an integral.

The integral used to compute $F(\omega )$ is obtained by a limiting argument applied to the complex Fourier coefficient $c_n$ .

As before, the double arrow indicates this relation is one-to-one (unique).

The $\mathcal{F}T$ will, in general, have a real part and an imaginary part. Hence, some authors will denote the $\mathcal{F}T$ as $F(j\omega )$ .

Examples of $\mathcal{F}T$

Of particular instructive interest is the derivation of the $\mathcal{F}T$ of $\cos ({\omega }_{0}t)$ . Suppose we ask what function $f_1(t)$ has an impulse as its $\mathcal{F}T$ ? That is,

$$F_1(\omega )=\delta (\omega -{\omega }_0)\leftrightarrow f_1(t)=?$$

The answer is that such a function will have to be given by the inverse $\mathcal{F}T$

$$f_1(t)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega -{\omega }_0)e^{j\omega t}\text{d}\omega$$

and due to the sampling property of the $\delta (\omega )$ function we find the integral yields the complex function

$$f_1(t)=\frac{1}{2\pi }{e}^{j{\omega }_{0}t}=\frac{1}{2\pi }(\cos ({\omega }_{0}t)+j\sin ({\omega }_{0}t))$$

By the same token, then, we can construct the complex conjugate of $f_1(t)$

$$\begin{array}{rl}{F}_2(\omega )& =\delta (\omega +{\omega }_0)\leftrightarrow f_2(t)=\frac{1}{2\pi }{e}^{-j{\omega }_{0}t}\\ f_2(t)& =\frac{1}{2\pi }(\cos ({\omega }_{0}t)-j\sin ({\omega }_{0}t))\end{array}$$

Upon adding $f_1(t)$ and $f_2(t)$ , we get the result that a real cosine function of time is equivalently represented in the frequency domain by a pair of symmetrically located delta functions

$$\cos ({\omega }_{0}t)\leftrightarrow \pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$$

This $\mathcal{F}T$ pair is a key result that makes modulation technology possible for transmitting signals as radio waves in communication systems.

As an illustrative modulation example, the audio signal $a(t)\leftrightarrow A(\omega )$ containing frequency components up to $20\mathrm{kHz}$ is multiplied with a carrier signal $c(t)=\cos (2\pi f_{c}t),$ where $f_c=100\mathrm{MHz}$ . The $\mathcal{F}T$ of the transmitted signal $a(t)c(t)$ is

$$a(t)c(t)\leftrightarrow \pi [A(\omega +{\omega }_0)+A(\omega -{\omega }_0)]$$

At the receiving end, the received signal is again multiplied by a local oscillator tuned to the carrier frequency. Assuming perfect transmission and reception, the resulting signal contains a copy of the transmitted signal $a(t)$ plus another copy shifted to $2f_c$

$$r(t)=a(t)c^2(t)=\frac{1}{2}a(t)-\frac{1}{2}a(t)\cos (4\pi f_c(t))\leftrightarrow R(\omega )$$

A sketch of $R(\omega )$ reveals that the content at $2f_c$ is nominally filtered out using a low-pass filter with cutoff at $20\mathrm{kHz}$ , thus recovering the transmitted signal $a(t)$ .

A second suggestive application example connects $\mathcal{F}T$ s to frequency response plots, or Bode plots, and filtering. Consider the series RC circuit of Figure 2 with capacitor voltage $v_c(t)$ governed by the first-order differential equation obtained from equation (7)

$$(p+1)v_c(t)=v_s(t)$$

The impulse response, that is, when $v_s(t)=\delta (t)$ , is equivalent to obtaining the response of the circuit to a sinusoidal input $v_s(t)=A\cos (\omega t)$ at every possible frequency value in one single test. Clearly impossible to accomplish in the lab since the $\delta (t)$ function cannot be physically generated. On paper, however, the impulse response and its $\mathcal{F}T$ are

$$v_s(t)=g(t)=e^{-t}\leftrightarrow G(\omega )=\frac{1}{j\omega +1}$$

Then, the graphs of the magnitude and angle of $G(\omega )$ as functions of frequency $\omega$

$$|G(\omega )|=\frac{1}{\sqrt{1+{\omega }^2}};\mathrm{\Theta }(\omega )=-ta{n}^{-1}(\omega )$$

constitutes the so-called Bode plot, which is a tool of choice in filtering applications (see Appendix A, note 2).

