Equalizer Variants for Discrete Multi-Tone (DMT ... · all i, no matrix inversion, but eigenvalue calculation required ⇒ high complexity due to extensive matrix operations ⇒ Lee,

Equalizer Variants for Discrete Multi-Tone(DMT) Modulation

Steffen Trautmann

Infineon Technologies Austria AG

Summer Academy @ Jacobs University, Bremen, 2007-07-03

Outline

1 Introduction to Multi-Carrier SystemsA Short History on Multi-Carrier SystemsThe Multi-Carrier Transmission Scheme

2 Discrete Multi-Tone ModulationThe Frequency-Domain Equalizer (FEQ)The Cyclic Prefix (CP)Inter-Symbol / Inter-Channel Interference (ISI/ICI)

3 ISI/ICI Suppressing Equalization MethodsThe Time-Domain Equalizer (TEQ)The Per-Tone Equalizer (PTEQ)The Generalized DMT Principle (GDMT)The Extended Per-Tone EqualizerOther Methods

4 Conclusions and Outlook

Steffen Trautmann DMT Equalization 2007-07-03 2 / 45

Introduction to Multi-Carrier Short History

Introduction to Multi-Carrier (MC)Systems

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split a wide-band, frequency-selective channel into large numberof small-band, assumed flat subchannels for simplifiedequalizationrobust to small-band interferers and impulse noiseoptimal power and bit allocation according to channelcharacteristics → best suited for DSM


Introduction to Multi-Carrier Short History

A Short History on MC Systems

algorithms known since the 1960’sfirst Multi-Carrier modems in conventionalfrequency-multiplex technology: analog filters for bandseparation, poor bandwidth efficiencystaggered QAM (SQAM) systems: close to 100 percentbandwidth efficiency, overlap at f3dB → constant sum spectrum,interference to the direct neighbors, orthogonality due toalternating in-phase and quadrature modulation, less than 20subcarrierslarge number of subcarriers with Orthogonal Multi-Carrier(OMC): band pass filters with si-like spectrum, complexmodulated filterbank with rectangular prototype filter →OFDM, DMTDiscrete Wavelet Multi-Tone (DWMT): cosine-modulatedfilterbanks, inter-channel interference reduced to a minimum


Introduction to Multi-Carrier The Basic Scheme

The Multi-Carrier Transmission Schemediscrete MC system, P ≥ M

FM−1(z)↑ P

↑ P

↑ P

u1(k)

u0(k)

uM−1(k)

G0(z)

G1(z)

GM−1(z)

c(n)

r(n)

↓ P

↓ P

↓ P uM−1(k)

u1(k)

u0(k)F0(z)

F1(z)z−1

equivalent polyphase representation of the basis filters

↑ P

P/S-Converter

r(n)

y(n)

u0(k)↓ P

↓ P

↓ P

F′p(z)

uM−1(k)

u1(k)

uM−1(k)

GTp (z)

u0(k)

u1(k)z−1

z−1

z−1

↑ P

↑ Pv(n)

z−1

z−1

z−1

z−1

c(n)

S/P-Converter


Introduction to Multi-Carrier The Basic Scheme

The MC Transmission Scheme (cont’d)→ Combining polyphase representation of channel impulse

response and P/S, S/P converters to circular polyphase matrixC(z) plus additional delay

C(z) =

26664C0(z) z−1CP−1(z) · · · z−1C1(z)C1(z) C0(z) z−1C2(z)

.... . .

...CP−1(z) CP−2(z) · · · C0(z)

37775→ Introducing redundancy: P = M + L with L ≥ 0 (L = 0 → critical

sampling)

u0(k)

u1(k)

uM−1(k)u(k) v(k)

v0(k)

v1(k)

vP−1(k)

z−1C(z)

u0(k)

u1(k)

uM−1(k)u(k)

yP−1(k)

y0(k)

y(k)

F(z)

r0(k)

y1(k)

r1(k)

rP−1(k)

G(z)


DMT The Frequency-Domain Equalizer (FEQ)

Discrete Multi-Tone (DMT) ModulationMulti-Carrier transmission scheme based on IDFT/DFT → basicstructure similar to Orthogonal Frequency Division Multiplexing(OFDM) for wireless transmission

CPCP

channeltransmission

add discard

u(k)

v0(k)

vM+Lg−1(k)

vLg(k)

vLg−1(k)

vLg+1(k)

C

νM−1(k)

ν1(k)

ν0(k)yLg(k)

yLg−1(k)

y0(k)

yM+Lg−1(k)

yLg+1(k)

ED

u0(k)

u1(k)

uM−1(k)

FG

u0(k)

u1(k)

uM−1(k)

u(k)

eM−1,M−1

e11

e00

IDFT

P/S

conv

erte

r

c(n)

DFT

S/P

conv

erte

r

application examples: ADSLx, VDSLx, WLAN, 3G, DVB-T, DAB,UWB, Powerline, ...


