By Marvin Frerking

An engineer's creation to innovations, algorithms, and developments in electronic sign Processing. This lucidly written source makes large use of real-world examples because it covers all of the vital layout and engineering references.

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This can be seen starting with Eq. 45) for the DFT. 52) Then, performing the outer conjugation gives the expression N-\ LX" k=O (k) w;r Digital Signal Processing Concepts 33 and, of course, X**(k) = X(k). Therefore, Eq. 52) gives the same result as Eq. 51) except for the scale factor N. FAST FOURIER TRANSFORMS The FFT is simply a more efficient way of computing the DFT. The result is identical. Therefore, we will simply discuss the mechanics of the computation here. The FFT can be used to compute the DFT provided the value ofN is chosen to be a power of 2 for radix two FFTs (N = 2k, where k is an integer).

Thus, a value for k = 7 corresponds to a frequency o 4 I 8 INDEX VALUE. 18 Real part ofFFT output for real input signal I 12 16 Digital Signal Processing Concepts 29 (7/l6)FS' while a value k = 9 corresponds to (-7/16)Fs. If the imaginary part of the output were shown, the plot would exhibit odd symmetry about the value k = 8. It should be noted that if the input is real and has even symmetry about the center value n = N/2, the output is purely real while an input function exhibiting odd symmetry about the center value n = NI2 gives a purely imaginary result.

We will often use the frequency domain fonn F(f) since, from a system designers viewpoint, frequency is usually defined in hertz rather than in radians per second. This is also more convenient when we are perfonning graphical convolution since, from Eq. 26) System studies using digital signal processing often require the use of several of the properties of Fourier transfonns. It is, therefore, appropriate to review the properties of interest so they will be clearly in mind when we apply them. The first of these is the property of differentiation.

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