Fourier transform 3

Discrete Fourier transform in real form (real DFT)

The previous chapter left a transformation of the implementation of the problem, in fact, this is a two-way, positive and inverse transformation, we first understand some of the frequency of knowledge four kinds of representations about frequency in the frequency domain1, the serial Number representation method: According to the time Domain signal sample number takes 0 ~N/2, uses this method in the procedure to obtain each kind of frequency amplitude value more directly, because the frequency value and the array ordinal is one by one corresponds: X[k], the value range is 0~ n/22, the fractional notation, Take 0~0.5:x[based on the proportional value of the sample number of the signal in the time domain? ] ，？ =k/n value range is 0-0.53, with radian value to represent, put? Multiplied by a 2π to get a radian value, this representation is called natural frequency x[w],w=2π? =2πk/n, the value range is 0~π4, expressed in Hertz second, the basic function of DFTCk[i] = cos (2πki/n) sk[i] = sin (2πki/n) where k represents the frequency of each positive cosine wave, such as 2 for two complete periods in 0 to N lengths, 10 that is 10 and a period, such as: the amplitude of each waveform is how to calculate is the DfT is more important place, more difficult to understand. We first synthesize the original signal (inverse DFT) of the decomposed positive cosine wave. three, synthetic operation methodThe DfT synthesis equation (synthesis of the original time signal, frequency-to-time, reverse transformation) is not familiar with this formula in the first section of the Fourier series is very similar to see the following formula: No matter how the first formula is, we just have to clear the Fourier series, You can use the first formula as a tool. The IM x[k] and re x[k in the DfT synthesis equation are not the same as the previously mentioned IM X[k] and re x[k], and here is the conversion method (for an explanation of this formula, see below): But when K equals 0 and N/2, the calculation of the real part is to use the following equation: The n in the four above is the total number of midpoint in the time domain, and k is the ordinal number from 0 to N/2. Why do you want to convert it like this? This can be understood from the spectral density (spectral density), such as a spectrogram:, this is a spectrum map, the horizontal axis represents the frequency size, the ordinate represents the amplitude size, the original signal length is n (here is 32), after the DfT converted 17 frequency spectrum, The spectral density represents how much amplitude per unit of bandwidth, so how is the bandwidth calculated? See, in addition to two, the remaining points of the width is 2/n, the width is the bandwidth of each point, two points of the bandwidth is 1/n, and]im x[k and re x[k] represents the spectral density, that is, the amplitude of each unit bandwidth, but the amplitude of the 2/n (or 1/n) bandwidth, so Should be the 2/n (or 1/n) of IM x[k] and re x[k]. Spectral density is like the physical density of the material, each point in the original signal is like a mixture, the mixture is composed of different density of matter, the mixture contains the quality of each substance is the same, in addition to the maximum and minimum two density of the material,    In this way we can get the density of the mixture by adding up the density of each substance, and the mass of the mixture is the unit mass, so the obtained density value is the same as the mass value of the mixture. If the DFT results have been obtained, then the inverse conversion, that is, the synthesis of the original signal, then the following steps can be converted: 1, based on the above four formulas calculated value: 2, and then based on the DfT synthesis equation to obtain the original signal data four, decomposition operation methodThere are three different ways to do DFT: one is to solve by simultaneous equations, from the point of view of algebra, to find n unknown value from n known value, need n simultaneous equations, and n simultaneous equations must be linear independent, but this method of calculation is very large and very complex, it is rarely used The second method is to use signal correlation (correlation) to calculate, this is the method we will introduce later, the third method is a fast Fourier transform (FFT), which is a very creative and revolutionary method, because it greatly improves the speed of operation, Makes the Fourier transform can be widely used in the computer, but this algorithm is based on the Fourier transform of the complex form, it is the n-point signal decomposition into a length of n frequency domain, this is not the same as the real-domain DFT transformation We are now doing, and this method is more difficult to understand, here we first do not understand, Once you understand the complex DFT, take a look at the FFT. One thing is very important, that is, the results of the three methods are the same, after the practice has proved that, when the frequency domain length is 32 o'clock, the use of correlation method to calculate the best efficiency, or FFT algorithm efficiency is higher. Now let's look at the correlation algorithm. Using the first method, the correlation of the signal (correlation) can detect the known signal from the noise background, we can also use this method to detect whether the signal wave contains a certain frequency signal wave: a signal to be detected by multiplying another signal wave, a new signal wave, Then the new signal wave all the points are added, from the result of the addition can be judged the similarity of the two signals. such as:      above A and B two figure is to detect the signal wave, figure A is clearly seen as a 3-period sine wave, the signal of Figure B does not see whether there is a sine or cosine signal, the figure C and D are a 3-period sine wave signal, figures E and F are a, b two graph with C, d two after multiplying the result, the average value of all points of figure E is 0.5, indicating that signal a contains a sine signal of 1 amplitude C, but the average of all points of figure F is 0, indicating that signal B does not contain signal d. This is the method by which signal correlation is used to detect whether a signal is contained.       The second method: Accordingly, I can also by the input signal and each frequency of the positive cosine signal multiplication (correlation operation), so that the original signal and each frequency of the degree of association (that is, the sum of the size), the result is we want the Fourier transform results, The following two equations are our calculation method:      The second formula has a minus sign in order to maintain the consistency of the plural, we know that in the calculation of IM x[k] There is a minus sign, so this is only a form of the problem, and there is no practical meaning, You can also go to the minus sign.X[K] and does not add a minus sign when calculating im. One thing that must be understood here is an orthogonal concept: Multiply two functions, and if the sum of each point in the result is 0, you can think of the two functions as orthogonal functions. To ensure that the correlation algorithm is correct, the function of the signal multiplied by the original signal must be orthogonal, and we know that all sine or cosine functions are orthogonal, which we can prove by means of a simple high number of knowledge, so we can use the associated method to separate the original signal from the sine cosine signal. Of course, other orthogonal functions are also present, such as: square wave, triangular waves and other forms of pulse signal, so the original signal can be decomposed into these signals, but this is only said can do, but it is useless.       This is just a discrete Fourier transform on the real domain, in which the plural form is used, but it is an alternative form, and it has no practical meaning, and in reality it is generally used in the form of discrete Fourier transforms of the plural, And the fast Fourier transform is based on the complex discrete Fourier transform to design the algorithm, in the back we first to review the contents of the complex, and then understand the real domain discrete Fourier transform based on the understanding of the complex form of discrete Fourier transform

Fourier transform 3