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Copyright 2003 Texas Instruments. All rights reserved.

DSP C5000

Chapter 19

Fast Fourier Transform

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Copyright 2003 Texas Instruments. All rights reserved.ESIEE, Slide 2

Discrete Fourier Transform

Allows us to compute an approximation of

the Fourier Transform on a discrete set offrequencies from a discrete set of timesamples.

Where k are the index of the discretefrequencies and n the index of the time samples

dtetxfX

f tj

2

)()(

1,,1,0for

21

0

NkenxkX

nN

kjN

n

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DFT Computation

We can write the DFT:

We need:

N(N-1) complex +

N2complex

NjN

n

eNknxkX

2

N

kn-

N

1

0

Wwith1,,1,0forW

N + x128 16256 16384

1024 1047552 1048576

4096 16773120 16777216

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Fast Fourier Transform 1 of 3

Cooley-Tukey algorithm:

Based on decimation, leads to a factorization ofcomputations.

Let us first look at the classical radix 2decimation in time.

This particular case of the algorithm requiresthe time sequence length to be a power of 2.

First we split the computation between odd andeven samples:

12nk-

N

12/

0

k2n-

N

12/

0

W12W2

N

n

N

n

nxnxkX

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Butterfly This leads to basic building block of the

FFT, the butterfly.

TFD N/2

TFD N/2

x(0)

x(2)

x(N-2)

x(1)

x(3)

x(N-1)

X(0)

X(1)

X(N/2-1)

X(N/2)

X(N/2+1)

X(N-1)

W0

W1

WN/2-1

-

-

-

We need:

N/2(N/2-1) complex + for eachN/2 DFT.

(N/2)2 complex for eachDFT.

N/2 complex at the input ofthe butterflies.

N complex + for the butter-flies.

Grand total:

N2/2 complex +

N/2(N/2+1) complex

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9/32Copyright 2003 Texas Instruments. All rights reserved.ESIEE, Slide 9

Recursion If N/2 is even, we can further split the computation of

each DFT of size N/2into two computations of half sizeDFT. When N=2rthis can be done until DFT of size 2 (i.e.

butterfly with two elements).

x(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

W80

W81

W82

W83

-

-

-

--

-

-

-

-

-

-

-

W80

W80

W81

W81

W80

W80

W80

W80

W80=1

1ststage2ndstage3rdstage

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Number of Operations

If N=2r, we have r=log2(N) stages. For each

one we have: N/2complex (some of them are by 1).

Ncomplex +.

Thus the grand total of operations is:

N/2log2(N)complex .

Nlog2(N)complex +.

N + x

128 896 448

1024 10240 5120

4096 49152 24576

These counts can be compared with the ones for the DFT

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Shuffling the Data, Bit Reverse Ordering

At each step of the algorithm, data are split

between even and odd values. This results inscrambling the order.

x(0)

x(1)

x(2)x(3)

x(4)

x(5)

x(6)

x(7)

x(0)

x(2)

x(4)x(6)

x(1)

x(3)

x(5)

x(7)

x(0)

x(4)

x(2)x(6)

x(1)

x(5)

x(3)

x(7)

Recursion of the algorithm

index addres s

000 000

100 001

010 010

110 011

001 100

101 101

011 110

111 111

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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 0then we add N/2 toobtain the next address. If there is a carry, itpropagates towards the least significant bit.

When the data arrive in natural order, they are

scrambled in this way.

n Address[x(n+1)] Address[x(n)]

000 RCP(000+100)=100 000

001 RCP(100+100)=010 100

010 RCP(010+100)=110 010

011 RCP(110+100)=001 110

100 RCP(001+100)=101 001

101 RCP(101+100)=011 101

110 RCP(011+100)=111 011

111 111

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Algorithm Parameters 1 of 2

The FFT can be computed according to thefollowing pseudo-code:

For each stage

For each group of butterfly For each butterfly

compute butter f ly

end

end

end

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Algorithm Parameters 2/2

The parameters are shown below:

