TAILIEUCHUNG - FIR Filters - DFT

For the implementation of band pass filter as per the specification implementation of 161 order is necessary. Implementation of such a higher order using simple DA architecture requires ROM size of 2161 which is very large and practically not feasible to implement on FPGA. Even though it is possible to reduce the size of the ROM by utilizing symmetrical property of coefficients | Chapter Discrete Fourier Transform The Fourier series FS introduced in Chap. 1 links the continuous-time domain to the discrete-frequency domain and the Fourier transform FT links the continuous-time domain to the continuous-frequency domain. The discrete-time Fourier transform DTFT introduced in Sec. links the discrete-time domain to the continuous-frequency domain. In this chapter we examine the discrete Fourier transform DFT which links the discrete-time and discrete-frequency domains. A complete treatment of the design and coding of DFT algorithms can fill volumes see Brigham 1975 Burrus and Parks 1984 Nussbaumer 1981 . Rather than attempt complete coverage of DFTs this chapter presents only those aspects that are germane to the design of digital filters. Coverage of the so-called fast algorithms for implementation of the DFT is limited to one specific type of algorithm along with an examination of the computational savings that fast algorithms can provide. Discrete Fourier Transform The discrete Fourier transform and its inverse are given by X m x n e i2 mnFT m 0 1 . . . N - 1 n 0 N - 1 N - 1 Y cos 2nmnFT j X xinJ sin 2nmnFT n 0 n 0 N- 1 x n X X m ei2nmnFT n 0 1 . . N - 1 m 0 X X m cos 2nmnFT j X A m sin 2nmnFT m 0 m 0 It is a common practice in the DSP literature to bury the details of Eqs. 137 138 Chapter Eight and by defining ej2n N ej2nFT and rewriting and as N- 1 X m X Kn n - 0 x n AI1X m m 0 Since the exponents in and differ only in sign another common practice in writing DFT software is to write only a single routine that can evaluate either or depending upon the value of an input flag being equal to 1 or 1. Back in the olden days when memory and disk space were expensive this was a big deal but these days having two separate routines may pay for itself in terms of clarity execution speed and simplified calling sequences. Parameter selection In designing a DFT for a particular .

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