A Reduced-Complexity Fast Algorithm for Software ...

EURASIP Journal on Applied Signal Processing 2002:9, 961–974c© 2002 Hindawi Publishing Corporation

A Reduced-Complexity Fast Algorithm for SoftwareImplementation of the IFFT/FFT in DMT Systems

Tsun-Shan ChanVXIS Technology Corporation, Hsin-chu, Taiwan, ROCEmail: [email protected]

Jen-Chih KuoDepartment of Electrical Engineering, Graduate Institute of Electronics Engineering, National Taiwan University,Taipei 106, Taiwan, ROCEmail: [email protected]

An-Yeu (Andy) WuDepartment of Electrical Engineering, Graduate Institute of Electronics Engineering, National Taiwan University,Taipei 106, Taiwan, ROCEmail: [email protected]

Received 31 August 2001 and in revised form 15 May 2002

The discrete multitone (DMT) modulation/demodulation scheme is the standard transmission technique in the application ofasymmetric digital subscriber lines (ADSL) and very-high-speed digital subscriber lines (VDSL). Although the DMT can achievehigher data rate compared with other modulation/demodulation schemes, its computational complexity is too high for cost-efficient implementations. For example, it requires 512-point IFFT/FFT as the modulation/demodulation kernel in the ADSLsystems and even higher in the VDSL systems. The large block size results in heavy computational load in running programmabledigital signal processors (DSPs). In this paper, we derive computationally efficient fast algorithm for the IFFT/FFT. The proposedalgorithm can avoid complex-domain operations that are inevitable in conventional IFFT/FFT computation. The resulting soft-ware function requires less computational complexity. We show that it acquires only 17% number of multiplications to computethe IFFT and FFT compared with the Cooly-Tukey algorithm. Hence, the proposed fast algorithm is very suitable for firmwaredevelopment in reducing the MIPS count in programmable DSPs.

Keywords and phrases: FFT, IFFT, DMT, software implementation.

1. INTRODUCTION

Recent progress of Internet access has a strong demand onhigh-speed data transmission. To overcome the transmissionbottleneck over the conventional twisted-pair telephonelines, several sophisticated modulation/demodulationschemes have been proposed, including carrierless-amplitude-phase (CAP) modulation [1], discrete multitonemodulation (DMT) [2, 3, 4, 5] and QAM technology [6].Among these advanced modulation schemes, the DMT canachieve highest transmission rate since it incorporates lotsof advanced DSP techniques such as dynamic bit allocation,multidimensional tone encoding, frequency-domain equal-ization, and so forth. As a consequence, the DMT has beenchosen as the physical layer transmission standard by theADSL standardization committee.

One major disadvantage of the DMT scheme is its high

computational complexity. In particular, the large blocksize of the IFFT/FFT consumes lots of computing powerin running programmable DSPs [7]. In [8], we have con-sidered a cost-efficient lattice VLSI architecture to realizethe IFFT/FFT in integrated circuits. In this paper, we pro-pose computationally efficient fast algorithms to run theIFFT/FFT function in software implementation such as pro-grammable DSP processors (DSPs). By making use of thesymmetric/antisymmetric properties of the Fourier trans-form, we first decompose the IFFT/FFT into a combinationof two new real-domain transform kernels—the ModifiedDCT and Modified DST. These two transform functions areused to replace the complex-domain IFFT/FFT. Then we em-ploy the divide-and-conquer approach in [9] to derive novelrecursive algorithms and butterfly architectures for the mod-ified DCT DST.

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962 EURASIP Journal on Applied Signal Processing

Modulator

X(0)

X(1)

X(2)

X(0)

X(1)

X(2)

Encodedcomplexsymbols

(from encoder)

X(N − 1)X(N)

2N-pointIFFT

Parallel/Serial

Con

jugate

X(2N−2)X(2N−1)

y(n)Channel

y(n)

Demodulator

X(0)

X(1)

X(2)

X(0)

X(1)

X(2)

Serial/Parallel

2N-pointFF

T X(N−1)

X(2N−2)X(2N−1)

Discard

Demodulatedcomplexsymbols

(to decoder)

...

......

......

Figure 1: The IFFT/FFT block diagram in the DMT system.

The new scheme can avoid redundant complex-domainof the IFFT/FFT. That is, it involves only real-valued opera-tions to compute the IFFT/FFT. Hence, we can avoid the spe-cial data structure in software programming to run complex-domain addition/multiplication operations in computingthe IFFT/FFT. In addition, our analysis shows that we needonly 17% and multiplications in computing the IFFT andFFT compared with Cooly-Tukey algorithm [10]. The lowcomputational complexity as well as real-domain operationsmakes it very suitable for firmware coding in DSPs, whichhelps to save the MIPS counts. Also, the DSP program canbe written in recursive form which requires less ROM/RAMprogram storage space to implement the IFFT/FFT.

The rest of this paper is organized as follows. Section 2shows the derivation of the IFFT algorithm. In Section 3, thederivation of the FFT algorithm is discussed. The computa-tion complexity comparison is shown in Section 4. The finiteprecision effect of our algorithm is also discussed. Finally, weconclude our work in Section 5.

