1. Chapter 19 Fast Fourier Transform
DSP C5000
Chapter 19
Fast Fourier Transform

2. Discrete Fourier Transform
Allows us to compute an approximation of the Fourier Transform on a discrete set of frequencies from a discrete set of time samples.
Where k are the index of the discrete frequencies and n the index of the time samples

3. Inverse Discrete Fourier Transform
The inverse formula is:
Where, again, k are the index of the discrete frequencies and n the index of the time samples.
We have the following properties:
Discrete time periodic spectra
Periodic time discrete spectra

4. DFT Computation We can write the DFT: We need: N(N-1) complex ‘+’
Each complex addition requires two real additions.
Each complex multiplication requires 4 real multiplications and 3 real additions.

5. Fast Fourier Transform 1 of 3
Cooley-Tukey algorithm:
Based on decimation, leads to a factorization of computations.
Let us first look at the classical radix 2 decimation in time.
This particular case of the algorithm requires the time sequence length to be a power of 2.
First we split the computation between odd and even samples:

6. Fast Fourier Transform 2 of 3
Using the following property:
The DFT can be rewritten:
For k=0, 1, …, N-1

7. Fast Fourier Transform 3 of 3
Using the property that:
The entire DFT can be computed with only k=0, 1, …,N/2-1.
and

8. Butterfly
This leads to basic building block of the FFT, the butterfly.
x(0)
TFD N/2
X(0)
We need:
N/2(N/2-1) complex ‘+’ for each N/2 DFT.
(N/2)2 complex ‘×’ for each DFT.
N/2 complex ‘×’ at the input of the butterflies.
N complex ‘+’ for the butter-flies.
Grand total:
N2/2 complex ‘+’
N/2(N/2+1) complex ‘×’
x(2)
X(1)
x(N-2)
X(N/2-1)
x(1) W0
TFD N/2
X(N/2) -
x(3) W1
X(N/2+1) -
x(N-1)
WN/2-1
X(N-1) -

9. Recursion
If N/2 is even, we can further split the computation of each DFT of size N/2 into two computations of half size DFT. When N=2r this can be done until DFT of size 2 (i.e. butterfly with two elements).
3rd stage
2nd stage
1st stage
x(0)
X(0)
W80
x(4)
X(1) -
W80
x(2)
X(2) -
W80
W81
x(6)
X(3) - -
W80
x(1)
X(4) -
W80=1
W80
W81
x(5)
X(5) - -
W80
W82
x(3)
X(6) - -
W80
W81
W83
x(7)
X(7) - - -

10. Number of Operations
If N=2r, we have r=log2(N) stages. For each one we have:
N/2 complex ‘×’ (some of them are by ‘1’).
N complex ‘+’.
Thus the grand total of operations is:
N/2 log2(N) complex ‘×’.
N log2(N) complex ‘+’.
These counts can be compared with the ones for the DFT

11. Shuffling the Data, Bit Reverse Ordering
At each step of the algorithm, data are split between even and odd values. This results in scrambling the order.
Recursion of the algorithm

12. Reverse Carry Propagation
This scrambling when we use radix 2 FFT can be obtained by the Reverse Carry Propagation (RCP) algorithm.
We start with address 0 then we add N/2 to obtain the next address. If there is a carry, it propagates towards the least significant bit.
When the data arrive in natural order, they are scrambled in this way.

13. Algorithm Parameters 1 of 2
The FFT can be computed according to the following pseudo-code:
For each stage
For each group of butterfly
For each butterfly
compute butterfly
end

14. Algorithm Parameters 2/2
The parameters are shown below:

15. Scaling The DFT computation for each k is :
To prevent overflow we need to have:
This is guaranteed provided a scale factor 1/N

16. Quantization Noise Quantization step is :
If words have b+1 bits and x(n) belongs to [-1,1]
If we assume that each real multiplication gives rise to a noise source of power
The total amount of noise power for each X(k) is given by

17. Signal to Quantization Noise Ratio (SQNR) DFT case
If we assume x(n) uniform in [-1,1], after scaling, variance of data become:
And because each X(k) comes from N summations:
SQNR for DFT with scaling is given by:
16 bits per word
ENOB: effective number of bits, gives the effective resolution given a SNR. Based on the assumption that 6dB of SNR equates to 1 bit of precision.

18. Signal to Quantization Noise Ratio (SQNR) FFT case 1 of 3
The computation of one X(k) requires N-1 butterflies:
N/2 butterflies of the 1st stage
N/4 butterflies of the 2nd stage
1 butterfly of the last stage
x(0)
X(0)
W80
x(4)
X(1) -
W80
x(2)
X(2) -
W80
W81
x(6)
X(3) - -
W80
x(1)
X(4) -
W80
W81
x(5)
X(5) - -
W80
W82
x(3)
X(6) - -
W80
W81
W83
x(7)
X(7)

19. Signal to Quantization Noise Ratio (SQNR) FFT case 2 of 3
Butterfly computation
To prevent overflow, we only need to scale the inputs of each butterfly by 1/2
Xn(l)
Xn+1(l) W
Xn(k)
Xn+1(k) -
1/2
Because we have r stages, the global scaling factor for one output is:
Xn(l)
Xn+1(l)
1/2 W
Xn(k)
Xn+1(k) -

20. Signal to Quantization Noise Ratio (SQNR) FFT case 3 of 3
Quantization noise source at one stage is attenuated by scale factors of all the following stages.
This gives an equivalent noise source of:
for each output.
SQNR for FFT with scaling is given by:
16 bits per word

21. FFT Algorithm with Block Floating Point Scaling
Input data in bit reverse order
Set-up for next stage
More groups ?
Set-up for next group
More stages ?
Set-up for next butterfly
Scale inputs of butterfly (×1/2)
End of algorithm
Compute butterfly
More butterflies ?

22. Case Study ‘C54x Use of audioFFT* inBuffer PCM3002 ADC pipRx Block
processing
pipTx
PCM3002 DAC
outBuffer
(*) this program is the same as \ti\examples\dsk5416\bios\audio except for some slight modifications that will be emphazised when necessary

23. Audio Program Block processing is echo() function in audio.c :
In the original program: Data from input stream are copied directly in output stream.
In the modified program :
The input stream is split between left and right in two separate buffers.
Each buffer is processed
Bit-reverse scrambling
Forward transform
Bit reverse scrambling
Inverse transform
Resulting left and right buffers are interleaved in the output stream.

24. DSPLIB functions
The DSP Library (DSPLIB) is a collection of high-level optimized DSP function modules for the ‘C54x and ‘C55x DSP platform.
‘C54x DSPLIB functions for FFT computation

25. Block Processing 1 of 3
Echo( )
Include and declarations

26. Block Processing 2 of 3
Dsplib functions calls

27. ‘C54x DSPLIB Bit Reversal

28. Block Processing 3 of 3
Linker .cmd file

29. ‘C54x DSPLIB FFT

30. To Run the Build Process
Project options

31. To Change the Size of the Processing Buffer
Change buffer length declaration in echo function (audio.c)
Change the call to cfft and cifft according to the size of the FFT (must be hard coded)
Change buffer alignment in linker .cmd file

32. Follow on Activities for TMS320C5416 DSK
Application 8 for the the TMS320C5416 DSK uses the FFT as a spectrum analyzer to display the power in an audio signal at various frequencies.
Rather than using the optimized library DSPLIB for the FFT, it uses a C code version that is slower, but can be stepped through line by line using Code Composer Studio.