11 Projects for MATH319 Linear Systems: Instructions

Topics. Each student should select one topic to write about. The list P1-P24 below are recommended; students who are unsure of what to write about, or wish to select another topic, should discuss with the lecturer. All of the topics relate to the subject matter of MATH319, although they variously emphasize linear algebra, analysis and require varying amounts of computing skill to carry out. Some projects have particular applications, as indicated after the titles. In some cases, the outlines are the starting point for the project, and it is expected that the student will go beyond the bare outline. A report which simply follows the outline or reproduces results and examples from the given sources is unlikely to gain a high mark. The sources listed are intended to help the student start, but please read them with care. The notation and terminology may be different from that of the lectures, in which case explain the notation in the report.

Report. The report should be structured so as have an introduction which describes the context of the problem. The main body of the report should describe the solution to the problem under consideration, and the conclusions. There should be a bibliography, listing the sources used.

Presentation. In the module, there are various descriptions of linear systems in terms of block diagrams, $(A,B,C,D)$, differential equations, transfer functions, Nyquist plots etc. Think: what is the best way to present any given idea? Explain what information each diagram conveys. You may wish to type-set calculations, using the methods of MATH240, and produce plots via computer. The final report should fit together as a cohesive unit, so fit the components together.

Submission. The report should be submitted electronically as a single pdf to the MOODLE portal for MATH319. If you have several elements in the report, such as diagrams, graphs, calculations etc, then produce these in pdf format and then combine them into a single pdf by using AcroreadPro or otherwise. The submitted pdf should have a title such as JamesClerkMaxwellMATH319P13.pdf including your name. Including all graphs, calculations, computer codes, the bibliography and so on, the report should be less than 8 sides of A4. A report which consists mainly of text can be much shorter, say 5 pages of A4.

Marking. The marker will award a letter grade A-F according to the criteria stated in the Mathematics and Statistics Part II handbook, and provide the student with a paragraph of feedback supporting the grade allocation, within four weeks.

 MATH319 Project topics with outlines

Chapter 1: Linear systems and their description: P9, P11

Chapter 2: Solving Linear Systems by Matrix Theory: P1, P4, P7, P19, P20, P24

Chapter 3: Laplace transforms: P8, P10, P18. P23

Chapter 4: Stability of MIMO in terms of transfer functions: P3, P5, P6, P14, P17, P22

Chapter 5: Feedback Control: P2, P12, P13, P15, P16, P21

Similar project titles P1-P20 were available in 2017/8, and P21-24 have been added. The number of students who offered projects on each topic is given beside the title, so that 7/51 means 7 projects from the class of 51 students. P1 Inverse Systems 3/51

Source: J.P. Hespana, Linear Systems Theory, (Princeton and Oxford, 2009), lecture 19.

A radio transmitter has an input $u$ and an output $y$. The signal $y$ is broadcast, and received by a listener, who wishes to recover $u$.

(i) Consider a SISO

$$\frac{dx}{dt}=Ax+Bu$$

$$y=Cx+Du.$$

Solve this system for $u$ in terms of $y$ and $x$, and derive a linear system for the receiver with input $y$, state $x$ and output $u$.

(ii) The simplest MIMO system is

$$y=Du.$$

Show that one can solve for $u$ in terms of $y$ if and only if $D$ is an invertible square matrix.

(iii) Suppose that the transmitter is modeled by a MIMO

$$\frac{dx}{dt}=Ax+Bu$$

$$y=Cx+Du$$

with transfer function

$$T(s)=D+C{(sI-A)}^{-1}B.$$

Solve the $MIMO$ for $u$ in terms of $x$ and $y$. Show that the receiver is modelled by a linear system

with transfer function

$$R(s)=D^{\times }+C^{\times }{(sI-A^{\times })}^{-1}{B}^{\times }.$$

(iv) Write down $R(s)$ in terms of $A,B,C,D$ and prove that

$$I=R(s)T(s).$$

P2 PID Controllers 2/51

Source: R.C. Dorf and R.H. Bishop, Modern Control Systems (Pearson) 12th Edition, section 7.6.