2. REDUCED-COMPLEXITY IFFT ALGORITHM

2.1. The IFFT derivation

The IFFT/FFT block diagram in the DMT system is showedin Figure 1. At the transmitter side, to ensure the IFFT gen-erates only real-valued outputs, the inputs of the IFFT in theDMT standard have the constraint [11],

X(0) = X(N) = 0,

X(k) = X∗(2N − k) for k = 1, 2, . . . , N − 1,(1)

where X(k)�= Xr(k) + j · Xi(k) are encoded complex sym-

bols. As defined in [12, Chapter 9], the IFFT of a finite-lengthsequence of length 2N is

x(n) = 12N

·[ 2N−1∑

k=0X(k)W−nk

2N

], for n = 0, 1, . . . , 2N − 1,

(2)

where

Wnk2N

�= exp(− j

2πnk2N

)= cos

2πnk2N

− j sin2πnk2N

. (3)

By decomposing n into the first half and the second half, (2)becomes

x(n) = 12N

·[ N−1∑

k=0X(k)W−nk

2N +2N−1∑k=N

X(k)W−nk2N

]. (4)

Next, by substituting (3) into (4), and using (1), we can sim-plify (4) as (see Appendix A)

x(n) = 1N·[ N−1∑

k=0Xr(k) cos

2πnk2N

−N−1∑k=0

Xi(k) sin2πnk2N

]

= 1N· [MDCT(n)−MDST(n)

],

for n = 0, 1, . . . , 2N − 1.(5)

A Reduced-Complexity Fast Algorithm for Software Implementation of the IFFT/FFT in DMT Systems 963

Xr(0)

Xr(1)

...

Xr(N−2)

Xr(N−1)

Even-oddindex

mapping

Xr(0)

Xr(2)

...

Xr(N − 4)

Xr(N − 2)

Xr(1)

Xr(3)...

Xr(N − 5)

Xr(N − 3)

Xr(N − 1)

+

...

+

+

N/2-pointMDCTg(n)

N/2-pointMDCTh′′(n)

xr(N−1)

−xr(N−1)+

+...

xr(N−1)

−xr(N−1)+

+

Γ0

Γ1

...

ΓN/2−2

ΓN/2−1

...

+

+

...

+

+

+

+

+

...

+

+

+

+

−

−

−

MDCT (0)

MDCT (1)

...

MDCT (N/2 − 2)

MDCT (N/2 − 1)

Special case

MDCT (N/2)

MDCT (N/2 + 1)

MDCT (N/2 + 2)

...

MDCT (N − 1)

Oddsummation

Injecteditems

Γn = 1/2Cn2N

n : 0 ∼ N/2 − 1

N-point MDCT

Xr(k) 1-pointMDCT

MDCT (n) Xr(k) MDCT (n)

Figure 2: N-point MDCT(n) butterfly structure, where 1-point MDCT is the minimum-sized processing block.

From (5), we can see that the computation of the IFFTis decomposed into two real-valued operations. One is adiscrete cosine transform DCT-like operation with Xr(k),k = 0, 1, 2, . . . , N − 1, as the inputs. The other is a dis-crete sine transform DST-like operation with Xi(k), k =0, 1, 2, . . . , N − 1, as the inputs. We will name the firstterm Modified DCT (MDCT), and the second term Modi-fied DST (MDST). Note that the MDCT and MDST involveonly real-valued operators. Furthermore, it can be shownthat

MDCT(n) =MDCT(2N − n), for n = 0, 1, . . . , N − 1,(6)

MDST(n) = −MDST(2N − n), for n = 0, 1, . . . , N − 1.(7)

Hence, we can focus on computingMDCT(n) andMDST(n)for n = 0, 1, . . . , N − 1. Then, expand the results for n = N +1, N +2, . . . , 2N−1. For the special cases of n = 0 and n = N ,the MDCT and MDST can be simplified as

MDCT(0) =N−1∑k=0

Xr(k) cos2π0k2N

=N−1∑k=0

Xr(k),

MDST(0) =N−1∑k=0

Xi(k) sin2π0k2N

= 0,

MDCT(N) =N−1∑k=0

Xr(k) cos2πNk

2N=

N−1∑k=0

Xr(k)(−1)k,

MDST(N) =N−1∑k=0

Xi(k) sin2πNk

2N= 0,

(8)

respectively. These simple relationships can help us to saveadditional computation complexity.

2.2. MDCT/MDST operations of the IFFT

From the preceding discussion, we can see that the imple-mentation issue of the IFFT is to realize MDCT and MDSTin a cost-efficient way. Then, we can just combine the re-sults of the MDCT and MDST to obtain the IFFT resultsbased on (5). Here, we first consider the implementation


of the MDCT. We follow the derivation in [9] and define

Cnk2N

�= cos(2πnk/2N). Then, the MDCT can be written as

MDCT(n) =N−1∑k=0

Xr(k)Cnk2N , for n = 0, 1, . . . , N − 1. (9)

Decompose the MDCT into even and odd indices of k, then(9) can be rewritten as

MDCT(n) = g(n) + h′(n), for n = 0, 1, . . . ,N

2− 1, (10)

where

g(n)�=

N/2−1∑k=0

Xr(2k)Cn(2k)2N =

N/2−1∑k=0

Xr(2k)CnkN ,

h′(n) �=N/2−1∑k=0

Xr(2k + 1)Cn(2k+1)2N .

