Algorithms of Scientiﬁc ComputingFast Real DFT Computation of a real-valued DFT using complex FFT...

Algorithms of Scientific Computing

FFT on Real Data

Michael BaderTechnical University of Munich

Summer 2016

DFT and Symmetry – Outlook

INPUT TRANSFORM

real symmetry fn ∈ R → Real DFT (RDFT)

even symmetry fn = f−n → Discrete Cosine Transform (DCT)

odd symmetry fn = −f−n → Discrete Sine Transform (DST)

“QUARTER-WAVE” INPUT TRANSFORM

even symmetry fn = f−n−1 → QW-DCT

odd symmetry fn = −f−n−1 → QW-DST

Michael Bader | Algorithms of Scientific Computing | FFT on Real Data | Summer 2016 2

Real-valued DFT (RDFT)Consider real-valued input data fn ∈ R, i.e.: f ∗n := fn = fn, then:

Fk =1N

N2∑

n=− N2 +1

fne−i2πnk/N =1N

N2∑

n=− N2 +1

fn

(cos

(2πnk

N

)− i sin

(2πnk

N

)).

Properties:

• Re {Fk} = 1N

N2∑

n=− N2 +1

fn cos( 2πnk

N

), Im {Fk} = − 1

N

N2∑

n=− N2 +1

fn sin( 2πnk

N

)• only N independent, real coefficients necessary, since:

F ∗k =1N

∑f ∗n{ω−nk

N

}∗=

1N

∑fnω−n(−k)N = F−k

Recall:{ω−nk

N

}∗={

e−i2πnk/N}∗ = cos(−2πnk/N)− i sin(−2πnk/N)

= cos(2πnk/N) + i sin(2πnk/N) = · · · = ωnkN


Real DFT (2)

Map N values to N coefficients (and vice versa):{f− N

2 +1, . . . , f0, . . . , f N2

}DFT ⇓ ⇑ IDFT

{F0,Re{F1}, Im{F1}, . . . , Re{F N

2−1}, Im{F N2−1},F N

2

}Note: real and imaginary parts of F−k correspond to those of Fk


Real DFT (3)

Situation:• only N real input values (as all N imaginary parts are 0)• only N independent, real output values (coefficient components)

due to symmetry F−k = F ∗k

Wanted→ new transformation:

N real input values→ N distinct real coefficient components

Hence: Insert symmetry F−k = F ∗k in IDFT!


Real DFT (4)Definition of “Real discrete Fourier transform” (RDFT)

• formulation 1

Fk =1N

N2∑

n=− N2 +1

fn

(cos

(2πnk

N

)− i sin

(2πnk

N

)), k = 0, . . . ,

N2

• formulation 2

Re{Fk} =1N

N2∑

n=− N2 +1

fn cos(

2πnkN

), k = 0, . . . ,

N2

Im{Fk} = − 1N

N2∑

n=− N2 +1

fn sin(

2πnkN

), k = 1, . . . ,

N2− 1


Inverse Real DFTGoal: compute a real representation of the DFT (i.e., no e−i2πnk/N ).

We start with an input vector (Fourier coefficients) of length 2N, then:

fn =N∑

k=−N+1

Fk ei2πnk/2N

= F0 +N−1∑k=1

(Fk ei2πnk/2N + F−k e−i2πnk/2N

)+ FNei2πnN/2N

= F0 +N−1∑k=1

(Fk ei2πnk/2N +

{Fk ei2πnk/2N

}∗)+ FNeiπn

= F0 + 2N−1∑k=1

Re{

Fk ei2πnk/2N}+ FNeiπn

= F0 + 2N−1∑k=1

(Re{Fk} cos

(πnkN

)− Im{Fk} sin

(πnkN

))+ FN cos (πn)


Inverse Real DFT (2)

Set ak := 2 Re{Fk} and bk := −2 Im{Fk} (but a0 := Re{F0} andaN := Re{FN}) to get :

fn = a0 +N−1∑k=1

(ak cos

(πnkN

)+ bk sin

(πnkN

))+ aN cos (πn)

“Real inverse discrete Fourier transform”

Using the (real-valued) formula for Fk :

ak =1N

N∑n=−N+1

fn cos(πnkN

), bk =

1N

N∑n=−N+1

fn sin(πnkN

)

but: a0 = 12N

∑. . . and aN = 1

2N

∑. . .


Inverse Real DFT – Compare with Textbooks

Alternate (probably more frequent) notation in textbooks:Set ak := 2 Re{Fk} and bk := −2 Im{Fk} to get

fn =12

a0 +N−1∑k=1

(ak cos

(πnkN

)+ bk sin

(πnkN

))+

12

aN cos (πn)

Using the (real-valued) formula for Fk :

ak =1N

N∑n=−N+1

fn cos(πnkN

), bk =

1N

N∑n=−N+1

fn sin(πnkN

)Differences:• no extra definition of a0 and aN

• but factors 12 in formula for fn → not as nicely related to interpolation


Real-valued Trigonometric Interpolation

Interpretation of the real DFT as an interpolation problem:• 2N ansatz functions:

gk (x) := cos(kx) k = 0, . . . ,Nhk (x) := sin(kx) k = 1, . . . ,N − 1

• 2N supporting points: xn :=2πn2N

=πnN

n = −N + 1, . . . ,N

• 2N interpolation conditions:

fn = a0 +N−1∑k=1

(ak cos

(πnkN

)+ bk sin

(πnkN

))+ aN cos (πn)

