Deep understanding of Fourier transform (repost)

An in-depth understanding of Fourier transform on Friday, October 05, 2007

Topic discussion 4: Discussion on Fourier transformation [highlights] prize collection: let's discuss the content related to Fourier Transformation: 1. the purpose, meaning, and application of the transformation. Difference and correlation between Fourier series and Fourier Transformation 3. Continuous Fourier transformation, Discrete Time Fourier transformation, discrete Fourier transformation, Series Fourier transformation, their definitions, differences, and connections. 3. The essence of High-Speed Fourier transformation, the differences and connections between frequently used algorithms, and their respective advantages. 4. FFT Application Discussion: 1. transformation is a variable function that converts a time variable function into a corresponding transform domain, which makes the operation simple and easy to process. Transform Domain transformations include ft (with the frequency domain characteristics as the main research object), LT and ZT (focus on pole and zero point analysis), dtft, DFT, FFT, dtwt and so on. 2. Fourier transformation is a non-periodic signal as the Fourier series (FST) of periodic signals. Fourier series-Periodic Signal, Fourier transformation-non-Periodic Signal 3. Non-cyclic continuity -- ft -- continuous non-periodic Continuous Period -- FST -- non-periodic discretization Non-periodic discretization -- dtft -- continuous period Discrete Period -- DFT -- Periodic discretization Discrete Fourier Transform (DFT) and sequential Fourier transform (dtft) are all related to the z transformation. dtft is the z transformation on the unit circle, while DFT is the uniform sampling of the z transformation on the unit circle. 4. The essence of High-Speed Fourier Transform (FFT) is "divide and conquer". Some items are merged using symmetry, periodicity, and reduction, and the DFT sequence is decomposed into short sequences to reduce the number of operations, increase the computing speed. 5. High-Speed Fourier transformation is widely used. Wherever Fourier transformation can be used for analysis, synthesis, and transformation, it can be implemented by using the FFT algorithm and digital computing technology. FFT is widely used in digital communication, voice analysis, image processing, and matching filtering. **************************************** **************************************** **************************************** **************************************** ************** The time domain cannot be seen clearly. The frequency domain may be simple. Because of the reciprocal relationship between T and F, the samples on T will be infinite on F, and vice versa. The relationship between macro and micro. ------------------------------------------------------------------------------------------------------------------------------- From the perspective of filter points, the complex transform is equivalent to filtering the signal using a filter group with different Q values of the same bandwidth, while the filter group with constant Q is a wavelet analysis. Fourier transform (FT) is a form of transformation from time domain to frequency domain. It is widely used in acoustics, telecommunications, power systems, signal processing, and other fields. We hope to implement spectrum analysis or other work on the computer. The computer's requirement for signals is that both the time domain and frequency domain must be discrete and must be finite. While Fourier transform (FT) can only process continuous signals, DFT is born in response to such a need. It is the representation of Fourier transformation in discrete fields. However, in general, the calculation of DFT is very large. Before the first high-speed Fourier Transform Algorithm FFT was proposed in 1965, its application field had been difficult to expand. The FFT was proposed to bring the implementation of DFT closer to real-time. The applications of DFT have also been rapidly expanded. In addition to high speed requirements, the FFT algorithm can basically meet the requirements of industrial applications. Because other operations of digital signal processing can be implemented by DFT, the FFT algorithm is an important cornerstone of digital signal processing. Euclidean's understanding of Fourier transformation Fourier variation is the Orthogonal Decomposition of signals. e ^ The form of the output signal remains unchanged after JWT passes through the current time-unchanged system, this is of great significance both in theory and in practice. After the emergence of digital signals, the high-speed FFT of DFT realizes the computer's signal processing and improves its useful value. Fourier series is a special form of Fourier transformation, and the signal it processes is periodic. Assume that a period of a periodic signal is taken as a finite signal in the time domain, and its transformation can be used to obtain the series form. In Zheng Junli's "Signals and Systems", he spoke very thoroughly. The discrete Fourier transformation is the same as the sequential Fourier transformation, Continuous Fourier transform (FT) is continuous in the time and frequency domains (periodic signal transformation in the frequency domain discrete), Discrete Time Fourier transform (dtft) in the time domain discrete, the frequency domain continuous and periodic, discrete Fourier Transform (DFT) is a sample of titanium. In my opinion The Gini Fourier series is generally interpreted as an infinite series in which signals can be expanded into linear combinations of orthogonal functions. Fourier transformation is the digital Fourier processing of analog signals, so that the computation after processing is more convenient. Physical Aspects Fourier transform is a concept of density function. It is a continuous spectrum, including all frequency components from zero to infinite height. The frequency of each frequency component is not harmonic. There is another way of saying, from elsewhere. 