(11)

Define h(n)�= 2Cn

2Nh′(n). Following the derivation in Lee’s

algorithm [9], we can find

MDCT(n) = g(n) + h′(n) = g(n) +1

2Cn2N

h(n). (12)

That is,

N−1∑k=0

Xr(k)Cnk2N︸︷︷︸

N-point MDCT

=N/2−1∑k=0

Xr(2k)CnkN︸︷︷︸

N/2-point MDCT, g(n)

+1

2Cn2N

N/2−1∑

k=0

[Xr(2k + 1) + Xr(2k − 1)

]CnkN︸︷︷︸

N/2-point MDCT, h′′(n)

+ Xr(N − 1)(−1)n︸︷︷︸injected item

,

for n = 0, 1, . . . ,N

2− 1.

(13)

On the other hand, by replacing index nwith (N−n) in (12),it can be shown that

MDCT(N − n) = g(n)− h′(n) = g(n)− 12Cn

2Nh(n). (14)

The special case MDCT(N/2) needs to be computed sepa-rately, which can be simplified as

MDCT(N

2

)=

N−1∑k=0

Xr(k)Ck(N/2)2N =

N−1∑k=0

Xr(k) coskπ

2. (15)

The mapping of (13), (14), and (15) is shown in Figure 2. Aswe can see, theN-point MDCT is decomposed into twoN/2-

pointMDCT (g(n) and h′′(n)) plus some pre-processing andpost-processing modules. Then we can apply the techniqueof divide-and-conquer to recursively expand the N/2-pointMDCT until 1-point MDCT is formed. That is, we repeatthe decomposition in (10) and (11) until N = 1.

Next, we consider the recursive implementation of the

MDST. We define Snk2N�= sin (2πnk/2N). As with the deriva-

tion in (10), (11), (12), (13), and (14), we can find

MDST(n) =N/2−1∑k=0

Xi(2k)SnkN

+1

2Cn2N

N/2−1∑k=0

[Xi(2k + 1) + Xi(2k − 1)

]SnkN ,

MDST(N − n) = −N/2−1∑k=0

Xi(2k)SnkN

+1

2Cn2N

N/2−1∑k=0

[Xi(2k + 1) + Xi(2k − 1)

]SnkN ,

for n = 0, 1, . . . ,N

2− 1.

(16)

It is worth noting that the injected item is zero in the MDST.Besides, the MDST also has a special case for index N/2 as

MDST(N

2

)=

N−1∑k=0

Xi(k)Sk(N/2)2N =

N−1∑k=0

Xi(k) sinkπ

2. (17)

The mapping of the MDST structure in Figure 3 is similar tothe MDCT structure, except that minimum processing blockis 2-point MDST (see Figure 3) and the injected items do notexist in theMDST implementation. That is, we repeat the de-composition in (16) untilN = 2. Note that the 1-pointMDSTis always equal to zero.

2.3. Overall IFFT computation procedures

The overall IFFT computation flow is shown in Figure 4.It consists of the MDCT/MDST operations and a post-processing operation. The operations in Figure 4 are as fol-lows:

(1) set the butterfly operation to MDCT mode;(2) Xr(k), k = 0, 1, . . . , N − 1, are first fed into the but-

terfly architecture to obtain the MDCT(n), for n =0, 1, . . . , N − 1;

(3) the post-processing operation expands the N-pointMDCT outputs to 2N-point MDCT using the sym-metric property in (6);

(4) set the butterfly operation to MDST mode;(5) repeat the computation in Steps 2 and 3 using Xi(k),

k = 0, 1, . . . , N−1 as inputs, and obtain the MDST(n),for n = 0, 1, . . . , N − 1;

(6) the post-processing operation expands the N-pointMDST outputs to 2N-point MDST by using the an-tisymmetric property in (7);


Xi(0)

Xi(1)

...

Xi(N−2)

Xi(N−1)

Even-oddindex

mapping

Xi(0)

Xr(2)

...

Xi(N − 4)

Xi(N − 2)

Xi(1)

Xi(3)...

Xi(N − 5)

Xi(N − 3)

Xi(N − 1)

+

+

+

...

N/2-pointMDSTg(n)

N/2-pointMDSTh′′(n)

...

...

Γ0

Γ1

ΓN/2−2

ΓN/2−1

...

+

+...

+

+

−

−

−

+

+

+

+

+

+

+

...

MDST (0)

MDST (1)

...

MDST (N/2 − 2)

MDST (N/2 − 1)

Special case

MDST (N/2)

MDST (N/2 + 1)

MDST (N/2 + 2)

...

MDST (N − 1)

Oddsummation

Γn = 1/2Cn2N

n : 0 ∼ N/2 − 1

N-point MDST

Xi(k0)

Xi(k1)

2-pointMDST

MDST (n0)

MDST (n1)

Xi(k0)

Xi(k1)

MDST (n0)

MDST (n1)

Figure 3: N-point MDST(n) butterfly structure, where 2-point MDST is the minimum-sized processing block.

(7) based on (5), we combine the MDCT and MDST re-sults together with the scaling operation (which isachieved by shifting right by log2(N) bits) to obtainthe IFFT results. This is done in the post-processingoperation.

