Understand the discrete Fourier transform (four). Complex form discrete Fourier transform)

Understanding of discrete Fourier transform (d)------complex form of discrete Fourier transform the discrete Fourier transform is very ingenious to use the complex method, making Fourier transform transformation more natural and concise, it is not simply to use the replacement method to use the complex Number, but to analyze the problem completely from the plural point of view, which is completely different from the real DFT.One, to represent a positive cosine function as a plural formThe positive cosine function can be represented as a plural form by Euler equation: cos (x) = 1/2 E J (-X) + 1/2 ejx sin (x) = J (1/2 e J (-X)-1/2 ejx) from this equation can be   To see that if the positive cosine function is represented as a complex number, they become positive cnoidal by positive and negative frequencies, whereas a positive cnoidal composed of positive and negative frequencies can be expressed in the form of a plural. We know that in the real Fourier transform, its spectrum is 0 ~π (0 ~ n/2), but it cannot represent the spectrum of-π~ 0, and it can be predicted that if the positive cosine is expressed in the plural form, the negative frequency can be included.Second, The variables before and after the transformation are considered as plural formsThe plural form Fourier transforms the original signal x[n] as a signal in the plural, in which the real number part represents the original signal value, the imaginary number is divided into 0, and the transformation result X[k] is also a plural form, but the imaginary part here has the value. In order to look at the original signal in the plural view, it is the key to understanding the Fourier transform of the plural form (if a complex function is learned, it may be better understood that the x[n] is considered a complex variable and then the same transformation as the real number is to the plural variable).Third, a correlation algorithm for complex numbers (forward Fourier transform)   from the real Fourier transform can be known, we can multiply the original signal by an orthogonal function of the form of the signal, and then the sum, and finally we can get the original signal contained in the orthogonal function signal component. Now that our original signal is plural, we're going to get the signal component of the plural, and we can multiply it by multiplying it into a complex orthogonal function. The answer is yes, the positive cosine functions are orthogonal functions, after becoming the plural in the following form, it is still an orthogonal function (the definition of the orthogonal function can be easily proved): cos x + j sin x, cos x–j sin x, ...   Here we use the second formula above Correlation summation, why use the second formula? As we'll know later, the sine function is transformed into a negative sine function, where we add a minus sign so that we get the positive sine wave, so it's easy to get the complex form of the DFT forward transformation equation:               This equation is easy to get Euler transformation formula:                  In fact, we use Euler transform for the convenience of expression, we still use the positive cosine expression more when we solve the problem.          for the above equation, we need to be clear about the following aspects (also different from the real number of DFT): 1, X[k], x[n] are plural, but x[n] the imaginary parts are composed of 0 , the real part represents the original signal; 2, K value range is 0 ~ N-1 (can also be expressed as 0 ~ 2π), where 0 ~ n/2 (or 0 ~π) is the positive frequency part, N/2 ~ N-1 (π~ 2π) is the negative frequency part, because of the symmetry of the positive cosine function, so We represent –π~ 0 as π~ 2π, which is for computational convenience. 3, where the J is an inseparable part of the just like a variable in an equation, you can't just take it off, it's completely different after you get rid of it, but we know that in real-world DFT, J is just a symbol, minus J, and the meaning of the whole equation is unchanged. 4, the following graph is the spectrum of a continuous signal, But the discrete spectrum is similar to this, so it does not affect our problemAnalysis of:                    The spectrum of the            above puts the negative frequencies on the left to cater to our thinking habits, But in the actual implementation we usually move it behind the positive spectrum. As can be seen from the above figure, the positive cnoidal in the Time-domain (the positive cnoidal used to compose the original signal) are divided into positive, the two components of a negative frequency, based on the previous scaling factor in this equation, are 1/n (or 1/2π) rather than 2/n, because the spectrum is now extended to the 2π, But adding plus or minus two frequencies is 2/n and restored to the form of the real number DFT, which can be seen more clearly in the following description. Since the complex DFT generates a complete spectrum, each point in the original signal is made up of positive, negative two frequency combinations, so the bandwidth of each point in the spectrum is the same, are 1/n, relative to the real DFT, both ends of the bandwidth than the other points less than half of the bandwidth; The spectral characteristics of complex DFT are periodic:-N /2 ~ 0 is the same as N/2 ~ N-1, the real-domain spectrum is symmetrical (representing the cnoidal spectrum), and the virtual-domain spectrum is singularly symmetric (representing the sine wave spectrum).  Four, Reverse Fourier transformSuppose we have obtained the spectrum x[k of the plural form], now we want to revert it to the original signal in the plural form x[n], of course, we should multiply x[k] by a complex number, then sum it up, then get the original signal x[n], this with x[k] Multiply the complex number first let us think of the above is the correlation calculation of the complex number: cos (2πkn/n) –j sin (2πkn/n), but the minus is in fact, in order to make the reverse Fourier transform sine function into a positive symbol, because the operation of the imaginary number J is special, So that the sine function that was supposed to be positive turns into a negative sine function (we see this from the derivation later), so the minus sign here is just to correct the effect of the symbol, we can remove the minus sign when we reverse DFT, so we get the inverse DFT transformation equation:

