Learning Fourier Series

[gobi1990]

Hi,

I am learning fourier series, and i have couple of doubt to ask.

As per my understanding FS is for analyzing periodic signal in Frequency domain. In Frequency domain we can able to separate each frequency with its amplitude which we cant do in time domain. With Use of this Frequency domain we found lot of advantage in removing noise, unwanted signal, etc..

My doubt is in Trignometric FS general formula why sin and cos term were present. Actually every signal is made up of sum of different frequency/Amplitude of sine waves, so only sin is the player in forming waves but why cos term is present in trigonometric FS Formula.

Thanks Much!!

 

[andre_luis]

Both sin and cos represents the same function, but one shifts the argument of the other by ±PI/2. Depending on what physical phenomen you want to obtain the model, the formula may become simplest if you chose one or another trigonometric function.

 

[gobi1990]

Hi, Thanks for the reply.

If sin and cos represents the same function then why both term mentioned separately in the formula, instead we can just put another sin in place of cos or why need of same function twice in formula. Why we need to put cos term badly and why sin term is mentioned every where in texts book while forming signals not cos function.

Thanks!!

 

[CataM]

Agree with Andre.

Is helpful for us to write the periodic function as a sum of sines and cosines because it is intuitive for us and better understand it. Sine is a odd function and cosine is a even function, so, if you have a even periodic function then you know that there must only appear sums of even functions = sums of cosines.

 

[andre_luis]

If sin and cos represents the same function then why both term mentioned separately in the formula

What formula are you talking about ?

 

[CataM]

why sin term is mentioned every where in texts book while forming signals not cos function.

AC monophase voltage is produced by AC generators as a sine wave and not a cosine wave.

 

[FvM]

In case you are only interested in the magnitude respective power spectrum of a signal, the phase of spectral components is irrelevant. But if you want to reconstruct the periodic time domain signal represented by the fourier series, phase matters. The same magnitude spectrum gives different waveforms if you change the phase relation between components (or change between sine (0° phase) and cosine (90° phase) for specific fourier components).

You can use the Falstad fourier applet to see the effect. https://www.falstad.com/fourier/

 

[gobi1990]

Hi Thanks for the reply.

To account also for even signal we are using cos function.I understand it now. Thanks

Andre

Below is the formula. I just understood that. Thanks

 

[andre_luis]

AC monophase voltage is produced by AC generators as a sine wave and not a cosine wave.

An additional advantage of using the sin function to represent the mathematical model of phenomens is that for small angles we can elliminate all other components keeping just the one with smallest order, due to the fact that sin(x)~x when x is near to 0.

 

[gobi1990]

AC monophase voltage is produced by AC generators as a sine wave and not a cosine wave.

Then can we say it like, AC Generators produce odd signal not even signal.? Am i forming it correctly?

If sin and cos are same function then why AC voltage is preferred to produce in sin but not in cos.?

Thanks!.

 

[c_mitra]

This is a initial value or boundary value problem. Let me try.

A given function in the time domain f(t) can be converted into a frequency domain function. This is a transformation, one function f(t) is being converted into another function F(w). The w and t must be related and the relation is w.t=1 and the Fourier transformation is integral (f(t).exp(-i.w.t).dt) producing the final function F(w).

Now you can see that this is a complex transformation and every such transformation has two components: one is real part and the other is imaginary part. In polar coordinate, we can say that one denotes the amplitude and the other the phase. Engineers call that the Fourier transformation gives the result in quadrature.

Most commonly we use the modulus (that will be sqrt(An*An+Bn*Bn)) and the result is called the power spectrum. But the phase information is lost.

The "moral" of this story is that any well behaved function can be the decomposed into a set of periodic functions (sin(nx) and cos(nx), in your example) and both are needed for the accurate description (you can write sin(nx+phi-n) but that is less elegant). sin(nx) and cos(nx) are the quadrature components.

 

[CataM]

If sin and cos are same function then why AC voltage is preferred to produce in sin but not in cos.?

There is nothing we prefer or do not prefer, it is the way it is. AC monophase generator gives you sine wave and nothing more i.e. without phase shift. (Because of laws of electromagnetics..)

 

[andre_luis]

any well behaved function can be the decomposed into a set of periodic functions (sin(nx) and cos(nx), in your example) and both are needed for the accurate description (you can write sin(nx+phi-n) but that is less elegant). sin(nx) and cos(nx) are the quadrature components.

Well said.

Another example that illustrates the samething: In order to use just 1 trigonometric function, one could represent the terms above mentioned function

broken link removed​

as

A₀ + Σ( A_n.cos(nx) + B_n.√(1 - cos² (nx) )​

But as you said it is a much less ellegant formulation.

 

[FvM]

Then can we say it like, AC Generators produce odd signal not even signal.?

Of course you can't. The assignment makes only sense if you can define an origin in time and a respective symmetry. That's impossible (or meaningless) for a steady state signal.

I neither agree with the statement that single phase generators are producing sine not cosine waves. If at all it's a convention of generator models.

 

[KlausST]

Hi,

View attachment 128161

A is the "real" part and B is the "imaginary" part of a complex vector.
n represents the frequency

In an carthesic coordinate system you can draw a vector from (0/0) to (A/B) and you see the phase (angle) and the amplitude (vector length) of your given frequency ("n").

X-Axis = real axis = cos axis
Y-axis = imaginary axis = sin axis

Klaus

 

[CataM]

The assignment makes only sense if you can define an origin in time and a respective symmetry. That's impossible (or meaningless) for a steady state signal.

The basic definition of AC single phase generation we assume (and set) the origin of time in manner that there is no initial phase.

I agree with you that if other initial phase is assumed then it is possible to produce cosine wave because of the trigonometric relation of sines and cosines but the derivate from Faraday-Lenz's law will always give you a sine (in this special case).

 

[c_mitra]

sin(omega_t+phi)=cos(phi)*sin(omega_t)+sin(phi)*cos(omega_t)

But there is a deeper reason: it has to do with the basis set. sin(omega_t) and cos(omega_t) are orthogonal as sin(m*x) and sin(n*x) (provided m and n are not equal).

So this is an expansion with respect to the basis set. also: exp(j*theta)=cos(theta)+j*sin(theta)

 

[gobi1990]

Now you can see that this is a complex transformation and every such transformation has two components: one is real part and the other is imaginary part.

Why complex quantity(i or j) presents in FT/FS? Is it because of our sample signal is complex or this FS/FT is a complex transform.

Engineers call that the Fourier transformation gives the result in quadrature.

I am not able to understand this. Please elaborate this.

Thanks!!

 

[FvM]

but the derivate from Faraday-Lenz's law will always give you a sine (in this special case).

Arbitrary as well, I think. It all depends on how you define the rotor angle.

 

[c_mitra]

Why complex quantity(i or j) presents in FT/FS? Is it because of our sample signal is complex or this FS/FT is a complex transform.

It is a complex transformation; as I mentioned F(w)=integral of (f(t)*exp(j*w*t))

Fourier transformation gives you two functions: one real and another imaginary.

In Laplace transformation, the j is missing. But the w is considered complex quantity (for other reasons). Both these transformations are similar.

I am not able to understand this. Please elaborate this.

The two components are orthogonal; one real function gets two orthogonal components under Fourier transformation. A realtime signal (V(t)) can be detected in quadrature - both frequency and phase- over some interval. The terminology is common in signal processing.