2.4. Matrix notation of theMDCT/MDST

In this section, we present the matrix notation of the pro-posed fast IFFT algorithm. The matrix form can help to seethe divide-and-conquer nature of our approach. By follow-ing the notation in [13], we rewrite Xr(k) and MDCT(n) as

[Xr(k)N

]�=[Xr(0) Xr(1) · · · Xr(N − 1)]T

, (18)[MDCT(n)N

]�=[MDCT(0) MDCT(1) · · · MDCT(N − 1)

]T,

(19)

respectively. Then (9) can be represented as

[MDCT(n)N

] = [TN,MDCT][Xr(k)N

], (20)

where [TN,MDCT] denotes the transform kernel matrix of theMDCT operation. Next, the injected items of (13) can be

represented as

Injected = [ON/2]Xr(N − 1), (21)

where

[ON/2

] = [1 −1 1 −1 1 · · · −1]T

. (22)

We define the odd-summation matrix as

[LN/2

] =

1 0 0 · · · 0 01 1 0 · · · 0 00 1 1 · · · 0 0...

......

. . ....

...0 0 0 · · · 1 1

(23)

and the scaling matrix as

[ΦN/2

] = diag{

12Cn

2N

}, for n = 0, 1, . . . ,

N

2− 1. (24)

The special case of the MDCT in (15) can be represented as

MDCT(N

2

)= [SN][Xr(k)N

], (25)


Xi(0)

Xi(1)

...

Xi(N − 2)

Xi(N − 1)

Phase IIfor

MDST

Xr(0)

Xr(1)

...

Xr(N − 2)

Xr(N − 1)

Phase Ifor

MDCT

...

N-pointMDCT/MDST

+

+

...

+...

+

+...

+

+

R

R

R

R

R

R

R

�

�

�

�

�

�

�

�

x(0)

x(1)

...

x(N/2)

...

x(N − 1)

Special casex(N)

x(N + 1)

...

x(3N/2)

...

x(2N − 1)

Shift left log2(N) bits

Post-processing(Expanding circuit)

Figure 4: The proposed IFFT architecture.

where

[SN] = [1 0 −1 0 1 · · · 0

]. (26)

Based on (12), (13), (14), (21), (22), (23), (24), (25), and(26), the [TN,MDCT] can be expressed in the matrix form as

[TN,MDCT

] =

[TN/2

] [ψN,MDCT

][TN/2

][JN/2

] [ψN,MDCT

][JN/2

] , (27)

where [ψN,MDCT] = [ΦN/2]([LN/2][TN/2] + [ON/2])‖[SN ] and[JN/2] denotes the opposite-diagonal identity matrix. We canalso represent (20) and (27) in the recursive form as shownin Figure 5. Following the above derivations, the matrix no-tation of transform kernel of the MDST can be derived as

[TN,MDST

] =[ [

TN/2] [

ψN,MDST]

−[TN/2][JN/2

] [ψN,MDST

][JN/2

]], (28)

where [ψN,MDST] = [ΦN/2][LN/2][TN/2][SN ]. Note that theMDST is similar to the MDCT except that there is no in-jected items. Also, the special case matrix can be modifiedas

[SN] = [0 1 0 −1 0 · · · −1

]. (29)

The block diagram of the MDST in the matrix form isshown in Figure 6.

3. REDUCED-COMPLEXITY FFT ALGORITHM

3.1. The FFT derivation

At the receiver side (see Figure 1), the 512-point FFT is usedto demodulate the received signals, which is given by

X(k) =2N−1∑n=0

x(n)Wnk2N , for k = 0, 1, . . . , 2N − 1, (30)

where

Wnk2N

�= exp(− j

2πnk2N

)= cos

2πnk2N

− j · sin 2πnk2N

. (31)

Note that x(n), n = 0, 1, . . . , 2N−1, are real-valued numbers.Hence, (30) can be rewritten as

X(k)=2N−1∑n=0

x(n) cos2πnk2N

− j2N−1∑n=0

x(n) sin2πnk2N

=MDCT(k)− j ·MDST(k), for k = 0, 1, . . . , 2N − 1.(32)

Equation (32) shows that the computation of the FFT is


Xr(0)

Xr(1)

...

Xr(N − 2)

Xr(N − 1)

Even-oddindex

mapping

Even

Odd

[TN/2]

[TN/2][LN/2]

Xr(N − 1) [QN/2]

[SN ]

+ [ΦN/2]

++

+

++

−[JN/2]

MDCT (n)N

Figure 5: Block diagram of the MDCT operation in matrix form.

Xi(0)Xi(1)

...

Xi(N − 2)

Xi(N − 1)

Even-oddindex

mapping

Even

Odd

[TN/2]

[TN/2][LN/2]

[SN ]

[ΦN/2]

+ +

+

++− [JN/2]

MDST (n)N

Figure 6: Block diagram of the MDCT operation in matrix form.

decomposed into a combination of two real-domainkernels—MDCT(k) and MDST(k). Both MDCT and MDSTuse x(n), n = 0, 1, . . . , 2N − 1, as the inputs. Hence, weonly employ two real-valued kernels (MDCT and MDST),thus no complex-valued operations are required in com-puting the FFT. In addition, in the DMT system, the lowerN-point FFT outputs are conjugate-symmetric to the up-per N-point outputs. We are only interested in N-point datafor k = 0, 1, . . . , N − 1. Hence, we can neglect the outputsX(k), for k = N,N + 1, . . . , 2N − 1.

3.2. MDCT/MDST operations of the FFT

In (32), the transform kernels are 2N-point MDCT(k) andMDST(k). Here, we propose a novel approach to further re-duce the computational complexity. Hence, we only need toperform N-point MDCT/MDST.

