discussing waveforms, frequencies, FFT, and so on...

[KlausST]

Hello to all,

In another post we unintentionally started a discussion on waveforms
Especially how a squarewave forms when it is lowpass filtered. How to interpret the resulting waveform, its frequency components, phase shift of the frequency components, the mathematical background like FFT and so on.

There are several points to discuss:
1) In a square wave FFT says there are the odd harmonics with decreasing amplitude and no phase shift (in relation to zero cross).
Are they really and truly present in the square wave, or are they a mathematical construct for building the square wave in a theoretical sense?

2) What is the difference of a lowpass filter and an integrator, if any?

3) Are there guidelines to interpret (filtered) waveforms just by viewing them?

Any other corresponding questions are welcome.

I'm looking forward to a warm discussion.
Klaus

 

[BradtheRad]

Hi, thanks for starting this thread.

Many concepts are connected in this topic.

Some concepts seem paradoxical. Not easy to get an intuitive grasp on.

Example, we can make a square wave easily, without needing to know anything about generating sine waves. We just switch a DC source on and off.

It might have led me to claim, therefore, that a square wave only contains sinewaves in a theoretical sense.

I was just playing with Falstad's fourier analyzer. I start with a square wave. I reduce the sine wave harmonics, one by one. It is indeed possible to watch the effect, as each sine wave is subtracted out. What remains is a thin squiggle. (The same happens when I start with a triangle wave.)

This implies that those higher frequencies really were present in the original waveform.

However suppose we feed the square wave to an oscilloscope. There is no indication that any sine harmonics are present. We see the simple straight lines of the square wave. Hardly a hint that it contains hidden complexities.

 

[ahsan_i_h]

What is the proof of fourier series ? I mean how Mr. Jean-Baptiste Joseph Fourier (1768–1830) came to a conclusion that any periodic function can be expressed as a combination of many sine waves ?

 

[BradtheRad]

What is the proof of fourier series ? I mean how Mr. Jean-Baptiste Joseph Fourier (1768–1830) came to a conclusion that any periodic function can be expressed as a combination of many sine waves ?

It does seem as though Mr. Fourier made a huge jump in logic, doesn't it?

There is a hint of the idea, if we start with a sine wave, and add a bit of 3x harmonic. It is startling to see how that creates a rounded sort of square wave.

I imagine Mr. Fourier wondered what would happen if he were to try the same thing (by hand, on paper). Seeing the result he could have then added a bit of 5x harmonic. The progression, continued, sharpens the right-angle corners.

It is mystifying to me that he would have the insight to conclude the same thing applies to all waveforms. For instance the sawtooth especially is very un-sinelike.

 

[mrinalmani]

Well I disagree that Mr. Fourier made mere imaginations to deduce such a spectacular theory.
Refer to "Advanced Engineering Mathematics by Erwin Kreszig", you will get a hint of the inherent relation between, power series, complex frequency transforms and Fourier series. In short, there's a lot of theory related to this rather than merely "a snake grabbing its own tail to make a circle!"

Unlike complex numbers, which are conceptual and must be transformed into real quantities using the Euler's identity, Fourier series are real and represent real quantities. Look at the theory behind Fourier series and it will at once become apparent how real they are. Just like a sine function can be expanded into infinite algebric terms, a simple algebric periodic function can be expanded into infinite sine terms.

Just because a square wave is easier to express in algebric terms, it would be incorrect to conclude that it cannot be expressed trigonometrically or that its trigonometric expression is only conceptual. Similarly, Just because a sine wave is easier to express in trigonometric terms, it would be incorrect to conclude that it cannot be expressed algebraically or that its algebraic expression is only conceptual.

Also, Fourier series are not limited only to periodic functions. The series may be found for any piece-wise continuous function in a given interval.