In digital signal processing, the frequency-domain analysis of discrete-time signal is an important phenomenon to perform. This process includes the conversion of time-domain sequence to an equivalent frequency-domain representation. The tools Discrete Fourier transform (DFT) and Discrete-time Fourier transform (DTFT) are used in this conversion. DFT is the better version of DTFT as problems that occur in DTFT are rectified in DFT. In this article, we will see the difference between DFT and DTFT.
Difference between DFT and DTFT – Comparison Table
The following table summarises the comparison between DFT and DTFT.
| Basis of comparison | DFT | DTFT |
| Full form | DFT stands for Discrete Fourier Transform. | DTFT stands for Discrete-time Fourier Transform. |
| Expression | $X(k)=\sum_{n=0}^{N-1}x(n)e^{-\frac{j2\pi kn}{N}}$ | $X(e^{j\omega })=\sum_{n=-\infty }^{\infty}x(n)e^{-j\omega n}$ |
| Range | The range of a DFT sequence is finite. The range is from 0 to N-1. A DFT sequence contains only positive frequencies. | The range of a DTFT sequence is infinite. It starts from negative infinity (-∞) to positive infinity (∞) and contains both positive and negative frequencies. |
| Continuity | Non-continuous sequence | Continuous sequence |
| Number of frequency components | A DFT sequence provides less number of frequency components as compared to DTFT. | A DTFT sequence provides more number of frequency components as compared to DFT. |
| Periodicity | A DFT sequence has periodicity, hence called periodic sequence with period N. | A DTFT sequence contains periodicity, hence called periodic sequence with period 2π. |
| Calculation | The DFT can be calculated in computers as well as in digital processors as it does not contain any continuous variable of frequency. The calculation is confined in a finite range of frequency. | The calculation of DTFT on computers or digital signal processors is always a problem as the DTFT deals with the infinite length signals and contains the functions of continuous variables such as frequency of the signals. |
| Computer implementation | Suitable | Not suitable |
| Applications | Applications – Used in image processing. | Applications – Used in the analysis of samples of a continuous function. |
What is DFT?
DFT is a computational tool that stands for Discrete Fourier Transform. To convert a time-domain discrete signal to its equivalent frequency domain response, DFT is used.
Mathematically, for a discrete time-domain signal x(n), its equivalent Fourier Transform is calculated as:
\[X(\omega )=\sum_{n=-\infty }^{\infty}x(n)e^{-j\omega n}\]
The discrete Fourier Transform of the sequence x(n) becomes:
\[X(k)=\sum_{n=0}^{N-1}x(n)e^{-\frac{j2\pi kn}{N}}\]
where, k = 0, 1, 2,…,N-1
Here, the Fourier Transform is calculated for a set of N equally spaced discrete frequencies. Hence, the sequence X(k) is called the Discrete Fourier Transform of the sequence x(n). The application of DFT is more significant nowadays as it provides the user with an increase in computational accuracy without any additional increase in computing time.
What is DTFT?
DTFT is an infinite continuous sequence that stands for Discrete-time Fourier Transform. The Discrete-time Fourier Transform provides the frequency domain (ω) representation for absolutely summable signals. This transformation is only defined for infinite length signals that are functions of a continuous variable of frequency. Hence, the Discrete-time Fourier Transform of a signal cannot be obtained by computers and digital signal processors.
Mathematically, the DTFT of a periodic signal x(n) becomes:
\[X(e^{j\omega })=\sum_{n=-\infty }^{\infty}x(n)e^{-j\omega n}\]
Summary
In the field of digital signal processing, the DFT is the rectified and practical version of DTFT. From the expressions of DFT and DTFT, we came to know that, DTFT contains some values of DFT. Both the DFT and DTFT will be the same and coincide if the length of the DFT sequence becomes infinite with the same frequency as the DTFT sequence.
Author
Sunmoni Gohain
NIT Silchar
It would be nice if a recursive formula would be included converting the DFT to the DTFT. Such a formula is very informative and useful.