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Discrete Time Fourier Transform or Discrete Fourier Transfor

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claudiocamera

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Hi Folks,

Studying DFT I came across a difference in definition of synthesis and analysis equations by Oppenheim Schafer and Haykin Van Veen's books.

Haykin defines: x[n] =Σ X[K]e^(jknΩ) and X[K]=1/N Σ x[n]e^(-jknΩ)

Oppenheim defines: x[n] =1/N Σ X[K]e^(jknΩ) and X[K]= Σ x[n]e^(-jknΩ)

Notice that there is a switching in the factor 1/N in the definitions above. Somebody could help me explaining why they use the factor 1/N differently ?

Are both of them correct ? What the factor 1/N means? If either way is correct, why ?

Thanxs in advance.
 

Re: Discrete Time Fourier Transform or Discrete Fourier Tran

To really understand why this is so, you'll have to take a step back and analyze the mathematical basis of continuous fourier transform. The FT is based on fourier series, which states that any periodic signal (of period T0, alternatively, frequency f0) comprises of the summation of infinite number of sinusoids, with magnitudes given by
Xk = (1/T0)∫x(t)exp(-2Πjkf0t)
To find the fourier series coefficient for an aperiodic signal, we assume it to be periodic with T0=∞ (or alternatively f0=0). You can immediately see that this presents the problem that all the spectral coefficients is ZERO! So we use spectral density instead of fourier coefficient, where X(f)=Xk * f0. This forms your basic understanding of continuous fourier transform.

But in the discrete case, the sampling window is finite, and we make a finite sample periodic. Therefore, we have to use the fourier coefficient and not the spectral coefficient. Referring back to the above, the period of one set of samples is T0 = N * Ts. We often presume Ts=1, to reduce DFT to the form you are familiar with.

Referring back to the forms you were given, it basically falls back to the idea of whether X[k] refers to fourier coefficents or spectral density.
 
Dear Checmate,

According to your explanation the correct set of equations would be x[n] =Σ X[K]e^(jknΩ) and X[K]=1/N Σ x[n]e^(-jknΩ) wich is given by Haykin.

As Haykin was the first book I studied this subject I undestood exactly in the way you presented.

The problem came when I was studing FFT in the Ifeachor and Jervis book " Digital Signal Processing - A practical approach" it provides the equation in the same way Oppenhein does. These presentations don't match what is in the Haykin book, the factor 1/N is switched from analysis to synthesis equation as I wrote before.

So, my doubt is wheter the factor N or 1/N deppends on the approach given, or whether one of the approach in the books quoted above is incorrect.
 

Re: Discrete Time Fourier Transform or Discrete Fourier Tran

checkmate said:
Referring back to the forms you were given, it basically falls back to the idea of whether X[k] refers to fourier coefficents or spectral density.
Like what I've mentioned, both books have different definitions of X[k]. Haykin defines X[k] to be the fourier series coefficients. Oppenheim defines X[k] to be the spectral density coefficients. Both are correct.
Just pick a form in which you are more comfortable in, but make sure that you stick to the same convention throughout.
Generally, I'd prefer Oppenheim's representation.
 

It's just a normalization factor, so that you get

x[n]-(forward)->X[k]-(inverse)->x[n]

Therefore, it doesn't matter whether you put 1/N at the forward, at the inverse, or even split it up to 1/sqrt(N) for both forward and inverse.
 

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