CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 1

Colorado Technical University EE 443 – Communication 1

Lab 1: MATLAB Project – Signal Spectrum August 2010

Loren Schwappach

ABSTRACT: This lab report was completed as a course requirement to obtain full course credit in EE443, Communication 1 at Colorado Technical University. Given two time domain signals this lab report uses MATLAB to examine the frequency content of signals. All of the code mentioned in this lab report was saved as a MATLAB m-file for convenience, quick reproduction, and troubleshooting of the code. All of the code below can also be found at the end of the report as an attachment, as well as all figures.

If you have any questions or concerns in regards to this laboratory assignment, this laboratory report, the process used in designing the indicated circuitry, or the final conclusions and recommendations derived, please send an email to LSchwappach@yahoo.com. All computer drawn figures and pictures used in this report are of original and authentic content.

I. INTRODUCTION

MATLAB is a powerful program and is useful in the

visualization of mathematics, physics, and applied

engineering. In this lab exercise MATLAB will be used to

determine the frequency content of a composite sinusoidal

signal (to include its magnitude and power spectrum), and

the frequency content of a pulse defined by a rectangular

function (to include its magnitude, energy spectrum, and

autocorrelation).

First: Given the following signal:

(1)

Use MATLAB to plot the magnitude and power spectrum of

the signal.

Second: Given the following signal:

(2)

Use MATLAB to plot the magnitude and energy spectrum of

the signal.

Third: Given the second equation:

Use MATLAB to plot the autocorrelation of the signal.

II. PROCEDURE / RESULTS

To complete the first task the following code was used in MATLAB to produce and plot the composite sinusoidal function in the time domain:

function Lab1Prob1 = Comm1Lab2Problem1() %Function name for calling in MATLAB % Colorado Technical University % EE 443 - Communications I % Lab 1 - MATLAB Project - Signal Spectrum % By Loren K. Schwappach % Uses centeredFFT() for obtaining a two-sided spectrum %--------------------------- % Task #1 Magnitude and Power Spectrum for: y=5cos(2*pi*400*t)+5cos(2*pi*700*t) % Composite sinusodial wave f1 = 400; %frequency of the first sinusoidal wave (400Hz) a1 = 5; %amplitude of the first sinusoidal wave (5V) f2 = 700; %frequency of the second sinusoidal wave (700Hz) a2 = 5; %amplitude of the second sinusoidal wave (5V) fs = 25*f2; %sampling frequency (25*highest freq) (17.5kHz) ts = 1/fs; %sampling interval (57us) t1 = 0:ts:1-ts; %time vector (0:57us:999.943ms) y = (a1*cos(2*pi*f1*t1) + a2*cos(2*pi*f2*t1)); %composite sinusoidal wave % Plot of sinusodial wave in time domain timePlot = figure; %gives graph window a name and keeps it available plot (t1(1:176),y(1:176)); %plots sinusodial wave in time domain title('Composite Sinusodial y(t)=5cos(2*pi*400*t1)+5cos(2*pi*700*t2)'); xlabel('Time (s)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 2

Figure 1: Composite sinusoidal wave in the time domain

y(t)=5cos(2*pi*400*t)+5cos(2*pi*700*t).

You can see form figure 1 above that this is indeed a

composite sinusoidal result of equation 1.

Next, the following code was added to plot the

sinusoidal wave’s magnitude in the frequency domain: % Plot of sinusodial wave (Magnitude) in frequency domain [YfreqDomain,YfrequencyRange] = centeredFFT(y,fs); %Uses centeredFFT function freqPlot = figure; %gives graph window a name and keeps it available stem(YfrequencyRange,abs(YfreqDomain)); %Creates stem graph for magnitude spectrum title('Magnitude of y(t) in frequency domain -> Y(f)') xlabel('Freq (Hz)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([-800,800,-1,3]); %defines axis [x(min),x(max),y(min),y(max)]

Figure 2: Composite sinusoidal wave's magnitude in the

frequency domain, Y(f).

As illustrated by figure 2 above, you should observe

that the composite sinusoidal results in two positive frequency

impulses in the frequency domain, each at their previous

individual frequencies (400Hz and 700Hz), and at half of their

original individual amplitudes (5/2=2.5) as expected.

Next, the following code was added to plot the power

spectrum for the composite sinusoidal:

% Power Spectrum of sinusodial wave

% Note: Power = ((A^2)/2T))

% The LCM of (1/400) and (1/700) is (1/280,000) so T =

(1/280,000)

Ey=((abs(YfreqDomain).*abs(YfreqDomain)))*(1/(2*(1/2800

00)));

powerPlot = figure;

stem(YfrequencyRange,Ey); %Creates stem graph for

magnitude spectrum

title('Power spectrum of Y(f)');

xlabel('Freq (Hz)'); %adds xlabel to graph

ylabel('Power (Watts)'); %adds ylabel to graph

grid; %turns on grid

axis([-800,800,-1,1000000]); %defines axis

[x(min),x(max),y(min),y(max)]

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 3

Figure 3: Power Spectrum of Y(f).