X[n] = x[k] (cos (2πkn/n) + j sin (2πkn/n))

Now we're going to analyze this equation and we'll see that the formula actually gets the same result as the real Fourier transform. Let's change the x[k] First:

X[k] = Re X[k] + j Im X[k]

This allows us to transform the x[n] again, such as:

X[n] = (Re x[k] + j Im x[k]) (cos (2πkn/n) + j sin (2πkn/n))

= (Re x[k] cos (2πkn/n) + j Im x[k] cos (2πkn/n) +

           J Re X[k] sin (2πkn/n)-  Im x[k] sin (2πkn/n))

       = (Re x[k] (cos (2πkn/n) + j sin (2πkn/n)) +   --------------------- (1)             Im x[k] (-sin (2πkn/n) + j cos (2πkn/n)) &nbs P ---------------(2) then we split the original equation into two parts, the first part is multiplied by the spectrum in the real domain, and the second part is multiplied by the spectrum in the virtual domain, and according to the spectrogram we can know that the Re x[k] is a symmetric variable, and Im x[k is a singularly symmetric variable. , that is, re x[k] = re x[-K] im x[k] =-Im x[-k] But K's range is 0 ~ N-1, 0~N/2 represents positive frequency, n/2~n-1 represents negative frequency, in order to express the convenience we can use the n/2~n-1 with K, so that in the 0 to the sum of the N-1 (1) and (2) respectively have N/2 K and K, and for (1) are: Re x[k] (cos (2πkn/n) + j sin (2πkn/n)) + Re x[-K] (cos (-2) πkn/n) + j sin (-2πkn/n)) according to the symmetry and the nature of the trigonometric function, the upper-type reduction is obtained: Re x[k] (cos (2πkn/n) + j sin (2πkn/n)) + Re x[K] (cos (2πkn/n)-J Sin (2πkn/n) The final result of this formula is: 2 Re x[K] cos (2πkn/n) and considering that there is a proportional coefficient 1/n in the Re x[K] equation, the 1/n is multiplied by 2, so the result is not the same as that in the real number DFT.   for (2) type, in the same way, we can also get the result:-2 Im x[k] sin (2πkn/n note that the upper-style front more than a negative symbol, which is due to the specificity of the transformation of imaginary numbers, of course, we certainly can not add the sine function of the negative symbol with the cosines, fortunately, We are using the cos (2πkn/n) –j sin (2πkn/n) to calculate the correlation, get the Im X[k] has a negative symbol, in this final result, there is no negative sign for the sine-chord function, which is why the negative sign is used for the imaginary part in the calculation of correlation (I think this may be a place where the plural form of DFT is in the ointment, giving people a sense of patchwork).   From the above analysis, it can be seen that the real Fourier transform and the complex Fourier transform, in the inverse transformation of the result is the same, but is the same. Attached: Word document download Address: http://download.csdn.net/source/444234