(cmp. exercises)


Fast Real DFT

Computation of a real-valued DFT using complex FFT is inefficient:

• N redundant components computed (symmetry)• complex arithmetics with lots of real/imaginary parts being 0

Possibilities to improve the efficiency:1. compute two real DFTs from one complex FFT2. compute a real DFT of length 2N from one complex FFT of length N3. “compact” real FFT – use symmetry of the data directly in the algorithm


Two Real DFTs from one complex FFT

Idea: for real-valued gn and hn, compute DFT of fn := gn + ihn:

Fk =1N

∑n

(gn + ihn)ω−nkN

Comparison with coefficients Gk and Hk of the two real DFTs:

Gk =1N

∑n

gnω−nkN Hk =

1N

∑n

hnω−nkN

Due to linearity of the Fourier transform:

Fk = Gk + iHk


Two Real DFTs from one complex FFT (2)

Since gn and hn are real data, we have the following symmetry:

Gk = G∗−k Hk = H∗−k .

Hence, we get for F ∗−k :

F ∗−k = (G−k + iH−k )∗ =

(G∗−k + i∗H∗−k

)= Gk − iHk .

Together with Fk = Gk + iHk , we obtain

Gk =12(Fk + F ∗−k

)and Hk = − i

2(Fk − F ∗−k

).


Two real DFTs from one complex FFT – Algorithm

Algorithm to compute two real DFTs:(1) set fn := gn + ihn

(2) compute Fk from FFT (using a library, e.g.)(3) compute Gk and Hk according to

Gk =12(Fk + F ∗−k

)and Hk = − i

2(Fk − F ∗−k

)⇒ “half” the costs compared to using complex FFT

but: additional operations for pre- and postprocessing


Real DFT of length 2N from complex FFT oflength N

Compute DFT of a real-valued vector (f−N+1, . . . , fN):

Fk =1

2N

N∑−N+1

fnω−nk2N for k = −N

2+ 1, . . . ,

N2

Split up in gn := f2n and hn := f2n−1; leads to butterfly scheme:

Fk = 12

(Gk + ωk

2NHk

),

Fk±N = 12

(Gk − ωk

2NHk

)for k = −N

2 + 1, . . . , N2 , respectively.


Real 2N-DFT from complex N-FFT

Now: compute Gk and Hk (two real DFTs) from one complex FFT→ applied to zn := gn + ihn = f2n + i f2n−1:

Gk = 12

(Zk + Z ∗−k

)and Hk = − i

2

(Zk − Z ∗−k

)Combine both schemes to:

Fk = 14 Zk

(1− iωk

2N

)+ 1

4 Z ∗−k

(1 + iωk

2N

), k = 0, . . . , N

2

Fk+N = 14 Zk

(1 + iωk

2N

)+ 1

4 Z ∗−k

(1− iωk

2N

), k = −N

2 + 1, . . . ,0


Real 2N-DFT from complex N-FFT – Algorithm

Algorithm for a real 2N-DFT:(1) set zn := f2n + i f2n−1

(2) compute Zk from FFT applied on zn (using a library, e.g.)(3) compute Fk according to

Fk = 14 Zk

(1− iωk

2N

)+ 1

4 Z ∗−k

(1 + iωk

2N

), k = 0, . . . , N

2

Fk+N = 14 Zk

(1 + iωk

2N

)+ 1

4 Z ∗−k

(1− iωk

2N

), k = −N

2 + 1, . . . ,0

⇒ Complexity determined by complex N-FFT

plus: additional operations for pre- and postprocessing


Compact Real FFT

Compute DFT of a real-valued vector (f−N+1, . . . , fN):

Fk =1

2N

N∑−N+1

fnω−nk2N for k = 0, . . . ,N

Split up in gn := f2n and hn := f2n−1; leads to butterfly scheme:

Fk = 12

(Gk + ωk

2NHk

)for k = 0, . . . , N

2 ,

Fk+N = 12

(Gk − ωk

2NHk

)for k = −N

2 + 1, . . . ,0 .

Gk and Hk are coefficients of a real-valued DFT of length N; hence:

Gk = G∗−k and Hk = H∗−k for k = 0, . . . , N2 − 1


Compact Real-valued FFT (2)

Use symmetry of Gk and Hk for the computation of Fk :

Fk = 12

(Gk + ωk

2NHk

)fur k = 0, . . . , N

2 ,

FN−k = 12

(G−k − ω−k

2N H−k

)= 1

2

(Gk − ωk

2NHk

)∗for k = 0, . . . , N

2 − 1

⇒ Computation of Fk (for k = 0, . . . ,N) reduced to the computation of Gkand Hk (for k = 0, . . . , N

2 , respectively).

“Edson’s algorithm” (1968)


Algorithms of Scientiﬁc ComputingFast Real DFT Computation of a real-valued DFT using complex FFT...

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