For a SISO with $y=KGu-Ky$, a particularly useful controller is

$$K(s)=as+b+\frac{c}{s},$$

where $a,b,c$ are complex constants; we refer to these as:

The advantages of $K$ are that such a $K$ is easy to manufacture, and for a suitable choice of $a,b,c$, the controller $K$ stabilizes

$$\frac{1}{1+KG},\frac{G}{1+KG},\frac{K}{1+KG},\frac{KG}{1+KG}.$$

(i) Suppose that $G(s)=1/(\alpha s^2+\beta s+\gamma )$ for some real $\alpha ,\beta ,\gamma$. Show that

$$\frac{1}{1+KG}=\frac{p(s)}{q(s)}$$

where $p(s)$ and $q(s)$ are cubic polynomials.

The characteristic equation is

$$q(s)=0.$$

We can vary one parameter of $a,b,c$ at a time, and the roots move along curves in the complex plane.

P3 Root locus method 0/51

Sources: A. Tewari, Modern Control Design John Wiley, Chapter 7;

R.C. Dorf and R.H. Bishop Modern Control Systems Pearson 12th Edition.

Let $p(s)$ and $q(s)$ be coprime complex polynomials, where degree of $p(s)$ is less than or equal to the degree of $q(s)$, and consider a transfer function $G(s)=p(s)/q(s)$. Then we consider a simple feedback system with design parameter $\kappa$ and transfer function

$$\frac{G}{1+\kappa G}=\frac{p(s)}{q(s)+\kappa p(s)};$$

the characteristic equation is

$$q(s)+\kappa p(s)=0.$$

The root locus method involves plotting the roots of the characteristic equation as the design parameter $\kappa$ varies. More specifically, we, suppose that $p(s)$ and $q(s)$ have real coefficients, and wish to determine $\kappa$ such that either:

P4 Popov–Belevitch–Hautus test for controllability 0/51

Source: J.P. Hespanha Linear System Theory, Princeton University Press; sections 12.3 and 14.4

Let $A$ be a $n\times n$ complex matrix, with transpose $A^T$, and let $B$ be a $n\times 1$ column matrix; let

$$V=\{\sum _{j=0}^n{a}_j{A}^{j}B;a_j\in 𝐂;j=0,\mathrm{\dots },n-1\}.$$

Prove that the following conditions are equivalent:

P5 The Nyquist Stability Criterion 6/51

Source: G.F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems Sixth Edition, (Pearson, 2010), Section 6.3.

The basic result is the Nyquist Stability Criterion, as discussed in the notes.

(iii) Discuss examples such as

$$G(s)=\frac{1}{{(s+1)}^2(s+2)}.$$

(iv) Discuss examples such as

$$G(s)=\frac{1}{s{(s+1)}^2},$$

where $G(s)$ has a pole on the imaginary axis; you will need to modify the contours to avoid this pole.

P6 The theory of Nyquist’s criterion 0/51

Source: I. Stewart and D.O. Tall, Complex Analysis: the hitchhiker’s guide to the plane, (Cambridge University Press, 1983), page 231

J.C.. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory

Let $T(s)=q(s)/p(s)$ where $p(s)$ and $q(s)$ are complex polynomials with degree of $q(s)$ less than the degree of $p(s)$, and suppose that $p(s)$ has all its zeros in LHP. Suppose that the Nyquist contour $\{T(ı\omega ):-\mathrm{\infty }\le \omega \le \mathrm{\infty }\}$ does not pass through or wind around $-1$. We aim to prove that $T/(1+T)$ is a strictly proper and stable rational function.

(i) Show that there exists $M_1$ such that

$$|T(s)|\le \frac{M_1}{1+|s|}$$

and that there exists $M_2>0$ such that

$$|T^{\prime }(s)|\le \frac{M_2}{1+{|s|}^2}$$

for all $s\in RHP$.

(ii) Let $S_r$ be the semicircle in the right half plane $S_r:z=r{e}^{ı\theta }$ for $-\pi /2\le \theta \le \pi /2$. Deduce from (i) that

$${\int }_{S_r}\frac{T^{\prime }(s)ds}{1+T(s)}\to 0$$

as $r\to \mathrm{\infty }.$

(iv) Deduce that

$${\int }_{{\gamma }_r}\frac{T^{\prime }(s)ds}{1+T(s)}=0$$

for all sufficiently large $r$.

P7 Oscillations of particles on a string 0/51 Applications to Physics

Sources: T.W.B Kibble and F.H. Berkshire, Classical Mechanics (Fourth Edition) (Longman, 1996) sections 11.5, 11.6.

