Equalizer Variants for Discrete Multi-Tone (DMT) Modulation

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1 Equalizer Variants for Discrete Multi-Tone (DMT) Modulation Steffen Trautmann Infineon Technologies Austria AG Summer Jacobs University, Bremen,

2 Outline 1 Introduction to Multi-Carrier Systems A Short History on Multi-Carrier Systems The Multi-Carrier Transmission Scheme 2 Discrete Multi-Tone Modulation The Frequency-Domain Equalizer (FEQ) The Cyclic Prefix (CP) Inter-Symbol / Inter-Channel Interference (ISI/ICI) 3 ISI/ICI Suppressing Equalization Methods The Time-Domain Equalizer (TEQ) The Per-Tone Equalizer (PTEQ) The Generalized DMT Principle (GDMT) The Extended Per-Tone Equalizer Other Methods 4 Conclusions and Outlook Steffen Trautmann DMT Equalization / 45

3 Introduction to Multi-Carrier Short History Introduction to Multi-Carrier (MC) Systems u 0(t) g 0(t) f 0(t) ^u 0(t) u 1(t) g 1(t) r(t) f 1(t) ^u 1(t) v(t) c(t) y(t) u M 1(t) g M 1(t) f M 1(t) ^u M 1(t) split a wide-band, frequency-selective channel into large number of small-band, assumed flat subchannels for simplified equalization robust to small-band interferers and impulse noise optimal power and bit allocation according to channel characteristics best suited for DSM Steffen Trautmann DMT Equalization / 45

4 Introduction to Multi-Carrier Short History A Short History on MC Systems algorithms known since the 1960 s first Multi-Carrier modems in conventional frequency-multiplex technology: analog filters for band separation, poor bandwidth efficiency staggered QAM (SQAM) systems: close to 100 percent bandwidth efficiency, overlap at f 3dB constant sum spectrum, interference to the direct neighbors, orthogonality due to alternating in-phase and quadrature modulation, less than 20 subcarriers large number of subcarriers with Orthogonal Multi-Carrier (OMC): band pass filters with si-like spectrum, complex modulated filterbank with rectangular prototype filter OFDM, DMT Discrete Wavelet Multi-Tone (DWMT): cosine-modulated filterbanks, inter-channel interference reduced to a minimum Steffen Trautmann DMT Equalization / 45

5 Introduction to Multi-Carrier The Basic Scheme The Multi-Carrier Transmission Scheme discrete MC system, P M u 0 (k) u 1 (k) P P G 0 (z) G 1 (z) c(n) r(n) z 1 F 0 (z) F 1 (z) P P û 0 (k) û 1 (k) u M 1 (k) P G M 1 (z) F M 1 (z) P û M 1 (k) equivalent polyphase representation of the basis filters u 0 (k) u 1 (k) G T p (z) P P r(n) v(n) c(n) z 1 z 1 y(n) P z 1 P z 1 F p (z) û 0 (k) û 1 (k) u M 1 (k) P z 1 z 1 z 1 P û M 1 (k) P/S-Converter S/P-Converter Steffen Trautmann DMT Equalization / 45

6 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 matrix C(z) plus additional delay 2 C 0(z) z 1 C P 1 (z) z 1 3 C 1 (z) C 1 (z) C 0(z) z 1 C 2(z) C(z) = C P 1 (z) C P 2 (z) C 0(z) Introducing redundancy: P = M + L with L 0 (L = 0 critical sampling) r 0 (k) v 0 (k) v 1 (k) y 0 (k) y 1 (k) u 0 (k) r 1 (k) û 0 (k) u 1 (k) G(z) z 1 C(z) F(z) û 1 (k) r P 1 (k) u M 1 (k) u(k) v P 1 (k) v(k) y P 1 (k) y(k) û M 1 (k) û(k) Steffen Trautmann DMT Equalization / 45