The $\mathcal{L}T$

It turns out the $\mathcal{F}T$ is not able to handle an important set of functions of interest in E, for example, exponentially unstable signals $f(t)=e^{\sigma t},\sigma >0$ . The reason these functions are of interest is that they appear in the solution of differential equations governing the performance of a system, even if there is no input signal, that is, the system responds only to the internal energy reflected in the initial conditions. In these cases, the system is deemed to be unstable, an undesirable condition in most applications, and an important performance criterion in automatic control system design.

The technical reason that unstable signals do not have a $\mathcal{F}T$ is that the time integral in equation (8) does not converge when $f(t)$ grows without bounds as $t\to \mathrm{\infty }$ . Pierre-Simon Laplace (1749–1827) was a French mathematician and astronomer who realized how to overcome this difficulty by setting the complex variable $s=\sigma +j\omega$ and expanding the set of complex sinusoids $\{e^{j\omega t}\}$ in the $\mathcal{F}T$ to the set of complex exponentials $\{e^{st}=e^{\sigma t}{e}^{j\omega t}\}$ used to define the $\mathcal{L}T$ as follows:

$$F(s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-st}dt\leftrightarrow f(t)=\frac{1}{2\pi j}{\int }_{\sigma -j\mathrm{\infty }}^{\sigma +j\mathrm{\infty }}F(s)e^{st}ds$$

(9)

In a sense, the introduction of $e^{\sigma t}$ forces the integral convergence to a meaningful function $F(s)$ in a region. The integration used to recover $f(t)$ from $F(s)$ requires some additional knowledge of complex variables outside the scope of these notes. As it is done in all applied work, though, a method called partial fraction expansions identifies the corresponding $f(t)$ from a given $F(s)$ by relying on standard tables of $\mathcal{L}T$ pairs.

Technically, the relation (9) defines the bilateral or two-sided $\mathcal{L}T$ because it considers the time interval $[-\mathrm{\infty },+\mathrm{\infty }]$ . In E applications, we are interested in the time-domain performance of causal systems, we invariably set $t\ge 0$ . The notation used to describe functions of time $f(t)$ that exist only for positive time is $f(t)u_s(t)$ , where $u_s(t)$ is the step function equal to zero for $t<0$ and equal to one for $t\ge 0$ . This notation is convenient, it restricts the time limits of integration to the interval $[0^{-},\mathrm{\infty }]$ , and defines the unilateral $\mathcal{L}T$ or simply the $\mathcal{L}T$ .

A few of the most common signals used in E system analysis and their (unilateral) $\mathcal{L}T$ s are listed in Appendix B (Table 1).

Table 1.

A brief list of common unilateral Laplace transform pairs.

$f(t)$	$F(s)$
$\delta (t)$	1
$u_s(t)$	$\frac{1}{s}$
$t{u}_s(t)$	$\frac{1}{s^2}$
$\frac{1}{2}{t}^2{u}_s(t)$	$\frac{1}{s^3}$
$e^{\sigma t}{u}_s(t)$	$\frac{1}{s-\sigma }$
$e^{\sigma t}\cos (\omega t)u_s(t)$	$\frac{s-\sigma }{(s-\sigma {)}^2+{\omega }^2}$
$e^{\sigma t}\sin (\omega t)u_s(t)$	$\frac{\omega }{(s-\sigma {)}^2+{\omega }^2}$

An application of $\mathcal{L}T$ s

A chief interest in $\mathcal{L}T$ s lies in their application to solving linear, constant-coefficient, ordinary differential equations or DFQ. As a motivating example, consider solving the second-order DFQ governing a mass, spring, damper system

$$\frac{d^{2}x(t)}{d{t}^2}+3\frac{\text{d}x(t)}{dt}+2x(t)=f(t)$$

where $x(t)$ is the mass position and $f(t)$ is an externally applied force. Furthermore, assume the system is initially at rest, that is, all initial conditions are zero $x(0)=0$ and $(dx/dt)(0)=0$ . Using the differential operator notation $p=\text{d}/\text{d}t$ , one can write the OTF

$$\frac{x(t)}{f(t)}=\frac{1}{p^2+3p+2}=\frac{1}{(p+1)(p+2)}$$

(10)

