Chapter 9

An Ordered Radix-2 DIT FFT

This chapter presents the sixth variant of the FFT algorithm, which applies decimation-in-time FFT to transform naturally ordered input time series to naturally orderedoutput frequencies. As expected, this ordered DITNN FFT corresponds to the orderedDIFNN FFT presented in Chapter 6. Readers of this chapter are thus assumed tobe familiar with the concept of combining permutation with buttery computation inthe ordered DIF FFT, which allows the implementation of repeated permutation ofintermediate results without extra accesses to memory. Since it would be helpful tolook at the same from a dierent angle, this chapter begins with the following genericdescription of the DIT FFT algorithm, which is correct regardless of where the inputelement xm, m = 0, 1, . . . , N 1, and its derivatives are stored in array a.

Algorithm 9.1 The radix-2 DIT FFT algorithm in terms of binary subscripts.

begink := n 1 Assume problem size N = 2nwhile k 0 do Combine two subproblems

Apply Cooley-Tukey buttery computation

to combine all pairs of elements of x whose

binary subscripts dier in bit ikk := k 1

end whileend

Accordingly, the two in-place DIT FFT algorithms presented in Chapters 7 and 8and the ordered DITNN FFT algorithm presented in this chapter must all implementexactly the same computation, and they each can be understood by relating the loca-tions where xm and its derivatives are stored in the input and the working arrays asshown in Table 9.1 for the familiar N = 8 example.

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Recall that x(3)i2i1i0 = Xi0i1i2 . Therefore, the result b[i0i1i2] = x(3)i2i1i0

from theDITNN algorithm ensures that the output is in natural order. Based on Table 9.1, theordered DITNN FFT can immediately be described in full for the N = 8 example inTable 9.2, which can be compared with Table 7.2 for the DITNR FFT and Table 8.2for the DITRN FFT.

The eect of combining buttery computation with permutation from array a to b,or vice versa, is reected by the shorthand notation

i2i1i0i2i1i0

i12i0

i012

a b a b.

Observe that the twiddle factors remain the same for all three DIT FFT algorithms,

i.e., they are

0N = 1, i20N , i1i2N .

Table 9.1 Relating the NR, RN, and NN variants of the DIT FFT.

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Table 0.2 The NN DIT FFT applied to a N = 8 example. Cooley-Tukey

)utterfly Combine

with Permutation

Stage 1: i z i l io t

t Stage 2: i 1 7 2 i o

1 Stage 3: i 0 7 1 7 2

Actual Modification &

Permutation from a to b

or Vice Versa

b[O] = a[O] + wO,a[4]

b[ l ] = u[1] + w5a[5] b[4] = u[O] - w ; u [ ~ ]

b[5] = ~ [ l ] - w;a[5] b[2] = u[2] + w;a[6] b[6] = a[2] - w;a[6] b[3] = u[3] + w$a[7] b[7] = a[3] - w;a[7] u[O] = b[O] + wLb[2] a[4] = b[O] - &b[2] u[1] = b[ l ] + w;b[3] ~ [ 5 ] = b[ l ] - &b[3] a[2] = b[4] + w%b[6] ~ [ 6 ] = b[4] - w;b[6] a[3] = b[5] + wgb[7] a[7] = b[5] - w$b[7] b[O] = a[O] + w;a[l]

b [ l ] = 421 + w:a[3]

b[2] = a[4] + w:u[5]

b[4] = a[O] - w;o[l]

b[5] = a[2] - w t a [ 3 ]

b[6] = ~ [ 4 ] - w ; u [ ~ ] b[3] = u[6] + wLo[7] b[7] = a[6] - wLa[7]

Butted]

Groups

Group 0

( 4 Pairs)

Group 0

( 2 Pair4

k Binary Address Based Notation

to Show the Combined Effect

a[Oiaio] = b[iZOiO] + w$Ob[iz l i o ] o[lizio] = b[i20io] - w y b [ i 2 1 i o ]

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9.1 Deriving the (Ordered) DITNN FFT From Its

Recursive Denition

For completeness, the derivation of the ordered DIT FFT from its recursive denitionis also depicted for the N = 8 example in Figures 9.19.4.