7 DMT The Frequency-Domain Equalizer (FEQ) Discrete Multi-Tone (DMT) Modulation Multi-Carrier transmission scheme based on IDFT/DFT basic structure similar to Orthogonal Frequency Division Multiplexing (OFDM) for wireless transmission u 0(k) u 1(k) IDFT v 0(k) add CP v Lg 1(k) v Lg (k) v Lg+1(k) P/S converter c(n) transmission channel S/P converter y 0(k) discard y Lg 1(k) CP y Lg (k) y Lg+1(k) DFT ν 0(k) ν 1(k) e00 e11 û 0(k) û 1(k) u M 1(k) v M+Lg 1(k) y M+Lg 1(k) ν M 1(k) em 1,M 1 û M 1(k) u(k) G C F E D û(k) application examples: ADSLx, VDSLx, WLAN, 3G, DVB-T, DAB, UWB, Powerline,... Steffen Trautmann DMT Equalization / 45

8 DMT The Cyclic Prefix (CP) The Cyclic Prefix (CP) repeat last L g samples at the beginning of the symbol v(k 1) v(k) v(k+ 1) v(n) c(n) y(n) y(k 1) y(k) y(k+ 1) n n 0 = Lc 1 n Cyclic Prefix Intersymbol Interference Perfect equalization in a noise-free environment, if L g L c 1 (Lg - length of cyclic prefix, Lc - length of channel impulse response) Strong Intersymbol and Intercarrier Interference (ISI/ICI) if above criterion is not fulfilled Steffen Trautmann DMT Equalization / 45

9 DMT ISI/ICI Transfer function of a DMT system v0(k) y0(k) u0(k) add CP vlg 1(k) vlg (k) discard ylg 1(k) CP ylg (k) ν0(k) e00 û0(k) u1(k) IDFT vlg+1(k) P/S converter c(n) transmission channel S/P converter ylg+1(k) DFT ν1(k) e11 û1(k) um 1(k) vm+lg 1(k) ym+lg 1(k) em 1,M 1 νm 1(k) ûm 1(k) u(k) G C F E D û(k) transfer function in polyphase and block matrix notation: Û(z) = E(z) F(z) z 1 C(z) G(z) U(z) û(k) = E F C G u (n) (k 1) (n number of interfering symbols) Steffen Trautmann DMT Equalization / 45

10 DMT ISI/ICI Inter-Symbol/Inter-Carrier Interference H {}}{ û(k) = } E F {{ C G} u (n) (k 1) T Sufficient CP: L g L c 1 Insufficient CP: L g < L c 1 n = 1 no ISI cyclic prefix sequential convolution with channel impulse response becomes virtually cyclic no ICI n > 1 (severe) ISI no longer cyclic convolution (severe) ICI Steffen Trautmann DMT Equalization / 45

11 DMT ISI/ICI ISI/ICI cont d Equalizer Matrix for Sufficient CP Equalizer Matrix for Insufficient CP ν 0 (k) ν 1 (k) e 00 e 11 ν M 1 (k) T = E H = E D! = I e M 1,M 1 E û 0 (k) û 1 (k) û M 1 (k) D diagonal matrix, solution E = D 1 with e ii = 1/C(e j2πi/m ) Efficient equalization with only one complex multiplication per tone T = E H =! [ } 0 0 {{ I 0 0 } ] n Solution only if E takes neighboring symbols into account E very big and no sparse structure ν 0 (k) ν M 1 (k) ν 0 (k 1) ν M 1 (k 1)? E û 0 (k) û 1 (k) û M 1 (k) Steffen Trautmann DMT Equalization / 45

12 DMT ISI/ICI Extracting ISI/ICI Part from Channel Matrix C Assumption: no Cyclic Prefix, L c M, no pre-cursor ISI from the preceding symbol û(k) E F C H G u(k 2) u(k 1) c(n) C 0 C 1 = C 0 + C 1 + C 0 C 0 C C cycl C err Steffen Trautmann DMT Equalization / 45

13 DMT ISI/ICI Extracting ISI/ICI Part (cont d) C can be split into ideal, cyclic part C cycl and ISI/ICI error part C err C = [C 0 C 1 ] = [0 M (C 0 + C 1 )] + [C }{{} 0 ( C 0 )] }{{} C cycl Substitution into transfer function E H = E F C G = E F (C cycl + C err ) G! = [ 0 I ] C err Steffen Trautmann DMT Equalization / 45

