Matrix Computations

ELEC 206

Computer Applications for Electrical Engineers

Dr. Ron Hayne

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Dot Product

Scalar formed by sum of products of vector elements

dot_product = sum(A.*B); dot(A,B);

n

iiiba

1

BA

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Example

Find Total Mass num_items = [ 3 5 2 1 ] mass_items = [3.5 1.5 .79 1.75]

item_totals = num_items .* mass_items total_mass = sum(item_totals)

total_mass = dot(num_items,mass_items)

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Matrix Multiplication

Value in position (i,j) is dot product of row i of the first matrix with column j of the second matrix Inner dimensions must be the same (conformable)

A = [2 5 1;0 3 -1] (2 x 3) B = [1 0; -1 4; 5 2] (3 x 2) C = A * B (2 x 2)

C11 = sum(A(1,:).*B(:,1)')

N

kkjikji bac

1,

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Matrix Powers

Square of each element of a matrix A.^2

Square of a matrix A^2 A*A

Other powers A^3 A*A*A ...

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Other Matrix Functions

Matrix Inverse Product of matrix and it's inverse yields identity matrix AA-1 = I and A-1A = I B = inv(A) I = A*B

Determinant Scalar computed from entries of a square matrix e.g. (2 x 2): |A| = a1,1a2,2 - a2,1a1,2

det(A)

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Special Matrices

Matrix of Zeros zeros(3) zeros(3,2)

Matrix of Ones ones(3) ones(3,2)

Identity Matrix eye(3)

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Systems of Linear Equations

System of 3 equations with 3 unknowns 3x1 + 2x2 - x3 = 10

-x1 + 3x2 + 2x3 = 5

x1 - x2 - x3 = -1

Matrix representation AX = B

111

231

123

A

3

2

1

x

x

x

X

1

B 5

10

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Solving Simultaneous Equations

Solution Using the Matrix Inverse AX = B A-1AX = A-1B IX = A-1B X = A-1B

MATLAB Solution X = inv(A)*B

Better MATLAB Solution using Left Division X = A\B Uses Gaussian elimination (without forming the inverse)

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Example

Mesh Analysis Problem Statement

Find mesh currents for a given circuit Input/Output Description

Four resistance valuesThree mesh current values

Three voltage values

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Example

Hand ExampleZero voltage results in zero currentSee example in circuits text

Algorithm Development Input resistance values Input voltage source valuesForm resistance matrixForm voltage matrixCompute current matrix

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MATLAB Solution

R1 = input('Input the value of R1: ');

R2 = input('Input the value of R2: ');

R3 = input('Input the value of R3: ');

R4 = input('Input the value of R4: ');

V1 = input('Input the value of V1: ');

V2 = input('Input the value of V2: ');

V3 = input('Input the value of V3: ');

resistance = [R1+R2, -R2 , -R1 ;

-R2 , R2+R3, -R3 ;

-R1 , -R3 , R1+R3+R4]

voltage = [V1; -(V2+V3); V3]

I = inv(resistance)*voltage

Ialt = resistance\voltage

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Testing

R1 = 1; R2 = 2; R3 = 3; R4 = 4; V1 = 7; V2 = 3; V3 = 9;

I1 = 2; I2 = -1; I3 = 1;

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Signal-to-Noise Ratio

Signal Power (Amplitude) power = sum(x.^2)/length(x)

Signal Power (Variance and Mean) power = std(x)^2 + mean(x)^2

Signal Power (Sinusoid) x = A*sin(2*pi*t) power = A^2/2

Signal-to-Noise Ratio SNR = (signal power)/(noise power)

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Random Numbers

Uniform Random Numbers rand('seed',n), rand(n), rand(m,n) Interval 0 to 1 Interval a to b

x = (b - a)*r + a;

Gaussian Random numbers randn('seed',n), randn(n), randn(m,n) Normal Distribution

Mean = 0, Standard Deviation = 1.0 Modified Distribution

Mean = b, Standard Deviation = a x = a*r + b;

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Random Noise

Uniform Noise rand(1,n) Interval -a to +a

mean = 0 variance = a2/3

x = 2*a*r - a

Gaussian Noise randn(1,n) Standard Deviation = a

mean = 0 variance = a2

x = a*r

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Sinusoid plus Uniform Noise

% Sine plus Uniform Noise

t=0:0.01:2;

noise_u=2*rand(1,201)-1;

sine=4*sin(2*pi*t);

s_u=sine+noise_u;

power_sine=sum(sine.^2)/length(sine)

power_noise_u=sum(noise_u.^2)/length(noise_u)

SNR_u=power_sine/power_noise_u

figure(1)

plot(t,s_u)

figure(2)

hist(noise_u)

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Sinusoid plus Gaussian Noise

% Sine plus Gaussian Noise

t=0:0.01:2;

sine=4*sin(2*pi*t);

noise_g=1/sqrt(3)*randn(1,201);

s_g=sine+noise_g;

power_sine=sum(sine.^2)/length(sine)

power_noise_g=sum(noise_g.^2)/length(noise_g)

SNR_g=power_sine/power_noise_g

figure(3)

plot(t,s_g)

figure(4)

hist(noise_g)

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Problem Solving Applied

Signal Generation with Noise Problem Statement

Generate a sinusoidal signal with a given amplitude and addition of uniform noise with a specified signal-to-noise ratio

Input/Output Description

Signal Plot

Signal and Noise Power

SNR

Sine Wave Amplitude

Desired SNR

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Problem Solving Applied

Hand ExampleSNR = (signal power)/(noise power)signal power = A2/2noise power = a2/3

Algorithm DevelopmentPrompt for A and SNRCompute aGenerate sine and noiseCompute powers and SNRPlot sine plus noise

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MATLAB Solution

% Sine plus Uniform Noise at SNR

A = input('Enter amplitude of sinusoid: ');

SNR = input('Enter desired signal-to-noise ratio: ');

a=sqrt(1.5*A^2/SNR);

t=0:0.01:2;

sine=A*sin(2*pi*t);

noise_u=2*a*rand(1,201)-a;

power_sine=sum(sine.^2)/length(sine);

power_noise_u=sum(noise_u.^2)/length(noise_u);

SNR_u=power_sine/power_noise_u

s_u=sine+noise_u;

plot(t,s_u)

title('Sinusoid with Uniform Noise')

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Summary

Matrix Computations Systems of Simultaneous Equations Signal-to-Noise Ratio

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Test #4 Review

Programming in MATLAB Problems with two variables Input / Output Functions Control Structures

Matrix Computations Systems of Simultaneous Equations Signal-to-Noise Ratio