Note Set #26• FFT Algorithm: Divide & Conquer Viewpoint• Reading: Sect. 8.1.2 & 8.1.3 of Proakis & Manolakis

EEO 401 Digital Signal Processing

Prof. Mark Fowler

2/15

The previous note set’s FFT development was somewhat ad hoc.Here we develop a more formalized & generalized approach that can be used to develop other FFT approaches.

To illustrate the basic ideas consider the case of an N-pt DFT where N is not prime can be factored into two integer factors: N LM=

Can always zero-pad out to appropriate number

We now can use either of two mappings from 2-D indices l,m to the actual time index n:

n Ml m= + n l mL= +“Row-Wise Input Mapping” “Column-Wise Input Mapping”

We use one or the other of these mappings to convert the input to a matrix and then take DFTs along either the rows or columns.

We now can use either of two mappings from 2-D indices p,q to the actual DFT result index k: k Mp q= + k p qL= +

“Row-Wise Output Mapping” “Column-Wise Output Mapping”

Divide & Conquer Approach

3/15

n Ml m= +

n l mL= +

“Row-Wise Input Mapping”

“Column-Wise Input Mapping”

DFTs DownColumns

DFTs AcrossRows

4/15

Now to illustrate how to use this machinery… Use• Column-wise input mapping• Row-wise output mapping

1

0

[ ] [ ]N

knN

nX k x n W

−

=

= ∑n l mL= +k Mp q= +1

( )( )

0 0

[ , ] [ , ]M L

Mp q l mLN

m lX p q x l m W

−+ +

= =

= ∑∑

1 mq plNmp N L N MN

MLmp mLq Mpl lqN N N N

W WW

W W W W== =

=

1

0 0

[ , ]

[ , ]

[ , ]

[ , ] [ , ]L M

lq mq plN N L N M

l m

F l q

G l q

X p q

X p q W x l m W W−

= =

=

=

∑ ∑

Compute M-pt DFTs of Rows

Apply Twiddle Factors

Compute L-pt DFTs of Colns

5/15

Illustration of this Approach

Although this LOOKS more complicated… it is actually more efficient!

Compute M-ptDFTs of Rows

Apply Twiddle Factors

Compute L-ptDFTs of Colns

6/15

Application to Develop Dec-In-Time Radix-2 FFT Let N = 2vWe apply the divide-and-conquer approach with M = N/2 & L = 2

n l mL= + “Column-Wise Input Mapping”

DFTs AcrossRows

Compute N/2-pt DFTs of Rows

Apply Twiddle Factors

Compute 2-pt DFTs of Colns

Now repeat this for each N/2-pt DFT… Etc., Etc., Etc.

x[0] x[2] x[4] x[ N – 2]

x[1] x[3] x[5] x[ N – 1]

7/15

Application to Develop Dec-In-Frequency Radix-2 FFT N = 2vWe apply the divide-and-conquer approach with M = 2 & L = N/2

n l mL= + “Column-Wise Input Mapping”

DFTs Across Rows

x[0]

x[1]

x[N/2 - 1]

x[N/2]x[N/2+1]

x[N–1]

g1[n]

g2[n]

kNW

X[0]X[2]

X[N–2]

X[1]X[3]x[0]

x[1]

x[N/2 – 1]

x[2]

x[N/2]

x[N/2 + 1]

x[N/2 + 2]

x[N – 1]

2-pt DFT

N/2-pt DFT

Compute N/2-pt DFTs of Columns

Apply Twiddle Factors

Compute 2-pt DFTs of Rows

Now repeat this for each

N/2-pt DFT… Etc., Etc., Etc.

8/15

N = 8 First Stage of Dec-in-Freq FFT

9/15

Complete Dec-in-Freq FFT for N = 8 DFT Values are in Bit Reversed Order!