14 DMT ISI/ICI Extracting ISI/ICI Part (cont d) With C cycl,red = C 0 + C 1 and G as a diagonal block in G, separation into two sub-systems possible I: E F C cycl,red G! = I II: E F C err G! = [ 0 0 ] Equation system I describes an ideal, distortion-free DMT system Equation system II eliminates ISI/ICI caused by C err Steffen Trautmann DMT Equalization / 45

15 TEQ The Time-Domain Equalizer (TEQ) FIR filter applied before receiver FFT to shorten effective length of the channel impulse response Channel c(t) Receiver A/D c(n) TEQ c(n) b(n) DFT 6 x c(n) Sample Index n c(n) b(n) Sample Index n filters at sampling rate complexity! mandatory for at least ADSL, determines performance for shorter loops optimal adaptation of the TEQ coefficients in the SNR sense too costly Steffen Trautmann DMT Equalization / 45

16 TEQ TEQ - MMSE Method Channel A(z) C(z) = z B(z) Noise N(z) TEQ ˆB(z) X(z) Remaining Error E(z) min Delay z z X(z) TIR Â(z) adopt delay, TEQ filter ˆB(z) and Target Impulse Response (TIR) Â(z) to minimize the minimum mean square error between the two branches MSE = E[ e(k) 2 ] with e(k) E(z) first introduced by Falconer and Magee (1973) for channel shortening of a ML receiver Steffen Trautmann DMT Equalization / 45

17 TEQ TEQ - MMSE, Variants Chow and Cioffi (1992) first applied for MC, to prevent trivial solution Unit-Tap Constraint (UTC) was introduced: â T e i = 1 with â = [â 0 â 1 â LTIR 1] T UTC requires search over all i, matrix inversions and multiplications Al-Dhahir and Cioffi (1996) showed that the Unit-Energy Constraint (UEC): â T â = 1 always leads to equal or smaller MSE than UTC, no search over all 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 L TIR L TEQ Steffen Trautmann DMT Equalization / 45

18 TEQ TEQ - MMSE, Adaptation Methods Falconer, Magee, 1973: proposed LMS for adaptation, low complexity but slow Chow, Cioffi, Bingham, 1993: Frequency-Domain LMS and Frequency Domain Division, alternating between FD and TD, still slow convergence Melsa, Younce, Rohrs, 1996: ARMA modelling of the channel, LS or RLS solution, efficient variant: translation to 2-channel AR model and solved with Levinson-Wiggins-Robinson (LWR) algorithm Wang, Adali, 1999: solve for MSE completely in the frequency domain, weight factor for each tone to exclude unused tones from optimization Steffen Trautmann DMT Equalization / 45

19 TEQ TEQ - MSSNR Method effective channel impulse response incl. TEQ c eff (n) = c(n) h TEQ (n) c eff = [c 0 c 1 c Lc+L TEQ 1] T split into kernel segment c win and pre and post cursor c wall minimize ISI contributing energy c T wall c wall, i.e. maximize the Shortening SNR SSNR [db] = 10 log 10 c T win c win c T wall c wall originally proposed by Melsa, Younce and Rohrs (1996), optimal solution optimal solution, involves Cholesky decomposition, matrix inversion, Eigenvalue calculation not suited for application, serves as a reference Steffen Trautmann DMT Equalization / 45

20 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 L TEQ is factorized into L TEQ 1 filters w i of length 2 w i = [1, g i ] T g i s iteratively optimized using two different approaches: Divide-and-Conquer by Cancellation (DCC) and Divide-and-Conquer by Minimization (DCM) nearly optimal solution computationally efficient, suitable for practical implementation odd behavior: energy concentration in first TEQ filter taps Steffen Trautmann DMT Equalization / 45

21 TEQ TEQ - MGSNR Method bit rate of a DMT system: N 1 ( b DMT = log SNR ) i Γ i=0 with SNR = Γ [ N 1 i=0 ( ) b DMT = N log SNR Γ ( 1 + SNR ) ] 1 N i Γ 1 GSNR maximize geometric mean GSNR of the SNR i over all used tones Al-Dhahir and Cioffi (1996) proposed suboptimal solution (due to some approximations) Henkel, and Kessler (2000) take external noise into account, optimal TIR may be longer than cyclic prefix Arslan, Evans and Kaiei (2001) derive more efficient Minimum-ISI method from nonlinear optimization problem all MGSNR methods far too complex for implementation Steffen Trautmann DMT Equalization / 45

