Fourier transform - Other possible wave forms

[k.udhay]

Hi,

I am totally a non-math guy. I had to attend a training (on automobile noise signals) that had a session discussed about Fourier Transform (FT). Let me pl. write down what I understand:

"The noise signal observed at any point in the transmission line can be formed using a sum of many sine waves. The one having the lowest frequency is called the fundamental frequency or first harmonic. All the other waves are multiples of first harmonic. As we extend the series till infinity (ie. till infinite no. of harmonics), the exact wave generated by transmission noise can be created"

Assuming what I understand is right, I have one question here to ask - Is there a possibility that the output noise of transmission be generated by other combination of waves that is different from Fourier Series (FS)? If my question was confusing, pl. let me try to explain with this analogy:

The output noise signal of transmission be replaced with a number 10. When FT is applied on this no., FS is 5+2.5+1.25+0.625 etc. But the same no. 10 can be produced by 5+3+2 as well. Does a similar possibility exist for waves as well? Thanks.

 

[joshthekid]

I think that you are unclear about what the numbers mean. The numbers are the amplitudes of several different waves, evaluated at a certain x) that when added together give an overall amplitude of 10. A Fourier expansion act on the whole waveform not just any particular single point. For example say I compose an overall waveform f(x)=2sin(k1*x)+3sin(k2*x)+15sin(k3*x) where k1,k2,and k3 represent different periods where ki=n*k1 where n is an integer. When a Fourier transform is applied it gives you the amplitude for each sinusoid the function is made up of. Thus for the example given if you were to graph the FT k would be on the x-axis and the amplitude of each component on the y. So if x=k1 y=2, x=k2 y=3, and x=k3 y=15. So for your example measure of 10 represents the amplitude of each sinusoidal component

 

[olivermsun]

"The noise signal observed at any point in the transmission line can be formed using a sum of many sine waves. The one having the lowest frequency is called the fundamental frequency or first harmonic. All the other waves are multiples of first harmonic. As we extend the series till infinity (ie. till infinite no. of harmonics), the exact wave generated by transmission noise can be created"

...I have one question here to ask - Is there a possibility that the output noise of transmission be generated by other combination of waves that is different from Fourier Series (FS)?

Yes. Any family of functions could be used so long as each new function adds new "information."

Sines and cosines are especially convenient because it's easy to generate new, independent sine functions (just use a different harmonic frequency). It's also easy to do math with sinusoidal functions because the rate of change (and the area under the curve) is also sinusoidal!

 

[FactChecker]

Is there a possibility that the output noise of transmission be generated by other combination of waves that is different from Fourier Series (FS)?

No. Your understanding is pretty good, but a couple of things need to be explained.

1) There is no frequency that is not included in the construction of the FS. The FS tells you how much of every frequency is in the original noise signal. There is no way to combine frequencies and get one that the FS has missed. You identify the first harmonic frequencies because those are the most important ones in the noise. If another frequency is not included in the FS, that is because it is not in the noise signal.

2) Each fundamental frequency is important and all multiples of that frequency will show up (usually. see note 3). So you only have to identify the lowest in the series. So if you see a 10 Hz frequency, there will also be a 20Hz, 30Hz, etc. But there can be other frequencies in the noise signal. The same system can have a mode that vibrates at 12Hz, 24Hz, 36Hz, etc. Both the 10Hz and the 12Hz series will show up in the FS.

3) It is possible to suppress frequencies. There might be a harmonic set of frequencies like 10Hz, 20Hz, 30Hz, ... where the higher frequencies are suppressed. Maybe everything over 100 Hz is suppressed for some reason. It is also possible for the low frequencies to be suppressed, or even for some in the middle to be suppressed without the higher or lower being suppressed. So there is no hard and fast rule about that.

 

[olivermsun]

No. Your understanding is pretty good, but a couple of things need to be explained.

1) There is no frequency that is not included in the construction of the FS. The FS tells you how much of every frequency is in the original noise signal. There is no way to combine frequencies and get one that the FS has missed.

I took the OP's question to be "Is there another sum of waveforms that could be used to describe the noise?" In that case it certainly is true that there are other series besides Fourier Series.

If the OP was asking whether there is more than one Fourier Series that can describe the same noise waveform, then no. The set of coefficients is unique.

 

[FactChecker]

I took the OP's question to be "Is there another sum of waveforms that could be used to describe the noise?" In that case it certainly is true that there are other series besides Fourier Series.

Oh. I didn't think of that interpretation. It does seem more likely.

 

[olivermsun]

I'm not sure which meaning was actually intended. Maybe the OP can clarify. :)