10/15

Butterfly Structure: DiT vs DiF

Butterfly Structure: Dec-in-Time

Butterfly Structure: Dec-in-Freq

11/15

N=8 D-in-Time FFT w/ BR InputsCan be done “in-place”

N=8 D-in-Time FFT w/ BR OutputsCan be done “in-place”

N=8 D-in-Time FFT w/ both sides “Normal Order”Can NOT be done “in-place”

Diagrams from R. Lyon, Understanding Digital Signal Processing, 3rd Ed., Prentice-Hall, 2011

3 Different Configurations of D-in-Time FFT

Writes over Data Needed Later!!

12/15

N=8 D-in-Freq FFT w/ BR InputsCan be done “in-place”

N=8 D-in-Freq FFT w/ BR OutputsCan be done “in-place”

N=8 D-in-Freq FFT w/ both sides “Normal Order”Can NOT be done “in-place”

Diagrams from R. Lyon, Understanding Digital Signal Processing, 3rd Ed., Prentice-Hall, 2011

3 Different Configurations of D-in-Freq FFT

Writes over Data Needed Later!!

13/15

Implementation Issues

• We’ve looked at two radix-two methods.- Other radices: 4 & 8- split radix (2 and 4)

• In-Place computation requires only 2N memory locations- But complicates the indexing & control operations- Doubling the memory to 4N locations can be advantageous

o Reduces complexity of indexing & controlo Allows natural ordering for both input and output

• In general, many factors come into play when determining best method- Parallelism, HW vs SW, fixed-point vs floating-point, etc.

• Also… no need to develop distinct IFFT algorithm

*1 12 / * 2 /

0 0

1 1[ ] [ ] [ ]N N

j kn N j kn N

n nx n X k e X k e

N Nπ π

− −−

= =

= = ∑ ∑ { } { }{ }*

1[ ] [ ]IDFT X k conj DFT X kN

=

14/15

Two Tricks for Real-Valued Signals

1. Efficient DFT of two Real-Valued Signals

Let x1[n] and x2[n] be real-valued signals, each length N

1 2[ ] [ ] [ ]x n x n jx n+

Then we have* *

1 2[ ] [ ] [ ] [ ][ ] & [ ]

2 2x n x n x n x nx n x n

j+ −

= =

Form:

Thus{ } { } { } { }* *

1 2

[ ] [ ] [ ] [ ][ ] & [ ]

2 2DFT x n DFT x n DFT x n DFT x n

X k X kj

+ −= =

But… DFT{ x*[n]} = X*[N – k]

* *1 2

1 1[ ] [ ] [ ] & [ ] [ ] [ ]2 2

X k X k X N k X k X k X N kj

= + − = − −

15/15

2. Efficient DFT of 2N-pt Real-Valued Signal

1 2[ ] [2 ] & [ ] [2 1]x n g n x n g n +

Let g[n] be a real-valued signal of length 2N

Then define:

And: 1 2[ ] [ ] j [ ]x n x n x n= +

From Trick #1 we have

* *1 2

1 1[ ] [ ] [ ] & [ ] [ ] [ ]2 2

X k X k X N k X k X k X N kj

= + − = − −

But using ideas from Dec-in-Time FFT we know1 1

2 (2 1)2 2

0 01 1

2 (2 1)1 2 2 2

0 0

[ ] [2 ] [2 1]

[ ] [ ]

N Nnk n k

N Nn nN N

nk n kN N

n n

G k g n W g n W

x n W x n W

− −+

= =

− −+

= =

= + +

= +

∑ ∑

∑ ∑

So then we get1 2 2

1 2 2

[ ] [ ] [ ], 0,1, , 1

[ ] [ ] [ ], 0,1, , 1

kN

kN

G k X k W X k k NG k N X k W X k k N

= + = −

+ = − = −

So computing one N-pt DFT of x[n] gets us the 2N-pt DFT of g[n]!

EEO 401 �Digital Signal Processing�Prof. Mark FowlerSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15