Signal Spectra Question

[noesis]

First off, great site!

Now my question:

I used to think that the frequencies obtained by a Fourier Series or Transform from a signal in the 'time domain' were simply a consequence of our mathematical system. In other words, it is a consequence of the fact that sinusoids are used to recreate or synthesize the signal.

It's just an approximation technique using an orthogonal set; and could've been done through another orthogonal set.

Since I viewed it as a consequence of our mathematical system, I did not see time signals having actual frequencies in nature.

This seemed evident to me as well in terms of a simple square signal. Due to the inverse relationship between bandwidth and time duration, a 'larger rectangle' had smaller frequency than a 'smaller rectangle.' But clearly both rectangles have NO frequency...they're just boxes!

Now that I am studying modulation, specifically amplitude modulation, I see that signals are analyzed via their Fourier Transforms and that actual filters (say a low-pass filter) are used to extract the signals again in demodulation.

This contradicts my original thoughts and so something is wrong. Frequency is being used here in practice, so there must be some link between a frequency spectra and the signal in reality.

I think of passing a gate signal, i.e., the last signal on this page: www.eas.slu.edu/Earthquake_Center/NM/INSTRUMENT/Q330/bq330.png , through a low pass filter and I don't see how any of it would ever be attenuated since it has no frequency.

Yet somehow we make use of a signal's frequency content in modulation! What's going on?

Thank you very much for any and all help.

[micky]

Both of your interpretations are true and they are not mutually incompatible. Specifically:

1. Yes, you can expand any waveform in terms of any orthonormal functions you want. These can be sinusoids or anything else.

2. Yes, the expansion you select (such as Fourier series) is "real" in the sense that if you have a device that can convert your waveform into those functions (in this case a spectrum analyzer or your ear or some such) then you will be able to "see and feel" those components of the chosen orthonormal functions (Fourier series components in this case). If you had a device that could somehow convert the waveform into some other orthonormal functions (say, Bessel series), you would also be able to "see" those components. In the case of Fourier series, they seem more "real" only because humans are conditioned to think in terms of individual frequency components.

So, in summary, both of your interpretations are correct and they are not incompatible with each other.

Hope this helps.

Michael

[uptonogoode]

Square pulses DO actually contain THOSE frequencies revealed by the Fourier transform. And not just the magnitude, those frequency components are present at exactly the PHASE revealed by the complex Fourier.

In the lab, I can take 10 SEPERATE signal generators, Program each one to output a perfect sine wave at the same magnitude and phase of the first 10 spectral components in a square wave. Combine the 10 outputs them together in a summing junction or power combiner, and presto, you ahve a square wave. (Sig gens have to be phase locked together via their 10 MHz reference inputs.)

Of course, it wont be a perfect square wave because Fourier says you need an infinite number of sine wave components to make a perfect square wave. But 10 is a very good approximation.

--Upton O Goode

[eeaamirsultan]

hi:
aamir here new to this forum:
my project is based on time and frequency synchronization of OFDM ...for that reason Charan Langton I need your kind assistance...if you can provide me with some text , books , etc that can help me in this regard.
with regards
aamir sultan