EE3417 Lab Session

Week 1

• Turn on your computer. Open MATLAB

• The slides are uploaded in Blackboard

• Quiz on June 12 (Week 2)• Syllabus: Week 1, 2

• My TA hours•

• Email: mdahsan.habib@mavs.uta.edu

MATLAB Overview• When MATLAB launched, Command Window Appears• Command prompt(>>) in Command Windows to accept instruction or input• Objects → Data• Objects are placed in MATLAB workspace

>> a = 4; b = 3+2j; c = a * b;• whos → Another way to view workspace

>> whos• who → short version of whos -> reports only the names of workspace objects

>> who• clear a → remove specific variables (a) from the workspace

>> clear a• clc → clears the command window

>> clc• help -> most important and useful command for new users

>> help whos• exit -> terminates MATLAB

Algebra of Complex Number• Complex number: z = a + jb

• Re z = a; Im z = b• Complex number expressed in polar coordinates (r,θ)

• a = rcos θ, b = rsin θ, z = r(cos θ+ jsin θ)• Euler Formulae:

• ejθ=cos θ + jsin θ, z = r ejθ

• z = |z|ej∠z

• |z| = r = √(a2+ b2)• ∠z = θ = tan-1 (b/a), π≥ θ ≥-π

• Conjugate of z, z* = a – jb = r e-jθ= |z|e-j∠z

• zz* = (a+jb)(a-jb) = a2 + b2 = |z|2

• Useful Identities (7)• e∓jnπ = -1, n → odd integer ⇨ e∓j(2n+1)π = -1, n → integer• e∓j2nπ = 1, n → integer

Complex Number – A common mistakez1 = a +jb

z2=-a -jb

θ1

θ2

∠ z1 = tan-1(b/a) = θ1

∠ z2 = tan-1(-b/-a) = θ2

∠ z2 ≠ ∠ z1

∠ z2 = θ2 = θ1 - 180

z1 = -a +jb

z2=a -jb

θ1

θ2

∠ z1 = tan-1(b/-a) = θ1

∠ z2 = tan-1(-b/a) = θ2

∠ z1 ≠ ∠ z2

∠ z1 = θ1 = 180 + θ2

Example B.1 (9)

Complex Number - MATLAB• Matlab predefines i = j =

>> z = -3-j4• real and imag operators extract real and imaginary components of z.

>> z_real = real(z)>> z_imag = imag(z)

• Modulus or Magnitude of a complex number•

>> z_mag = sqrt(z_real^2+z_imag^2)• |z|2 = zz*

>> z_mag = sqrt(z*conj(z))>> z_mag = abs(z)

• Angle of a complex number>> z_rad = atan2(z_mag, z_real)

• atan2 -> two-argument arc-tangent function; ensures the angle reflects in the proper quadrant.>> z_rad = angle(z)

• MATLAB function pol2cart number polar form to Cartesian form• z = 4 e-j(3π/4)

>> [z_real, z_imag] = pol2cart(-3*pi/4,4)

Complex Number - Exercise

• Determine z1z2 and z1/z2 if z1 = 3+j4 and z2 = 2+3j>> Verify your results using MATLAB• Convert your results from Cartesian coordinate to Polar coordinate>> Verify your results using MATLAB function pol2cart

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MATLAB - Vector Operation• Vectors of even valued integers

>> k = 0:2:11• Negative and noninteger step sizes

>> k = 11:-10/3:0• If step size not speified, value of one assumed

>> k = 0:11• In MATLAB, ascending positive integer indices specify particular vector elements.

>> k(5), k(1:4),• Vector representation to create signals

• 10 Hz sinusoid described by f(t) = sin(2π10t+π/6) when 0≤t<0.2>> t = 0:0.0004:0.2-0.0004; f = sin(2*pi*10*t+pi/6); f(1)

• Find the three cube roots of minus 1, • →

>> k = 0:2;>> w = exp(j*(pi/3 + 2*pi*k/3))

Exercise (56)

• Find the 100 cube roots of minus 1?

