Can You Create a Single Equation for Multiple Time-Dependent Cosine Functions?

[ACLerok]

Given:
x(t) =
cos(2*pi*f1*t) 0 <= t < 4
cos(2*pi*f2*t) 4 <= t < 8
cos(2*pi*f3*t) 8 <= t < 12

f1, f2, and f3 are given as well

Would it be possible to combine all three of these conditions into one convenient equation or am I just dreaming? I tried using Matlab to concatenate the three graphs, but it won't give me an equation of it.

 

[HallsofIvy]

I'm not sure what you mean by a "convenient" equation. What you have is a perfectly good function, You could use the "Heaviside step function" which is defined by H(x)= 0 if x<= 0, H(x)= 1 if x> 0.

Then f(t)= cos(2pi f1 t)+ ((cos(2pi f2 t)- cos(2pi f1 t))H(t-4)+ (cos(2pi f3 t)- cos(2pi f2 t))H(t-8).

for 0<= t<= 4, both t-4< 0 and t-8< 0 so both H(t-4) and H(t-8) are 0 and only the first term, cos(2pi f1 t), is non zero. For 4< t<= 8, t-4> 0 so H(t-4)= 1 but H(t-8) is still 0: f(t)= cos(2pi f1 t)+ cos(2pi f2 t)- cos(2pi f1 t)= cos(2pi f2 t). If 8< t<= 12, both t-4> 0 and t-8>0 so both H(t-4) and H(t-8) are 1. f(t)= cos(2pi f1 t)+ cos(2pi f2 t)- cos(2pi f1 t)+ cos(2pi f3 t)- cos(2pi f2 t)= cos(2pi f3 t).

But is cos(2pi f1 t)+ ((cos(2pi f2 t)- cos(2pi f1 t))H(t-4)+ (cos(2pi f3 t)- cos(2pi f2 t))H(t-8) better than what you have?

 

[Architeuthis Dux]

Yes, it is possible to combine all three conditions into one equation. This is known as a piecewise function, where different equations are used for different intervals of the independent variable (in this case, t). The combined equation would look something like this:

x(t) = cos(2*pi*f1*t), 0 <= t < 4
cos(2*pi*f2*t), 4 <= t < 8
cos(2*pi*f3*t), 8 <= t < 12

This equation can be written in a more compact form using the Heaviside step function, which is defined as:

H(t) = 0, t < 0
1, t >= 0

Using this function, the combined equation would be:

x(t) = cos(2*pi*f1*t) * H(t) + cos(2*pi*f2*t) * H(t-4) + cos(2*pi*f3*t) * H(t-8)

This equation takes into account the different intervals for t and uses the appropriate cosine function for each interval. This can be easily implemented in Matlab as well.

So, you are not dreaming and it is possible to combine all three conditions into one equation.