inf3470/4470: digital signal processing matlab introduction · inf3470/4470: digital signal...

INF3470/4470: Digital Signal ProcessingMATLAB Introduction

Carl Inge C. Nilsen

January 2007

Carl Inge C. Nilsen INF3470/4470: Digital Signal ProcessingMATLAB Introduction

MATLAB Facts

MATLAB = MATrix LABoratory

Optimized for matrix operations

Every operation must be described in matrix/vector terms

Programming/scripting capabilities

Implicit variable declaration

Plotting/visualization capabilities

Includes toolboxes and many functions


Performing Calculations and Using Variables

A simple calculation:

25*3 - 10^7 + 7/pi

Same, using variables:

TMP1 = 25*3;TMP2 = 10^7;TMP3 = 7/pi;RESULT = TMP1 - TMP2 + TMP3;

Numerical precision:

format long;RESULT = TMP1 - TMP2 + TMP3;


Creating Vectors and Matrices

A 1× 4 row vector:

V = [1, 0, -13, 29];

A 4× 1 column vector:

V = [1; 0; -13; 29];

A 2× 2 matrix:

A = [1, 0; -13, 29];

The N × N identity matrix:

I = eye(N);

An M × N matrix of zeroes:

M0 = zeros(M,N);

An M × N matrix of ones:

M1 = ones(M,N);


Matrix Dimensions

View the dimensions of a variable:

h = [1, 2, 3, 4];size(h)

gives number of rows and columns:

ans =1 4

A scalar is a 1x1 matrix:

h = 13:size(h)

gives

ans =1 1

Get the number of rows/columns:

num_cols = size(h,2); num_rows = size(h,1);Carl Inge C. Nilsen INF3470/4470: Digital Signal ProcessingMATLAB Introduction

Matrix/Vector Operations

Transpose:

v = v.’;

Transpose conjugate (hermitian):

v = v’;

The inverse of a matrix:

A_inv = inv(A);


Matrix product (number of columns in B equals number ofrows in C):

A = BC , [A]ij =N∑

n=1

[B]in[C ]nj

A = B*C;

Element-wise (Hadamard) product (dimensions of B and Care identical):

A = B � C , [A]ij = [B]ij [C ]ij

A = B.*C;


Calling Functions

To see a list of return values and parameters:

help <function name>

Most functions can be called with (one) vector argument, andreturn a vector with the function employed on each element:

y = acos([-1.0, -0.5, 0.0, 0.5, 1.0]);

The variable y above contains:

[acos(-1.0), acos(-0.5), acos(0.0), acos(0.5), acos(1.0)]


Scalar Functions in One Variable

We want to find, usually to plot, for instance

f (x) = 2x2 − 3x + 14, x ∈ [xmin, xmax ]

In MATLAB, we can find the 1× N vector of samples:

~f = [f (x1), f (x2), · · · , f (xN)]

= [f (xmin), f (xmin + ∆), f (xmin + 2∆), · · · , f (xmax −∆), f (xmax)]

where ∆ = (xmax − xmin/(N − 1).First, define the vector of sampling points [x1, · · · , xN ]:

x_min = 0;x_max = 10;N = 100;x = [x_min:(x_max - x_min)/(N - 1):x_max];

Then, use the vector in the expression:

f = 2*x.*x - 3*x + 14;

Note: The use of the .* operator.Carl Inge C. Nilsen INF3470/4470: Digital Signal ProcessingMATLAB Introduction

Iterations vs. Vector-Operations

Alternative approach:

f = [];

x_min = 0;x_max = 10;N = 1024;delta_x = (x_max - x_min)/(N - 1);

for t=1:Nx = x_min + (t - 1)*delta_x;f = [f, 2*x^2 - 3*x + 14];

end

Same result, but time consuming!