1: (Time Domain) the frequency spectrum of the periodic signal is discrete; that is, the frequency of the discrete time signal (time) series is periodic. 2: Fourier transform is mainly used for continuous time signals, and discrete time signals can also be used. Digital Signals (discrete time signals) mainly use discrete ft, because it is easy to perform numerical operations. 3: The Discrete ft is equivalent to the FT sample in the frequency field, and the transformed ft is also a discrete sequence in the frequency field. This is more conducive to numeric operations. 4: A Finite Long sequence can be regarded as a cycle of a periodic sequence. Therefore, there is no essential difference between a finite long sequence and a periodic sequence (actually the same ). In this way, both in the time domain and in the frequency domain can be expressed (finite length ). At the same time, you can also perform FFT. From a mathematical point of view, discrete Fourier transformation is a special van der matrix transformation, because such a matrix can be decomposed, there is a high-speed algorithm. 1. The concept of Fourier analysis first came from the study of Fourier's periodic function. Through Fourier series, we can expand the periodic function into an infinite series form. More than one hundred years later, with the development of power, electronics, and computer technology, Fourier analysis has been widely used. In my opinion, the idea of transformation is simply to recognize signals from different angles, and there are also different transformation methods for different applications. Another closely related to transformation is the concept of convolution. 2. Fourier series expands the infinite series of periodic signals based on trigonometric or exponential functions. Assume that the period of the periodic function is infinite, and the limit of the Fourier series is obtained. In addition to signal differences, for Fourier series, the signal spectrum (derived from the concept of spectrum in physics) is obtained, while Fourier transformation produces the signal spectrum density. Of course, after the introduction of the impact function, the Fourier series can be unified in Fourier transformation. 3. Fourier series (FS) corresponding time-domain continuous period signal Fourier transform (FT) corresponding time-domain continuous non-Periodic Signal Discrete Fourier series (DFS) discrete periodic signals in the corresponding time domain Discrete Time Fourier transform (dtft) corresponding time-domain discrete non-Periodic Signal The discrete Fourier transform (DFT) is more accurate. After a discrete non-periodic signal (a sequence of N points long) is extended into a periodic signal, the main value range of the Fourier series is obtained, so it is an approximate transformation, but this method is convenient for computer computing, and then there will be a high-speed algorithm, that is, high-speed Fourier Transformation (FFT) Bytes ------------------------------------------------------------------------------------------------------------------------------- DFT/FFT is a useful tool for converting linear convolution into cyclic convolution. The conversion of convolution into a product relationship is the starting point of most high-speed signal processing, and is almost always prosperous ------------------------------------------------------------------------------------------------------------------------------- The FFT application is used in the final setup. In Signal Analysis, Fourier transform can easily find out the amplitude and phase spectrum of each frequency component in the messy signal. The amplitude spectrum can represent the energy of the corresponding frequency, while the phase spectrum can represent the phase characteristics of the corresponding frequency. This is applicable to physiological electrical signal analysis and radar signals. Bytes ------------------------------------------------------------------------------------------------------------------------------- FT represents the signal in another domain. To determine each point in the f space, we should not only observe a point in the T space, but also observe all the points in the T space to determine the intensity of the vibration in the f space (that is, the value of the spectrum) bytes ------------------------------------------------------------------------------------------------------------------------------- TD-SCDMA The midamble code channel is expected to use the time domain Circular Convolution equivalent to the frequency domain point multiplication feature, and uses the FFT Uppch checks for filtering and loop-related, Using FFT ------------------------------------------------------------------------------------------------------------------------------- For continuous time-period signals, the Fourier series is the Fourier transform samples of the non-cyclic signals after one of its periods, the dtft of the discrete signal obtained after the Continuous Time Signal samples can be seen as a simulation of the original Continuous Time Fourier transformation on the horizontal axis-a periodic extension after the digital frequency transformation. Discrete Fourier transformation can be seen as the same interval between dtft and the main value range (0 to 2 * PI ------------------------------------------------------------------------------------------------------------------------------- I noticed this post today and talked about my views on continuous signals: For the infinite time domain, the infinite continuous