Why sine wave is the basic building block signal to make other signals?

[dexterdev]

Why sine wave is the basic building block signal to make other signals?
Not any other wave.

-Devanand T

 

[milesyoung]

Sines and cosines are not the basic building blocks to make other signals.

You can decompose a periodic function into sines and cosines as a Fourier series but sines and cosines are not the only functions that can serve as a basis for it. There's a great deal of math involved, but it basically relates to the orthogonality of certain sets of functions:

http://mathworld.wolfram.com/FourierSeries.html

 

[skeptic2]

Think for a second about where sines and cosines come from. They are the Y and X coordinates of a point that revolves along the unit circle. The equivalent for a squarewave would a point following a unit square. With just a pair of X and Y values, it is not possible to determine the exact phase of the point. While it is possible to create mathematical functions using other waveforms, such as squarewaves, it would be much more difficult to make electronic filters for those waveforms.

 

[dijkarte]

Why sine wave is the basic building block signal to make other signals?
Not any other wave.

The only reason I can think of is:

Ancient Greek scientists used to think that all matter is comprised of four elements of earth, air, fire and water. Is this valid now?

But no the building block of waves according to now science is the complex exponential infinite series.

 

[milesyoung]

But no the building block of waves according to now science is the complex exponential infinite series.

I can only assume you mean the complex form of the Fourier series. I could select any other set of functions that form a complete orthogonal system and call that set the 'building blocks' of waves. There is no unique answer as to what waves "are made of".

 

[ZVdP]

As said, you can decompose a wave into other functions as well, but why do we use sine waves (most of the time) then ?

Take a look at a linear system with 1 input and 1 output. The linearity means:
F(ax)=aF(x)
F(x₁+x₂)=F(x₁)+F(x₂)
where a is a scalar.

This means that to characterise the system, we only need to know the response to a set of basis functions (f_i) , since any other function can be written as a sum of these functions.
g=\sum_i a_if_i\Rightarrow F(g)=\sum_i a_i F(f_i)
This shows we only need to know the functions F(f_i) (for each i) to know the output for every input signal, g.

This is where the sine waves become interesting. It can be shown that sine waves are eigenfunctions of a linear system, meaning that if I apply a sine wave at the input of a linear system, the output will also be a sine wave of the same frequency, but with scaled amplitude and shifted phase:
F(sin(ωx))=A*sin(ωx+\varphi)

This means that we only need two numbers, A and \varphi, to uniquely characterise the response to a sine wave.
The characterisation of the entire system reduces therefore to 1 complex function H(ω) which gives us the A and \varphi) value at each frequency ω as a complex number Ae^-j\varphi.

If we would use another set of functions to decompose our wave into, we would require an entire set of functions to characterise the system (one function for each function in the set), compared to just 1 function in case of sine waves.

 

[carlgrace]

As said, you can decompose a wave into other functions as well, but why do we use sine waves (most of the time) then ?

We use sine waves simply because they make the math easier*. Generally, things are actually decomposed into complex exponentials most often in practice (simply for convenience) but as milesyoung correctly said, there are actually an infinite number of acceptable basis functions out there. None of them are as easy to work with as sines and cosines, though. There are specialized cases where you choose to decompose signals differently, however.

*why are they easy? One big reason: exponential is its own derivative, and d/dt(cos(t)) = -sin(t)

 

[ZVdP]

We use sine waves simply because they make the math
*why are they easy? One big reason: exponential is its own derivative, and d/dt(cos(t)) = -sin(t)

Ah yes, I forgot that's a useful property as well. My post concerned LTI system theory, while the derivative property can be very useful indeed in wave mechanics (eliminating the partial derivatives).

 

[dexterdev]

Thank you guys for the replies.