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.