Simple FFT function library

Transcription

1 Simple FFT function library Publication Date: 2013/11/1 XMOS 2013, All Rights Reserved.

2 Simple FFT function library 2/8 Table of Contents 1 Simple FFT Function Library Overview Resource Usage and Performance Complex FFT Real FFT API Sine arrays FFT functions Programming Guide

3 1 Simple FFT Function Library I THIS CHAPTER Overview Resource Usage and Performance API Programming Guide 1.1 Overview DSP transformations perform functions such as FFTs or DFTs; they transform between the time and frequency domains. There are shelves of textbooks that cover filtering; a two-page summary is necessarily incomplete. Typical characterisations of transformations include: The number of points. This is the number of smaples on which to run the transform. Whether it requires a forward or a backwards transform. Type of filter (for example, this module only implements a one-dimensional FFT) As a rule of thumb, computational requirements are O(n log n) in the number of points. Double the number of points doubles the time that an FFT takes plus a bit more. Typically, overflow leads to chaos. umbers are typically rounded to the nearest value, with 0.5 being rounded up. This FFT module implements a set of functions that can be used to compute the FFT of a set of complex data points, or the inverse. 1.2 Resource Usage and Performance Complex FFT The table below shows the number of thread cycles for the bit twiddling, and for performing either an forward or an inverse FFT. The maximum rate shows the rate for performing a twiddle, a forward FFT, a twiddle, and an inverse FFT on a single 50 MIPS thread: Points Thread cycles Max ratefft+inv Error Memory Twiddle FFT Hz 3-4 bits 1 KB Hz 5-6 bits 2 KB log 2/ log /2 bits 1 KB +

4 Simple FFT function library 4/8 The memory required comprises roughly 1K of code, and then extra space for a sin() lookup table. It excludes storage for the data points, which amounts to eight bytes per point (four bytes real part, and four bytes imaginary part). ote that there is a trade-off between rounding and accuracy - the inverse FFT can be made more accurate (by half to one extra bit) at a cost of 5% extra instructions. This is marked in the code. The errors are indications only, maximum errors will be higher and are value dependent Real FFT The table below shows the number of thread cycles for performing either an forward or an inverse FFT. The maximum rate shows the rate for performing a forward FFT, and an inverse FFT on a single 50 MIPS thread: Points Thread cycles Max rate Error Memory 2 x Hz 4 bits 1 KB 2 x Hz 6 bits 2 KB 2 x 29 log 1.8/ log /2+1 bits 1 KB + ote that the rate is slightly lower but that this computes two real FFTs simultaneously, and is hence 1.8 times faster than its complex counterpart. The memory required comprises roughly 1K of code, and then extra space for a sin() lookup table. It excludes storage for the data points, which amounts to eight bytes per point (four bytes real part, and four bytes imaginary part). (Even though the FFT is on real data, the answer is still in the complex domain) The real FFT uses the ocmplex FFT code, and the same tradeoff between runtime and accuracy exists. The real code is half a bit less accurate than its complex counterpart. 1.3 API Sine arrays The include file fft.h defines a set of arrays that are to be used with the FFT functions, called sine_8[], sine_16[],..., sine_8192[]. Depending on the number of points, pick the appropriate array and pass it to fftforward and fftinverse as required FFT functions void ffttwiddle(int re[], int im[], int ) This function twiddles the arrays around prior to computing an FFT. A calling sequence for a forward FFT involves ffttwiddle() followed by fftforward(), and for an inverse FFT it involves ffttwiddle() followed by fftinverse(). In some cases twiddling can be avoided, for example when computing a convolution.

5 Simple FFT function library 5/8 re im real part of each input point imaginary part of each input point number of points. Must be a power of 2, both re and im should be long void fftforward(int re[], int im[], int, int sine[]) This function computes a forward FFT. The complex input array is supplied as two arrays of integers, with numbers represented as fixed-point values. The number of points must be a power of 2, and the array of sine values should contain a quarter sine-wave. Use one of sine_ provided in sine.h. The function does not perform a bit-twiddle - if required then ffttwiddle() should be called beforehand. re im real part of each input point imaginary part of each input point number of points. Must be a power of 2, both re and im should be long sine array of /4+1 sine values, each represented as a sign bit, and a 31 bit fraction. 1 should be represented as 0x7fffffff. Arrays are provided in sine.h; for example, for a 1024 point FFT use sin_1024. void fftinverse(int re[], int im[], int, int sine[]) This function computes an inverse FFT. The complex input array is supplied as two arrays of integers, with numbers represented as fixed-point values. The number of points must be a power of 2, and the array of sine values should contain a quarter sine-wave. Use one of sine_ provided in sine.h. The function does not perform a bit-twiddle - if required then ffttwiddle() should be called beforehand. re im sine real part of each input point imaginary part of each input point number of points. Must be a power of 2, both re and im should be long array of /4+1 sine values, each represented as a sign bit, and a 31 bit fraction. 1 should be represented as 0x7fffffff. Arrays are provided in sine.h; for example, for a 1024 point FFT use sin_1024.

6 Simple FFT function library 6/8 void ffttworealsforward(int re1[], int re2[], int im1[], int im2[], int, int sine[]) This function computes the FFT of two real sequences in one go. It uses a nifty trick () that enables one to use a single complex FFT to compute two real FFTs simultaneously. The real inputs should be in the first two real arrays, the output is in the real and imaginary arrays (the output of a real FFT is still a complex number). re1 re2 im1 im2 sine array of first set of real numbers on which to compute FFT, on output this array stores the real part of the complex FFT on this set of numbers. array of second set of real numbers on which to compute FFT, on output this array stores the real part of the complex FFT on this set of numbers. imaginary parts of complex FFT of first array imaginary parts of complex FFT of second array number of points array of /4+1 sine values, each represented as a sign bit, and a 31 bit fraction. 1 should be represented as 0x7fffffff. Arrays are provided in sine.h; for example, for a 1024 point FFT use sin_1024. void ffttworealsinverse(int re1[], int re2[], int im1[], int im2[], int, int sine[]) This function computes the inverse FFT on two sets of complex data that are known to result in real numbers only in one go. It uses a nifty trick () that enables one to use a single complex inverse FFT to compute two real inverse FFTs simultaneously. The outputs are in the two real arrays, the imaginary arrays are unchanged.

7 Simple FFT function library 7/8 re1 re2 im1 im2 sine real part of first set of complex numbers on which to compute inverse FFT real part of second set of complex numbers on which to compute inverse FFT imaginary part of first set of complex numbers on which to compute inverse FFT imaginary part of second set of complex numbers on which to compute inverse FFT number of points array of /4+1 sine values, each represented as a sign bit, and a 31 bit fraction. 1 should be represented as 0x7fffffff. Arrays are provided in sine.h; for example, for a 1024 point FFT use sin_ Programming Guide Below is an example calling sequence: nclude " fft.h" t main ( void ) { int re [8], im [8]; for ( int i = 0; i < 8; i ++) { // Fill re and im. } ffttwiddle (re, im, 8); fftforward (re, im, 8, sine_8 ); // Modify re and im, which are in the frequency domain ffttwiddle (re, im, 8); fftinverse (re, im, 8, sine_8 ); // and back to the time domain

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