When many people start to learn digital signal processing, they are completely unclear about the concepts and functions of various Fourier transforms, especially discrete Fourier transforms. IC Xiuzhen Institute has sorted out various knowledge points about DTFT and DFT on the Internet. Let's take a look at the difference between DTFT and DFT.
The difference between DTFT and DFT is that they have different meanings, different properties, and different uses.
1. Different meanings: DTFT is discrete-time Fourier transform, and DFT is discrete Fourier transform.
2. The nature is different: the frequency in the graph after DTFT transformation is generally continuous (except for special functions such as cos(wn), which are shock strings after transformation), while DFT is equal interval sampling of DTFT, which is a discrete point .
3. Different uses: DFT is completely developed in response to the development of computer technology, because if there is no computer, you can use DTFT analysis to see the frequency response. In order to adapt to computer calculations, you must use discrete values, because computers cannot handle continuous value.
DTFT and DFT
1. DTFT is discrete time Fourier transform, and DFT is discrete Fourier transform.
2. The frequency in the graph after DTFT transformation is generally continuous (except for special functions such as cos(wn), which are shock strings after transformation), and DFT is equal interval sampling of DTFT, which is a discrete point. It can be seen from the representation that its function is expressed as X(k), while the function of DTFT is expressed as X(exp(jw)). (Here it is mainly highlighted that DFT is the equal interval sampling of DTFT. The frequency response after DTFT transformation is generally continuous, and the frequency response after DFT transformation is discrete.) 3.
DTFT takes 2pi as the period. And the sequence X(k) of DFT is of finite length.
4. DTFT is represented by the weighted sum of the complex exponential sequence {exp(-jwn)}, and DFT is equal interval sampling. Since it is equal interval, what is the interval? There is an important parameter in DFT is N. We generally say that how many DFT operations, this point is N (the length of the discrete sequence), and the sampling interval is to divide the unit element into N intervals for sampling, and go around the circle, (2 pi)/N is the interval (this should be obvious, a circle is 2 pi, divided into N equal parts, just like cutting a cake on our birthday).
5. Both DTFT and DFT can represent the information of the original sequence. Because computers are mainly used for calculations now, discrete values must be used to participate in calculations. Therefore, DFT is widely used in engineering. DFT also has a fast algorithm, that is, FFT.
DTFT is for people, DFT is for machines, and DFT is the frequency domain sampling of DTFT.
Before talking about DFT, let's recall several previous Fourier transforms.
1. Continuous time periodic signal: processing time continuous and periodic signal, its frequency domain is discrete and non-periodic.
2. Continuous-time non-periodic signal: The processing time is continuous but not periodic, and its frequency domain is continuous and non-periodic.
3. Discrete-time non-periodic signal: The processing time is discrete and non-periodic, and its frequency domain is continuous and periodic.
4. Discrete-time periodic signal: The processing time is discrete and has periodic signal, and its frequency domain is discrete and has periodicity.
Theoretically speaking, these four transformations already cover the types of signals we can encounter, so why introduce DFT additionally? Formally, DFT is very similar to the transformation of discrete-time periodic signals. Why?
First of all, we noticed that in digital signal processing, we are exposed to discrete-time signals, so the first two continuous-time Fourier transforms are not used here. In addition, one of the main points of digital signal processing is to discuss the processing methods and algorithm design of digital signals. The processing methods mentioned here are not only artificial and analytical processing methods, but also processing methods that can be used by machines. way. Where are the limitations of the machine?
Machines cannot express an infinitely long sequence, nor can they express continuous frequency domain features. For general discrete-time signals, it is really good to use DTFT directly, which is very convenient for us to analyze the frequency domain characteristics of the signal, but the problem is that this set of machines cannot be used. That's why we need DFT, that is to say, DTFT is for people and DFT is for machines.
The so-called introduction of DFT, I think it can be divided into two points, one is truncation, and the other is (frequency domain) sampling. The need to truncate is because the machine cannot represent infinitely long sequences, and can only process finitely long sequences, which is easier to understand. Regarding sampling, it is the focus of understanding DFT. We mentioned earlier that the Fourier transform (DTFT) of a discrete non-periodic sequence is continuous in the frequency domain. This continuous frequency domain feature cannot be expressed by a machine, so we need to sample it.
Also, due to the periodicity in the frequency domain, it is only necessary to sample an interval of 2pi length. So how many points should be taken? Similar to the practice of Nyquist sampling theorem, we can conclude that the number of sampling points M≥N (N represents the length of the sequence), and we only need to take M=N for convenience. From this, the two motivations for introducing DFT are clear: it is to truncate an infinite-length sequence into a finite-length sequence, and then perform DTFT in the frequency domain
So why is the form of DFT similar to the Fourier transform form of discrete-time periodic signals? Note that a finite sequence can be turned into a periodic signal through periodic extension, so the similarity between them is self-evident. However, it should be noted that DFT can be used for finite-length sequences, but the Fourier transform of discrete-time periodic signals can only deal with periodic signals, which is an essential difference.
The above are the knowledge points about DTFT and DFT shared by IC Xiuzhen Academy. If you want to know more, just follow it.