equation for the sum of sinusoids

Hello,

I am attempting to create a sum of sines waveform using 4 different frequencies, with the same amplitude of 1, and all with a phase of 0. Sampling rate of 60 with a duration of 30 seconds. Through reading different forums I came up with the code below, however I'm not sure I understand the pi/3, pi/4, etc. used in the x1-x4 equations below. Can anyone explain why this is included? or if there is a better more simple formula than what I have below.

Fs = 60;

Ts = 1/60;

t = 0:Ts:30

x1 = 1*cos(2*pi*.73*t+pi/3);

x2 = 1*cos(2*pi*1.33*t-pi/4);

x3 = 1*cos(2*pi*1.93*t+pi/5);

x4 = 1*cos(2*pi*2.93*t-pi/6);

x = x1 + x2 + x3 + x4;

u = numel(x);

M = zeros(u,6);

M(:,3) = x;

M(:,(1:6)~=3);

M

figure(2)

subplot(5,1,1)

h = plot(t,x1); box off; grid off

xlabel('Time(s)');

ylabel('Amplitude');

subplot(5,1,2)

h = plot(t,x2); box off; grid off

xlabel('Time(s)');

ylabel('Amplitude');

subplot(5,1,3)

h = plot(t,x3); box off; grid off

xlabel('Time(s)');

ylabel('Amplitude');

subplot(5,1,4)

h = plot(t,x4); box off; grid off

xlabel('Time(s)');

ylabel('Amplitude');

subplot(5,1,5)

h = plot(t,x); box off; grid off

xlabel('Time(s)');

ylabel('Amplitude');

Answer 1

[Adam Danz]

"I'm not sure I understand the pi/3, pi/4, etc. used in the x1-x4 equations below. Can anyone explain why this is included?"

A quick tutorial

figure; 
t = 0 : 1/60 : 30;
plot(cos(t), 'k-')
hold on

The plot produced above is your base sinusoid. To change the frequency, multiply t by a value greater than one to increase the frequency or a value less than one to decrease the frequency.

plot(cos(t * 1.2), 'b-')  %increase the freq.

To change the phase, add or subtract values from t. Adding values to t will shift the sinusoid leftward; subtractive values will shift if rightward. Try the block below.

figure
plot(cos(t), 'k-')
hold on
plot(cos(t + pi/6), 'b:')  %shift leftward

You're working in radians so pi/6 is 30 degrees. If you shift it 360 degrees (2*pi in radians) the black and blue curves will overlap.

In your code (below), t is multiplied by 2*pi*.73 for whatever reason which changes the frequency of your sinusoid. The result is then added to pi/3 which shifts the phase. The reason the values are in fractions of pi is because you're working in radians.

x1 = 1*cos(2*pi*.73*t+pi/3); 

"is a better more simple formula than what I have below"

That entirely depends on what you're doing. If you're just trying to create some sinusoids and their parameters don't matter, you could replace the 2*pi*.73 with whatever value that equals.