Understanding discrete Fourier transforms (i.) Transformation of the origin of Fourier)

Understanding Discrete Fourier transforms (i.)                     ------The origin of Fourier transform       On the Fourier transform, whether it is a book or on the Internet 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 understand from the perceptual. Recent. I occasionally see an electronic book on digital signal processing from the Internet, a writer named Steven W. Smith, Ph.D, written by a foreigner, is very plain, with seven chapters that are specifically about the Fourier transform of discrete signals. Although it is an English document. I still bite the bullet. Read about the Fourier transformation of the relevant content, see the feeling of enlightened, here to get the understanding from me to share with you. Hope very much by the Fourier transform confused friends can get a little revelation, this e-book is free, interested friends can also download from the Internet to look at the URL is: http://www.dspguide.com/pdfbook.htm       To understand the Fourier transform. Really need some patience, don't think about how the Fourier transform is changing, of course. Also need a certain higher mathematics foundation. The most important is the series transformation, in which Fourier series transformation is the basic formula of Fourier transform.

  one, Fourier transform the proposed           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 censors voted to publish the paper, Lagrange firmly objected, for nearly 50 years. Lagrange insists that Fourier's method cannot represent angular signals, such as the slope of discontinuous variation in square waves. The French Science society succumbed to Lagrange's prestige. It denies the work of Fourier. Fortunately, Fourier had other things to do, he joined the political movement, with Napoleon's expedition to Egypt. After the French Revolution, fear of being pushed to the guillotine has 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. But. We can use a sinusoidal curve to express it very close to it. There is no energy difference between approaches to two representations, and Fourier is right based on this. Why do we use a sinusoidal curve instead of the original? If we can also use square or triangular wave to replace it. The method of decomposition of the signal is infinite, but the purpose of decomposition of the signal is to more easily deal with the original signal. It is much simpler to use positive cosines to represent the original signal. Because the sine cosine possesses the properties that the original signal does not have: sinusoidal fidelity. After a sine-cosine curve signal is input, the output is still a sine-cosine curve. Only the amplitude and phase may change, but the frequency and wave shape are 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:

1		Fourier transform (Fourier Transform)
2	periodic continuous signal	Fourier series (Fourier series)
3
4	Periodic discrete signals	Discrete Fourier transform (discrete Fourier Transform)

       Is the legend of four original signals:  These four Fourier transforms are for positive infinity and negative infinity signal. That is, the length of the signal is infinitely large. We know that this is not possible for computer processing, so is there a finite Fourier transform for the length? No. As the positive cosine wave is 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 as a signal of infinite length. Can extend the signal infinitely from the left and right, the extended part is expressed in zero. Such This signal can be viewed 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. What we're going to learn here is discrete signals, which we don't discuss for continuous signals. Since the computer can only handle discrete numerical signals, our aim is to use the computer to process the signals.

But for non-cyclical signals, we need to use an infinite number of sine curves of different frequencies to represent them. This is impossible for a computer to achieve. So for the discrete signal transformation only has the discrete Fourier transform (DFT) ability to be applied, for the computer only has the discrete and the finite length data ability to be processed. Other types of transformations are only used in mathematical calculus. We can only use the DfT method in front of the computer, and we should understand the DfT method in the back.

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 both real and complex methods. The real-number method is best understood, but the complex method is relatively complex. It is necessary to understand the theory of complex numbers, but, assuming that the real DFT is understood, then it is easier to understand the complex Fourier transform, 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 is a mathematical transformation. However, the function transformation is different from the function transformation, which conforms to the mapping criterion. For discrete digital signal processing (DSP). There are many transformations: Fourier transform, Laplace transform, Z transform, Hilbert transform, discrete cosine transform, and so on, these all extend the definition of function transformation, agree that the input and output have a variety of values, simply say the transformation is a bunch of data into a bunch 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. It is possible to decompose this signal into 9 cosine and 9 sine waves (a signal of length n can be decomposed into a n/2+1 sine cosine signal, which is why?). Combined with the following 18 positive Yu Yingtu, I think it is not difficult to understand the accuracy of the computer processing. A signal of length n can only be n/2+1 a different frequency. More often than the range of precision the computer can handle, 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 side is the frequency domain signal representation method, the forward transition from left to right (Forward DfT), and the right-to-left representation of the reverse conversion (inverse DFT). An array of amplitude values for the signal at each point in time, in lowercase x[], with uppercase x[] representing an array of sub-values for each frequency, due to the n/2+1 frequency. So the array length is n/2+1,x[] and the array is divided into two types. One is the different frequency amplitude values that represent cosine wave: Re x[]. There is also a different frequency amplitude value of the sine wave: Im x[],re is the meaning of real, im is the meaning of the imaginary number (Imagine), using the plural representation of the positive cosine wave together to represent, but here we do not consider the complex number of other functions. The only way to remember the combination of the method is to make it easier to express (we know that in the back, the Fourier length transformation of the plural form is n. instead of n/2+1).

Transform the next section we'll look at the detailed method of true Fourier.

Understanding discrete Fourier transforms (i.) Transformation of the origin of Fourier)