A collection of $N$ particles is placed in a straight line, and all particles are connected to their nearest neighbours by elastic bands, so that the first and last particles are connected by elastic bands to rigid supports at either end. The displacement of the particle $j$ from rest is given by $x_j$, where the $x_j$ satisfy the systems of differential equations $x_0=0$, $x_{N+1}=0$ and

$$\frac{d^2{x}_k}{d{t}^2}=x_{k+1}+x_{k-1}-2x_k\mathit{\hspace{1em}\hspace{1em}}(k=1,\mathrm{\dots },N).$$

(i) Introduce a $(N\times N)$ matrix $A$ to describe this as $\frac{d^{2}X}{d{t}^2}=AX$, where $X$ is the $(N\times 1)$ column vector

$$X=\left[\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_N\hfill \end{array}\right)\right].$$

(ii) Let $p\in \{1,\mathrm{\dots },N\}$ and consider

$$x_k=\sin \frac{kp\pi }{N+1}\cos {\omega }_{p}t\mathit{\hspace{1em}\hspace{1em}}(k=0,\mathrm{\dots },N+1).$$

Show that this gives a solution of the differential equation when

$${\omega }_p=2\sin \frac{\pi p}{2(N+1)}\mathit{\hspace{1em}\hspace{1em}}(p=1,\mathrm{\dots },N).$$

(iii) Deduce the values of the eigenvalues of $A$.

P8 Bessel Filters 0/51

Source: A. Ambardar Analog and Digital Signal Processing, PWS Foundations in Engineering Series, (ITP, 1995) Section 10.8.1

The Bessel polynomials are defined by the recurrence relation:

$$Q_0(s)=1,$$

$$Q_1(s)=s+1;$$

$$Q_n(s)=(2n-1)Q_{n-1}(s)+s^2{Q}_{n-2}(s)$$

where $(n=2,3,\mathrm{\dots })$ The corresponding Bessel controller is

$$H_n(s)=\frac{Q_n(0)}{Q_n(s)}.$$

(i) Write a recurrence formula in MATLAB or R to generate the Bessel polynomials, and compute several of them.

(ii) Obtain the polar decomposition for $s=i\omega$ of the Bessel controller $H_n$, as in

$$H_n(i\omega )={\mathrm{\Gamma }}_n(\omega )e^{i{\varphi }_n(\omega )}$$

and plot the gain ${\mathrm{\Gamma }}_n(\omega )$ and phase ${\varphi }_n(\omega )$ for the first few $n$.

(iii) Define the spherical Bessel functions $j_n$ and $y_n$ by

$$j_0(x)=\frac{\sin x}{x},y_0(x)=\frac{\cos x}{x}$$

and the formula

$$j_n(x)={(-x)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n{j}_0(x)\mathit{\hspace{1em}\hspace{1em}}(n=1,2,\mathrm{\dots }),$$

$$y_n(x)=-{(-x)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n{y}_0(x)\mathit{\hspace{1em}\hspace{1em}}(n=1,2,\mathrm{\dots }).$$

Compute the first few of these.

(iv) Show that

$$Q_n\left(\frac{1}{i\omega }\right)=\frac{\omega e^{i\omega }}{i^n}\left(-y_n(\omega )-i{j}_n(\omega )\right)$$

for the first couple of $n$.

P9 Fraunhofer diffraction of light 1/51 Applications to Physical Optics.

Source: H.J. Pain, The Physics of Vibrations and Waves, Fourth Edition, (Wiley, 1993) page 367-372

We consider $N$ equal monochromatic sources of light arranged in a straight line and with each source separated from its neighbour by distance $f$. Let $\lambda$ be the wavelength of the light. When we view the array from a distance much greater than $Nf$, and at a small angle $\theta$ to the line perpendicular to the array, the observed intensity $I$ is

$$I=I_0\frac{{\sin }^2(N\beta (\theta ))}{{\sin }^2(\beta (\theta ))},$$

where $\beta (\theta )=(\pi f/\lambda )\sin \theta$ and $I_0$ is a constant.

(i) Show that

$$1\ge \frac{\sin \beta }{\beta }\ge \frac{2}{\pi }$$

for , and discuss the largest value that $I$ can take.

(iv) Plot the graphs of

$$\frac{{\sin }^2\beta }{{\beta }^2}$$

and

$$\frac{{\sin }^2\beta }{{\beta }^2}\frac{{\sin }^2(N\beta )}{{\sin }^2(\beta )}$$

for a few small $N$.

