Case Western Reserve University

Discrete and Continuous Time Signals

Lab 1

Jared Bell

1/22/2012

2.3.1 Analytical Calculation

1.

∫0

2π

sin2 (5t ) dt

∫0

2π 12

(1−cos (2∗5 t ))dt

12∫02π

1dt−¿ 12∫02π

cos (10 t ) dt ¿

12t ¿02 π−

12∗1

10(sin (10 t ) ¿0

2π)

( 2π2 −12∗0)− 1

20(sin (10∗2π )−sin (10∗0 ))

( 2π2 −0)− 120

(0−0)

∫0

2π

sin2 (5t )dt=π

2.

∫0

1

etdt

∫0

1 ddte t dt

e t ¿01

e1−e0

e−1

∫0

1

etdt=e−1=1.718

2.3.2 Displaying Continuous-Time and Discrete-Time Signals in Matlab

The stem plot uses evenly spaced integers between 0 and 60 to show the discrete time

version of the function sin(n/6). The plot of the continuous time function shows the same

function, except in continuous time. These are both accurate graphs. However, the third graph

shows the continuous time function with the integers spaced by 10. As a result this graph

shows undersampling, and is not an accurate representation of the function.

2.3.3 Numerical Computation of Continuous-Time Signals

As the value of N approached 100, the Riemann sum stabilized to a value similar to the

one calculated in the Analytical Calculations. That is for the function sin(5t)2 it approached π,

and for the function e(t) it approached e – 1, or 1.72. The only unusual behavior in the

Riemann sum function were the two spots in the Riemann sum of the sin function graph, where

the sum was approximately 0. I’m assuming that this is due to the rectangles being too large to

accurately approximate the integral. I believe that this is why I(5) = I(10) = 0, however on my

graph it appears to be slightly shifted. When N = 5 in my code, f(t) alternates perfectly between

0 and 1, and the same when N = 10. When N = 6 is when f(t) becomes very small, which is why

when it is multiplied by delta x it approaches 0.

2.4 Processing of Speech Signals

2.5 Attributes of Continuous-Time Signals

Min = -0.1918

Max = 1

Energy = 2.0658

I chose to start at one and go to 100, as it was a broad range that allowed me to see the

signal very easily. However, the graph can probably be truncated at around 50-60, as that is the

point where the signal becomes a shallow sinusoidal centered around 0. So the approximate

start and stop values should be 1 and 60, but I chose 100 for the visual representation of said

sinusoid.

2.6 Special Functions

2.7 Sampling

Changing the sample time lost the information of the sine wave. With Ts = 1/3 the sine

wave became similar to a triangle wave. With Ts = 1/2 it became a line. These first two changes

in the sampling time demonstrate undersampling. With Ts = 10/9, it became just one peak of

the sine wave. The graph was slightly oversampled, but that is hard to tell due to the

constraints of the axes.

2.8 Random Signals

As the value of n increased, the average values settled to their mean values. Essentially

with the increase in n, the average value was statistically more like to be the mean since it is a

Gaussian distribution. This allows the viewer to easily tell the difference between two different

random signals depending on their means.

2.9 2-D Signals

The surface plot is visually more appealing to me, as it allows you to visualize and work

in 2-D space easier. The image gives you the same information, but in 2-D. I personally would

not want to work with the image, but I believe that format is easier to use for Fast Fourier

Transforms, as I have used them for that before.