22 TEQ TEQ - Recent Methods Arslan, Lu, Clark, Evans, 2001: (Modified) Matrix pencil design method Farhang-Boroujeny, Ding, 2001: Eigen approach Martin, Johnson, Ding, Evans, 2003: Symmetric maximum shortening SNR for further information have a look at: bevans/projects/adsl/dmtteq/dmtteq.html Steffen Trautmann DMT Equalization / 45

23 PTEQ The Per-Tone Equalizer [Van Acker et al] T-1 z -1 z -1 P P y(( kp+l z -1 g ) z -1 M P M-point FFT w 0,1 w n,1 w 0,2 w 0,T M/2 w n,2 w n,t z -1 P=M+L g z -1 y(( k+1) P-1) P transfer T-tap TEQ operation to FD requires T-times sliding FFT efficient realization with single FFT and subsequent difference terms separate T-tap FEQ for each tone increased memory requirements similar complexity compared to TEQ Steffen Trautmann DMT Equalization / 45

24 l with impulse response [h 0, h 1, eeds to be shortened to x nonzecan aim at an impulse response x-2, b x-1, 0, 0, 0, ] with a zero tion delay, or one can aim at [0, x-1, b x, 0, 0, ] or [0, 0, b 2, b 3, PTEQ (cont d), 0, ], and so on (i.e., with a chronization delay). For each ynchronization delay, one can MSE-optimal TEQ, and then ansmission capacities for these re 5 shows a typical plot (for stream transmission) of capacionization delay for a fairly large lays (dotted line). Clearly, the TEQ initialization shows very behavior, where an optimal setsynchronization delay is not ed beforehand. Again, this indiexhaustive search (over a large lays) is called for. Unfortunatexhaustive search generally canrded in a short modem startup Mostly, an educated guess is optimal synchronization delay, nly one TEQ is computed and d, the outcome obviously being ependent on the initial guess. an be shown that the unit norm ields the minimal MSE, other that result in a larger MSE can capacity. In particular, imposing on the behavior of the TIR at used for data transmission can formance. equency-domain equalization ented next shows much smoother le behavior (see, e.g., the full line s based on a separate MSE optiach tone. A small MSE for a parenerally corresponds to a large apacity for this tone. Per-tone inirefore corresponds much better to y optimization, as witnessed by its exhibit some deficiencies, ISI/ICI Suppression leaving some room PTEQfor improvement. In this section an alternative receiver structure is described based on so-called per-tone equalization, where an MSE optimization is performed for each carrier separately, leading to improved as well as more predictable and reproducible performance. Even though in this structure every single tone is given its own distinct and optimal equalizer, the computational requirement is roughly kept at the same level (it can even be smaller than the computational requirement in a TEQ-based modem), provided an appropriate initialization scheme (based on equalizer training) is included. The memory requirement obviously increases significantly, but this is not considered a major implementation N-times more coefficients to be initialized tone grouping efficient RLS-based method initialization method proposed, LMS for updates during runtime larger T always improves performance insensitive to delay parameter Rate (b/s) x FEQ 8 FEQ 32 TEQ Delay (samples) Figure 5. TEQ and FEQ performance. Steffen Trautmann DMT Equalization / 45

25 GDMT Extracting ISI/ICI Part (cont d) With C cycl,red = C 0 + C 1 and G as a diagonal block in G, separation into two sub-systems possible I: E F C cycl,red G! = I II: E F C err G! = [ 0 0 ] Equation system I describes an ideal, distortion-free DMT system Equation system II eliminates ISI/ICI caused by C err Steffen Trautmann DMT Equalization / 45

26 GDMT Decomposing Equalizer Matrix E Under assumption that K tones not used decomposition of equalizer matrix into E = E 1 + E 0 unused used unused used unused used unused used E N used carriers K unused carriers K = M N = + E 1 E 0 E 1,red E 0,red E 0 contains K columns of unused tone output samples E 1 contains N = M K columns of used tone output samples Steffen Trautmann DMT Equalization / 45

27 GDMT Perfect Equalization Condition After elimination of zero rows and columns equation system I independent from E 0 E 1,red F 1,red C cycl,red G! red = I N }{{} D red solution for I with E 1,red = D 1 red deg. of freedom in E 0,red used for solving II! Solution independent from channel frequency response at the unused carrier positions! Steffen Trautmann DMT Equalization / 45