P10 Initial value theorem for Laplace transforms 0/51

Source: I.N. Sneddon, The Use of Integral Transforms (McGraw-Hill, 1972), pages 185-7

Let $f$ be continuous on $[0,\mathrm{\infty })$ and suppose that $f$ satisfies $(E)$. Then the Laplace transform $\widehat{f}$ satisfies

$$f(0)=\underset{s\to \mathrm{\infty }}{lim}s\widehat{f}(s).$$

(i) Show that

$$s\widehat{f}(s)-f(0)={\int }_0^{\mathrm{\infty }}\left(f(x/s)-f(0)\right)e^{-x}𝑑x.$$

(ii) Given $\epsilon >0$, and $M,\alpha >0$ such that

$$|f(t)|\le M{e}^{\alpha t}\mathit{\hspace{1em}\hspace{1em}}(t\ge 0),$$

consider $s>2\alpha$ and $R>0$. Show that

$$\left|{\int }_R^{\mathrm{\infty }}(f(x/s)-f(0))e^{-x}dx\right|\le 4M{e}^{-R/2}\mathit{\hspace{1em}\hspace{1em}}(s>2\alpha ).$$

(iii) Now choose and fix $R$ so large that Show that there exists $s_0$ such that

$$\left|{\int }_0^R\left(f(x/s)-f(0)\right)e^{-x}𝑑x\right|\le \epsilon$$

for all $s>s_0$.

(iv) Deduce that for all $s>\max \{s_0,2\alpha \}$,

$$|s\widehat{f}(s)-f(0)|\le \left|{\int }_0^{\mathrm{\infty }}\left(f(x/s)-f(0)\right)e^{-x}𝑑x\right|\le 2\epsilon .$$

(v) Discuss examples of this result, for instance with $f$ rational or

$$f(x)=\sum _{j=1}^n(a_j\cos (b_{j}x)+c_j\sin (d_{j}x)).$$

(vi) Discuss the validity of the formula

$$\underset{p\to 0+}{lim}p\widehat{f}(p)=f(\mathrm{\infty }).$$

P11 Invertibility of matrices with stable rational entries 1/51

Source: J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, Chapter 3.

Let $𝐂[s]$ be the space of complex polynomials and $𝐂(s)$ the space of complex rational functions. Let $A(s)\in M_n(𝐂[s])$ be a $n\times n$ matrix with entries that are polynomials in $s$.

P12 Stabilizing by rational controllers 1/51

Source: J.C. Doyle, B.A. Francis and A.R. Tannenbaum, Feedback Control Theory, Chapter 5.

Consider a SISO with plant described by a rational transfer function $G$. We wish to find a rational controller $K$ such that

$$\frac{1}{1+KG},\frac{G}{1+KG},\frac{K}{1+KG},\frac{KG}{1+KG}$$

are all stable, so the system is internally stable.

(i) Consider

$$G(s)=\frac{1}{(s-1)(s+2)}.$$

Express $G=N/M$ as a quotient of coprime stable rational functions .

(iii) By considering $XM+YN=1$ and

$$K=\frac{Y+MQ}{X-NQ},$$

describe all of the controllers that internally stablize the system.

P13 Circles of constant gain 0/51

Source: N.S. Nise, Control Systems Engineering second edition, (Benjamin 1995), section 10.9.

Consider a SISO system with transfer function $R$ which is fed back through a simple feedback loop to give transfer function

$$T=\frac{R}{1+R}.$$

Let $T=u+iv$ and $R=p+iq$, where $u,v,p$ and $q$ are all real.

P14 May–Wigner Law 0/51 Applications to Biology

Source: J.P. Hespana, Linear Systems Theory, (Princeton and Oxford, 2009), page 78.

(i) Consider the continuous time system

$$\frac{dX}{dt}=AX$$

where $A$ is a $n\times n$ real matrix. Suppose that there exist $\mu >0$ and positive definite matrices $P$ and $Q$ such that

$$A^{T}P+PA+2\mu P=-Q.$$

(i) Show that all the eigenvalues of $\mu I+A$ have negative real parts.

(iii) Let $J$ be a $(n\times n)$ matrix and suppose that the eigenvalues of $J$ lie inside the circle with centre $0$ and radius $\sigma \sqrt{n}$, where $\sigma >0$ is constant. Let $\nu$ be real and consider the linear system

$$\frac{dX}{dt}=-\nu X+JX.$$

Show that the system is stable if

P15 Wind turbine speed control 11/51

Source: R.C. Dorf and R.H. Bishop Modern Control Systems, Twelfth Edition, (Pearson, 2011), Example 7.13.