1st stage 2nd stage 3rd stage Last stage

Node

Spacing 1 2 3 N/2

Butterfliesper group

1 2 3 N/2

Number of

groups N/2 N/4 N/8 1

Twiddle

factor

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Scaling

The DFT computation for each k is :

To prevent overflow we need to have:

This is guaranteed provided a scale factor 1/N

kn-

N

1

0

W

N

n

nxkX

1kX

1W1

kn-

N

1

0

N

n

nxN

kX

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Quantization Noise

Quantization step is :

If words have b+1bits and x(n)belongs to[-1,1]

If we assume that each real multiplicationgives rise to a noise source of power

The total amount of noise power for eachX(k) is given by

b

2

2

e

NrN

rb

et 2

2/222

logwith3

24

Si Q i i i i (SQ )

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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:

2

2

3

1

Nx

NX

3

12

rbdBSQNRt

X 66log10)(2

2

N SQNR(dB) ENOB

128 48 81024 30 5

4096 18 3

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.

16 bits per word

Si l Q i i N i R i (SQNR)

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Signal to Quantization Noise Ratio (SQNR)FFT case 1 of 3

The computation of one X(k) requires N-1 butterflies:

x(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

W80

W81

W82

W83

-

-

--

-

-

-

-

-

W80

W80

W81

W81

W80

W80

W80

W80

N/2 butterfliesof the 1ststage

N/4 butterfliesof the 2ndstage

1 butterfly ofthe last stage

Si l Q i i N i R i (SQNR)

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Butterfly computation

To prevent overflow, we only need to scale theinputs of each butterfly by 1/2

Xn(l)

-W

Xn(k)

Xn+1(l)

Xn+1(k)kXWlXkX

kXWlXlX

nnn

nnn

1

1

Xn

(l)

-W

Xn(k)

Xn+1

(l)

Xn+1(k)

1/2

1/2

Because we have rstages, the global

scaling factor for one output is:

N

r

1

2

1

Si l t Q ti ti N i R ti (SQNR)

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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:

3

22

2

112

3

222

2brb

t

336log10)(2

2

rbdBSQNRt

X

N SQNR(dB) ENOB

128 66 111024 57 9,5

4096 51 8,5

16 bits per word

FFT Al ith ith Bl k Fl ti P i t

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FFT Algorithm with Block Floating PointScaling

Input data in bit reverse order

Set-up for next stage

Set-up for next group

Set-up for next butterfly

Scale inputs of butterfly (1/2)

More butterflies ?

More groups ?

More stages ?

End of algorithm

Compute butterfly

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Case Study C54x

Use of audioFFT*

(*) this program is the same as \ti\examples\dsk5416\bios\audio except for some slight

modifications that will be emphazised when necessary

Block

processing

PCM3002ADC

PCM3002DAC

pipRx

pipTx

inBuffer

outBuffer

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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 twoseparate buffers.

Each buffer is processed

Bit-reverse scrambling

Forward transform

Bit reverse scrambling

Inverse transform

Resulting left and right buffers are interleaved in theoutput stream.

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DSPLIB functions

The DSP Library (DSPLIB) is a collection of

high-level optimized DSP function modules forthe C54x and C55x DSP platform.

C54x DSPLIB functions for FFT computation

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Block Processing 1 of 3

Echo( )

Include and declarations

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Block Processing 2 of 3

Dsplib functions calls

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C54x DSPLIB Bit Reversal

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C54x DSPLIB FFT

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To Run the Build Process

Project options

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To Change the Size of the Processing Buffer

Change buffer length declaration in echofunction (audio.c)Change the call to cfft and cifftaccording to the size of the FFT (must be hard coded)

Change buffer alignment in linker .cmd file

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Follow on Activities for TMS320C5416 DSK

Application 8 for the the TMS320C5416 DSK

uses the FFT as a spectrum analyzer todisplay the power in an audio signal atvarious frequencies.

Rather than using the optimized library

DSPLIB for the FFT, it uses a C code versionthat is slower, but can be stepped through lineby line using Code Composer Studio.

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