As illustrated by figure 3 above, you should observe

that the power spectrum of the composite sinusoidal is much

higher in value than the amplitude of Y(f)’s magnitude. This

is due to the fact that the signals power is derived by squaring

Y(f)’s amplitude and dividing the result by 2*(1/T) where T is

the least common multiple of (1/400) and (1/700). Thus the

power ends up being very high since (1/T) is very small.

To complete the second task the following code was used in MATLAB to produce and plot the rectangular pulse function: % Task #2 Magnitude and Energy Spectrum for: x(t) = 2*rect(t/.002) % Note: By definition rect(t/x)=u(t+x/2)-u(t-x/2) so.. % x(t) = 2u(t+1e-3)-2u(t-1e-3) A = 2; t0 = -2e-3; tf = 2e-3; ts = (tf-t0)/1000; %(4us) fs = 1/ts; %(250kHz) t=[t0:ts:(tf-ts)]; %(-2ms:4us:2ms) x = A*rectpuls(t/2e-3); % Plot of x(t)=2*rect(t/.002) in time domain rTimePlot = figure; plot(t,x); %Creates stem graph for magnitude spectrum title('x(t)=2*rect(t/.002) in time domain'); xlabel('time (s)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([t0,tf-ts,0,2.1]); %defines axis [x(min),x(max),y(min),y(max)]

Figure 4: Rectangular pulse function x(t).

As shown by figure 4 above, the MATLAB code

correctly produces the rectangular pulse required by the

equation 2:

Next knowing that a rectangular pulse results in a sinc function the following code was added to plot the rectangular pulse’s magnitude in the frequency domain: % Plot of x(t)=2*rect(t/.002) (Magnitude) in frequency

domain

[XfreqDomain,XfrequencyRange] = centeredFFT(x,fs);

%Uses centeredFFT function

rFreqPlot = figure; %gives graph window a name and keeps it

available

plot(XfrequencyRange,abs(XfreqDomain)); %Creates stem

graph for magnitude spectrum

title('Magnitude of x(t) in frequency domain -> X(f)');

xlabel('Freq (Hz)'); %adds xlabel to graph

ylabel('Amplitude'); %adds ylabel to graph

grid; %turns on grid

axis([-10000,10000,0,1.1]); %defines axis

[x(min),x(max),y(min),y(max)]

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 4

Figure 5: Magnitude of x(t) in the frequency domain X(f).

As produced by figure 5 above, you should observe

that the rectangular pulse function equates to a sinc function in

the frequency domain as expected. However, since the

magnitude of the function was required the sinc function never

drops below zero.

Next the following code was used to graph the energy spectrum of the rectangular pulse:

% Energy Spectrum of x(t)=2*rect(t/.002)

Ex = (abs(XfreqDomain).*abs(XfreqDomain));

rEnergyPlot = figure;

plot(XfrequencyRange,Ex); %Creates stem graph for

magnitude spectrum

title('Energy spectrum of X(f)');

xlabel('Freq (Hz)'); %adds xlabel to graph

ylabel('Amplitude'); %adds ylabel to graph

grid; %turns on grid

axis([-10000,10000,0,1.1]); %defines axis

[x(min),x(max),y(min),y(max)]

Figure 6: Energy spectrum of X(f)

As seen by figure 6 above, the energy spectrum

(energy spectral density) of the rectangular pulse is merely the

frequency domain result squared. This results in a much

smaller amplitude since the frequency domain amplitude is

already <1 (something small^2 = something even smaller).

To complete the final task the following code was used in MATLAB to plot the autocorrelation of the rectangular pulse x(t):

% Task #3 Plot the autocorrelation for the rect function in task

2.

% Uses task #2's variables and functions.

Rxx=xcorr(x); % Estimate its autocorrelation

rEnergyPlot = figure;

plot(Rxx); % Plot the autocorrelation

title('Autocorrelation function of x(t)=2*rect(t/.002)');

xlabel('lags');

ylabel('Autocorrelation');

grid;

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 5

Figure 7: Autocorrelation of x(t).

As seen by figure 7 above, the autocorrelation of the

rectangular pulse results in a linear pyramid. The

autocorrelation function is a measure of the similarity between

x (t) and its delayed counterpart x ( ).

III. CONCLUSIONS

. MATLAB is a great utility for representing complex

concepts visually and can easily be manipulated to show

signals in various formats. This lab project was successful in

demonstrating MATLABs powerful features in a quick and

easy method, and demonstrating how MATLAB can be used

for displaying the frequency contents of signals.

REFERENCES

[1] Haykin, S., “Signals and Systems 2nd

Edition” McGraw-Hill, New York, NY, 2007.

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 6

Figure 8: Composite sinusoidal wave in the time domain y(t)=5cos(2*pi*400*t)+5cos(2*pi*700*t).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 7

Figure 9: Composite sinusoidal wave's magnitude in the frequency domain, Y(f).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 8

Figure 10: Power Spectrum of Y(f).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 9

Figure 11: Rectangular pulse function x(t).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 10

Figure 12: Magnitude of x(t) in the frequency domain X(f).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 11

Figure 13: Energy spectrum of X(f)

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 12

Figure 14: Autocorrelation of x(t).