signal in the frequency domain, that is, the most general signal, Use Fourier transform to analyze it (of course, it is necessary to meet the conditions of Fourier transformation ). For finite continuous signals in the time domain, it can also be analyzed using Fourier transform, However, the representation of Fourier series is much more concise. Fourier series decomposition can be understood as a signal in Sample in the frequency field. That is, the time domain Fourier series decomposition corresponds to the frequency domain regression sample. For a finite continuous signal in the frequency domain, it can also be analyzed using Fourier transform, However, it is much simpler to use the time-domain attention sample for interpolation, which is actually in the frequency domain. The signal is decomposed by Fourier series. That is, the time domain attention sample corresponds to the Fourier series decomposition in the frequency domain. Bytes ------------------------------------------------------------------------------------------------------------------------------- 1. for Fourier series, whether continuous or discrete signals, a set of orthogonal functions (positive intersection) are used to calculate the weighted sum of them to approximate the original periodic signal. Generally, the orthogonal set of the Continuous Time Fourier series has an infinite number of functions, because the Discrete Time orthogonal functions are periodic, if the period is n, there are only N functions in the orthogonal set of Discrete Time Fourier series. The weighting coefficient used in the weighted summation process forms the coefficient spectrum of the periodic signal. For the continuous cycle signal, the coefficient spectrum is non-cyclic. For the discrete cycle signal, the coefficient spectrum is in the cycle of N. 2. fourier transformation represents a transformation relationship between the time domain and the frequency domain of the signal. We can derive from the Fourier series expression that is not very strict. The frequency spectrum of the Continuous Time Signal is non-cyclical, however, the frequency of discrete-time signals is extended by a period of 2 * pi. In addition, we can see that the coefficient of Fourier series is the sampled value of the non-cyclic Signal Spectrum of the corresponding main value range; in other words, the frequency spectrum of a non-periodic signal is the envelope of the Fourier series coefficient of the signal obtained from the periodic continuation of the signal. The values of the two are equal on the samples. It is worth noting that the Fourier transformation of a periodic signal is an integer multiple of its fundamental frequency, and the weighting coefficient is exactly the coefficient of the Fourier series of the signal. 3. Relationship between dtft and DFT We know that the spectrum of the Fourier transform (dtft) of an n-point discrete time series is a continuous function that is extended in the cycle of (2 * Pi). We know from the samples theorem that, if the time domain is sampled, the frequency domain cycle is extended. Similarly, if the frequency domain is sampled, the time domain will also be periodically extended. Discrete Fourier Transform (DFT) is based on this theory. It is used in the frequency domain to extract N points (the same as the number of sequential points) in a period, so as to discretize the signal spectrum, obtain an important relationship: the discrete time signal of a n point can be uniquely identified by a n point sequence in the frequency domain. This is what is revealed by the DFT expression. Bytes ------------------------------------------------------------------------------------------------------------------------------- In my opinion, the Fourier transformation is used for non-periodic signals to obtain the continuous spectral density function NW-> W. The linear system and signal of B P. lathi are described in detail ------------------------------------------------------------------------------------------------------------------------------- The fu sequence transformation is derived from the Fu sequence. The fu sequence is a series of orthogonal trigonometric functions that all periodic functions (signals) can be decomposed, the paybyte series corresponding to the periodic function is its spectrum function, that is, the isolated spectral line. To analyze non-cyclic functions, the concept of spectral density is introduced, that is, the spectral function of non-cyclic signals is infinitely small, but the spectral density has a value. In this way, the non-cyclic signal is regarded as a periodic signal with an infinite cycle length, and f (t)/T is introduced, that is, the spectral density function of the non-cyclic function. In order to unify the concept, the concept of the impulse function is introduced. In this way, the periodic signal can also have a Fourier transformation, and its spectral density function is impulse. The paybyte transform plays an important role in the analysis of continuous time signals. It is used to analyze the frequency components of signals or process signals in the frequency domain. After the concept of frequency domain is referenced, the combination of communication and mathematics becomes closer. The development of communication is actually the development of mathematics. As for the discrete Fourier transform, it is actually an analysis and processing of the digital signal transformation to the frequency domain. It plays a significant role in digital signal processing. Digital signal processing is completely dependent on the device situation for Signal Processing in the simulation period, and can be directly processed through computation. For example, a digital filter only calculates the incoming digital signal using the system coefficient. After