Appendix: Code

Main

%% Header % Jared Bell% EECS 313% Lab 1 - Discrete and Continuous Time Signals% 1/22/12 %% 2.3.2 Displaying Continuous Time and Discrete Time Signals in Matlab % This section establishes n as a series of evenly spaced integers (by 2)% between 0 and 60, then plots them according the the y function.n = 0:2:60;y = sin(n/6);subplot(3,1,1)stem(n,y)title('Discrete Time');xlabel('Time');ylabel('y(n)'); % This section does the same as the previous section, however it uses plot% instead of stem, to be used similar to a continuous time function. This% is an accurate extrapolation of the data.n1 = 0:2:60;z = sin(n1/6);subplot(3,1,2)plot(n1,z)title('Continuous Time');xlabel('Time');ylabel('y(n)'); % This section does the same as the previous, however it shows% undersampling due to the integer points being spaced by 10. As a result,% this is a very inaccurate graph of the sin function.n2 = 0:10:60;w = sin(n2/6);subplot(3,1,3)plot(n2,w)title('Continuous Time Undersampled');xlabel('Time');ylabel('y(n)'); pause %% 2.3.3 Numerical Computation of Continuous-Time Signals % functions are called to calculate riemann sum based off of N I = 1:100; % presetting vector length to save memory

J = 1:100; for Ncounter = 1:100 I(Ncounter) = integ1(Ncounter); J(Ncounter) = integ2(Ncounter); % calling htem both to save for loops end subplot(2,1,1)plot(I)xlabel('N rectangles');ylabel('Area');title('Riemann sum of sin(5t)^2'); subplot(2,1,2)plot(J)xlabel('N rectangles');ylabel('Area');title('Riemann sum of exp(t)'); % Note that as N approaches 100 the value of the area becomes approximately% the value solved for in the analytical solutions, that is pi and e - 1% (1.72). pause %% 2.4 Processing of Speech Signals audio_vector = auread('speech.au'); % loads the speech filefigure;plot(audio_vector);title('Speech.au File'); % plotted and labeled sound(audio_vector); % this plays the file pause %% 2.5 Attributes of Continuous-Time Signals t = 0:100;plot(signal1(t)); % Every integertitle('Signal1(t)');xlabel('Time');ylabel('y(t)'); pause t2 = 0:2:100;plot(signal1(t2)); % Every other integer t3 = 0:2:50;plot(signal1(t3)); % Every other integer to 50

minSignal = minSignal1(t); % calls functions to return min and max valuesmaxSignal = maxSignal1(t);energySignal = energySignal1(t); pause %% 2.6 Special Functions t_sinc = linspace(-10*pi,10*pi);t_rect = linspace(-2,2); y = sinc(t_sinc);z = (abs(t_rect)<=0.5); subplot(2,1,1)plot(y);subplot(2,1,2)plot(z);xlabel('Time');ylabel('f(t)');title('F(t) vs Time'); pause a1 = 0.8;a2 = 1.0;a3 = 1.5; n = -20:20; ya1 = (a1.^n).*(heaviside(n) - heaviside(n-10)); % calculates the functionya2 = (a2.^n).*(heaviside(n) - heaviside(n-10));ya3 = (a3.^n).*(heaviside(n) - heaviside(n-10)); orient tall; % plots the functionssubplot(3,1,1)plot(ya1);ylabel('Time');xlabel('y(t)');title('a = 0.8'); subplot(3,1,2)plot(ya2);ylabel('Time');xlabel('y(t)');title('a = 1.0'); subplot(3,1,3)plot(ya3);ylabel('Time');xlabel('y(t)');title('a = 1.5');

pause n2 = -1:10;w = pi/4; z1 = cos(w*n).*(a1.^n).*(heaviside(n)); % same, calculates the functionsz2 = cos(w*n).*(a2.^n).*(heaviside(n));z3 = cos(w*n).*(a3.^n).*(heaviside(n)); orient tall; % plots the functionssubplot(3,1,1)plot(z1);ylabel('Time');xlabel('z(t)');title('a = 0.8'); subplot(3,1,2)plot(z2);ylabel('Time');xlabel('z(t)');title('a = 1.0'); subplot(3,1,3)plot(z3);ylabel('Time');xlabel('z(t)');title('a = 1.5'); pause %% 2.7 Sampling Ts1 = 1/10; % establishes constantsTs2 = 1/3;Ts3 = 1/2;Ts4 = 10/9; ns1 = 0:100;ns2 = 0:30;ns3 = 0:20;ns4 = 0:9; xn1 = sin(2*pi*Ts1*ns1); % calculates functionsxn2 = sin(2*pi*Ts2*ns2);xn3 = sin(2*pi*Ts3*ns3);xn4 = sin(2*pi*Ts4*ns4); subplot(4,1,1)plot(xn1)ylabel('Time');xlabel('x(n)');title('Ts = 1/10');axis([0,100,-1,1]);

