computation of dfttwins.ee.nctu.edu.tw/courses/dsp_08/dsp9.pdf · nctu ee 1 computation of dft •...

DSP (2007) Computation of DFT

NCTU EE 1

Computation of DFT

• Efficient algorithms for computing DFT – Fast Fourier Transform. (a) Compute only a few points out of all N points (b) Compute all N points

• What are the efficiency criteria? Number of multiplications Number of additions Chip area in VLSI implementation

DFT as a Linear Transformation • Matrix representation of DFT

Definition of DFT:

1,,1,0,)(1)(

1,,1,0,)()(1

0

1

0

−==

−==

∑

∑−

=

−

−

=

NnWkXN

nx

NkWnxkXN

k

knN

N

n

knN

K

K

where

Let ,

)1(

)1()0(

,

)1(

)1()0(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=

NX

XX

Nx

xx

NNMM

Xx

and

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−−−

−

−

)1)(1()1(2)1(

)1(242

12

1

11

1111

NNN

NN

NN

NNNN

NNNN

N

WWW

WWWWWW

L

MLMMM

L

L

L

W

Thus,

NN

NNN

NNN

N

NN

XW

XWxxWX

*

1

1IDFTpoint -

point DFT-

=

==

−

Because the matrix (transformation) NW has a specific structure and because k

NW has par-

ticular values (for some k and n), we can reduce the number of arithmetic operations for computing this transform.


NCTU EE 2

Example 3] 2 1 0[][ =nx

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

jj

jj

WWWWWWWWW

WWWWWWWWWWWWWWWW

111111

111111

111

1111

14

24

34

24

04

24

34

24

14

94

64

34

04

64

44

24

04

34

24

14

04

04

04

04

04

4W

Only additions are needed to compute this specific transform. (This is a well-known radix-4 FFT)

Thus, the DFT of ][nx is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−+−

==

j

j

22222

6

444 xWX

Fast Fourier Transform -- Highly efficient algorithms for computing DFT • General principle: Divide-and-conquer • Specific properties of k

NW

Complex conjugate symmetry: *)( knN

knN WW =−

Symmetry: kN

NkN WW −=+ 2

Periodicity: kN

NkN WW =+

Particular values of k and n: e.g., radix-4 FFT (no multiplications) • Direct computation of DFT

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]∑

∑−

=

−

=

⎭⎬⎫

⎩⎨⎧

+⋅+⋅−⋅

=

−=⋅=

1

0

1

0

Re][ImIm][ReIm][ImRe][Re

1,,1,0 ,][][

N

nkn

Nkn

N

knN

knN

N

n

knN

WnxWnxjWnxWnx

NkWnxkX K

For each k, we need N complex multiplications and N-1 complex additions. 4N real multiplications and 4N-2 real additions.

We will show how to use the properties of kNW to reduce computations.

• Radix-2 algorithms: Decimation-in-time; Decimation-in-frequency • Composite N algorithms: Cooley-Tukey; Prime factor • Winograd algorithm • Chirp transform algorithm


NCTU EE 3

Radix-2 Decimation-in-time Algorithms -- Assume N-point DFT and ν2=N

Idea: N-point DFT 2

N -point DFT 4N -point DFT

4N -point DFT

2N -point DFT 4

N -point DFT

4

N -point DFT

Sequence: ]1[]2[]3[]2[]1[]0[ −Nxnxxxxx LL

Even index: ]2[]2[]0[ −Nxxx L Odd index: ]1[]3[]1[ −Nxxx L

2/2/

2222

12

0

12

0

)12(2

12

odd

2

even

1

0

]12[]2[

][][

1,,1,0 ,][][

NN

jN

j

N

N

r

N

r

krN

rkN

rn

n

knN

rn

n

knN

N

n

knN

WeeW

WrxWrx

WnxWnx

NkWnxkX

===

++=

+=

−==

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

−

=

−

=

+

+==

−

=

∑ ∑

∑∑∑

ππ

Q

4342143421

K

][][

]12[]2[][

point DFT-2

12

02/

point DFT-2

12

02/

kHWkG

WrxWWrxkX

kN

N

N

r

rkN

kN

N

N

r

rkN

+=

++= ∑∑−

=

−

= 444 3444 2144 344 21


NCTU EE 4

Comparison: (a) Direct computation of N-point DFT (N frequency samples): ~ 2N complex multiplications and 2N complex adds

