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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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