DMT The Cyclic Prefix (CP)

The Cyclic Prefix (CP)

repeat last Lg samples at the beginning of the symbol

nn

y(k)

Lc−10

n

v(n) c(n) y(n)

y(k +1)y(k−1)v(k +1)v(k)v(k−1)

∗ =

Intersymbol InterferenceCyclic Prefix

Perfect equalization in a noise-free environment, if

Lg ≥ Lc − 1

(Lg - length of cyclic prefix, Lc - length of channel impulse response)

Strong Intersymbol and Intercarrier Interference (ISI/ICI) ifabove criterion is not fulfilled


DMT ISI/ICI

Transfer function of a DMT system

CPCP

channeltransmission

add discard

u(k)

v0(k)

vM+Lg−1(k)

vLg(k)

vLg−1(k)

vLg+1(k)

C

νM−1(k)

ν1(k)

ν0(k)yLg(k)

yLg−1(k)

y0(k)

yM+Lg−1(k)

yLg+1(k)

ED

u0(k)

u1(k)

uM−1(k)

FG

u0(k)

u1(k)

uM−1(k)

u(k)

eM−1,M−1

e11

e00

IDFT

P/S

conv

erte

r

c(n)

DFT

S/P

conv

erte

r

→ transfer function in polyphase and block matrix notation:

U(z) = E(z) · F(z) · z−1 · C(z) ·G(z) ·U(z)•|◦

u(k) = E · F · C ·G · u(n)(k− 1)

(n – number of interfering symbols)


DMT ISI/ICI

Inter-Symbol/Inter-Carrier Interference

u(k) = E ·H︷︸︸︷

F · C ·G︸︷︷︸T

·u(n)(k− 1)

Sufficient CP: Lg ≥ Lc − 1 Insufficient CP: Lg < Lc − 1

n = 1 → no ISIcyclic prefix → sequentialconvolution with channelimpulse response becomesvirtually cyclic → no ICI

n > 1 → (severe) ISIno longer cyclic convolution→ (severe) ICI


DMT ISI/ICI

ISI/ICI cont’dEqualizer Matrix for

Sufficient CPEqualizer Matrix for

Insufficient CP

T = E ·H = E ·D != I

ν0(k)

E

ν1(k)

νM−1(k)

e00

e11

eM−1,M−1

uM−1(k)

u1(k)

u0(k)

D diagonal matrix,solution E = D−1

witheii = 1/C(e j2πi/M)Efficientequalization withonly one complexmultiplication pertone

T = E ·H != [ 0 · · · 0 I 0 · · · 0︸︷︷︸n

]

Solution onlyif E takesneighboringsymbols intoaccountE very bigand no sparsestructure

?E

u0(k)

uM−1(k)νM−1(k−1)

ν0(k−1)

νM−1(k)

ν0(k)

u1(k)


DMT ISI/ICI

Extracting ISI/ICI Part from ChannelMatrix C

Assumption: no Cyclic Prefix, Lc ≤ M, no pre-cursor→ ISI from the preceding symbol

=

Ccycl

C1C0

C

C0 +C1 +

Cerr

C0 −C0

c(n)

u(k−2)

u(k−1)G

CFEu(k)

H


DMT ISI/ICI

Extracting ISI/ICI Part (cont’d)

⇒ C can be split into ideal, cyclic part Ccycl and ISI/ICIerror part Cerr

C = [C0 C1] = [0M (C0 + C1)]︸︷︷︸Ccycl

+ [C0 (−C0)]︸︷︷︸Cerr

⇒ Substitution into transfer function

E ·H = E · F · C ·G = E · F · (Ccycl + Cerr) ·G != [ 0 I ]


DMT ISI/ICI


⇒ With Ccycl,red = C0 + C1 and G′ as a diagonal block inG, separation into two sub-systems possible

