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Fast Fourier Transforms

Quote of the Day

On two occasions I have been asked, Pray, Mr.

Babbage, if you put into the machine wrong figures,will the right answers come out? I am not ablerightly to apprehend the kind of confusion of ideas

that could provoke such a question.

Charles Babbage

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice HallInc.

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Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 2

Discrete Fourier Transform

The DFT pair was given as

Baseline for computational complexity:

Each DFT coefficient requires

N complex multiplications

N-1 complex additions

All N DFT coefficients require N2complex multiplications

N(N-1) complex additions

Complexity in terms of real operations 4N2real multiplications

2N(N-1) real additions

Most fast methods are based on symmetry properties

Conjugate symmetry

Periodicity in n and k

1N

0k

knN/2jekX

N

1]n[x

1N

0n

knN/2je]n[xkX

knN/2jnkN/2jkNN/2jnNkN/2j eeee nNkN/2jNnkN/2jknN/2j eee

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The Goertzel Algorithm

Makes use of the periodicity

Multiply DFT equation with this factor

Define

With this definition and using x[n]=0 for nN-1

X[k] can be viewed as the output of a filter to the input x[n]

Impulse response of filter:

X[k] is the output of the filter at time n=N

1ee k2jNkN/2j

1N

0r

nNrN/2j1N

0r

rnN/2jkNN/2j e]r[xe]r[xekX

r

rnkN/2jk rnue]r[xny

Nnk

nykX

nue knN/2j

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The Goertzel Filter

Goertzel Filter

Computational complexity

4N real multiplications

2N real additions

Slightly less efficient than the direct method

Multiply both numerator and denominator

1kN2

jk

ze1

1

zH

21

1k

N

2j

1k

N

2j

1k

N

2j

1k

N

2j

k

zzN

k2cos21

ze1

ze1ze1

ze1zH

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Second Order Goertzel Filter

Second order Goertzel Filter

Complexity for one DFT coefficient Poles: 2N real multiplications and 4N real additions

Zeros: Need to be implement only once 4 real multiplications and 4 real additions

Complexity for all DFT coefficients Each pole is used for two DFT coefficients

Approximately N2real multiplications and 2N2real additions

Do not need to evaluate all N DFT coefficients Goertzel Algorithm is more efficient than FFT if

less than M DFT coefficients are needed

M < log2N

21

1k

N

2j

k

zzN

k2cos21

ze1zH

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Decimation-In-Time FFT Algorithms

Makes use of both symmetry and periodicity

Consider special case of N an integer power of 2

Separate x[n] into two sequence of length N/2 Even indexed samples in the first sequence

Odd indexed samples in the other sequence

Substitute variables n=2r for n even and n=2r+1 for odd

G[k] and H[k] are the N/2-point DFTs of each subsequence

1N

oddn

knN/2j1N

evenn

knN/2j1N

0n

knN/2j e]n[xe]n[xe]n[xkX

kHWkG

W]1r2[xWW]r2[x

W]1r2[xW]r2[xkX

kN

12/N

0r

rk2/N

kN

12/N

0r

rk2/N

12/N

0r

k1r2N

12/N

0r

rk2N

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Decimation In Time

8-point DFT example usingdecimation-in-time

Two N/2-point DFTs

2(N/2)2complex multiplications

2(N/2)2complex additions

Combining the DFT outputs

N complex multiplications

N complex additions

Total complexity

N2/2+N complex multiplications

N2/2+N complex additions

More efficient than direct DFT

Repeat same process

Divide N/2-point DFTs into Two N/4-point DFTs

Combine outputs

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Decimation In Time Contd

After two steps of decimation in time

Repeat until were left with two-point DFTs

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Decimation-In-Time FFT Algorithm

Final flow graph for 8-point decimation in time

Complexity:

Nlog2N complex multiplications and additions

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Butterfly Computation

Flow graph constitutes of butterflies

We can implement each butterfly with one multiplication

Final complexity for decimation-in-time FFT

(N/2)log2N complex multiplications and additions

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In-Place Computation

Decimation-in-time flow graphs require two sets of registers

Input and output for each stage

Note the arrangement of the input indices Bit reversed indexing

111x111X7x7X011x110X3x6X

101x101X5x5X

001x100X1x4X

110x011X6x3X010x010X2x2X

100x001X4x1X

000x000X0x0X

00

00

00

00

00

00

00

00

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Decimation-In-Frequency FFT Algorithm

The DFT equation

Split the DFT equation into even and odd frequency indexes

Substitute variables to get

Similarly for odd-numbered frequencies

1N

0n

nkNW]n[xkX

1N

2/Nn

r2nN

12/N

0n

r2nN

1N

0n

r2nN W]n[xW]n[xW]n[xr2X

12/N

0n

nr2/N

12/N

0n

r22/NnN

12/N

0n

r2nN W]2/Nn[x]n[xW]2/Nn[xW]n[xr2X

12/N

0n

1r2n2/NW]2/Nn[x]n[x1r2X

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Decimation-In-Frequency FFT Algorithm

Final flow graph for 8-point decimation in frequency

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