A wind turbine consists of an electrical generator, a rotor with blades whose pitch can be adjusted to suit the wind conditions and gears to increase the speed of rotation of the turbine to suit the electrical generator. The transfer function of the turbine is

$$G(s)=\frac{4.2(s-827)(s^2-5.4s+194)}{(s+0.2)(s^2+0.1s+482)},$$

(i) Find the poles of $G$, and discuss stability.

(iii) Let $K(s)=b+c/s$ be a $PI$ controller; find

$$\frac{1}{1+GK},\frac{G}{1+GK},\frac{K}{1+GK},\frac{GK}{1+GK}.$$

and determine the poles.

P16 PID control of wind turbines for clean energy 2/51

Source: R.C. Dorf and R.H. Bishop Modern Control Systems, Twelfth Edition, (Pearson, 2011), Example 9.10.

A wind turbine consists of an electrical generator, a rotor with blades whose pitch can be adjusted to suit the wind conditions and gears to increase the speed of rotation of the turbine to suit the electrical generator. The transfer function of the turbine is

$$G(s)=\frac{1}{\tau s+1}\frac{K{\omega }^2}{s^2+2\zeta \omega s+{\omega }^2},$$

where the constants are $\omega ,\tau ,\zeta >0$ and $K$ is real, possibly negative.

(ii) We introduce a $PID$ controller

$$V(s)=as+b+\frac{c}{s},$$

for some $a,b,c$. Obtain the transfer function

$$L(s)=\frac{1}{1+V(s)G(s)}$$

and the related transfer functions

$$\frac{V(s)}{1+V(s)G(s)},\frac{G(s)}{1+V(s)G(s)},\frac{V(s)G(s)}{1+V(s)G(s)}.$$

(iii) Discuss stability of these transfer functions, using algebraic criteria.

P17 EVAD for cardiological illness 6/51 Applications to Medicine

Source: R.C. Dorf and R.H. Bishop Modern Control Systems, Twelfth Edition, (Pearson, 2011), AP9.11.

Medical patients with cardiological illness can have their heartbeats regulated by a electric ventricular assist device, which has input electrical power and output blood flow. The pump has transfer function $G(s)=e^{-sT}$, where $T>0$ is a constant, and the controller has the form

$$K(s)=\frac{a}{s(s+b)}$$

where $a,b>0$.

(i) Find the transfer functions

$$\frac{1}{1+K(s)G(s)},\frac{G(s)}{1+K(s)G(s)},\frac{K(s)}{1+K(s)G(s)},\frac{K(s)G(s)}{1+K(s)G(s)},$$

for instance when $T=1$, $a=5$, $b=10$.

P18 Square waves 3/51

Source: B.P. Conrad, Differential Equations: A Systems Approach, (Prentice-Hall, 2003), pages 306-9

We consider the system

$$y^{\prime \prime }+k^{2}y=u(t),$$

$$y(0)=y^{\prime }(0)=0,$$

where $k>0$ is constant and $u(t)$ is a bounded input.

(iii) Consider the input

$$u(t)=\{\begin{array}{cc}1\text{ for }t\in [0,1]\cup [2,3]\cup [4,5]\cup \mathrm{\dots }\hfill & \\ -1,\text{ for }t\in (1,2)\cup (3,4)\cup (5,6)\cup \mathrm{\dots };\hfill & \end{array}$$

which is a square wave, and its graph looks like the top of the curtain wall of Lancaster castle. Compute the Laplace transform of $u$.

P19 Phase portraits 5/51

Source: B.P. Conrad, Differential Equations: A Systems Approach, (Prentice-Hall, 2003), section 8.1

We consider the system

$$\frac{dX}{dt}=AX$$

where $X$ is a real column and $A$ is the real matrix

The eigenvalues of $A$ are sometimes called characteristic values.

When one plots $(x(t),y(t))$ is the usual $(x,y)$ plane, the resulting graph is called a phase portrait. The shape depends upon the matrix $A$, and is traditionally described as follows:

P20 The Cayley–Hamilton Theorem 6/20

Source: H. Eves, Elementary Matrix Theory, (Dover, 1980), pages 199-203.

This exercise shows that every square matrix satisfies its characteristic equation. Let $A$ be a complex $(n\times n)$ matrix and let $\text{adj}(A)$ be the adjugate of $A$.