I: E · F · Ccycl,red ·G′ != I

II: E · F · Cerr ·G != [ 0 0 ]

→ Equation system I describes an ideal, distortion-freeDMT system

→ Equation system II eliminates ISI/ICI caused by Cerr


ISI/ICI Suppression TEQ

The Time-Domain Equalizer (TEQ)FIR filter applied before receiver FFT to shorten effective lengthof the channel impulse response

A/D DFTChannelc(n)c(t) c(n)∗b(n)

TEQ

Receiver

0 200 400 600 800−2

−1

0

1

2

3

4

5

6x 10

−3

Sample Index n

c(n) ⇒

0 200 400 600 800−0.01

−0.005

0

0.005

0.01

0.015

Sample Index n

c(n)

∗b(n

)filters at sampling rate → complexity!mandatory for at least ADSL, determines performance forshorter loopsoptimal adaptation of the TEQ coefficients in the SNR sense toocostly



TEQ - MMSE Method

B(z)

A(z)z−∆X(z)

Noise N(z)

X(z)

Delay

ChannelC(z) = z−∆ A(z)

B(z)

z−∆

E(z) → minRemaining Error

TIR

TEQ

adopt delay ∆, TEQ filter B(z) and Target Impulse Response(TIR) A(z) to minimize the minimum mean square errorbetween the two branches MSE = E[|e(k)|2] with e(k) ◦−−• E(z)first introduced by Falconer and Magee (1973) for channelshortening of a ML receiver



TEQ - MMSE, VariantsChow and Cioffi (1992) first applied for MC, to prevent trivialsolution Unit-Tap Constraint (UTC) was introduced:

aTei = 1 with a = [a0 a1 · · · aLTIR−1]T

UTC requires search over all i, matrix inversions andmultiplications

Al-Dhahir and Cioffi (1996) showed that the Unit-EnergyConstraint (UEC):

aTa = 1

always leads to equal or smaller MSE than UTC, no search overall i, no matrix inversion, but eigenvalue calculation required

⇒ high complexity due to extensive matrix operations⇒ Lee, Chow and Cioffi (1995) proposed circulant correlation

matrices → matrix operations using DFT/IDFT, but LTIR ≥ LTEQ



TEQ - MMSE, Adaptation Methods

Falconer, Magee, 1973: proposed LMS for adaptation, lowcomplexity but slowChow, Cioffi, Bingham, 1993: Frequency-Domain LMS andFrequency Domain Division, alternating between FD and TD,still slow convergenceMelsa, Younce, Rohrs, 1996: ARMA modelling of the channel,LS or RLS solution, efficient variant: translation to 2-channel ARmodel and solved with Levinson-Wiggins-Robinson (LWR)algorithmWang, Adali, 1999: solve for MSE completely in the frequencydomain, weight factor for each tone to exclude unused tonesfrom optimization



TEQ - MSSNR Method

effective channel impulse response incl. TEQceff(n) = c(n) ∗ hTEQ(n) → ceff = [c0 c1 · cLc+LTEQ−1]T split intokernel segment cwin and pre and post cursor cwall

minimize ISI contributing energy cTwallcwall, i.e. maximize the

Shortening SNR

SSNR [dB] = 10 · log10cT

wincwin

cTwallcwall

originally proposed by Melsa, Younce and Rohrs (1996), optimalsolutionoptimal solution, involves Cholesky decomposition, matrixinversion, Eigenvalue calculationnot suited for application, serves as a reference



TEQ - MSSNR, DCC and DCM

based on Divide-and-Conquer Ansatz, introduced by Lu, Clark,Arslan and Evans (2000)Divide-and-Conquer: TEQ filter of length LTEQ is factorized intoLTEQ − 1 filters wi of length 2

wi = [1, gi]T

gi’s iteratively optimized using two different approaches:Divide-and-Conquer by Cancellation (DCC) andDivide-and-Conquer by Minimization (DCM)nearly optimal solutioncomputationally efficient, suitable for practical implementationodd behavior: energy concentration in first TEQ filter taps



