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### Introduction

SYSC5603 (ELG6163) Digital Signal Processing Microprocessors, Software and Applications

Miodrag Bolic

Objective

- FFT Introduction
- Some FFT algorithms
- FFT on PDSP
- FFT floating to fixed-point conversion
- Hardware implementation of FFT

FFT for TMS320x67 with 2 buffers

Buffer (ping)

Destination address 1

count

Source

address

Serial

EDMA

Port

FFT

Buffer (pong)

Processing

Destination address 2

event (internal timer 1 is selected)

count

Switch address at the completion of a

transfer

FFT Fixed point - Xilinx

- Performing the calculations with no scaling and carrying computation
- The growth of the fractional bits created from the multiplication are truncated after the multiplication.
- The width of the output will be the (input width + number of stages + 1).
- For example, a 1024-pt transform with an input of 16 bits consisting of 1 integer bit and 15 fractional bits, will have an output of 27 bits with 12 integer bits and 15 fractional bits.
- Scaling at each stage using a fixed-scaling schedule
- Scaling automatically using block-floating point

[Xilinx05]

Block-floating point

- The computation is fixed-point
- After every addition there is an overflow test
- If the overflow is detected the array is divided by ½
- The number of division is counted to determine the scale factor
- SNR depends on how many overflows occurs

Butterfly with Scaling multipliers

[Oppenheim98]

Sequential FFT-Xilinx core

[Xilinx05]

Pipelined FFT-Xilinx core

[Xilinx05]

Pipelined FFT architecture

• Radix-2 multipath delay commutator (R2MDC)

• Radix-2 single-path delay feedback (R2SDC)

• Radix-4 multipath delay commutator (R4MDC)

• Radix-4 single-path delay commutator (R4SDC)

• Radix-4 single-path delay feedback (R4SDF)

• Radix-22 single-path delay commutator (R22SDC)

[Li03]

Radix-2 multipath delay commutator

- The total number of delay elements is 4 + 2 + 2 + 1 + 1 = 10 for the 8-point FFT.
- The utilization of the butterfly and the multiplier is 50%

[Li03]

Radix-2 single-path delay feedback

- The total number of delay elements is N – 1=N/2 + N/4 +... + 1

[Li03]

Requirements

- Requirement
- Transform length is 1024
- Transform time is less than 40 ms (continuously)
- Continuous I/O
- 25.6 Msamples/sec. throughput
- Complex 24 bits I/O data
- Steps in designing
- Architecture selection
- Partitioning
- Scheduling
- Word length selection
- RTL model generation
- Validation of models

[Li03]

Resource analysis

- Computation time for the 1024-point FFT
- The number of butterfly operations for Radix2
- Assume 1 clock cycle per Butterfly
- The minimum number of Butterflies is
- This is optimal with the assumption that ALL data are available to ALL stages, which is impossible for continuous data streams. Each butterfly has to be idle for 50% in order to reorder the incoming data.

[Li03]

Resource analysis

- The solution: the number of butterflies is 10
- The number of complex multipliers is 9
- Memory length for Radix-2 single-path delay feedback is N-1

[Li03]

RAM Based Commutator

- A dual-port memory is required since the read and write operation must be performed in one clock cycle.

[Li03]

Complex multiplier

[Li03]

Altera radix-4 butterfly

[Oppenheim98]

References

[Altera05] Altera, FFT MegaCore Function User Guide, DSP Literature, 2005.

[Li03] W Li, Studies on implementation of low power FFT processors, Thesis, Linköpings University, 2003

[Oppenheim98] A. V. Oppenheim, R. W. Schafer, Discrete-time signal processing, 2nd edition, Prentice Hall, 1998.

[Xlinx05] Xilinx, “Fast Fourier Transform v3.2”,DS260 August 31, 2005

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