Hi,

I want to know how to use Fourier series (exponential one) in cont. time signa.l
In practical how it work??

 

Too many cobwebs in my ossified cranium for me to be of any help with the Fourier Series, but I can tell you that you will heave a big sigh of relief when Laplace Transforms are finally introduced in class as the simpler, more efficient alternative for wave-form expression.

 

I am learning b'coz it is in my course!!

 

I promise to offer no further asides. Talking too much has always been my undoing.

 

Again, wikipedia is your friend.

 

Give some idea..............

 

The wikipedia articles are quite comprehensive, complete with animations.

http://en.wikipedia.org/wiki/Square_waves

 

In Fourier if we compress signal by factor a then in freq. domain it increase by a factor?? what does this mean & how??

 

reply................

 

I'm not into calculus, which this is a branch of. Most of the formula are already out there. It will tell you what frequencies and their amplitudes (and phase relationships) for every waveform out there. The fact a square wave is rich in harmonics can be useful, but not always.

 

In Fourier if we compress signal by factor a then in freq. domain it increase by a factor?? what does this mean & how??

I am not sure I know what you mean here. The Fourier transform of a signal should result in a series (Fourier series) that breaks down that signal into its sinusoidal constituents and their amplitudes.

Are you talking about compressing the original signal in amplitude (reducing the amplitude of the signal) or reducing it in period (increasing it's frequency)? Or what? Do you have a documents or expression you can post?

 

Hi again,

I am talking about scaling property, how does it work in real??

 

It is a simple percentage of whatever the PP level of the original square wave. I'm not sure if this is what you're asking though.

 

It is a simple percentage of whatever the PP level

What is this PP level??

My question is of scaling property by factor a.

 

Peak to Peak, the voltage from the + peak to the - peak.

 

Don't get confused between a fourier SERIES and a fourier TRANSFORM.

The fourier series basically expresses a function in terms of an infinite series of sine waves (harmonics) in the TIME DOMAIN that all add up to give you the function in the TIME DOMAIN.

The fourier TRANSFORM (either continuous or discrete) is the frequency domain representation of the signal. I.e. it represents the sum of all the harmonics (in the FREQUENCY domain) which together form the overall signal in the FREQUENCY domain. Transform this back into the TIME domain, and you get your normal signal. If you decrease the amplitude of the signal, you automatically decrease the magnitudes of all the harmonics in the time or frequency domain. This all comes out in the maths, since the fourier coefficient (scaling factor) is dependant on the magnitude of the signal in the time domain.

 

So, how decreasing amplitude by factor a, increases it freq. domain by a??

 

Oh... hmmm I actually don't know. I was obviously mistaken. Well, at least I know that changing the time domain amplitude changing the frequency domain amplitude. You'll have to go through the maths to see why.

 

Please, give a small example how scaling property in really work??

 

Okay, I think I know what you mean. Your talking about scaling in the time domain of the original function. Right?

Given:

\( f(t) \ has \ a \ Fourier \ transform \ of \ F(\omega)\)

Then:

\( f(at) \ \Rightarrow \ \frac{F(\frac{\omega}{a})}{|a|} \)

Right?


Meaning: Where a > 1, you are compacting f() over time and are therefore increasing the frequencies in the frequency domain. Higher frequencies carry more energy, so they need to be reduced by the 1/(|a|) factor for the total energy to be the same. If a < 1, then the opposite is true.

Example: A trombone player plays a note given by:

\( f(t) \ which\ is\ made\ up\ of\ the\ frequency\ distribution\ given\ by\ F(\omega)\)

Then he plays a note 1 octave higher, but not any louder, such that it is given by:

\( f(2t) \ which\ is\ made\ up\ of\ the\ new\ frequency\ distribution\ given\ by\ \frac{F(\frac{\omega}{2})}{2}\)

Which simply means that the new tone is made up of higher constituent frequencies of lower amplitude (but the same overall energy).

Does this help?