AN2768, Implementation of a 128-Point FFT on the MRC6011 Device

1 Basics of Fast Fourier Transform ............................. 2 1.1 Decimation in Time and in Frequency ......................2 1.2 Radix-2 and Radix-4 ..................................................4 1.3 Bit-reversal .................................................................6 2 MRC6011 Architecture Overview ............................ 6 2.1 Frame Buffer ..............................................................8 2.2 RC Array ....................................................................9 3 FFT on the MRC6011 Device .................................. 9 3.1 Data Organization for Parallel Computing ................9 3.2 Input Data Storage in Frame Buffer ..........................9 3.3 Twiddle Factor Storage in Frame Buffer .................10 3.4 Transposition for Bit Reversal .................................11 3.5 Post-Transpose Data Organization ..........................15 3.6 Parallel Butterfly Operations ...................................15 4 Fixed Point and Precision Issues ............................ 22 4.1 Input Data Analysis .................................................23 4.2 Butterfly Output Scaling ..........................................24 4.3 Rounding Operation on RC .....................................26 5 Performance and Error Analysis ............................. 26 6 Summary ................................................................. 28 7 References ............................................................... 29 The Fast Fourier Transform (FFT) is an efficient way to compute the Discrete-time Fourier Transform (DFT) by exploiting symmetry and periodicity in the DFT. Because it is so efficient, the algorithm is implemented on many DSPs and hardware platforms for real-time applications. FFT applications include not only DSP but also spectrum analysis, speech processing, and filter designs where filter coefficients are determined according to the frequency response of the filter. The frequency response of a filter can be obtained by taking the Discrete Fourier Transform of its impulse response. Conversely, given the frequency response samples in the frequency domain, the time domain impulse response can be computed by taking the inverse DFT. For any discrete time sequences, the frequency components or spectral components can be obtained by taking the FFT on the discrete time sequences.