# What is the difference between the DFS (Discrete Fourier Series) and DTFS (Discrete-Time Fourier Series)

I'm looking at two different books written by Oppenheim. In Discrete-Time Signal Processing (source 1) he defines the DFS to be:

where, $$W_N=e^{-j(2\pi/N)kn}$$ , while in Signals and Systems (source 2) he defines the DTFS to be:

Can someone explain why there is a difference? It appears to me that the only difference is that the DTFS coefficients are scaled by a factor of 1/N. Is the difference that the DFS is defined in terms of Fourier series coefficients and the DTFS is defined in terms of spectral coefficients? Do these differences arise from how the series are derived?

Thanks!

Edit

Sources

(1) Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing Second Edition, 1999

(2) Alan V. Oppenheim and Alan S. Willsky, Signals and Systems Second Edition, 1997

• Could you edit your question with the complete citation data for each book, including the coauthors and copyright date? Oppenheim is a very prolific author. He's been publishing for decades, and he is continually updating Signals and Systems. Feb 23 at 16:07
• Sure, good idea. No problem. Feb 23 at 17:28

That's primarily due to inconsistent naming.

Typically the term "Discrete Time Fourier Transform" (DTFT) is used for signals that are discrete in time but continuous in Frequency, which is DIFFERENT from how Oppenheim used in in source (2) (and hence it's probably a mistake on his part).

For a more common definition of the term see: https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform

There are four different types of Fourier Transform depending on whether the signal is discrete or continuous in either domain. Whenever a domain is discrete we use sums. When it's continuous, we use integrals. There are total of 4 combinations, so there are 4 different transforms. Note that discreteness in one domain implies periodicity in the other (and vice versa).

Unfortunately the naming of the different flavors isn't super consistent. Most common we have

Time                   Frequency                Name
Continuous/Aperiodic   Continuous/Aperiodic     FT   = Fourier Transform, or Continuous FT
Discrete/Aperiodic     Continuous/Periodic      DTFT = Discrete Time Fourier Transform
Continuous/Periodic    Discrete/Aperiodic       FS   = Fourier Series
Discrete/Periodic      Discrete/Periodic        DFT  = Discrete Fourier Transform


Unless I'm really suffering a brain fart, (8.11, 8.12) and (3.94, 3.95) are mathematically identical. Just take $$W^{kn}_N = e^{jk(2 \pi /N)n} \tag 1$$ and see for yourself.

Terminology changes, and Oppenheim has been publishing for a very long time. I suspect that the publication dates for the two books are quite different, and the intended audiences are also different (Signals and Systems is usually a 2nd-year book, quite a few of his earlier books are grad level).

I think the only difference is the term he used, and a deep dive into the history of the terms may reveal that he's being consistent with the terms use around the date of publication (or the DSP community is following his usage).

• Tim, there's a difference between the DFT and the DTFT. And there's a relationship between the two also. And I didn't downvote this answer. Feb 23 at 17:07
• I also didn’t realize Oppenheimer was a prolific DSP expert…lol! Feb 23 at 17:57
• The question is about things Oppenheim is naming the "Discrete Fourier Series" and the "Discrete-Time Fourier Series" -- which look identical to each other, and an awful lot like the DFT. I suspect this is all from the early days when names were yet to be solidified. Feb 23 at 18:55
• @Baddioes but not as prolific as Oppenheim. Thanks for noting that; edit made. Feb 23 at 18:56