(i) Expand the adjugate of $sI-A$ as

$$\text{adj}(sI-A)=B_{n-1}{s}^{n-1}+B_{n-2}{s}^{n-2}+\mathrm{\dots }+B_0$$

where $B_j$ are $(n\times n)$ complex matrices; expand the characteristic polynomial ${\chi }_A(s)$ as

$${\chi }_A(s)=det(sI-A)=s^n+p_{n-1}{s}^{n-1}+\mathrm{\dots }+p_{1}s+p_0.$$

(ii) Use the identity

$$det(sI-A)I=(sI-A)\text{adj}(sI-A)$$

to compare the coefficients of $s^n,s^{n-1},\mathrm{\dots },s,1$.

P21 Stabilizing the damped harmonic oscillator (new topic)

Source: G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Third Edition,

(MacMillan), page 108.

We consider the damped harmonic oscillator

$$\frac{d^{2}x}{d{t}^2}+\beta \frac{dx}{dt}+\gamma x=u$$

with $\beta >0$ and real $\gamma$.

P22 Nyquist by change of variable (new topic)

Source: I. Stewart and D.O. Tall, Complex Analysis: the hitchhiker’s guide to the plane, (Cambridge University Press, 1983), page 231

(i) Let

$$z=\frac{s-1}{s+1}.$$

Show that if and only if $\mathrm{\Re }s>0$. Deduce that $s\mapsto z$ takes the right half plane onto the unit disc .

(iv) Let $C(0,1)$ be the unit circle $\{z:|z|=1\}$, described once in the positive sense. Suppose that $Y(z)$ has no poles on $\{z:|z|=1\}$. Show that

$$\frac{1}{2\pi i}{\int }_{C(0,1)}\frac{Y^{\prime }(z)dz}{Y(z)}=Z-P$$

where $Z$ is the number of zeros of $Y(z)$ in $D(0,1)$, and $P$ is the number of poles of $Y(z)$ in $D(0,1)$, counted according to multiplicity.

P23 Transfer functions on the disc (new topic) Applications to digital music

Source: D.J. Benson, Music: A Mathematical Offering, (Cambridge University Press, 2007) pages 257-261 and 270-3.

(i) Let

$$z=\frac{s-1}{s+1}.$$

Show that if and only if $\mathrm{\Re }s>0$. Deduce that $s\mapsto z$ takes the right half-plane onto the unit disc .

(iv) Let $L$ be a positive integer, and let $a_j,\mu \in 𝐂$ where . Find the poles of

$$Y(z)=\frac{a_L{z}^L+a_{L-1}{z}^{L-1}+\mathrm{\dots }+a_0}{z^L-\mu }.$$

Find the poles of $Y(z).$

(v) Let $(A,B,C,D)$ be a SISO system. Show that

$$Y(z)=D+C{(I-zA)}^{-1}B$$

is stable if all the eigenvalues of $A$ are in $D(0,1)$. Note that the formula for $Y(z)$ has a different shape from the transfer functions considered in lectures.

P24 Discrete Fourier transform (new topic) Applications to digital music

Source: D.W. Kammler, A first course in Fourier Analysis (Prentice Hall, 2000) pages 239-246.

Source: D.J. Benson, Music: A Mathematical Offering, (Cambridge University Press, 2007) pages 257-264

Let $L$ be a positive integer and $\omega =e^{2\pi i/L}$.

(i) Show that

$$G=\{1,\omega ,{\omega }^2,\mathrm{\dots },{\omega }^{L-1}\}$$

gives a cyclic group of order $L$ under multiplication.

(ii) Let $k$ be an integer. Show that

$$\sum _{j=0}^{L-1}{\omega }^{jk}=\{\begin{array}{cc}L\text{ if }L\text{ divides }k\hfill & \\ 0,\text{ else }\hfill & \end{array}$$

(iii) Let

$$f(z)=\sum _{j=0}^{L-1}{a}_j{z}^j$$

Show that

$$a_{\mathrm{\ell }}=\frac{1}{L}\sum _{k=0}^{L-1}{\omega }^{-k\mathrm{\ell }}f({\omega }^k).$$

Deduce that any polynomial of degree less than $L$ is determined by its values on $G$.

(iv) Show that the matrix

$$U=\frac{1}{\sqrt{L}}{\left[{\omega }^{(j-1)(k-1)}\right]}_{j,k=1}^L$$

satisfies $U^{†}U=I.$