Recall our earlier discussion where we used the logarithm transformation or phasors to convert difficult operations to easier ones, albeit in a different domain. In much the same way, applying the $\mathcal{L}T$ on both sides of the differential equation, and using the first three $\mathcal{L}T$ properties in Appendix B (Table 2), the differential equation is transformed into an algebraic equation in the complex variable $s$ , where $X(s)$ and $F(s)$ are the $\mathcal{L}T$ of $x(t)$ and $f(t)$ , respectively:

$$(s^2+3s+2)X(s)=F(s)$$

This algebraic equation can be solved for $X(s)$ , yielding the nominal transfer function (NTF) as the ratio of the $\mathcal{L}T$ of the output to the $\mathcal{L}T$ of the input, with the system initially at rest

$$\frac{X(s)}{F(s)}=\frac{1}{s^2+3s+2}=\frac{1}{(s+1)(s+2)}$$

(11)

It is noteworthy that the NTF is obtained from the differential equation or the OTF by simply replacing the differential operator $p=\text{d}/\text{d}t$ by the complex number $s$ . In fact, the transformation $p\to s\to j\omega$ was used in the Fourier series example of the RC circuit.

Table 2.

A brief list of common unilateral Laplace transform properties.

$f(t)+g(t)\leftrightarrow F(s)+G(s)$

Linearity

$\frac{df}{dt}\leftrightarrow sF(s)-f(0^{+})$

Differentiation

$\frac{d^{2}f}{d{t}^2}\leftrightarrow s^{2}F(s)-sf(0^{+})-f^{\prime }(0^{+})$

Second-derivative

$\frac{d^{n}f}{d{t}^n}\leftrightarrow s^{n}F(s)-\sum _{k=1}^n{s}^{n-k}{f}^{k-1}(0^{+})$

$n$ th-derivative

$f(t-t_d)\leftrightarrow e^{t_{d}s}F(s)$

Time delay

$f(t)\otimes g(t)\leftrightarrow F(s)G(s)$

Convolution

Suppose that we wish to explore how the system responds to a unit step input force, that is, $f(t)=u_s(t)$ . From the table of $\mathcal{L}T$ , we identify $F(s)=1/s$ . Substituting into equation (11) and solving for $X(s)$

$$X(s)=\frac{1}{s(s^2+3s+2)}\leftrightarrow x(t)$$

where we indicate that $x(t)$ can be determined by the inverse $\mathcal{L}T$ operation. Next, the partial fraction expansion method rewrites $X(s)$ as the unique linear combination of terms

$$X(s)=\frac{1}{s(s^2+3s+2)}=\frac{1/2}{s}-\frac{1}{s+1}+\frac{1/2}{s+2}$$

each found in Appendix B (Table 1). Because of linearity, we inverse $\mathcal{L}T$ term by term to return to the time domain, and write the final answer

$$x(t)=(\frac{1}{2}-e^{-t}+\frac{1}{2}{e}^{-2t})u_s(t)$$

To conclude, we have solved a second-order DFQ using the $\mathcal{L}T$ as a tool that converts the DFQ into an algebraic equation. The process can be easily formalized to solve an $n$ th-order DFQ. Other circuit examples that were used to find a transfer function $G(p)$ can be brought in at this point to practice $\mathcal{L}T$ s (Appendix A, item 3).

The next section goes a step further and considers sampling of continuous-time functions leading to the $\mathcal{Z}T$ .

The $\mathcal{Z}T$

The ideal sampling process can be visualized in Figure 5, where the electronic switch or analog-to-digital converter closes every $T$ seconds to capture the signal value and produce a sequence of samples. The result is a sampled version $f*(t)=S(t)f(t)$ , where $S(t)$ is an ideal sampling function consisting of a train of equally spaced delta functions

$$S(t)=\sum _{k=0}^{\mathrm{\infty }}\delta (t-kT)$$

Figure 5.