TEQ - MGSNR Methodbit rate of a DMT system:

bDMT =N−1∑i=0

log2

(1 +

SNRi

Γ

)→ bDMT = N log2

(1 +

SNRΓ

)

with SNR = Γ

[N−1∏i=0

(1 +

SNRi

Γ

)] 1N

− 1

≈ GSNR

maximize geometric mean GSNR of the SNRi over all used tonesAl-Dhahir and Cioffi (1996) proposed suboptimal solution (dueto some approximations)Henkel, and Kessler (2000) take external noise into account,optimal TIR may be longer than cyclic prefixArslan, Evans and Kaiei (2001) derive more efficientMinimum-ISI method from nonlinear optimization problem

⇒ all MGSNR methods far too complex for implementation



TEQ - ”Recent” Methods

Arslan, Lu, Clark, Evans, 2001: (Modified) Matrix pencil designmethodFarhang-Boroujeny, Ding, 2001: Eigen approachMartin, Johnson, Ding, Evans, 2003: Symmetric maximumshortening SNR

⇒ for further information have a look at:http://users.ece.utexas.edu/ bevans/projects/adsl/dmtteq/dmtteq.html


ISI/ICI Suppression PTEQ

The Per-Tone Equalizer [Van Acker et al]

M-pointFFT

P

z-1

z-1

z-1

z-1

P P

z-1

z-1

w0,1

wn,1

w0,2 w0,T

wn,Twn,2

M/2

T-1

M

Py kP+L(( )g

y k+1 P(( ) -1)

P=M+Lg

transfer T-tap TEQ operation to FD→ requires T-times sliding FFT→ efficient realization with single FFT and subsequent ”difference”

termsseparate T-tap FEQ for each tone→ increased memory requirements

similar complexity compared to TEQSteffen Trautmann DMT Equalization 2007-07-03 30 / 45

ISI/ICI Suppression PTEQ

PTEQ (cont’d)N-times more coefficients to be initialized → ”tone grouping”efficient RLS-based method initialization method proposed, LMSfor updates during runtimelarger T always improves performanceinsensitive to delay parameter

IEEE Communications Magazine • May 2000 109

are the unit norm (for TEQ or TIR) or unit tapconstraints.

While the TEQ-based DMT-receiver struc-ture (Fig. 3) appears to have become the mostpopular receiver structure for digital subscriberline (DSL) applications, it is still recognized thatthe optimal TEQ parameter design problem is adifficult one, largely unsolved. Evidently, there isno direct relation between this time domainmean-square-error (MSE)-optimal channelshortening and the envisaged transmission capac-ity optimization. In particular, this often leads toodd behavior:• The least one can expect from the overall

procedure is that more equalizer coefficientslead to improved performance (i.e., highertransmission capacity). This appears not tobe the case with the MSE-based TEQ ini-tialization. The outcome (transmissioncapacity) is unpredictable, and sometimes,shorter equalizers are found to give betterresults. This already indicates that anexhaustive search and/or assisting heuristicsare called for.

• An important design parameter is the so-called synchronization delay (∆). When anFIR channel with impulse response [h0, h1,h2, h3, …] needs to be shortened to x nonze-ro taps, one can aim at an impulse response[b0, b1, …, bx-2, bx-1, 0, 0, 0, …] with a zerosynchronization delay, or one can aim at [0,b1, b2, …, bx-1, bx, 0, 0, …] or [0, 0, b2, b3,…, bx, bx+1, 0, …], and so on (i.e., with anonzero synchronization delay). For eachchoice of synchronization delay, one cancompute an MSE-optimal TEQ, and thencompare transmission capacities for theseTEQs. Figure 5 shows a typical plot (forADSL downstream transmission) of capaci-ty vs. synchronization delay for a fairly largerange of delays (dotted line). Clearly, theMSE-based TEQ initialization shows verynonsmooth behavior, where an optimal set-ting for the synchronization delay is noteasily selected beforehand. Again, this indi-cates that an exhaustive search (over a largerange of delays) is called for. Unfortunate-ly, such an exhaustive search generally can-not be afforded in a short modem startupprocedure. Mostly, an educated guess ismade of the optimal synchronization delay,and then only one TEQ is computed andimplemented, the outcome obviously beingvery much dependent on the initial guess.

• Whereas it can be shown that the unit normconstraint yields the minimal MSE, otherconstraints that result in a larger MSE cangive higher capacity. In particular, imposingconstraints on the behavior of the TIR atcarriers not used for data transmission canimprove performance.The all-frequency-domain equalization

approach presented next shows much smootherand predictable behavior (see, e.g., the full linein Fig. 5). It is based on a separate MSE opti-mization for each tone. A small MSE for a par-ticular tone generally corresponds to a largetransmission capacity for this tone. Per-tone ini-tialization therefore corresponds much better tooverall capacity optimization, as witnessed by its

not-odd behavior, where, for example, moreequalizer taps always lead to improved perfor-mance.

