Hi all

If we consider a periodic rectangular pulse train, to find its frequency spectrum, we can find its fourier coefficients; since it is made up of sine and cosine, we expect the spectrum to be discrete.

However if we choose to represent this pulse as an infinite sum of time shifted rect functions, applying Fourier transform and applying the time shifting property, we will end up with a summation of sinc * e^jwt terms which are clearly continuous.

My question is that for the same signal why would there be 2 totally different spectrums? What did I so wrongly?

Many thanks!

 

They are not two totally different spectrums. You did not do anything wrong.
What you are doing is creating two models of the same thing.

It is like painting a picture of what you see. One artist may paint using a full palette of colours. Another may paint by numbers.

One camera may use film technology and get infinite spacial and colour resolution.
A digital camera gives you finite spacial and colour resolution.

That is the analogy to continuous versus discrete Fourier transform.

 

They are not two totally different spectrums. You did not do anything wrong.
What you are doing is creating two models of the same thing.

It is like painting a picture of what you see. One artist may paint using a full palette of colours. Another may paint by numbers.

One camera may use film technology and get infinite spacial and colour resolution.
A digital camera gives you finite spacial and colour resolution.

That is the analogy to continuous versus discrete Fourier transform.

Hi

So am i right to say that this discrepancy is due to that in one case i am applying CFT, while the other DFT is being applied?

 

Yes. Note also that the Fourier transform is only one of many other types of transforms.

 

Yes. Note also that the Fourier transform is only one of many other types of transforms.

Ah i see... But 1 thing i still don't quite get... If the representations of the original signal is mathematically the same, how can the same fourier transform formula come up with 2 different answers?

 

Did you draw graphs of the two results and compare the two?

 

If the representations of the original signal is mathematically the same, how can the same fourier transform formula come up with 2 different answers?

They are different transform formulas, not the same. You can't use a key to open a combination lock. Each transform has its own inverse transform. The answers are different (although very similar) because the inverse formulas, which bring you back to the original signal, are different.

As MrChips says, you could graph them and see the similarity. They are not identical in the frequency domain because the continuous frequency function will look like an train of Dirac delta functions, while the discrete frequency function will be a train of Kronecker delta functions. The two are similar and related, but the former requires an integral relation for the signal representation, and the latter requires a summation relation for the signal representation. There are ways to show an equivalence of the two functions, but it will be clear that the areas of the Dirac delta functions are proportional to the magnitudes of the Kroneker delta functions.