equalizer variants for discrete multi-tone (dmt ... · all i, no matrix inversion, but eigenvalue...
TRANSCRIPT
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
Steffen Trautmann DMT Equalization 2007-07-03 4 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 5 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 7 / 45
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)
Steffen Trautmann DMT Equalization 2007-07-03 8 / 45
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, ...
Steffen Trautmann DMT Equalization 2007-07-03 10 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 12 / 45
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)
Steffen Trautmann DMT Equalization 2007-07-03 14 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 15 / 45
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)
Steffen Trautmann DMT Equalization 2007-07-03 16 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 17 / 45
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 ]
Steffen Trautmann DMT Equalization 2007-07-03 18 / 45
DMT ISI/ICI
Extracting ISI/ICI Part (cont’d)
⇒ 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
Steffen Trautmann DMT Equalization 2007-07-03 19 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 21 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 22 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 23 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 24 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 25 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 26 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 27 / 45
ISI/ICI Suppression TEQ
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
Steffen Trautmann DMT Equalization 2007-07-03 28 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 31 / 45
ISI/ICI Suppression GDMT
Extracting ISI/ICI Part (cont’d)
⇒ 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
Steffen Trautmann DMT Equalization 2007-07-03 33 / 45
ISI/ICI Suppression GDMT
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
Steffen Trautmann DMT Equalization 2007-07-03 34 / 45
ISI/ICI Suppression GDMT
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!
Steffen Trautmann DMT Equalization 2007-07-03 35 / 45
ISI/ICI Suppression GDMT
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!
Steffen Trautmann DMT Equalization 2007-07-03 36 / 45
ISI/ICI Suppression GDMT
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
Steffen Trautmann DMT Equalization 2007-07-03 37 / 45
ISI/ICI Suppression GDMT
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)
Steffen Trautmann DMT Equalization 2007-07-03 38 / 45
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
Steffen Trautmann DMT Equalization 2007-07-03 40 / 45
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
ISI/ICI Suppression Other Methods
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
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ISI/ICI Suppression Other Methods
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
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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
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