Understanding of Discrete Fourier Transformation (I) ------ the origin of Fourier Transformation: The description of Fourier transformation can be easily found in books or on the Internet, but it is mostly an article about it, it's too abstract. It's just a list of seemingly daunting formulas, making it hard for people to understand from their senses. Recently, I occasionally see an electronic book about digital signal processing on the Internet. It's Steven W. smith, pH. d. the seven chapters in this article are devoted to the Fourier transformation of discrete signals, although they are written by foreigners, I still read the relevant content about Fourier transform with my head, and I have a feeling of being very open. Here I will share my understanding with you, I hope many friends who are confused by Fourier transform can get some inspiration. This e-book is free of charge. If you are interested, you can download it from the Internet. The URL is: http://www.dspguide.com/pdfbook.htm to understand Fourier transformation, really need some patience, don't suddenly think How is the Fourier transformation transformed? Of course, it also requires a certain foundation of Higher Mathematics. The most important thing is the series transformation, in which the Fourier series transformation is the basic formula of Fourier transformation.
I. Proposal of Fourier TransformationLet's first look at why there is Fourier transformation? Fourier is the name of a French mathematician and physicist, formerly Jean Baptiste Joseph Fourier (1768-1830). Fourier is very interested in heat transfer, in 1807, I published a paper at the French Scientific Conference describing the use of sine curves to describe the temperature distribution. There was a controversial decision in this paper: any continuous cycle signal can be combined by a set of appropriate sine curves. Two of the people who reviewed the paper at the time were the famous mathematicians in history, Joseph Louis lags (1736-1813) and Laplace (Pierre Simon de Laplace, 1749-1827 ), when Laplace and other reviewers voted and wanted to publish this paper, he firmly opposed it. In the past 50 years, he insisted that the Fourier method could not represent a signal with edges and corners, for example, the slope of a non-continuous change occurs in a square wave. The French scientific society gave in to the prestige of rangean, denying the fruits of Fourier's work. Fortunately, Fourier had other things to do. He joined the political movement and went on an expedition to Egypt with Napoleon, after the French revolution, he had been escaping for fear of being pushed to the broken head. It was not until 15 years after his death that the paper was published. Who is right? Returns the right result from the use of the Laplace curve. The Sine Curve cannot be combined into a signal with edges and corners. However, we can use a sine curve to represent it very closely. There is no energy difference between the two Representation Methods. Based on this, Fourier is correct. Why should we replace the original curve with a sine curve? For example, we can also use square waves or triangular waves to replace them. The signal decomposition method is infinite, but the signal decomposition aims to process the original signal more simply. It is easier to use the positive cosine to represent the original signal, because the positive cosine has the properties that the original signal does not have: sine curve fidelity. After a positive cosine curve signal is input, the output is still a positive cosine curve. Only the amplitude and phase may change, but the frequency and wave shape are still the same. And only positive cosine curves have this property, so we don't need square waves or triangular waves to represent it.
Ii. Fourier transform ClassificationBased on the different types of original signals, we can divide Fourier transform into four types:
| 1 |
Non-periodic continuous signal |
Fourier transform) |
| 2 |
Periodic continuous signal |
Fourier Series) |
| 3 |
Non-periodic Discrete Signal |
Discrete Time Domain Fourier Transformation (Discrete Time Fourier Transform) |
| 4 |
Periodic Discrete Signal |
Discrete Fourier Transform (Discrete Fourier Transform) |
There are four original signal legends: these four Fourier Transformations are aimed at positive infinity and negative infinity signals, that is, the signal length is infinite, we know that this is impossible for computer processing. Is there any Fourier transformation for a finite length? No. Since the positive cosine wave is defined as from negative infinity to positive infinity, we cannot combine an infinite-length signal into a limited-length signal. In the face of such difficulties, the method is to express a signal with a limited length as an infinite signal, which can be infinitely extended from the left and right, and the extended part is represented by zero, this signal can be seen as a non-periodic splitting signal, and we can use the Discrete Time Domain Fourier transformation method. In addition, the signal can be extended by means of replication, so that the signal becomes a periodic split signal, and then we can use the Discrete Fourier Transform Method for transformation. Here we want to learn discrete signals. We will not discuss continuous signals. Because computers can only process discrete numerical signals, our goal is to use computers to process signals. However, for non-cyclical signals, we need to use an infinite number of sine curves with different frequencies, which is impossible for computers. Therefore, only discrete Fourier transform (DFT) can be used for discrete signal transformation. for computers, only discrete and finite-length data can be processed, other transformation types are only used in mathematical calculations. In the face of computers, we can only use the DFT method. What we need to understand later is the DFT method. Here we need to understand that we use periodic signals to solve this problem in a mathematical way. It is meaningless to consider where periodic signals are obtained or how they are obtained. Each Fourier transformation is divided into two methods: real number and plural number. It is best to understand the real number method. However, the plural method is much more complicated and requires understanding of the theoretical knowledge about the plural number, assume that we understand the real Discrete Fourier Transformation (real DfT), and then we can understand the complex Fourier transformation to make it easier. Therefore, we first put the Fourier transformation of the complex number aside, and first understand the real Fourier transformation, later, we will talk about the basic theory of the plural, and then understand the complex Fourier transformation based on the understanding of the real Fourier transformation. Also, although the transformation we want to talk about here is a mathematical transformation, it is different from the function transformation. The function transformation conforms to the one-to-one ing principle, for discrete digital signal processing (DSP), there are many transformations: Fourier transformation, Laplace transformation, z transformation, Hilbert transformation, discrete cosine transformation, and so on, all of which extend the definition of function transformation, agree that the input and output have multiple values. Simply put, the transformation is to convert a pile of data into a pile of data.
3. An example of real Discrete Fourier Transform (real DFT)First, let's look at a transformation example. It's a raw signal image: the signal length is 16, so we can break down this signal into nine cosine waves and nine sine waves (a signal with a length of N can be divided into n/2 + one cosine signal, why? Combined with the following 18 positive cosine graphs, I think it is not difficult to understand the accuracy of computer processing. A signal with a length of N can have at most N/2 + 1 different frequencies, an extra frequency is beyond the precision range that the computer can process. For example, the original signal can be obtained by adding all the above signals to 9 cosine signals: 9 sine signals, so how do we convert the nine different frequency signals separately? First, let's take a look at the above transformation results and how do we represent them in the program, let's take a look at the following demo: The left side of the sample shows the signals in the time domain, the right side shows the Signal Representation Method in the frequency domain, and the right side shows the forward conversion (Forward DFT ), from the right to the left, it indicates the reverse conversion (inverse DFT), and uses lowercase X [] to indicate the array of amplitude values of the signal at each time point, uppercase X [] indicates the array of sub-degrees of each frequency. Because there are n/2 + 1 frequencies, the length of this array is n/2 + 1, array X [] is divided into two types, one is to represent different frequency amplitude values of the cosine wave: RE x [], and the other is to represent different frequency amplitude values of the sine wave: im X [], RE is a real number (R (EAL), Im is the meaning of the imaginary number (IMAGINE). The cosine combines the positive cosine wave with the expression of the plural, but here we do not consider the other functions of the plural, just remember that it is a combination method to facilitate expression (we will know later that the length of the Fourier transformation in the form of the plural is N, rather than n/2 + 1 ). In the next section, we will look at the detailed methods of real-number Fourier transformation.
Understanding Discrete Fourier Transformation (1. The origin of Fourier transformation)
