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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).

Contents

## 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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