Understanding Discrete Fourier Transformation (4) ------ complex form discrete Fourier transformation uses the complex method very cleverly, making the Fourier transformation more natural and concise, it does not simply use the replacement method to use the plural, but analyzes the problem from the perspective of the plural, which is totally different from the real DFT.
I,
Returns the positive cosine of a number.Through the Euler's equation, the positive cosine function can be expressed as a complex number: cos (x) = 1/2 e j (-x) + 1/2 ejx sin (X) = J (1/2 e j (-x)-1/2 ejx) from this equation, we can see that if the positive cosine function is expressed as a complex number, they become positive and Cosine waves composed of positive and negative frequencies. On the contrary, a positive cosine wave composed of positive and negative frequencies can be expressed in the form of plural numbers. We know that in the real Fourier transformation, its spectrum is 0 ~ π (0 ~ N/2), but cannot represent-π ~ 0 spectrum, it is foreseeable that if the positive cosine is expressed as a plural form, the negative frequency can be included.
II,
Consider the variables before and after the transformation as the form of a plural number.In the plural form, Fourier transformation treats the original signal X [N] as a signal expressed in the plural form. The real part indicates the original signal value, and the imaginary part is 0, the transformation result X [k] is also in the form of a plural number, but the imaginary part here has a value. Here we need to look at the original signal from the plural point of view, which is the key to understanding the Fourier transformation of the complex form (if you have learned the complex variable function, it may be better understood, consider X [N] As a plural variable, and then perform the same transformation for the plural variable as the real number ).
III,
Correlation Algorithm for the plural (positive Fourier transformation)We can know from the real Fourier transformation that we can multiply the original signal by a signal in the form of an orthogonal function, and then calculate the sum, finally, we can obtain the component of the orthogonal function signal contained in the original signal. Now our original signal is changed to a complex number. Of course we want to obtain the signal component of the complex number. Can we multiply it by an orthogonal function in the form of a complex number? The answer is yes. The positive cosine function is an orthogonal function. After it becomes a complex number in the following form, it is still an orthogonal function (this can be easily proved by the definition of an orthogonal function ): cos x + J SiN x, Cos x-J SiN x ,...... Here we use the second sub-statement above for correlation summation. Why is the second sub-statement used ?, We will later know that the sine function is a negative sine function after the transformation in the virtual number. Here we add a negative number to make the final result a positive sine wave, according to this, we can easily obtain the DFT positive transformation equation in the complex form: This formula can easily obtain the Euler's transformation formula: in fact, we only use the Euler's transform type for the convenience of expression. We still use the positive cosine expression many times when solving the problem. For the equation above, we need to clarify the following aspects (also different from the real DFT): 1, x [K], X [N] are all plural, however, the imaginary part of X [N] is composed of 0, and the real part represents the original signal. 2. The value range of K is 0 ~ N-1 (can also reach 0 ~ 2 π), where 0 ~ N/2 (or 0 ~ π) is the positive frequency part, n/2 ~ N-1 (π ~ 2 π) is the negative frequency part. Because of the symmetry of the positive cosine function, we take-π ~ 0 indicates π ~ 2π, which is convenient for computing. 3. J is an inseparable component, just like a variable in an equation, which cannot be removed at will. After removal, the meaning is completely different, however, we know that in the real DFT, J is just a symbol, removing J, and the meaning of the entire equation remains unchanged; 4. It is a continuous signal spectrum, however, the discrete spectrum is similar to this, so it does not affect our analysis of the problem: the above spectrum map places the negative frequency to the left to cater to our thinking habits, but in actual implementation, we usually move it to the back of the positive spectrum. It can be seen that the positive cosine wave (used to form the positive cosine wave of the original signal) in the time domain is divided into two parts of the positive and negative frequencies in the spectrum of the complex DFT, based on the above formula, the proportional coefficient is 1/N (or 1/2 π), rather than 2/N, because the spectrum is now extended to 2 π, however, after adding the positive and negative frequencies, we get 2/n again and return it to the form of real-number DFT. This can be seen more clearly in the subsequent descriptions. Because the complex DFT generates a complete spectrum, each point in the original signal is composed of positive and negative frequencies. Therefore, the bandwidth of each point in the spectrum is the same, the values are 1/n. Compared with the real-number DFT, the bandwidth at both ends is half less than the bandwidth at other points. The spectrum characteristics of the complex DFT are cyclical:-n/2 ~ 0 and n/2 ~ The N-1 is the same, the real domain spectrum is symmetric (representing the cosine wave spectrum), the imaginary domain spectrum is odd symmetry (representing the sine wave spectrum ).