Ideal sampling.

The sampled version of $f(t)$ is written as follows:

$$\begin{array}{rl}f*(t)& =S(t)f(t)=(\sum _{k=0}^{\mathrm{\infty }}\delta (t-kT))f(t)\\ f*(t)& =\sum _{k=0}^{\mathrm{\infty }}f(kT)\delta (t-kT)\end{array}$$

and visualized as a train of equally spaced delta functions modulated by the amplitude of $f(t)$ at each sampling time $t=kT$ .

Formally, since $\delta (t-kT)\leftrightarrow e^{-kTs}$ (Appendix B, Table 2, Time delay entry 5), it follows that the $\mathcal{L}T$ of $f*(t)$ is

$$f*(t)=\sum _{k=0}^{\mathrm{\infty }}f(kT)\delta (t-kT)\leftrightarrow F*(s)=\sum _{k=0}^{\mathrm{\infty }}f(kT)e^{-kTs}$$

Notice that for any value of the complex number $s$ , the quantity $e^{Ts}$ is just another complex number, say $z=e^{Ts}$ . Hence, $F*(s)$ can be rewritten as

$$F*(s)=\sum _{k=0}^{\mathrm{\infty }}f(k)z^{-k};z=e^{Ts}$$

where it is customary to drop the $T$ in $f(kT)$ since it is understood that $f(29)$ really means the sampled value of $f(t)$ at time $t=29T$  s. Finally, the unilateral $\mathcal{Z}T$ of the sequence $f(k)$ emerges as

$$F(z)=\sum _{k=0}^{\mathrm{\infty }}f(k)z^{-k}=f(0)+f(1)z^{-1}+f(2)z^{-2}+\cdots$$

and $f(k)$ and $F(z)$ are a $\mathcal{Z}T$ pair $f(k)\leftrightarrow F(z)$ . Similar to the $\mathcal{L}T$ , there is a bilateral $\mathcal{Z}T$ definition that extends to negative integers $k=-1,-2,\dots -\mathrm{\infty }$ with applications in other fields that are beyond the scope of these lecture notes. In Appendix B (Tables 3 and 4), some useful properties of the $\mathcal{Z}T$ and commonly used $\mathcal{Z}T$ pairs are listed.

Table 3.

A brief list of common unilateral $\mathcal{Z}$ -transform ( $\mathcal{Z}T$ ) properties.

$f(k)+g(k)\leftrightarrow F(z)+G(z)$

Linearity

$f(k+1)\leftrightarrow z(F(z)-f(0))$

Advance by one

$f(k+2)\leftrightarrow z(z(F(z)-f(0))-f(1))=z^{2}F(z)-z^{2}f(0)-zf(1)$

Advance by two

$E^{-n}f(k)=f(k-n)\leftrightarrow z^{-n}F(z)$

Time delay

$f(k)\otimes g(k)\leftrightarrow F(z)G(z)$

Convolution

Table 4.

A brief list of common unilateral $\mathcal{Z}$ -transform ( $\mathcal{Z}T$ ) pairs.

$f(k)$	$F(z)$
$\delta (k)$	1
$u_s(k)$	$\frac{z}{z-1}$
$kT{u}_s(k)$	$\frac{Tz}{(z-1{)}^2}$
$e^{\sigma kT}{u}_s(k)$	$\frac{z}{z-e^{\sigma T}}$
$\cos (\omega kT)u_s(k)$	$\frac{z(z-\cos (\omega T))}{z^2-2z\cos (\omega T)+1}$
$\sin (\omega kT)u_s(k)$	$\frac{z\sin (\omega T))}{z^2-2z\cos (\omega T)+1}$

A filtering example

The finite sequence $y(k)=\{1,2,4,6,0,1\}$ has the $\mathcal{Z}T$

$$Y(z)=1+2z^{-1}+4z^{-2}+6z^{-3}+z^{-5}$$

which can be rewritten as a ratio of polynomials

$$Y(z)=\frac{z^5+2z^4+4z^3+6z^2+1}{z^5}=\frac{P(z)}{Q(z)}$$

The sequence $y(k)$ can be written from a ratio of polynomials $P(z)/Q(z)$ by simply doing long division and expressing the expansion in decreasing negative powers of $z$ . In filtering theory, a system with the above transfer function is called a finite impulse response filter because its impulse response, obtained by letting the input $u(k)=\delta (k)$ , is the finite sequence $y(k)=\{1,2,4,6,0,1\}$ .