FREQUENCY DOMAIN EQUALIZATIONPresent-day DSL receiver designs are based onthe TEQ approach presented in the previoussection. However, this approach is found toexhibit some deficiencies, leaving some room forimprovement. In this section an alternativereceiver structure is described based on so-calledper-tone equalization, where an MSE optimiza-tion is performed for each carrier separately,leading to improved as well as more predictableand reproducible performance. Even though inthis structure every single tone is given its owndistinct and optimal equalizer, the computation-al requirement is roughly kept at the same level(it can even be smaller than the computationalrequirement in a TEQ-based modem), providedan appropriate initialization scheme (based onequalizer training) is included. The memoryrequirement obviously increases significantly, butthis is not considered a major implementation

� Figure 4. A model of TEQ and TIR.

CIR, hk

Noise

TEQ, wk ek

Delay ∆ TIR, bk

� Figure 5. TEQ and FEQ performance.

–30–40

Rate

(b/

s)

Delay ∆ (samples)

0.5

0

1

1.5

2

2.5

3

–20 –10 0 10 20 30

FEQ 8FEQ 32TEQ 32

x 106


ISI/ICI Suppression GDMT


⇒ With Ccycl,red = C0 + C1 and G′ as a diagonal block inG, separation into two sub-systems possible

I: E · F · Ccycl,red ·G′ != I

II: E · F · Cerr ·G != [ 0 0 ]

→ Equation system I describes an ideal, distortion-freeDMT system

→ Equation system II eliminates ISI/ICI caused by Cerr



Decomposing Equalizer Matrix E

Under assumption that K tones notused decomposition of equalizermatrix into E = E1 + E0

E1,red E0,red

= +

E1 E0E

K = M−N

unus

ed

used

unus

ed

used

N used carriersK unused carriers

unused

used

unused

used

→ E0 contains K columnsof unused tone outputsamples

→ E1 containsN = M− K columnsof used tone outputsamples



Perfect Equalization Condition

⇒ After elimination of zero rows and columns equationsystem I independent from E0

E1,red · F1,red · Ccycl,red ·G′red︸︷︷︸

Dred

!= IN

solution for I with E1,red = D−1red → deg. of freedom in

E0,red used for solving II !

⇒ Solution independent from channel frequencyresponse at the unused carrier positions!



Perfect Equalization Condition (cont’d)

Solution for II exists if

K + Lg ≥ Lc − 1

⇒ combination of TD and FD redundancy → may bearbitrarily distributed

Special cases:→ K = 0: Traditional DMT/OFDM without usage of FD

redundancy→ Lg = 0: No cyclic prefix, but symbol-separate, perfect

equalization!



Generalized DMT [Trautmann et al]Transmitter similar to DMT, Receiver withslightly extended Equalizer → SparseMatrix ERedundancy may be arbitrarily distributedto either time-domain (length of cyclicprefix), or frequency domain (number ofunused subcarriers)

unus

ed

used

unus

ed

used

unused

used

unused

used

E

GIGI

channeltransmission

discardadd

u(k)

v0(k)

vM+Lg−1(k)

vLg(k)

vLg−1(k)

vLg+1(k)

C

yLg(k)

yLg−1(k)

y0(k)

yM+Lg−1(k)

yLg+1(k)

FG

u2(k)

u3(k)

0

0

0

e20

e21

e22

e33

E

eM−2,M−1

e2,M−1νM−1(k)

ν1(k)

ν0(k)

eM−2,M−2

u3(k)

u2(k)

uM−2(k)

eM−2,1

eM−2,0

u(k)

uM−2(k)

IDFT

P/S

conv

erte

r

c(n)

DFT

S/P

conv

erte

r



Complexity Reduction for GDMT?SVD decompositionOnly meaningful for the branches from the measurement tones

Singular ValueDecomposition

E0,red

VrUr

Sr

unused

desunu ∗

K/2

K/2

N/2N/2

r r

K

matrix Ered

GDMT equalizersymbol x(k)

equalizedv(k)

DFT output

Krr

N/2 · K mults.E0,red = U · S · V

r · (N/2+K) mults.