(b) Direct computation of 2

N -point DFT:

~ 2

2 ⎟⎠⎞

⎜⎝⎛ N complex multiplications and 2

2 ⎟⎠⎞

⎜⎝⎛ N complex adds

+ additional N complex multis and N complex adds

~ (Total:) 22

222 NNNN +=⎟

⎠⎞

⎜⎝⎛+ complex multis and adds

(c) N2log -stage FFT

Since ν2=N , we can further break 2

N -point DFT into two 4

N -point DFT and

so on.

At each stage: ~ N complex multis and adds

Total: ~ NN 2log complex multis and adds (--> NN2log

2)


NCTU EE 5

Number of points, N

Direct Computation: Complex Multis

FFT: Complex Multis

Speed Im-provement Factor

4 16 4 4.08 64 12 5.3

16 256 32 864 4,096 192 21.3

256 65,536 1,024 64.01024 1,048,576 5,120 204.8

Butterfly: Basic unit in FFT

Two multiplications:

One multiplication:


NCTU EE 6

In-place computations Only two registers are needed for computing a butterfly unit.

][][][

][][][

11

11

qXWpXqX

qXWpXpX

mr

Nmm

mr

Nmm

−−

−−

−=

+=

Advantage: less storage! In order to retain the in-place computation property, the input data are accessed in the

bit-reversed order. Note: The outputs are in the normal order (same as the “position”)

Position Binary equivalent Bit reversed Sequence index

6 110 011 3 2 010 010 2

Remark: Index 3 input data is placed at position 6.

We may also place the inputs in the normal order; then the outputs are in the bit-reversed order.


NCTU EE 7

If we try to maintain the normal order of both inputs and outputs, then in-place compu-tation structure is destroyed.


NCTU EE 8

Radix-2 Decimation-in-frequency Algorithms Dividing the output sequence ][kX into smaller pieces.

1,,1,0,)()(1

0−== ∑

−

=NkWnxkX

N

n

knN K

If k is even, rk 2= .

∑

∑

∑ ∑

∑ ∑

∑

−

=

−

=

+

−

=

−

=

⎟⎠⎞

⎜⎝⎛ +

−

=

−

=

−

=

⋅⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ ++=

⋅⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ ++=

==

⋅⎥⎦⎤

⎢⎣⎡ ++=

+←+=

−==

12

02

12

0

2

22]2[2

12

0

12

0

22

2

12

0

1

2

22

1

0

2][

2][

2][

)2(][][

12

,,1,0 ,][]2[

N

n

nrN

N

n

nrN

rnN

rNN

rnN

NnrN

N

n

N

n

Nnr

Nnr

N

N

n

N

Nn

nrN

nrN

N

n

knN

WNnxnx

WNnxnx

WWWW

WNnxWnx

NnnWnxWnx

NrWnxrX

Q

L

Similarly, if k is odd, 12 += rk .

∑−

=⋅⋅⎟

⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ +−=+

12

022

][]12[

N

n

nrN

nN WWNnxnxrX

⎪⎪

⎩

⎪⎪

⎨

⎧

⋅⋅⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ +−=+

⋅⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ ++=

∑

∑−

=

−

=

12

02

12

02

2][]12[

2][]2[

N

n

nrN

nN

N

n

nrN

WWNnxnxrX

WNnxnxrX

Let

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡ +−=

⎥⎦⎤

⎢⎣⎡ ++=

2][][

2][][

Nnxnxnh

Nnxnxng


NCTU EE 9

We can further break ]2[ rX into even and odd groups … Again, we can reduce the two-multiplication butterfly into one multiplication. Hence, the

computational complexity is bout NN2log

2. The in-place computation property holds if the

outputs are in bit-reversed order (when inputs are in the normal order).


NCTU EE 10

FFT for Composite N -- Cooley-Tukey Algorithm: 21NNN =

⎪⎪⎩

⎪⎪⎨

⎧

⎩⎨⎧

−≤≤−≤≤

+=

⎩⎨⎧

−≤≤−≤≤

+=

1010

:index Freq.

