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Discrete Time Fourier Transform (DTFT) vs Discrete Fourier Transform (DFT)

In this post, we will encapsulate the differences between Discrete Fourier Transform (DFT) and Discrete-Time Fourier Transform (DTFT).

What is DTFT?

DTFT stands for Discrete-Time Fourier Transform. We can represent it using the following equation. Read the equation carefully.

X(\omega )=\sum _{ n=-\infty }^{ \infty }{ x(n){ e }^{ -j\omega n } }

Here, the signal has a period of 2π.

What is DFT?

DFT stands for discrete Fourier Transform. We can represent it using the following equation.

X(k)=\sum _{ n=0 }^{ N-1 }{ x(n){ e }^{ \frac { -j2\pi kn }{ N } } }

Probably the only things that you can notice in this equation are the fact that the summation is over some finite series. Additionally, the exponential function seems to have gotten a bit more complicated. Let’s address what these differences actually translate to.

What is the difference between DFT and DTFT?

Another difference that you may have noticed is the fact that in DTFT, we are calculating for a quantity X(ω). X(ω) represents a continuous frequency domain.

DTFT DFT
DTFT is an infinite continuous sequence where the time signal (x(n)) is a discrete signal. DFT is a finite non-continuous discrete sequence. DFT, too, is calculated using a discrete-time signal.
DTFT is periodic DFT has no periodicity.
The DTFT is calculated over an infinite summation; this indicates that it is a continuous signal. The DFT is calculated over a finite sequence of values. This indicates that the result is non-continuous.
The ω in the exponential function is a continuous frequency variable. The continuous variable found in the DTFT (ω) is replaced with a finite number of frequencies located at 2πk/NTs. Here Ts is the sampling rate. In other words, if we take the DTFT signal and sample it in the frequency domain at omega=2π/N, then we get the DFT of x(n). In summary, you can say that DFT is just a sampled version of DTFT.
DTFT gives a higher number of frequency components. DFT gives a lower number of frequency components.
DTFT is defined from minus infinity to plus infinity, so naturally, it contains both positive and negative values of frequencies. DFT is defined from 0 to N-1; it can have only positive frequencies.
More accurate To improve the accuracy of DFT, the number of samples must be very high. However, this will, in turn, cause a significant increase in the required computational power. So it’s a trade-off.
DTFT will contain some of the values of DFT too. DTFT and DFT will coincide at intervals of omega=2ωk/N where k = 0,1,2…N-1.

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6 comments

  1. If x is an acceleration, based on the above relationships, X also is an acceleration.
    However, the FT X should be a velocity. Where is the problem ?

    Thank you

    1. Absolutely true. But that happens in the case of the Fourier transform. A forward Fourier transform adds /Hz to the units. However, in the case of discrete Fourier transform you are using a vector of samples and the answer will also be a vector of samples. There’s no way of knowing the units. So to summarize, you are correct in terms of Fourier transform which will distribute your units over frequency. But this does not apply to DFT. I hope that answers your question!

      1. Thank you for your prompt response. But the DFT can be used for a numerical calculation of the FT on computer. There is therefore a correction to be made on the result.

        1. In summary, and unless I am mistaken, the DFT
          – Is a function whose name is confusing,
          – Despite its name (discrete FOURIER TRANSFORM), does not allow to calculate (directly) the Fourier transform,
          – Is rather a theoretical mathematical tool than a tool directly usable by physicists, while the discrete form is suitable for computer calculations.
          Thank you for your very clear explanations.

          1. That is a good summary.
            – The name is not very confusing if you view DFT as a (Discrete) Fourier Transform of discrete sequence, rather than seeing it as discrete version of Fourier Transform.
            – You can say, it’s not primarily used to calculate Fourier Transform but you can actually have an approximation of continuous Fourier Transform from DFT. Here’s a nice article that discusses that:
            https://dspillustrations.com/pages/posts/misc/approximating-the-fourier-transform-with-dft.html
            It has lots of programming code but the conclusion gives you what you need.
            – Yes, DFT is used for computer calculations and is a pretty popular tool for frequency analysis in digital signal processing.

            1. We can therefore say that the above expression of DFT:
              – does not directly calculate the FT of a short signal (mechanical shock for example). To obtain it, divide this expression by the sampling frequency. The FT obtained is indeed a speed if the signal is an acceleration.
              -allows to calculate a development in Fourier series.

              We can therefore say that the above expression of DFT:
              – does not directly calculate the FT of a short signal (mechanical shock for example). To obtain it (approximation), divide this expression by the sampling frequency. The FT obtained is indeed a velocity if the signal is an acceleration.
              -allows to calculate a development in Fourier series (Fourier Transform of discrete sequence).

              Thank you for all your explanations and for the very interesting paper..