An application of $\mathcal{Z}T$ and $\mathcal{L}T$

Continuous-time signal reconstruction from a sequence of sampled values is a process depicted in Figure 6 using a sample-and-hold operation, that is, each sample is held for one time period until the next sample occurs. This is also called a zero-order hold (ZOH). Intuitively, the error introduced by the ZOH decreases as the sampling frequency increases. Mathematically, the reconstructed signal can be expressed as follows:

$$\begin{array}{rl}\underset{\_}{f}(t)& =f(0)u_s(t)+(f(T)-f(0))u_s(t-T)\\ & +(f(2T)-f(T))u_s(t-2T)+\cdots \\ & +(f(kT)-f((k-1)T))u_s(t-kT)+\cdots \end{array}$$

with $\mathcal{L}T$ (Appendix B, Table 1 entry 3 and Table 2, Time delay entry 5)

$$\begin{array}{rl}\underset{\_}{F}(s)& =\frac{f(0)}{s}+\frac{(f(T)-f(0))e^{-Ts}}{s}\\ & ++\frac{(f(2T)-f(T))e^{-2Ts}}{s}+\\ & +\cdots +\frac{(f(kT)-f((k-1)T))e^{-kTs}}{s}+\cdots \end{array}$$

Cleverly, this expression can be rewritten as

$$\underset{\_}{F}(s)=\frac{1-e^{-Ts}}{s}[f(0)+f(T)e^{-Ts}+f(2T)e^{-2Ts}+\cdots ]$$

and recognize the terms in square brackets add up to $F*(s)$ . Therefore, we can write the relation

$$\underset{\_}{F}(s)=\frac{1-e^{-Ts}}{s}F*(s)=G_{ZOH}(s)F*(s)$$

and visualize it in a block diagram as in Figure 7

Figure 6.

Signal reconstruction using a zero-order hold (ZOH).

Figure 7.

Zero-order hold (ZOH) transfer function.

Next, we extend this application to determining the transfer function for the block diagram shown in Figure 8. A continuous-time system or plant, for example, a DC motor and load, is modeled using the $\mathcal{L}T$ by the NTF $G_p(s)=Y(s)/U(s)$ . The armature voltage or control signal $u(t)$ is generated by a microprocessor-based controller. Hence, the input signal $e(t)$ representing an angular position error is sampled, producing the sequence $e(k)$ . The control algorithm is the transfer function $G_c(z)=U(z)/E(z)$ implemented by the microcontroller acting on the sequence $e(k)$ to produce the control sequence $u(k)$ . Finally, the ZOH (digital-to-analog converter) reconstructs the sequence $u(k)$ into the continuous-time signal $u(t)$ applied to the physical system $G_p(s)$ . The question before us is to determine the transfer function relating $y(k)$ and $e(k)$ :

$$G(z)=\frac{Y(z)}{E(z)}$$

(12)

Figure 8.

Block diagram of a continuous-time system controlled by a digital computer.

First, from Figure 8, the output $y(t)$ can be written as the triple convolution operation (Appendix B, Table 2 entry 6 and Table 3 entry 5)

$$\begin{array}{rl}y(t)& =g_p(t)\otimes g_{zoh}(t)\otimes u*(t)=y_1(t)\otimes u*(t)\\ & =y_1(t)\otimes \sum _{k=0}^{\mathrm{\infty }}u(kT)\delta (t-kT)\\ & =u(0)y_1(t)+u(T)y_1(t-T)+\\ & +u(2T)y_1(t-2T)+\cdots \end{array}$$

(13)

where $y_1(t)=g_p(t)\otimes g_{zoh}(t)$ is just a temporary signal.