subplot(4,1,2)plot(xn2)ylabel('Time');xlabel('x(n)');title('Ts = 1/3');axis([0,30,-1,1]); subplot(4,1,3)plot(xn3)ylabel('Time');xlabel('x(n)');title('Ts = 1/2');axis([0,20,-1,1]); subplot(4,1,4)plot(xn1)ylabel('Time');xlabel('x(n)');title('Ts = 10/9');axis([0,9,-1,1]); % Changing the sample time lost the information of the sine wave. With Ts% = 1/3 the sine wave became similar to a triangle wave. With it equal to% 1/2 it became a line. With it equal to 10/9, it became just one peak of% the sine wave. pause %% 2.8 Random Signals sig1 = random('norm',0,1,1000);sig2 = random('norm',0.2,1,1000); % random signals subplot(2,1,1)plot(sig1)title('Signal 1 Mean=0 Variance=1');subplot(2,1,2)plot(sig2)title('Signal 2 Mean=0.2 Variance=1'); pause ave1 = 1:1000;ave2 = 1:1000; for counter = 1:1000 % use a for loop to make a summation of the averages ave1(counter) = mean(sig1(1:counter)); ave2(counter) = mean(sig2(1:counter)); end figure;plot(ave1, 'k');

hold onplot(ave2, 'b');title('Average of the Random Signals Based on n');xlabel('n');ylabel('Average');legend('Average1', 'Average2');hold off pause %% 2.9 2-D Signals m = linspace(-50,50);n = linspace(-50,50); [x,y] = meshgrid(m,n); % generates arrays f = 255.*(abs(sinc(0.2.*x).*sin(0.2.*y))); % calculates function mesh(f); % mesh plot to display the signaltitle('Mesh plot of f(m,n)');pause image(f); % make imagetitle('Grayscale image of f(m,n)');colormap(gray(256)); % grayscale

Integ1

function [ I ] = integ1( N )%INTEG1 does a reimann sum approximation based off the number of rectangles%N t = linspace(0,2*pi,N); % makes evenly spaced based off number of rectangles delta = t(end)/N; ft = sin(5.*t).^2; % calculates height A = ft.*delta; % calculates area I = sum(A); % sums end

Integ2

function [ J ] = integ2( N )

%INTEG2 Riemann sum calculation based of the number of rectangles N t = linspace(0,1,N); % makes evenly spaced based off number of rectangles delta = t(end)/N; ft = exp(t); % calculates height A = ft.*delta; % calculates area J = sum(A); % sums end

MinSignalfunction [ minSignal ] = minSignal1(t)%MINSIGNAL1 - Calculates the minimum of a signal over a range of time% Takes input arguement and calculates the minimum, calls signal1 minSignal = min(signal1(t)); end

MaxSignalfunction [ maxSignal ] = maxSignal1(t)%MAXSIGNAL1 - Calculates the maximum of a signal over a range of time% Takes input arguement and calculates the maximum, calls signal1 maxSignal = max(signal1(t)); end

EnergySignalfunction [ energySignal ] = energySignal1(t)%ENERGYSIGNAL1 - Calculates the energy of a signal over a range of time% Takes input arguement and calculates the energy by intergrating and% calling signal1 absSignal = abs(signal1(t)); % This takes care of the absolute valuesignalSquared = absSignal.^2; % takes care of the squaringenergySignal = sum(signalSquared); % summation of the data end