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 13

%MATLAB CODE

function Lab1Prob1 = Comm1Lab2Problem1() %Function name for calling in MATLAB % Colorado Technical University % EE 443 - Communications I % Lab 1 - MATLAB Project - Signal Spectrum % By Loren K. Schwappach % Uses centeredFFT() for obtaining a two-sided spectrum

%---------------------------

% Task #1 Magnitude and Power Spectrum for: y=5cos(2*pi*400*t)+5cos(2*pi*700*t) % Composite sinusodial wave f1 = 400; %frequency of the first sinusodial wave (400Hz) a1 = 5; %amplitude of the first sinusodial wave (5V) f2 = 700; %frequency of the second sinusodial wave (700Hz) a2 = 5; %amplitude of the second sinusodial wave (5V) fs = 25*f2; %sampling frequency (25*highest freq) (17.5kHz) ts = 1/fs; %sampling interval (57us) t1 = 0:ts:1-ts; %time vector (0:57us:999.943ms) y = (a1*cos(2*pi*f1*t1) + a2*cos(2*pi*f2*t1)); %composite sinusodial wave

% Plot of sinusodial wave in time domain timePlot = figure; %gives graph window a name and keeps it available plot (t1(1:176),y(1:176)); %plots sinusodial wave in time domain title('Composite Sinusodial y(t)=5cos(2*pi*400*t1)+5cos(2*pi*700*t2)'); xlabel('Time (s)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid

% Plot of sinusodial wave (Magnitude) in frequency domain [YfreqDomain,YfrequencyRange] = centeredFFT(y,fs); %Uses centeredFFT function freqPlot = figure; %gives graph window a name and keeps it available stem(YfrequencyRange,abs(YfreqDomain)); %Creates stem graph for magnitude spectrum title('Magnitude of y(t) in frequency domain -> Y(f)') xlabel('Freq (Hz)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([-800,800,-1,3]); %defines axis [x(min),x(max),y(min),y(max)]

% Power Spectrum of sinusodial wave % Note: Power = ((A^2)/2T)) % The LCM of (1/400) and (1/700) is (1/280,000) so T = (1/280,000) Ey = ((abs(YfreqDomain).*abs(YfreqDomain)))*(1/(2*(1/280000))); powerPlot = figure; stem(YfrequencyRange,Ey); %Creates stem graph for magnitude spectrum title('Power spectrum of Y(f)'); xlabel('Freq (Hz)'); %adds xlabel to graph ylabel('Power (Watts)'); %adds ylabel to graph grid; %turns on grid axis([-800,800,-1,1000000]); %defines axis [x(min),x(max),y(min),y(max)]

%---------------------------

% Task #2 Magnitude and Energy Spectrum for: x(t) = 2*rect(t/.002) % Note: By definition rect(t/x)=u(t+x/2)-u(t-x/2) so.. % x(t) = 2u(t+1e-3)-2u(t-1e-3) A = 2; t0 = -2e-3; tf = 2e-3;

CTU: EE 443 – Communications 1: Lab 1: MATLAB Project – Signal Spectrum 14

ts = (tf-t0)/1000; %(4us) fs = 1/ts; %(250kHz) t=[t0:ts:(tf-ts)]; %(-2ms:4us:2ms) x = A*rectpuls(t/2e-3);

% Plot of x(t)=2*rect(t/.002) in time domain rTimePlot = figure; plot(t,x); %Creates stem graph for magnitude spectrum title('x(t)=2*rect(t/.002) in time domain'); xlabel('time (s)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([t0,tf-ts,0,2.1]); %defines axis [x(min),x(max),y(min),y(max)]

% Plot of x(t)=2*rect(t/.002) (Magnitude) in frequency domain [XfreqDomain,XfrequencyRange] = centeredFFT(x,fs); %Uses centeredFFT function rFreqPlot = figure; %gives graph window a name and keeps it available plot(XfrequencyRange,abs(XfreqDomain)); %Creates stem graph for magnitude spectrum title('Magnitude of x(t) in frequency domain -> X(f)'); xlabel('Freq (Hz)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([-10000,10000,0,1.1]); %defines axis [x(min),x(max),y(min),y(max)]

% Energy Spectrum of x(t)=2*rect(t/.002) Ex = (abs(XfreqDomain).*abs(XfreqDomain)); rEnergyPlot = figure; plot(XfrequencyRange,Ex); %Creates stem graph for magnitude spectrum title('Energy spectrum of X(f)'); xlabel('Freq (Hz)'); %adds xlabel to graph ylabel('Amplitude'); %adds ylabel to graph grid; %turns on grid axis([-10000,10000,0,1.1]); %defines axis [x(min),x(max),y(min),y(max)]

%---------------------------

% Task #3 Plot the autocorrelation for the rect function in task 2. % Uses task #2's variables and functions. Rxx=xcorr(x); % Estimate its autocorrelation rEnergyPlot = figure; plot(Rxx); % Plot the autocorrelation title('Autocorrelation function of x(t)=2*rect(t/.002)'); xlabel('lags'); ylabel('Autocorrelation'); grid;

%---------------------------

% end Comm1Lab1Problem1