Simple Plotting• MATLAB’s plot command

>> plot(t,f);• Axis labels are added using xlabel and ylabel

>> xlabel(‘t’); ylabel(‘f(t)’)• Plotting discrete points, 100 unique roots of w^100=-1

>> plot(real(w), imag(w), ‘o’);>> xlabel(‘Re(w)’); ylabel(‘Im(w)’);>> axis equal

Element by Element Operations• Multiplication, Division and Power• x = [5 4 6]; y = [1 2 3];

>> x_mul_y = x * y >> x_elem_mul_y = x.*y>> x_div_y = x/y>> x_elem_div_y = x./y>> x_elem_pow_y = x.^y

•Suppose h(t) = f(t)g(t) where g(t) = exp(-10*t)>> g = exp(-10*t);>> h = f.*g;>> plot (t,f,’-k’,t,h,’-b’);>> xlabel(‘t’); ylabel(‘Amplitude’);>> legend (‘f(t)’,’h(t));

h g(t)

Damped Sinusoid

Matrix Operation• Common Useful function

• eye(m) creates the m×m identity matrix>> eye(3)

• ones(m,n) creates the m×n matrix of all ones>> ones(2,3)

• zeros(m,n) creates the m×n matrix of all zeros>> zeros(3,2)

•Row vector>> r = [1 3 2];

•A 2×3 matrix>> A = [2 3; 4 5; 0 6];

Matrix Operation• Transpose

>> c= r’;• Concatenation

>> B = [c A];• Matrix inverse

>> D = inv(B);

• Matrix indices>> B(1,2)>> B(1:2,2:3)

• Colon can be used to specify all elements along a specified dimension

>> B(2,:)>> B(:,2)

Matrix Operation

Solve

• Ax = y; • x = A-1Ax = A-1y >> A = [1 -2 -3; -sqrt(3) 1 –sqrt(5); 3 –sqrt(7) 1]; >> y = [1; pi; exp(1)]; >> x = inv(A)*y

,

function handle• handle = @(arglist)anonymous_function• constructs an anonymous function and returns a handle to that function.• arglist is a comma-separated list of input arguments.• The statement below creates an anonymous function that finds the square of a number. • To execute the function associated with it

• fhandle(arg1, arg2, ..., argN)>> sqr = @(x) x.^2;>> a = sqr(5);

function handle• fplot function → Plot function between specified limits

• fplot(fun,limits);• fplot(sqr,[-10 20 -1 20]);

• Unit Step Function• step = @(x)(x>=0);• fplot(step,[-1 5 -1 5]);• step = @(x)(x-2>=0);• fplot(step,[-1 5 -1 5]);

-1 0 1 2 3 4 5-1

0

1

2

3

4

5

-1 0 1 2 3 4 5-1

0

1

2

3

4

5

Functions• Unit Pulse Function

>> u = @(x)and((x<=3),(x>=1));>> fplot(u,[-5 5 -1 3],10000);>> u = @(x)and(((x+2)<=3),((x+2)>=1));>> figure(); fplot(u,[-5 5 -1 3],10000);>> u = @(x)and(((2*x)<=3),((2*x)>=1));>> fplot(u,[-5 5 -1 3],10000);

• Piecewise function

>> f1 = @(x)(2*x)*and((x<=10),(x>=5));>> fplot(f1,[-10 20 -10 30]);

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.5

0

0.5

1

1.5

2

2.5

3

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.5

0

0.5

1

1.5

2

2.5

3

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.5

0

0.5

1

1.5

2

2.5

3

10

20

5 10

Eigenvalue and Eigenvector

For an (n×n) square matrix A, and vector x (x≠0) that satisfy the eqAx = λx ----- eq(i)

is an eigenvector and λ is the corresponding eigenvalue of A

• Q(λ) = |λI – A| = λn+an-1 λn-1+…+a1 λ+a0 λ0= 0• Q(λ)→ characteristic polynomial of matrix A.

• The n zeros of the polynomial are the eigenvalues of A.• Corresponding to each eigenvalue, there is an eigenvector that

satisfies eq(i)

Example B.13 (47)

Eigenvalue and EigenvectorApplications

Tacoma Narrow Bridge collapsing

• Constructed at 1940s.• Crashed by winds which set bridge oscillated at a frequency closed to its own natural frequency.• Natural frequency of the bridge is the eigenvalue of smallest magnitude of a system that models the bridge.• The eigenvalue of smallest magnitude of a matrix is the same as the inverse (reciprocal) of the dominant eigenvalue of the inverse of the matrix.

• Eigenvalues can also be used to test for cracks or deformities in a solid.

• Car designers analyze eigenvalues in order to damp out the noise so that the occupants have a quiet ride.