the signal is output to the system, the processed data is expressed in the time domain. In terms of understanding, the discrete Fourier transform is not exactly the same as the continuous-Signal Fourier transform. It mainly refers to the discrete-Signal Fourier transform, to-cycle extension, and peripheral convolution. In fact, the high-speed discrete Fu Ye transformation is an algorithm for the Fu Yi transformation, which is used to solve the problem that the discrete Fu Yi transformation has a large amount of computing and is not useful, this reduces the computational workload of the Fourier transformation by one or several orders of magnitude, which makes the Discrete Fourier transformation widely used. In addition, the emergence of FFT also overcomes a considerable number of computing problems, so that other computing can also be solved through FFT. Bytes ------------------------------------------------------------------------------------------------------------------------------- Meaning Fourier transform has uniqueness. The properties of Fourier transform reveal the inherent relationship between the time-domain characteristics of signals and the frequency-domain characteristics. The purpose of this paper is to discuss the properties of Fourier transform. Understand the internal connection of features; Evaluate F (ω) by nature; understand the application in the field of communication systems .------------------------------------------------------------------------------------------------------------------------------- Fu's series and Fu's transformation we are now familiar with the common expression of signal amplitude changes with time, for example, the amplitude of the sine signal changes with time according to the law of the sine function. On the other hand, if the amplitude, frequency, and phase of a sine signal are known, the waveform of the sine signal is also uniquely determined. According to this principle and Fourier series theory, all periodic signals meeting certain conditions can be decomposed into linear combinations of sine components of different frequencies, therefore, we use the frequency-amplitude and frequency-phase of each sine component to describe the periodic signal. This is called the Frequency Spectrum Representation of the periodic signal. With the in-depth study of the signal, we extend the Spectrum Representation of the periodic signal to the Spectrum Representation of the non-periodic signal, that is, the usual Fourier transformation. For periodic signals, the spectrum is generally expressed by Fourier series, while the coefficient of Fourier series is called the spectrum of signals. Fast Fourier trans formation A high-speed algorithm for finite discrete Fourier transform (DFT. FFT for short. A complex waveform can be decomposed into a series of harmonic waves. Aiming at this physical phenomenon, a set of effective research methods have been established and developed in mathematics, which is Fourier analysis. Fourier analysis using electronic computers mainly deals with the Fourier expansion of discrete functions, that is, the interpolation of trigonometric functions. One-dimensional DFT mainly applies an N-element array a (I) (I = ,..., N-1) through a linear transformation into an array of n elements X (I) (I = 0 ,... N,-1 ). Assuming that directly calculating all array elements requires about N2 multiplication and addition operations, when n is very large, the computational workload is amazing. In 1965, American culi and Tuji proposed a high-speed algorithm that can greatly reduce the number of computations, namely, the FFT algorithm. Its basic principle is to divide a transformation into the product of two transformations, by using the periodic nature of trigonometric functions, the original transformation formula is combined into a new formula to reduce the number of computations to the nlog2n magnitude. That is to say, the FFT algorithm is improved by N/log2n times than the DFT algorithm. For example, when n = 220, the speed is increased by about 50 thousand times. It can be seen that when n is very large, this is a great improvement. FFT technology has been successfully applied in spectral analysis, digital filtering, structural analysis, system analysis, image and signal processing, geophysical exploration, antenna, radar, satellite, medical, and many other technical fields. Lifecycle 1. the essence of these transformations is the same. They all decompose a complex signal in a Orthogonal System. The difference is that the basis of selection is different. fu's transformation selects the complex exponent and triangular basis, and wavelet transformation selects other bases. 2. the signal is parity between the time domain and the frequency domain. the periodicity and continuity of a domain correspond to the non-periodicity of another domain. For example, when the continuous signal of a periodic signal has an absolute product condition, the series can be expanded, the discrete non-periodic spectrum is obtained. 3. Contact and difference between DFT, dtft, DFS, and FFT DFT and FFT are an essential algorithm of DFT. DFS is the Discrete Fourier seriers, which expands the series of discrete period signals. DFT is the master value of DFS, and DFS is the periodic extension of DFT. Dtft is a pair of Discrete Time Fourier transformation, which is a pair of sequential ft to obtain a continuous cyclic spectrum. While DFT and FFT are a finite-length non-periodic discrete spectrum, not one. the Gini Fourier series is also an expression of periodic signals in the time domain, that is, the orthogonal series. It overlays waveforms of different frequencies. Fourier transformation is a complete analysis of the frequency domain. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------