Figure 9.1 The rst division step of the (ordered) DITNN FFT.

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Figure 9.2. Figure 9.3.

Figure 9.4.

Figure 9.2 Solve one half-size subproblem by (ordered) DITNN FFT.

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Figure 9.3 Solve one half-size subproblem by (ordered) DITNN FFT.

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Figure 9.4 Solve the entire problem by (ordered) DITNN FFT.

9.2 The Pseudo-code Program for the DITNN FFT

In contrast to the DIFNN FFT pseudo-code program given by Algorithm 6.1, the DITNNversion is given below as Algorithm 9.2. Note that the twiddle factors are assumed tobe stored in natural order, i.e., w[] = N , = 0, 1, . . . , N/2 1.

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Figure 9.2. Figure 9.3.

Algorithm 9.2 The (ordered) radix-2 DITNN FFT algorithm.

beginPairsInGroup := N/2 Initially: N/2 pairs in one groupNumOfGroups := 1Distance := N/2NotSwitchInput := truewhile NumOfGroups < N do Combine pairs in each group

if NotSwitchInput Array a contains input; array b contains outputL := 0for K := 0 to NumOfGroups 1 do

JFirst := 2 K PairsInGroupJLast := JFirst + PairsInGroup 1Jtwiddle := K PairsInGroup Assume w[] = NW := w[Jtwiddle] Same twiddle factor in each groupfor J := JFirst to JLast do

Temp := W a[J + Distance]b[L] := a[J ] + Tempb[L+N/2] := a[J ] TempL := L+ 1

end forend forNotSwitchInput := false

else Array b contains input; array a contains outputL := 0for K := 0 to NumOfGroups 1 do

JFirst := 2 K PairsInGroupJLast := JFirst + PairsInGroup 1Jtwiddle := K PairsInGroup Assume w[] = NW := w[Jtwiddle] Same twiddle factor in each groupfor J := JFirst to JLast do

Temp := W a[J + Distance]a[L] := b[J ] + Tempa[L+N/2] := b[J ] TempL := L+ 1

end forend forNotSwitchInput := true

end ifPairsInGroup := PairsInGroup/2NumOfGroups := NumOfGroups 2Distance := Distance/2

end whileend

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9.3 Applying the (Ordered) DITNN FFT to a

N = 32 Example

Generalizing the shorthand notation for N = 32, the following sequence represents allve stages of permutation and computation depicted in Figure 9.5.

i4i3i2i1i0i4i3i2i1i0

i34i2i1i0

i234i1i0

i1234i0

i01234

a b a b a b.

The corresponding twiddle factors are

0N = 1, i4000N , i3i400N , i2i3i40N , i1i2i3i4N ;and w[i1i2i3i4] = i1i2i3i4N is assumed in Algorithm 9.2.

By comparing Figure 9.6, where the butteries associated with a particular pairof resulting subproblems are shown without the cluttering of others, with the twounordered DIT FFT in Figures 7.6 and 8.5, one once again observes that

all three DIT algorithms treat exactly the same pairs of subproblems duringeach stage of the computation.

Thus, they all implement the same radix-2 DIT FFT algorithm.

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Figure 9.5 Butteries of the (ordered) DITNN FFT algorithm.

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Figure 9.6 Identifying the subproblem pairs in the (ordered) DITNN FFT.

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INSIDE the FFT BLACK BOX: Serial and Parallel Fast Fourier Transform AlgorithmsTable of ContentsPart II: Sequential FFT AlgorithmsChapter 9: An Ordered Radix-2 DIT FFT9.1 Deriving the (Ordered)DIT NN FFT From Its Recursive Definition9.2 The Pseudo-code Program for the DIT NN FFT9.3 Applying the (Ordered)DIT NN FFT to a N = 32 Example