# Why do we use the DFT instead of the DTFS? Or, why was the FFT algorithm built for the DFT instead of the DTFS?

As we know, both the DTFS (discrete-time Fourier series) and the DFT (discrete Fourier transform) are used to represent discrete-time periodic signals for all time (or the periodic extension of aperiodic signals in the time interval where the DTFS or DFT is computed) using a finite number of different harmonically related real discrete sinusoids or complex exponentials. The only difference between the DTFS and the DFT is a real factor. The FFT is just a faster way to compute the DFT.

If we compare the formulas for the CTFS (continuous-time Fourier series) and for the DTFS, we see they are analogous, the only main difference being that the DTFS requires a finite number of harmonics (as opposed to an infinite number as in the CTFS). On the other hand, if we compare the formulas for the CTFS and the DFT, besides the differences between the CTFS and the DTFS, we additionally see there is a scaling factor difference. Thus, the DTFS is more analogous to the CTFS than the DFT (if you disagree on this, please explain why). But why do we use more often the DFT than the DTFS? Equivalently, why was the FFT algorithm built for the DFT instead of the DTFS?

• "The only difference between the DTFS and the DFT is a real factor". Since that's true, how should we make sense of the question "why was the FFT algorithm built for the DFT instead of the DTFS?" I mean, what algorithmic difference would you expect? Jun 20 '21 at 12:41
• Said another way: why was both the DTFS and DFT invented? Why wasn't just one invented? Jun 20 '21 at 20:51
• What book are you reading that calls the Discrete Fourier Series "DTFS"? Jun 21 '21 at 1:47
• This question has been asked and answered before. For some reason, there is a false controversy when the mathematics are perfectly clear. Jun 21 '21 at 1:53
• @robertbristow-johnson First book is Signals and Systems (2nd ed.) by Alan V. Oppenheim, Alan S. Willsky & S. Hamid Nawab, on subsection 3.6.2, pages 212-213. Second book is Signals and Systems: Analysis Using Transform Methods and MATLAB® (3rd ed.) by Michael J. Roberts, on section 7.2, pages 307, 310, 311. Both have seem like good books so far for me. You can check the names in these screenshots. Jun 21 '21 at 5:17

If done consistently, scaling is somewhat arbitrary and the best method of scaling depends on your signals and application. Interestingly enough, neither DFT or DTFS get Parceval's theorem correctly. That requires using $$1/\sqrt{N}$$ for both forward and inverse transforms and also makes the transform matrices unitary.