Understanding Discrete Fourier transforms (i.) ------The origin of Fourier transform On the Fourier transform, whether it is books or online can be very easy to find about the Fourier transform descriptive narrative, but most are some of the mysterious articles, too abstract, is full of some people look at the list of the daunting formula, so that people are very difficult to get understanding from the perceptual, recently, I occasionally see an electronic book on digital signal processing from the Internet, which is called Steven W. Smith, Ph.D. written by foreigners, written very simple, there are seven chapters in the easy-to-understand Fourier transform of discrete signals, although it is an English document, I still bite the bullet to read about the Fourier transformation of the relevant content, see the feeling of enlightened, here I get from the understanding to share with you, hope very much by Fourier transform confused Friends can get a little revelation, this e-book is free, interested friends can also download from the Internet to see, the URL address is: http://www.dspguide.com/pdfbook.htm To understand the Fourier transform, do need a certain degree of patience, do not think of the Fourier transform is how to transform, of course, also need a certain higher mathematics foundation, the main is the series transformation, Fourier series transformation is the basis of Fourier transform formula.
The present of Fourier transform Let's see why there are Fourier transforms. Fourier is the name of a French mathematician and physicist named Jean Baptiste Joseph Fourier (1768-1830), Fourier, who was very interested in heat transfer and published a paper in 1807 at the French Science Society, The paper describes the description of the temperature distribution of the narrative using a sinusoidal curve, in the paper there is a controversial decision at the time: no matter what the continuous periodic signal can be composed of a set of appropriate sinusoidal combination. Two of the people who examined the paper were the historically famous mathematician Lagrange (Joseph Louis Lagrange, 1736-1813) and Laplace (Pierre Simon de Laplace, 1749-1827), When Laplace and other reviewers voted to publish the paper, Lagrange firmly objected that, for nearly 50 years, Lagrange insisted that Fourier's method could not represent angular signals, such as the slope of discontinuous variation in square waves. The French Science society succumbed to Lagrange's prestige, denied the work of Fourier, fortunately, Fourier has other things to do, he participated in the political movement, with Napoleon expedition to Egypt, after the French Revolution for fear of being pushed to the guillotine and have been running away. The paper was not published until 15 years after the Lagrange's death. Who's right? Lagrange is right: the sinusoidal curve cannot be combined into a signal with an angular edge. However, we can use a sinusoidal curve to express it very close, approximation to the two representations there is no energy difference, based on this, Fourier is right. Why do we use a sinusoidal curve instead of the original? If we can also use square or triangular wave to replace Ah, the decomposition of the signal is infinite, but the purpose of the decomposition signal is to more easily deal with the original signal. It is simpler to use positive cosines to represent the original signal, since the sine cosine possesses the properties that the original signal does not have: sinusoidal fidelity. A sine-cosine curve signal input, the output is still the positive cosine curve, only the amplitude and phase may change, but the frequency and wave shape is still the same. And only the sine-cosine curve has this property, and that's why we don't need a square or triangular pleiku.
ii. Classification of Fourier transform Based on the different types of original signals, we are able to divide the Fourier transform into four categories:
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Fourier transform (Fourier Transform) |
| 2 |
periodic continuous signal |
Fourier series (Fourier series) |
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| 4 |
Periodic discrete signals |
Discrete Fourier transform (discrete Fourier Transform) |
Is the legend of four original signals: These four kinds of Fourier transform are for positive infinity and negative infinity signal, that is, the length of the signal is infinite, we know this is not possible for computer processing, then there is no length of the Fourier transform it? No. Since positive cosine wave are defined as from negative infinity to positive infinity, we cannot combine a signal of infinite length into a signal of limited length. In the face of such difficulties, the method is to express a limited length of the signal to the infinite length of the signal, the signal can be infinitely extended from left to right, the extended portion is expressed in zero, so that the signal can be regarded as a non-periodic dissociation signal, we can use the discrete time domain Fourier transform method. Also, the signal can be extended by means of replication, so that the signal becomes a periodic dissociation signal, then we can use the discrete Fourier transform method to transform. Here we want to learn the discrete signal, for the continuous signal we do not discuss, because the computer can only handle discrete numerical signals, our goal is to use the computer to process the signal. But for non-cyclical signals, we need to use an infinite number of sine curves of different frequencies to represent, which is impossible for a computer to achieve. So for the discrete signal transformation only has the discrete Fourier transform (DFT) talent is applicable, for the computer only has the discrete and the finite length data ability to be processed, for the other transformation type only has in the mathematical calculus The talented use, in front of the computer we can only use the DFT method, What we need to understand later is the DfT method. It is important to understand that we use periodic signals for the purpose of solving this problem mathematically, and it is meaningless to consider where periodic signals are obtained or how they are obtained. Each Fourier transform is divided into real and complex two methods, for the real method is the best understanding, but the complex number method is relatively complex, need to understand the theory of complex numbers, but the assumption that the real DfT is understood, and then to understand the complex Fourier transform is easier, So we first put the Fourier transform of the complex number to one side, first to understand the real Fourier transform, in the following we will first talk about the basic theory of complex numbers, and then understand the real Fourier transform based on the understanding of the complex Fourier transform. Also, the transformation (transform) We are going to say here, although it is a mathematical transformation, is different from the function transformation, the function transformation conforms to the one-to-one mapping criterion, for discrete digital signal processing (DSP), there are many transformations: Fourier transform, Laplace transform, Z transform, Hilbert transform, Discrete cosine transforms, all of which extend the definition of a function transformation, agreeing that input and output have a variety of values, simply by transforming a bunch of data into a pile of data.
A sample of a real DFT with a discrete Fourier transform Let's look at a transformation instance, which is a raw signal image: The length of the signal is 16, so the signal can be decomposed into 9 cosine waves and 9 sine waves (a signal of length n can be decomposed into a n/2+1 sine cosine signal, why?). Combined with the following 18 positive Yu Yingtu, I think from the computer processing accuracy is not difficult to understand, a signal length of n, at most only can have n/2+1 a different frequency, and more frequency than the computer can handle the accuracy range), for example:9A cosine signal:9A sine signal: Add all the above signal can get the original signal, as to how to change the 9 different frequency signal, we first not urgent, first look at the above transformation results, in the program is how to express, we can look at the following demonstration sample diagram: The left side represents the signal in the time domain, the right is the frequency domain signal representation method, from left to right to represent the forward conversion (Forward DfT), right-to-left for reverse conversion (inverse DFT), with lowercase x[] to represent the signal at each point in time, the magnitude of the array of values, with uppercase x[] An array of sub-values representing each frequency, because of the n/2+1 frequency, the array length is n/2+1,x[] array is divided into two, one is to represent cosine wave different frequency amplitude values: Re x[], there is a sine wave of different frequency amplitude values: Im x[],re is real (real) , Im is the meaning of imaginary (Imagine), using the representation of complex numbers to represent the positive cosine wave, but here we do not consider the other functions of the complex number, just remember is a combination method, the purpose is to facilitate the expression (we will know in the following, the complex form of Fourier transform length is n, Rather than n/2+1). In the next section we will look at the detailed method of the real Fourier transform.
Understanding discrete Fourier transforms (i.) The origin of Fourier transform)