We first decompose input sequence into a symmet-ric sequence, xc(n), plus an antisymmetric sequence, xs(n),where

xc(n)�= 1

2

[x(n) + x(2N − n)

],

xs(n)�= 1

2

[x(n)− x(2N − n)

], for n = 1, 2, . . . , N − 1.

(33)

Hence, we have

x(n) = xc(n) + xs(n), (34)

x(2N − n) = xc(n)− xs(n), for n = 1, 2, . . . , N − 1. (35)

By substituting (34) and (35) into (30), we can simplify (30)as (see Appendix B)

X(k) ={x(0) + x(N)(−1)k

+ 2

[ N−1∑n=0

xc(n) cos2πnk2N

− jN−1∑n=0

xs(n) sin2πnk2N

]}

= {x(0) + x(N)(−1)k + 2[MDCT(k)− jMDST(k)

]},

for k = 0, 1, . . . , N − 1,(36)

where xc(0) = 0 and xs(0) = 0. Since the block size is reducedfrom 2N-point (see (32)) to N-point (see (36)).

Next, following the derivations of the IFFT in Section 2,we can have


MDCT(k) = g(k) +1

2Ck2N

h(k)

=N/2−1∑n=0

xc(2n)CnkN︸︷︷︸

N/2-point MDCT, g(k)

+1

2Ck2N

[ N/2−1∑n=0

[xc(2n + 1) + xc(2n− 1)

]CnkN︸︷︷︸

N/2-point MDCT, h′′(k)

+ xc(N − 1)(−1)k︸︷︷︸injected item

],

(37)

MDCT(N − k) = g(k)− 1

2Ck2N

h(k)

=N/2−1∑n=0

xc(2n)CnkN

− 1

2Ck2N

[ N/2−1∑n=0

[xc(2n + 1)

+ xc(2n− 1)]CnkN

+ xc(N − 1)(−1)k],

for k = 0, 1, . . . ,N

2− 1.

(38)

Similarly, for the MDST(k), we have

MDST(k) = g(k) +1

2Ck2N

h(k)

=N/2−1∑n=0

xs(2n)SnkN

+1

2Ck2N

N/2−1∑n=0

[xs(2n + 1) + xs(2n− 1)

]SnkN ,

(39)

MDST(N − k) = −g(k) + 1

2Ck2N

h(k)

= −N/2−1∑n=0

xs(2n)SnkN

+1

2Ck2N

N/2−1∑n=0

[xs(2n + 1) + xs(2n− 1)

]SnkN ,

for k = 0, 1, . . . ,N

2− 1.

(40)

The two special cases for index N/2 are

MDCT(N

2

)=

N−1∑n=0

xc(n) cosnπ

2,

MDST(N2)=

N−1∑n=0

xs(n) sinnπ

2.

(41)

The block diagram of theMDCT(k) is shown in Figure 7.The mapping of the MDST structure is similar to the MDCTstructure in Figure 7 except that minimum processing blockis 2-point MDST and the injected items do not exist in theMDST(k) implementation (see Figure 8). Then we can justcombine the MDCT(k) and MDST(k) outputs, followed byadding x(0) and x(N)(−1)k, to obtain the FFT results basedon (36).

3.3. Overall FFT computation procedures

The overall computation flow of the FFT is shown inFigure 9. The operations are as follows.

(1) The received signals x(n), n = 0, 1, . . . , 2N − 1, aredecomposed to xc(n) and xs(n), n = 0, 1, . . . , N − 1, throughthe pre-processing operation.

(2) In the first phase, the generated xc(n) are fed into re-cursive butterfly operation to obtain the MDCT(k) outputs.

(3) In the second phase, we repeat the computation byusing the xs(n) as inputs into recursive butterfly operation toobtain the MDST(k) outputs.

(4) We combine the MDCT(k) and MDST(k) resultsthen add x(0) and x(N)(−1)k together to obtain the FFT re-sults based on (36). This is done in the post-processing oper-ation.

3.4. Matrix notation of theMDCT/MDSTBased on (19), (20), (21), (22) (23), (24), (25), and (26), wecan represent (37), (38), and (39) as

[TN,MDCT

] =[ [

TN/2] [

ψN,MDCT][

TN/2][JN/2

] −[ψN,MDCT][JN/2

]], (42)

where [ψN,MDCT] = [ΦN/2]([LN/2][TN/2] + [ON/2])‖[SN ],

[TN,MDST

] =[ [

TN/2] [

ψN,MDST]

−[TN/2][JN/2

] [ψN,MDST

][JN/2

]], (43)

where [ψN,MDST] = [ΦN/2][LN/2][TN/2][SN ], of theMDCT(k)/MDST(k), respectively. The block diagramsof the MDCT(k) and MDST(k) are very similar to theMDCT(n) and MDST(n) in Section 2. The difference is thatit requires a pre-processing to compute the xc(n) and xs(n).The block diagrams of the MDCT and MDST are shown inFigures 10 and 11, respectively.

4. COMPLEXITY COMPARISON ANDFINITE-PRECISION EFFECT

4.1. Comparison of hardware complexityIn this section, we compare the computation complexity ofthe proposed algorithm with the traditional Cooly-Tukey


2N-point

x(0)

x(1)

...

x(2N−2)

x(2N−1)

Pre-processing

N-point

xc(0)

xc(1)

...

xc(N−2)

xc(N−1)

Even-oddindex

mapping

xc(0)

xc(2)

...

xc(N − 4)

xc(N − 2)

xc(1)

xc(3)...

xc(N − 5)xc(N − 3)

xc(N − 1)

+...