A Function in Two Variables

We want to evaluate...

f (x , y) = α1x2 + α2y

2 + α3xy + α4x + α5y + α6

x ∈ [xmin, xmax ], y ∈ [ymin, ymax ]

we can discretize/sample both x and y, but the output variable fmust contain all possible combinations(xi , yj), i = 1, · · · ,M, j = 1, · · · ,N.Create two M × N matrices, X and Y, so that...

Xij = xi ,Yij = yj

[X,Y] = meshgrid(x_min:(x_max - x_min)/(M - 1):x_max,y_min:(y_max - y_min)/(N -1):y_max);

f = ALPHA_1*X.*X + ALPHA_2*Y.*Y + ALPHA_3*X.*Y+ ALPHA_4*X + ALPHA_5*Y + ALPHA_6;

f(i,j) is equal to f (xi , yj).


Plotting

Plotting the pairs (i , f (xi )):

plot(f)

Plotting the pairs (xi , f (xi )):

plot(x,f);

Default plotting is linear interpolation (i.e. straight linesconnecting the points).


Alternative#1 - Stem-plot:

stem(f); % Use the sample numbers as x-axis valuesstem(x,f); % Use the correct x-axis values

Alternative#2 - Discrete points with different colors:

plot(x,f,’r*’); % Mark each point with a red asteriskplot(x,f,’go’); % Mark each point with a green circle

Alternative#3 - Different lines:

plot(x,f,’.-’); % Use a dot-dashed lineplot(x,f,’--’); % Use a dashed lineplot(x,f,’.’); % Use a dotted line


Plot title (header):

title(’This is the title of the plot!’);

Axis labels:

xlabel(’Frequency (\omega)’);ylabel(’Magnitude (dB)’);

Adjusting axes:

x_min = 0;x_max = 2500;y_max = 0;y_min = -20;axis([x_min, x_max, y_min, y_max]);


Each time you use plot or stem, the current contents of the plotare discarded.Create a new figure:

plot(x,f);figure;plot(x,g);

Plot several curves in one figure:

plot(x,f);hold on;plot(x,g);plot(x,h);


Plot several figures in one window:

ROWS = 2; COLUMNS = 2;

% Plot something in the upper left figuresubplot(ROWS, COLUMNS, 1);plot(x, f1);

% Plot something in the upper right figuresubplot(ROWS, COLUMNS, 2);plot(x, f2);

% Plot something in the lower left figuresubplot(ROWS, COLUMNS, 3);plot(x, f3);

% Plot something in the lower right figuresubplot(ROWS, COLUMNS, 4);plot(x, f4);


Plotting matrices

As an intensity image in 2D (autoscale to palette):

imagesc(A);

As a surface in 3D:

surf(A);


Creating Scripts

Writing comments:

% Declaring variables:f = 3.5E3; % Center frequencyc = 340; % Speed of sound

The if-test:

X = Y*Z;

if X >= 0% Do something

end


The for-loop:

for i=1:100% Do something

end

The while-loop:

X = 10;

while X>=0% Do somethingX = X - 1;

end


Creating Functions

A function (to be stored in func name.m):

function Y = func_name(X1, X2, X3)Y = X1 + 2*X2 - X3/pi;


A function should check for parameter validity:

function Y = vector_div(X1, X2)if size(X1) ~= size(X2)

error(’Input vectors must have identical dimensions.’);return;

end

if or(sum(size(X1) == 1) == 0, sum(size(X2) == 1) == 0)error(’Inputs must be vectors or scalars.’);return;

end

if sum(X2 == 0) > 0error(’Divide by zero.’);return;

end

Y = X1./X2;


A function can have many return values:

% Convert z = A + j*B to z = R*exp(j*THETA)function [R, THETA] = cart2pol(A, B)

R = sqrt(A.^2 + B.^2);

if A ~= 0THETA = atan(B./A);

elseTHETA = sign(B)*pi/2;

end


inf3470/4470: digital signal processing matlab introduction · inf3470/4470: digital signal...

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