Turn: https://blog.csdn.net/deepdsp/article/details/6630288
Fourier analysis is arguably one of the most important cornerstones of signal processing. However, Fourier series, Fourier transform, discrete-time Fourier transform, discrete Fourier transform and a little bit like a few like the concept, not only do a beginners often confusing, sometimes a bit let veteran confused.
Before the opening, a brief introduction about the Fourier analysis of gossip old news. In the early 19th century, the French mathematician, physicist Fourier heat conduction at the time of the study, proposed periodic signal can be decomposed into a superposition of many sine function. In 1807, he put this idea to write a paper voted for the French Academy of Sciences. I was responsible for the review of this article has two world famous French mathematician: Laplace and Lagrange. Review the results that can express the Laplace, Lagrange discontinuous signals such that the square wave signal can not be decomposed by a trigonometric function. French Academy of Sciences awed Lagrange Wal-Mart, and finally rejected the paper. Since then Fourier first busy with political activities, and later with Napoleon's expedition to Egypt, the French Revolution and then try against the fear of the guillotine, the Fourier simply too busy to attend this paper. After the paper was rejected 15 years, the French Academy of Sciences has published a paper, because the paper published at that time did not agree Lagrange died. It is this paper, laid a Fourier analysis on the status of the signal processing history.
1, Fourier series
Higher Mathematics have been known, in certain conditions are satisfied, any periodic signal can be decomposed into a superposition of sinusoidal signals. In mathematics, this decomposition is called a Fourier series. In the initial stages of learning signal processing, and it is from this concept, start typing into the Fourier world signal processing. In signal processing, a Fourier analysis of the signal is called a continuous cycle of Fourier series. At this time, before the Fourier analysis of the signal is periodic, continuous, after the result is discrete.
2, Fourier transform
For continuous signal, if the signal is not periodic, the Fourier analysis of its results is what happened? Non-periodic signal may be equivalent to a period of infinite periodic signal. Thus, starting from the Fourier series, limit the use of the concept, the Fourier analysis can be derived non-periodic signal, which is a Fourier transform. A long-winded, aperiodic Fourier analysis of the signal is called a continuous Fourier Transform. Prior to Fourier analysis, signal non-periodic, continuous, after the result is also continuous.
3, discrete-time Fourier transform
Fourier series and Fourier transform for continuous signals are concerned, then for digital signals, whether there is a corresponding Fourier analysis of it? The answer is yes, this is the discrete-time Fourier transform (DTFT) and Discrete Fourier Transform (DFT).
Aperiodic signal discrete Fourier analysis called Discrete-Time Fourier Transform. Prior to Fourier analysis, non-periodic signal, the discrete, after the result is continuous.
4, Discrete Fourier Transform
Periodic discrete signal discrete Fourier transform known as Fourier analysis. Prior to Fourier analysis, signal is periodic, discrete, after the result is discrete. If you follow the first three named analysis, discrete Fourier transform called the discrete Fourier series seems more appropriate. However, due to historical reasons, we used to call this Fourier analysis known as the discrete Fourier transform. Of course, implicit signals about whether DFT cyclical problems, there are also some controversy. Some believe it means default DFT analysis conducted discrete signal is periodic, while others do not have to think as a discrete signal cycle. The default take discrete signal periodic statement here, if the DFT seen as a sampling of the DTFT results in the frequency domain, then according to the theory of signaling system shows that the sampling frequency domain is equivalent to: mainly based on the following reasons periodic domain extension, so, naturally discrete signal becomes a cycle. In the actual analysis, is seen as the DFT DTFT sampling frequency domain result is reasonable.
This top four and Fourier analysis of concepts, the most important is DFT. Because the previous three analysis needs to assume that the signal in the time domain and frequency domain are infinitely long. Conceptually, though DFT also need time and frequency domain infinitely long, but because of the time and frequency domain are periodic, so the only information to a cycle. Further, since the computer and other digital equipment can process the digital signals, that is, whether it is a requirement domain or frequency domain, have to be discrete. Therefore, DFT occupies the most important position in practice. Fourier series, Fourier transform, discrete-time Fourier transform of these three concepts is more of a theoretical analysis.