+

+

N/2-pointMDCTg(k)

N/2-pointMDCTh′′(k)

xc(N−1)

−xc(N−1)

xc(N−1)

−xc(N−1)

+

+

+

+

...

Γ0

Γ1

...

...

++++...

++

++

+− ...+−+−

MDCT (0)

MDCT (1)

...

MDCT (N/2 − 2)

MDCT (N/2 − 1)

Special case

MDCT (N/2)

MDCT (N/2+1)

MDCT (N/2+2)

...

MDCT (N − 1)

Oddsummation

Injecteditems

Γk = 1/2Ck2N

k : 0 ∼ N/2 − 1

N-point MDCT

xc(n) 1-pointMDCT

MDCT(k) xc(n) MDCT(k)

Figure 7: N-point MDCT(k) butterfly structure, where 1-point MDCT is the minimum-sized processing block of the FFT module.

2N-point

x(0)

x(1)

...

x(2N−2)x(2N−1)

Pre-processing

N-point

xs(0)

xs(1)

...

xs(N−2)xs(N−1)

Even-oddindex

mapping

xs(0)

xs(2)...

xs(N − 4)

xs(N − 2)

xs(1)

xs(3)...

xs(N − 5)

xs(N − 3)

xs(N − 1)

+...

+

+

N/2-pointMDSTg(k)

N/2-pointMDSTh′′(k)

Γ0

Γ1

......

...

++++...

++++

+−...+−+−

MDST (0)

MDST (1)...

MDST (N/2 − 2)

MDST (N/2 − 1)

Special case

MDST (N/2)MDST (N/2 + 1)MDST (N/2 + 2)

...

MDST (N − 1)

Oddsummation

Injecteditems

Γk = 1/2Ck2N

k : 0 ∼ N/2 − 1

N-point MDST

xs(n0)

xs(n1)

2-pointMDST

MDST(k0)

MDST(k1)

xs(n0)

xs(n1)

MDST(k0)

MDST(k1)

Figure 8: N-point MDST(k) butterfly structure, where the 2-point MDST is the minimum-sized processing block of the FFT module.


2N-point

x(0)

x(1)

...

x(2N−2)x(2N−1)

Pre-processing

N-point N-point

xs(0), xc(0)

xs(1), xc(1)

......

xs(N−2), xc(N−2)xs(N−1), xc(N−1)

2ndphase

1stphase

N-pointMDCT(1st phase)MDST(2nd phase)

x(0) + x(N)(−1)k

+ RX(0)

......

x(0) + x(N)(−1)k

+ RX(N/2)

......

x(0) + x(N)(−1)k

+ RX(N − 1)

Post-processing

Figure 9: The proposed FFT architecture.

x(0)

x(1)

...

x(2N−2)x(2N−1)

Pre-processing

xs(0)

xs(1)

...

xs(2N−2)xs(2N−1)

Even-oddindex

mapping

Even

Odd [LN/2]

xc(N − 1)

[TN/2]

[TN/2]

[ON/2]

[SN ]

+ [ΦN/2]

++

+

+

−+ [JN/2]

MDCT (k)N

Figure 10: Block diagram of the MDCT in matrix form for the FFT operation.

x(0)

x(1)

...

x(2N−2)x(2N−1)

Pre-processing

xs(0)

xs(1)

...

xs(N − 2)

xs(N − 1)

Even-oddindex

mapping

Even

Odd [LN/2]

[TN/2]

[TN/2]

[SN ]

[ΦN/2]

++

+

+

−+ [JN/2]

MDST (k)N

Figure 11: Block diagram of the MDST in matrix form for the FFT operation.

algorithm. The corresponding butterfly architecture requireslog2(2N) stages in the 2N-point IFFT/FFT. Each stageconsists of N multiplications and 2N additions. Because

input sequences are complex data, the IFFT/FFT kernelsare complex in nature. Hence, it requires 4 real-valuedmultiplications and 2 real-valued additions for 1 complex


Table 1: Comparison of computational complexity for 2N-point IFFT/FFT.

IFFT FFT

Cooly-Tukey [10] Chan et al. Cooly-Tukey [10] Chan et al.

(O1) (O2) CR (O1) (O2) CR

N 4N log2 2N N log2N − 2N + 2 4N log2 2N N log2N − 2N + 2

256 9216 1538 0.169 9216 1538 0.169

512 20480 3586 0.175 20480 3586 0.175

1024 45056 8194 0.182 45056 8194 0.182

2048 98304 18434 0.188 98304 18434 0.186

4096 212992 40962 0.192 212992 40962 0.192

8192 458752 90114 0.196 458752 90114 0.196

(a) Number of multiplication operations.

IFFT FFT

Cooly-Tukey [10] Chan et al. Cooly-Tukey [10] Chan et al.

(O1) (O2) CR (O1) (O2) CR

N 6N log2 2N (9/2)N log2N +N + 1 6N log2 2N (9/2)N log2N +N

256 13824 9473 0.685 13824 9472 0.685

512 30720 21249 0.692 30720 21248 0.692

1024 67584 47105 0.697 67584 47104 0.697

2048 147456 103425 0.701 147456 103424 0.701

4096 319488 225281 0.705 319488 225281 0.705

8192 688128 487425 0.708 688128 487424 0.708

(b) Number of addition operations.

multiplication. Also, it takes 2 real additions to realize a com-plex addition. As a result, the direct approach requires a to-tal of 4N log2(2N) real multiplications and 6N log2(2N) realadditions. The large computation complexity are not suitablefor cost-effective realization of the IFFT/FFT modules in theDMT system.