IV,
Inverse Fourier TransformationAssume that we have obtained the spectrum X [k] in the form of a plural number, and now we want to restore it to the original signal X [N] in the form of a plural number. of course, we should multiply X [k] by a plural number, then sum it, and finally obtain the original signal X [N]. the complex number multiplied by X [k] first reminds us of the complex number for correlation calculation above: cos (2 π kN/N) -J sin (2 π kN/N), but the negative number is actually used to convert the sine function into a positive symbol when performing inverse Fourier transformation. Because of the special operation of the virtual number J, so that it should be a positive sine function to a negative sine function (we will see this point from the subsequent derivation), so the negative number here is only to correct the effect of the symbol, when performing reverse DFT, we can remove the negative number, so we get the inverse DFT transformation equation:
X [N] = x [k] (COS (2 π kN/n) + J sin (2 π kN/n ))
When we analyze this formula, we will find that this formula is the same as the real Fourier transformation. First, let's change X [k:
X [k] = Re x [k] + J im X [k]
In this way, we can perform another transformation on X [N], for example:
X [N] = (RE x [k] + J im X [k]) (COS (2 π kN/n) + J sin (2 π kN/n ))
= (RE x [k] cos (2 π kN/n) + J im X [k] cos (2 π kN/n) +
J re X [k] sin (2 π kN/N)-im X [k] sin (2 π kN/n ))
= (RE x [k] (COS (2 π kN/n) + J sin (2 π kN/n) + ------------------- (1)
Im X [k] (-sin (2 π kN/n) + J cos (2 π kN/n) ------------- (2) in this case, we divide the original equation into two parts. The first part is to multiply the spectrum in the real domain, and the second part is to multiply the spectrum in the virtual domain. Based on the spectrum graph, we can know that, re x [k] is an even symmetric variable, and Im X [k] is an odd symmetric variable, that is, RE x [k] = Re x [-K] Im X [k] =-im X [-K], but K ranges from 0 ~ N-1, 0 ~ N/2 indicates the positive frequency, n/2 ~ N-1 represents the negative frequency, in order to express convenience we put n/2 ~ The N-1 is represented by-K, so that in the summation process from 0 to the N-1 for (1) and (2) There are n/2 pairs of K and-K respectively and, for the (1) formula: RE x [k] (COS (2 π kN/n) + J sin (2 π kN/n )) + RE x [-K] (COS (-2 π kN/n) + J sin (-2 π kN/n) according to the properties of even symmetry and trigonometric function, the preceding formula is reduced to RE x [k] (COS (2 π kN/n) + J sin (2 π kN/n )) + RE x [k] (COS (2 π kN/N)-J sin (2 π kN/n) the result of the substatement is: 2 RE x [k] cos (2 π kN/n) Then considering that there is a proportional coefficient 1/N in the RE x [k] equation, multiply 1/N by 2, isn't this the same result as the formula sub in the real DFT? For the formula (2), we can also obtain the following result using the same method: -2 Im X [k] sin (2 π kN/n note that there is a negative symbol in front of the above formula, which is caused by the particularity of the virtual number transformation, of course, we certainly cannot add the sine function of the negative symbol to the cosine. Fortunately, we used cos (2 π kN/N)-J sin (2 π kN/N) after correlation calculation, the obtained im X [k] has a negative symbol. In this way, the normal string function does not have a negative symbol in the final result, this is why negative symbols are used in the imaginary part of correlation calculation (I think this may be a weakness in the complex form of DFT, which makes people feel like a piece of work ). From the above analysis, we can see that the result of the real Fourier transformation and the complex Fourier transformation is the same in the inverse transformation, but it is just the same thing. Appendix: Word documents: http://download.csdn.net/source/444234