Complexity reduced to r·(N/2+K)N/2·K (tends to r/K for N � K)


ISI/ICI Suppression Extended PTEQ

The Extended PTEQ [Vanbleu et al]extra FD redundancy with TN nulltones = TZ zero tones + TPpilot tones

M-pointFFT

P

z-1

z-1

z-1

z-1

P P

z-1

z-1

w0,1

wn,1

w0,2 w0,T

wn,Twn,2

T-1

M

Pw0, +1T w0,T+Tn

wn,T+Tnwn T, +1

w0, + +1T Tn w0,T+Tn+Tp

wn,T+Tn+Tpwn T Tn, + +1

Yk

N(1)

Y Tk N

N( )

Uk

P(1) U Tk P

P( )

T-1T-1T-1

Tp pilot symbols

TN nulltones

M/2

P=M+Lg

y k+1 P(( ) -1)

y kP+L(( )g

perfect equalization for IIR channels C(z) = A(z)B(z) =

PLA−1l=0 alz−lPLB−1l=0 blz−l

if

TN + Lg ≥ LA − 1 and T ≥ LB


ISI/ICI Suppression Other Methods

Other MethodsWindowing/Pulse Shaping

shape the rectangular-windowed DMT symbols at the edges toimprove selectivitywith some extra TD redundancy orthogonality can be keptalso helps for RFI cancellation and echo suppression at the US/DSband edgesstandardized for VDSLx

Alternative Transform Basesoverlapping basis filters → pulse shapingexample: Cos-Modulated Filterbanks (CMFB) → DWMT, ”wDSL”

S/P

DC

T

P/SP

AM

{L zeros

data

u0

u1

uM -1

u( )k

v( )k

S/P

ZF

/MM

SE

equal

izat

ion

IPA

M

P/S

data

c n( )

r n( )

channel

y( )k u( )k^

vP -1

v0

v1

poly

phas

efi

lter

s

redundant CosFB

superior ISI/ICI robustness, even without cyclic prefixSteffen Trautmann DMT Equalization 2007-07-03 42 / 45


Other Methods (cont’d)MIMO Equalizer

S/P

IDF

T P/S

QA

M

{L samplesdata

u0

u1

uM -1

u( )k v( )k

c n( )

r n( )

channel

û1

S/P

MM

SE

equal

izer

y( )k

IQA

M

P/S

data

u( )k^Fp’ z( )

ûM -1

û0

insufficient or no cyclic prefix, combine inverse transform andequalizer into large rectangular matrixno sparse structure when combined with DMT transmitter

Fractional MIMO Equalizerinstead of symbol-wise FFT, apply sliding FFT to the RX signalFFT outputs at sampling rate, followed by MIMO equalizer →special case: PTEQsimilar to bank of N parallel filters → SIMO structure



Other Methods (cont’d)DFE MIMO Equalizer

instead of cyclic prefix, use Decision-Feedback Equalizer (DFE) forfull equalizationdifficult to apply for DMT, since decoding after the FFTefficient structures proposed by Al-Dhahir and Cioffi (1995-1997)

Redundant Filterbanksgeneral zero-forcing filterbanks with introduced redudancychannel impulse response must be shorter than P = M + Lcomplexity

Transmitter Pre-Codingpre-distort transmit symbol according to channel characteristics tosimplify equalization at the receiver sideTomlinson-Harashima Pre-coding (THP) adopted for DMT withinsufficient cyclic prefix by Cheong and Cioffi (1998)increased transmit power, increased Crestfactor


Conclusions and Outlook

Conclusions and Outlook

DFT as the transform base for DMT/OFDM was notthe optimal choice in terms of spectral containment ofthe subchannels → Cyclic Prefix was introducedfurther extensions to the original scheme (TEQ,Windowing, ...) necessary to improve spectralselectivity of the subchannels and thus to reduceISI/ICIextended FEQ methods like GDMT and PTEQ areable to also incorporate other tasks like RFIsuppression, Echo cancellation, Windowing,Crestfactor reduction, ...

⇒ joint optimization to reduce the amount of requiredredundancy


Documents

Equalizer Variants for Discrete Multi-Tone (DMT ... · all i, no matrix inversion, but eigenvalue calculation required ⇒ high complexity due to extensive matrix operations ⇒ Lee,