Eigenvalue and EigenvectorMATLAB

• poly(A) to determine Characteristic Polynomial of matrix A (n×n)• Output is a row vector with n+1 elements that are the coefficients of the

characteristic polynomial>> A = [3 2 -2; -3 -1 3; 1 2 0]; >> poly(A)

• roots(C) computes the roots of the polynomial whose coefficients are the elements of the vector C.

>> roots(poly(A))

•The command in the form [V D] = eig(A) computes both the eigenvalues and eigenvectors of A.

>> [V D] = eig(A)

Partial Fraction Expansion• Rational function F(x) can be expressed as

= P(x)/Q(x)

• The function F(x) is improper if m≥n and proper if m<n• An improper function can always be separated into sum of a polynomial in x and a proper function.

• A proper function can be further expanded into partial fraction

Partial Fraction Expansion• Method of Clearing Fraction

•

• The Heaviside “Cover-up” Method•

• Repeated Factors of Q(x)• If a function F(x) has a repeated factor in its denominator,

Its partial expansion is given by

•

Example B.8

Example B.9

Example B.10

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Partial Fraction Expansions• Partial fraction expansion of rational function F(x) = B(x)/A(x)• MATLAB residue command. The basic form:

>> [R,P,K] = residue(B,A);• B → Polynomial coefficient of the numerator• A → Polynomial coefficient of the denominator• This vectors are ordered in descending powers of the independent variable• R → coefficient of each partial fraction• P → contain the corresponding roots of each partial fraction• For a root repeated r times, the r partial fractions are ordered in ascending powers.• K → the direct terms, when the ration function is not proper• They are ordered in descending powers of independent variable

Partial Fraction Expansions

Solve

>> [R,P,K] = residue([1 0 0 0 0 pi],[1 -sqrt(8) 0 sqrt(32) -4])

R = 7.8888 5.9713 3.1107 0.1112P = 1.4142 1.4142 1.4142 -1.4142K = 1.0000 2.8284

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Partial Fraction ExpansionExercise

• Compute by hand the partial fraction expansion of

• Verify your solution using MATLAB

MATLAB SCRIPT• The command window is to design and execute individual command one at a time

>> x = 1:10;>> y=log(x)>> plot(x,y)

• To automate the execution of many commands - matlab program/script

• MATLAB program - collection of MATLAB command and function stored in disk as a text file of type .m

• MATLAB editor – To run F5

• FOR loopclear all;x = (1:1000)';for k = 1:5

Y(:,k)=k*log(x);endplot(x,y)

MATLAB FUNCTION• Syntax

function [y1,...,yN] = myfun(x1,...,xM)

• Open a new script file and save it as eval_log_func.m

• Write a function that output multiple log functionsfunction eval_log_func(maxLoop)x = (1:1000)'for k = 1:maxLoop

y(:,k)=k*log(x);endplot(x,y)

• In Command window>> clear all;>> g = eval_log_func(10);

MATLAB Function - Exercise

• Write a function (N_root.m) that will calculate the Nth root of -1. Here, N is the input

• Write a script that will • vary N from 1:5• call N_root each time to find Nth roots • display the discrete Nth roots

Integral• syms: shortcut for creating symbolic variables and functions

• Syntax: syms var1 ... varN>> syms x y

• Symbolic Integration• Syntax: int(expr,var)• computes the indefinite integral of expr with

respect to var• Syntax : int(expr, var, a, b)• computes the definite integral of expr with

respect to var from a to b.• a and b: Number or symbolic expression, including

expressions with infinities.

Integral

>> syms x;>> int(-2*x/(1 + x^2)^2,x)

• >> int(x*log(1 + x), 0, 1)

• >> int((6*exp(-(x-2)))^2,2,Inf);

MATLAB FUNCTION• Syntax

function [y1,...,yN] = myfun(x1,...,xM)

• Open a new script file and save it as eval_log_func.m

• Write a function that output multiple log functionsfunction eval_log_func(maxLoop)x = (1:1000)'for k = 1:maxLoop

y(:,k)=k*log(x);endplot(x,y)

• In Command window>> clear all;>> g = eval_log_func(10);

MATLAB Function - Exercise

• Write a function (N_root.m) that will calculate the Nth root of -1. Here, N is the input

• Write a script that will • vary N from 1:5• call N_root each time to find Nth roots • display the discrete Nth roots