Next, the sampled version of $y(t)$ by a fictitious sampler at the output is

$$\begin{array}{rl}y(kT)& =u(0)y_1(kT)+u(T)y_1((k-1)T)\\ & +u(2T)y_1((k-2)T)+\cdots \end{array}$$

or

$$y(kT)\equiv u(kT)\otimes y_1(kT)\leftrightarrow Y(z)=U(z)Y_1(z)$$

Substituting $U(z)=G_c(z)E(z)$ into the last equation gives

$$Y(z)=G_c(z)Y_1(z)E(z)$$

(14)

The $\mathcal{L}T$ of the temporary signal $y_1(t)=g_p(t)\otimes g_{zoh}(t)$ is

$$Y_1(s)=G_p(s)\frac{1-e^{-Ts}}{s}=Y_2(s)-Y_2(s)e^{-Ts}$$

where $Y_2(s)=G_p(s)/s$ is a new temporary signal with inverse $\mathcal{L}T$ given by $y_1(t)=y_2(t)-y_2(t-T)$ . Sampling $y_1(t)$ and computing its $\mathcal{Z}T$ gives

$$\begin{array}{rl}{y}_1(kT)& =y_2(kT)-y_2((k-1)T)\leftrightarrow \\ Y_1(z)& =Y_2(z)-z^{-1}{Y}_2(z)=\frac{z-1}{z}{Y}_2(z)\end{array}$$

where

$$Y_2(z)=\mathit{\text{Z}}\left[\frac{G_p(s)}{s}\right]$$

is computed by taking the $\mathcal{Z}T$ of the impulse response of $G_p(s)/s$ .

Computationally, the steps needed to find $Y_2(z)$ are

Find $y_2(t)$ by taking the inverse $\mathcal{L}T$ of $Y_2(s)=G_p(s)/s$ .

Sample $y_2(t)$ to get $y_2(kT)$ .

Take the $\mathcal{Z}$ T of $y_2(kT)$ .

Therefore, $Y_1(z)$ is given by

$$Y_1(z)=\frac{z-1}{z}\mathit{\text{Z}}\left[\frac{G_p(s)}{s}\right]$$

which is substituted into equation (14) to obtain the formula we can use to compute the overall transfer function (12) in the discrete-time domain for the block diagram of Figure 8

$$Y(z)=\frac{z-1}{z}{G}_c(z)\mathit{\text{Z}}\left[\frac{G_p(s)}{s}\right]E(z)$$

(15)

For example, suppose the plant is an integrator $G_p(s)=1/s$ . Then,

$$Y(z)=\frac{z-1}{z}{G}_c(z)\mathit{\text{Z}}\left[\frac{1}{s^2}\right]E(z)$$

or

$$Y(z)=G_c(z)\frac{z-1}{z}\frac{Tz}{(z-1{)}^2}E(z)=G_c(z)\frac{T}{z-1}E(z)$$

Let the controller box implement a proportional control law $G_c(z)=K_c$ , which produces the difference equation $y(k)=y(k-1)+K_{c}Te(k-1)$ .

Finally, the block diagram in Figure 8 may be redrawn in a feedback block diagram as in Figure 9, with transfer function

$$Y(z)=\frac{G_c(z)G_1(z)}{1+G_c(z)G_1(z)H(z)}$$

where $G_1(z)$ is from equation (15)

$$G_1(z)=\frac{z-1}{z}\mathit{\text{Z}}\left[\frac{G_p(s)}{s}\right]$$

and $H(z)$ is a feedback filter. This is how the automatic digital control area starts, where the goal is to synthesize the controller box $G_c(z)$ to achieve a desired closed-loop performance.

Figure 9.

A closed-loop discrete-time system.

Conclusions

This educational article advocated a seamless transition of a specific sequence of topics to cover $\mathcal{F}T$ , $\mathcal{L}T$ , and $\mathcal{Z}T$ applications in the context of a signals and systems analysis course covered in the junior year of Electrical/Computer E and ET programs that lead to the B.S. degree accredited by ABET. The sequence is pedagogically sound for junior students, particularly for those who have not had a prior exposure to transform theory in other mathematics, E, or ET courses. The article is of value to students as a reading assignment, to E and ET educators who teach a “Linear Systems” junior-level course, and especially to textbook authors who may adopt and expand this presentation sequence.