day1module05

8/9/2019 day1module05

http://slidepdf.com/reader/full/day1module05 1/17

MODULE 5- MATLAB FOR SINE

WAVES PART II



SUM OF SINE WAVES OF DIFFERENT

FREQUENCIES

t=0:0.01:20; % Creates an array ta=sin(t); % Generates a sine wave with period for the elements of t

subplot(3,1,1) % Breaks the figure window into an 3X1 matrix of plots and

% chooses the first plot row wise

plot(t,a,'LineWidth',2) % Generates a plot of t vs a

grid on % Displays a grid on the plotaxis([0 20 -1.2 1.2])

xlabel('time') % Labels the x axis

ylabel('sin(t)') % Labels the y axis

title('Plot to show that the sum of sine waves of different frequencies does not…

result in a sine wave')% Displays a title for the plot

% Code continued on next page



b= sin(4*t);

subplot(3,1,2)

plot(t,b,'LineWidth',2) % Generates a plot of t vs c

grid on % Displays a grid on the plot

axis([0 20 -1.2 1.2])


ylabel('sin(4t)') % Labels the y axis

c= a + b;

subplot(3,1,3)

plot(t,c,'LineWidth',2) % Generates a plot of t vs c

grid on % Displays a grid on the plotaxis([0 20 -2.4 2.4])


ylabel('sin(t)+sin(4t)') % Labels the y axis



0 2 4 6 8 10 12 14 16 18 20

-1

0

1

time

s i n ( t )

Plot to show that the sum of sine waves of different frequencies does not result in a sine wave

0 2 4 6 8 10 12 14 16 18 20

-1

0

1

time

s i n ( 4 t )

0 2 4 6 8 10 12 14 16 18 20

-2

0

2

time

s i n ( t ) + s i n ( 4 t )



GENERATION OF A SQUARE WAVE FROM

SINE WAVES OF DIFFERENT FREQUENCIES

% Generation of square wave from sine waves of different frequencies

t=0:0.01:20;

y1 = sin(t);

plot(t,y1,'g','LineWidth',2)grid on


title('Generation of a square wave from sine waves') % Displays a title for the plot

hold on

y2 = sin(t) + sin(3*t)/3 + sin(5*t)/5 + sin(7*t)/7 + sin(9*t)/9;plot(t,y2,'k','LineWidth',2)



0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time

Generation of a square wave from sine waves



GENERATION OF A SAWTOOTH WAVE FROM

SINE WAVES OF DIFFERENT FREQUENCIES

clc;

clear;

close all;

t=0:0.001:10; % Creates an array t

N= length(t);

s = zeros(1,N);

for k=1:1:1001

s= -(sin(2*pi*k.*t)./k)+ s;

end

plot(t,s,'LineWidth',2)

grid on

title('Generation of Sawtooth waveform from Sine Waves of different frequencies')



0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Generation of Sawtooth waveform from Sine Waves of different frequencies



FFT IN MATLAB• The Fourier theorem states that any waveform can be duplicatedby the superposition of a series of sine and cosine waves.

• We did this while generating a square from the superposition of

sine waves of different frequencies!

• We use Fourier transform to perform the decomposition

• We’re going from time domain to frequency domain

• FFT( Fast Fourier transform) is an algorithm to find the Fouriertransform of an input sequence

• We use the MATLAB command ‘fft‘

>> g=[1 2 3 4 5];

>> fft(g)

ans =

15.0000 -2.5000 + 3.4410i -2.5000 + 0.8123i -2.5000 -

0.8123i -2.5000 - 3.4410i



SPECTRUM OF A PERIODIC(SINE) WAVE

clc;

clear;

close all;

N=2^13; % Number of samples

fmax=64; % Maximum frequency in Hz

fs=2*fmax; % Sampling Frequency

df=fs/N; % frequency spacing

dt=1./(df.*N); % time spacing

t=(1:N)*dt-dt; % time samplesf(1:N/2)=(1:N/2)*df-df; % Nonnegative frequency samples

f(N/2+1:N)=-fmax+(0:N/2-1)*df; % Negative frequency samples

f2=fftshift(f); % shifting the values in f so that it goes from –f max to +fmax




xt = sin(2*pi*10*t); % sine wave of frequency 4 in time domain

xf = fft(x).*dt; % sine wave in the frequency domain

xf = 20*log10(fftshift(abs(xf))); % converting to dB, taking the absolute value

% and shifting the values to correspond to the

% frequencies in f

plot(f2,xf,'LineWidth',2)

grid on

title('Spectrum of a Sine wave of one frequency')

xlabel('Frequency in Hz')

ylabel('Magnitude in dB')



-80 -60 -40 -20 0 20 40 60 80-350

-300

-250

-200

-150

-100

-50

0

50Spectrum of a Sine wave of one frequency

Frequency in Hz

M a g n i t u d e

i n d B



SPECTRUM OF SPEECH

• Type this in the MATLAB command window

>>demoai_fft



CHIRP SIGNAL

• A chirp is a signal in which the frequency increases ('up-chirp') or decreases

('down-chirp') with time.

• Code to generate and play a chirp signal :

clc;

clear

close all;

Fs = 4000;

T = 1;

dt = 1/Fs;

N = T*Fs; % Number of samples

df = Fs/N; % frequency spacingt = (1:N)*dt - dt; % time samples

f(1:N/2)=(1:N/2)*df-df; % Nonnegative frequency samples

f(N/2+1:N)=-fmax+(0:N/2-1)*df; % Negative frequency samples

f2=fftshift(f); % shifting the values in f so that it goes from –f max to +fmax




x = cos(f.*t);

wavplay(x,1/dt)plot(t,x)

axis([0 2.2 -1.2 1.2])

xf = fft(x).*dt;

xf = fftshift(abs(xf)); % Absolute value of the Fourier coefficientsfigure

stem(f,xf)

grid on

title(‘Spectrum of chirp signal’)



GUESS WHAT SIGNAL IS FORMED

WHEN YOU RUN THIS CODE

clc;

clear;

close all;

dt = 0.001;

t=0:dt:4-dt; % Creates an array t

N= length(t);

x = zeros(1,N);x(1,N/4:N/2) = 1:-4/N:0;

x(1,N/2:(3*N/4)) = 0:4/N:1;



% Find the Fourier Series Co-efficients:

X = zeros(1,N);

X = X + x(1);k=0:1:(N-1);

for n = 1:N-1

X = X + (x(n+1)*exp(-j*2*pi*k*n/N));

end

%Reconstruction Using Fourier Co-efficients

x_r = zeros(1,N);

x_r = x_r + X(1);

n = 0:N-1;

for k=1:10 % Change the value of k to change the number of coefficients used in reconstruction

x_r = (x_r + (X(k+1)*exp(j*2*pi*k*n/N)));end

%plot(x_r)

plot(t,x_r,'LineWidth',2)

grid on

day1module05

Documents

Transcript of day1module05