1010

:index Time

22

11211

22

11212

NkNk

kNkk

NnNn

nnNn

Remark: ),( 21 nnn ↔ and ),( 21 kkk ↔

Goal: Decompose N-point DFT into two stages:

1N -point DFT⊗ 2N -point DFT

( )( )

44444444 344444444 21

3214444 34444 21

43421321321

point-

1

0factor

twiddlepoint-

1

0212

1

0

1

0 1

212

1

0

1

0212

211

1

0

2

2

2

222

21

1

1

1

111

2

2

1

1

1221

222

21121

111

112

2

2

1

1

212211

][

][

][

][

10 ,][][

N

N

n

nkN

nkN

N

N

n

nkN

N

n

N

n

nkNNN

W

nNkN

nkN

W

nkNN

N

n

N

n

nnNkNkN

N

n

knN

WWWnnNx

WWWWnnNx

WnnNx

kNkX

NkWnxkX

nkN

nkN

∑ ∑

∑ ∑

∑ ∑

∑

−

=

−

=

−

=

−

=

−

=

−

=

++

−

=

⋅

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⋅⎥⎦

⎤⎢⎣

⎡⋅+=

⋅⋅⋅⋅+=

⋅+=

+=

−≤≤=

Procedure

(1) Compute 1N -point DFT: (row transform)

∑−

=⋅+=

1

021212

1

1

111

][],[N

n

nkNWnnNxknG

(2) Multiply twiddle factors: ],[],[~

121221 knGWknG nk

N ⋅=

(3) Compute 2N -point DFT: (column transform)

∑−

=⋅=+

1

012211

2

2

222

],[~][N

n

nkNWknGkNkX


NCTU EE 11

(Computation of N=15-point DFT by means of 3-point and 5-point DFTs.)


NCTU EE 12

Extension: νNNNN L21=

21 Ifpoint DFT-for tionsmultiplica ofnumber )(Let

NNNNN

=≡μ

⎪⎩

⎪⎨

⎧

⋅=

⋅

)(: transfrmcolumn 3.:factors twiddle2.

)(:ormrow transf 1.

21

21

12

NNNNN

NN

μ

μ

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

+⋅+⋅=

1)()()()()(

2

2

1

1

2112

NN

NNN

NNNNNNμμ

μμμ

In general, ⎟⎟⎠

⎞⎜⎜⎝

⎛−+= ∑

=

ννμμ

1)1()()(

i i

i

NNNN

In fact, the term ( )1−ν should be ( )2

1−ν because rearranging the butterfly structure

would make half of the branches becoming “1”.

Special Case: 221 ==== νNNN L

Radix-2: 221 ==== νNNN L and N2log=ν

( )2

1)( −= νμ NN multiplications because )2(μ requires no multiplications.

Radix-4: 421 ==== νNNN L and N4log=ν

( )2

1)( −= νμ NN multiplications because )4(μ requires no multiplications. This FFT

has fewer stages than Radix-2 ==> fewer multiplications.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

jj

jj

WWWWWWWWW

WWWWWWWWWWWWWWWW

111111

111111

111

1111

14

24

34

24

04

24

34

24

14

94

64

34

04

64

44

24

04

34

24

14

04

04

04

04

04

4W


NCTU EE 13

Inverse FFT IDFT: ∑

−

=

−⋅=1

0][1][

N

k

knNWkX

Nnx (*)

DFT: ∑−

=⋅=

1

0][][

N

n

nkNWnxkX

Hence, take the conjugate of (*) :

( )

( )

[ ])( DFT1

][1

][1

][1][

*

1

0

*

1

0

*

*1

0

*

kXN

WkXN

WkXN

WkXN

nx

N

k

knN

N

k

knN

N

k

knN

=

⋅=

⋅=

⎟⎠⎞

⎜⎝⎛ ⋅=

∑

∑

∑

−

=

−

=

−

−

=

−

Take the conjugate of the above equation:

[ ]( )[ ]( )**

**

)(FFT 1

)(DFT 1][

kXN

kXN

nx

=

=

Thus, we can use the FFT algorithm to compute the inverse DFT.

computation of dfttwins.ee.nctu.edu.tw/courses/dsp_08/dsp9.pdf · nctu ee 1 computation of dft •...

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