CH31(1) - CHAPTER 31 The Complex Fourier Transform Although...

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567 CHAPTER 31 Re X [ k ] 2 N j N 1 n 0 x [ n ] cos(2 B kn / N ) Im X [ k ] 2 N j N 1 n 0 x [ n ] sin(2 B kn / N ) EQUATION 31-1 The real DFT. This is the forward transform, calculating the frequency domain from the time domain. In spite of using the names: real part and imaginary part , these equations only involve ordinary numbers. The frequency index, k , runs from 0 to N /2. These are the same equations given in Eq. 8-4, except that the 2/ N term has been included in the forward transform. The Complex Fourier Transform Although complex numbers are fundamentally disconnected from our reality, they can be used to solve science and engineering problems in two ways. First, the parameters from a real world problem can be substituted into a complex form, as presented in the last chapter. The second method is much more elegant and powerful, a way of making the complex numbers mathematically equivalent to the physical problem. This approach leads to the complex Fourier transform , a more sophisticated version of the real Fourier transform discussed in Chapter 8. The complex Fourier transform is important in itself, but also as a stepping stone to more powerful complex techniques, such as the Laplace and z-transforms . These complex transforms are the foundation of theoretical DSP. The Real DFT All four members of the Fourier transform family (DFT, DTFT, Fourier complex numbers. Since DSP is mainly concerned with the DFT, we will use it as an example. Before jumping into the complex math, let's review the real DFT with a special emphasis on things that are awkward with the mathematics. In Chapter 8 we defined the real version of the Discrete Fourier Transform according to the equations: In words, an N sample time domain signal, , is decomposed into a set x [ n ] of cosine waves, and sine waves, with frequencies given by the N /2 % 1 N /2 % 1
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The Scientist and Engineer's Guide to Digital Signal Processing 568 index, k . The amplitudes of the cosine waves are contained in , while ReX [ k ] the amplitudes of the sine waves are contained in . These equations Im X [ k ] operate by correlating the respective cosine or sine wave with the time domain signal. In spite of using the names: real part and imaginary part , there are no complex numbers in these equations. There isn't a j anywhere in sight! We have also included the normalization factor, in these equations. 2/ N Remember, this can be placed in front of either the synthesis or analysis equation, or be handled as a separate step (as described by Eq. 8-3). These equations should be very familiar from previous chapters. If they aren't, go back and brush up on these concepts before continuing. If you don't understand the real DFT, you will never be able to understand the complex DFT. Even though the real DFT uses only real numbers,
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CH31(1) - CHAPTER 31 The Complex Fourier Transform Although...

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