The complexity comparison for 2N-point IFFT/FFT arelisted in Table 1. The complexity ratio (CR) is defined as

CR�= O2

O1, (44)

where O1 and O2 are the number of multiplications (or ad-ditions) in other fast algorithms and our approach, respec-tively. We can see that the complexity ratio of the multi-plication is only 17% for N = 256 compared with conven-tional IFFT/FFT. Table 1 also shows that our approach cangain more computation savings as N gets larger in the VDSLsystems [14].

4.2. Experiment results

There are lots of DSP processors on the market. Due tothe variety or hardware structure, coding styles, compli-ers, and so forth, we are not trying to do the detail op-timization for specific processors. On the other hand, wewould like to compare the proposed algorithm with Cooly-Tukey’s algorithm, which is a baseline of the FFT realiza-tion. The implementation platform is TI TMS320C54 eval-uation board, http://www.ti.com. Both algorithms are writ-ten in C language without any assembly-level program-ming tricks. During compilation, the TI C54X C com-plier is used without adding special compilation options,neither.

Table 2 shows the comparison of the proposed algorithmand the conventional FFT in terms of clock cycles. As we cansee, the proposed algorithm requires only about 30% clockcycles of the Cooly-Tukey’s. The result is very consistent withour observation in Table 1.

http://www.ti.com


Table 2: Comparison of clock cycle for Cooley-Tukey FFT and pro-posed recursive algorithm.

128-point 256-point 512-point

Cooley-Tukey FFT 16,485 37,118 82,347

Proposed 11,869 25,726 55,435

Clock cycle Ratio 28% 31% 33%

0 5 10 15 20 25 30 35 40 45 50Wordlength (B)

0

50

100

150

200

AveragedSN

R(dB)

Direct butterfly approachOur approach

(a)

0 5 10 15 20 25 30 35 40 45 50Wordlength (B)

0

50

100

150

200

AveragedSN

R(dB)

Direct butterfly approachOur approach

(b)

Figure 12: Averaged SNR versus wordlength for the 512-point (2Nvalue) (a) IFFT. (b) FFT.

4.3. Finite-precision effect

In fixed-point implementation of the IFFT/FFT kernels, it isimportant to consider the effects of finite register length inthe IFFT/FFT calculations (see [12, Chapter 9] and [15]). Tocompare the butterfly approach and our approach in fixed-point implementation, we conduct extensive computer sim-ulation by using MATLAB for finite-wordlength IFFT/FFTarchitecture. Figure 12 shows the SNR performance with as-signed wordlength B = 8, 16, 32 bits. We observe that theSNR performance with B =16 bits is good enough in prac-tical fixed-point implementations. From the simulation re-sults, we can see that the SNR performance of our approachis comparable to the traditional butterfly approach under thesame wordlength.

5. CONCLUSIONS

In this paper, we develop a computationally efficient fast al-gorithm for the software implementation of the IFFT/FFTkernel in the DMT system. We reformulate the IFFT/FFTfunctions so as to avoid complex-domain operations. Thecomplexity ratio of themultiplications is only 17% comparedwith the direct butterfly implementation approach. The pro-posed algorithm provides a good solution in reducing MIPScount in programmable DSP implementation for the appli-cations of the DMT transceiver systems.

APPENDICES

A. DERIVATION OF (4)

Decomposing (4) into the first half and second half with thefact that X(0) = X(N) = 0, (4) can be represented as

x(n) = 12N

·[ N−1∑

k=1X(k)W−nk

2N +2N−1∑k=N+1

X(k)W−nk2N

]. (A.1)

Use k′ = 2N − k to replace the variable in the second term.Then, we have

x(n)= 12N·[ N−1∑

k=1X(k)W−nk

2N +1∑

k′=N−1X(2N−k′)W−(2N−k′)n

2N

].

(A.2)

Because k′ is a dummy variable, we can rewrite (A.2) as

x(n) = 12N

·[ N−1∑

k=1X(k)W−nk

2N +N−1∑k=1

X(2N − k)W−(2N−k)n2N

]

= 12N

·[ N−1∑

k=1X(k)W−nk

2N +N−1∑k=1

X(2N − k)W−2Nn2N Wnk

2N

].

(A.3)

By using the facts that

W2Nn2N = 1,

Wnk2N = exp

(− j

2πnk2N

)= cos

2πnk2N

− j sin2πnk2N

,

W−nk2N = exp

(j2πnk2N

)= cos

2πnk2N

+ j sin2πnk2N

,

X(0) = X(N) = 0,

(A.4)

we can rearrange (A.3) to

x(n) = 1N·[ N−1∑

k=0

(Xr(k) cos

2πnk2N

− Xi(k) sin2πnk2N

)].

(A.5)


B. DERIVATION OF (30)

Equation (30) can be represented as

X(k) = x(0)+x(N)(−1)k+[ N−1∑

n=1x(n)Wnk

2N+2N−1∑n=N+1

x(n)Wnk2N

].

(B.1)

Use n′ = 2N − n to replace the variable in the second term.Then, we have

X(k) = x(0) + x(N)(−1)k

+

[ N−1∑n=1

x(n)Wnk2N +

1∑n′=N−1

x(2N − n′)Wk(2N−n′)2N

].

(B.2)

Because n′ is a dummy variable, we can rewrite (B.2) as

X(k) = x(0) + x(N)(−1)k

+

[ N−1∑n=1

x(n)Wnk2N +

N−1∑n=1

x(2N − n)Wk(2N−n)2N

]

= x(0) + x(N)(−1)k

+

[ N−1∑n=1

x(n)Wnk2N +

N−1∑n=1

x(2N − n)W2kN2N W−nk

2N

].

(B.3)

By using the fact thatW2kN2N = 1 and applying the assumption

of the input data in (35), we can rearrange (B.3) as

X(k) = x(0) + x(N)(−1)k

+ 2

[ N−1∑n=1

xc(n) cos2πnk2N

− jN−1∑n=1

xs(n) sin2πnk2N

].

(B.4)

ACKNOWLEDGMENT

T. S. Chan is with the VXIS Tech. Corp. Hsin-Chu, Taiwan,ROC. This work is supported in part by the National ScienceCouncil, ROC, under Grant NSC 87-2213-E-008-011.

REFERENCES

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[2] J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitonetransceiver system for HDSL applications,” IEEE Journal onSelected Areas in Communications, vol. 9, no. 6, pp. 895–908,1991.

[3] K. Sistanizadeh, P. Chow, and J. M. Cioffi, “Multi-tone trans-mission for asymmetric digital subscriber lines (ADSL),” inProc. IEEE International Conf. on Communications, vol. 2, pp.756–760, Geneva, Switzerland, 1993.

[4] I. Lee, J. S. Chou, and J. M. Cioffi, “Performance eval-uation of a fast computation algorithm for the DMT inhigh-speed subscriber loop,” IEEE Journal on SelectedAreas in Communications, vol. 13, no. 9, pp. 1560–1570,1995.

[5] T. N. Zogakis, J. T. Aslanis Jr., and J. M. Cioffi, “A coded andshaped discrete multitone system,” IEEE Trans. Communica-tions, vol. 43, no. 12, pp. 2941–2949, 1995.

[6] B. Daneshrad and H. Samueli, “A 1.6Mbps digital-QAM sys-tem for DSL transmission,” IEEE Journal on Selected Areas inCommunications, vol. 13, no. 9, pp. 1600–1610, 1995.

[7] B. R. Wiese and J. S. Chow, “Programmable implementationsof xDSL transceiver systems,” IEEE Communications Maga-zine, vol. 38, no. 5, pp. 114–119, 2000.

[8] A.-Y. Wu and T. S. Chan, “Cost-efficient parallel lattice VLSIarchitecture for the IFFT/FFT in DMT transceiver technol-ogy,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Pro-cessing, pp. 3517–3520, Seattle, Wash, USA, May 1998.

[9] B. G. Lee, “A new algorithm to compute the discrete cosinetransform,” IEEE Trans. Acoustics, Speech, and Signal Process-ing, vol. 32, no. 6, pp. 1243–1245, 1984.

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[14] T1E1.4/2000-013R3, “Very-high-speed digital subscriberlines (VDSL) metallic interface, part 3: Technical specificationof a multi-carrier modulation transceiver,” 2000.

[15] K. J. R. Liu, A.-Y. Wu, A. Raghupathy, and J. Chen,“Algorithm-based low-power and high-performance multi-media signal processing,” Proceedings of the IEEE, vol. 86, no.6, pp. 1155–1202, 1998, Special Issue on Multimedia Signal

Processing.

Tsun-Shan Chan was born in Chang-Hui,Taiwan, ROC, in 1973. He received hisM.S. degree in electrical engineering fromthe National Central University, Taiwan, in1998. During 1998–1999, he worked oncommunication applications in IndustrialTechnology Research Institute, Hsin-Chu,Taiwan. Since 1999, he has been serving asa system engineer of the video processingprojects in VXIS Technology Corporation.

Jen-Chih Kuo received his B.S. degree inelectrical engineering from the Nation Tai-wan University, Taiwan, in 2000. He is nowin the Graduate Institute of Electronics En-gineering of the same school. His researchinterests include VLSI architectures for DSPalgorithms, adaptive signal processing, anddigital communication systems.


An-Yeu (Andy) Wu received his B.S. de-gree from National Taiwan University in1987, and the M.S. and Ph.D. degrees fromthe University of Maryland, College Parkin 1992 and 1995, respectively, all in elec-trical engineering. During 1987–1989, heserved as a signal officer in the Army, Taipei,Taiwan, for his mandatory military service.During 1990–1995, he was a graduate teach-ing and research assistant with the Depart-ment of Electrical Engineering and Institute for Systems Researchat the University of Maryland, College Park. From August 1995 toJuly 1996, he was a Member of Technical Staff at AT&T Bell Labo-ratories, Murray Hill, NJ, working on high-speed transmission ICdesigns. From 1996 to July 2000, he was with the Electrical Engi-neering Department of National Central University, Taiwan. He iscurrently an Associate Professor with the Department of Electri-cal Engineering Department and Graduate Institute of ElectronicsEngineering of National Taiwan University, Taiwan. His researchinterests include low-power/high-performance VLSI architecturesfor DSP and communication applications, adaptive signal process-ing, and multirate signal processing.

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