28 GDMT Perfect Equalization Condition (cont d) Solution for II exists if K + L g L c 1 combination of TD and FD redundancy may be arbitrarily distributed Special cases: K = 0: Traditional DMT/OFDM without usage of FD redundancy L g = 0: No cyclic prefix, but symbol-separate, perfect equalization! Steffen Trautmann DMT Equalization / 45

29 GDMT Generalized DMT [Trautmann et al] Transmitter similar to DMT, Receiver with slightly extended Equalizer Sparse Matrix E Redundancy may be arbitrarily distributed to either time-domain (length of cyclic prefix), or frequency domain (number of unused subcarriers) unused used unused used unused used unused E used 0 0 u2(k) u3(k) IDFT v0(k) add GI vlg 1(k) vlg (k) vlg+1(k) um 2(k) 0 vm+lg 1(k) P/S converter c(n) transmission channel S/P converter y0(k) discard ylg 1(k) GI ylg (k) ylg+1(k) ym+lg 1(k) DFT ν0(k) ν1(k) νm 1(k) e21 e22 e33 e20 em 2,M 2 e2,m 1 em 2,0 em 2,1 em 2,M 1 û2(k) û3(k) ûm 2(k) u(k) G C F E û(k) Steffen Trautmann DMT Equalization / 45

30 GDMT Complexity Reduction for GDMT? SVD decomposition Only meaningful for the branches from the measurement tones equalized symbol ˆx(k) GDMT equalizer matrix E red DFT output v(k) K/2 unused N/2 E 0,red N/2 Singular Value K Decomposition N/2 K mults. E 0,red = U S V r U r r S r r r (N/2 + K) mults. K V r r K/2 desunu Complexity reduced to r (N/2+K) N/2 K (tends to r/k for N K) Steffen Trautmann DMT Equalization / 45

31 Extended PTEQ The Extended PTEQ [Vanbleu et al] y(( kp+l z -1 g ) z -1 extra FD redundancy with T N nulltones = T Z zero tones + T P pilot tones P w 0,1 T-1 z -1 z -1 P P w 0,2 w 0,T P=M+L g w 0, T+1 w 0,T+Tn T p pilot symbols U kp (1) U P k ( TP) w 0, T+ Tn+1 w 0,T+Tn+Tp M/2 M M-point FFT w n,1 w n,2 w n,t w nt, +1 w n,t+tn w nt, + Tn+1 w n,t+tn+tp z -1 Y k N(1) z -1 y(( k+1) P-1) P N Y ( T ) k N T N nulltones perfect equalization for IIR channels C(z) = A(z) P LA 1 B(z) = a l=0 l z l P LB 1 l=0 b l z l if T N + L g L A 1 and T L B Steffen Trautmann DMT Equalization / 45

32 Other Methods Other Methods Windowing/Pulse Shaping shape the rectangular-windowed DMT symbols at the edges to improve selectivity with some extra TD redundancy orthogonality can be kept also helps for RFI cancellation and echo suppression at the US/DS band edges standardized for VDSLx Alternative Transform Bases overlapping basis filters pulse shaping example: Cos-Modulated Filterbanks (CMFB) DWMT, wdsl data S/P u 0 u 1 u M -1 u( k) PAM DCT redundant CosFB L zeros{ polyphase filters v 0 v 1 v P -1 v( k) P/S channel rn () cn () S/P ZF/MMSE equalization y( k) u( ^ k) superior ISI/ICI robustness, even without cyclic prefix Steffen Trautmann DMT Equalization / 45 IPAM P/S data

33 Other Methods Other Methods (cont d) MIMO Equalizer data S/P QAM L samples{ u 0 u 1 IDFT P/S channel rn () cn () S/P MMSE equalizer û 0 û 1 IQAM P/S data u M -1 û M -1 u( k) v( k) y( k) F p z () u( ^ k) insufficient or no cyclic prefix, combine inverse transform and equalizer into large rectangular matrix no sparse structure when combined with DMT transmitter Fractional MIMO Equalizer instead of symbol-wise FFT, apply sliding FFT to the RX signal FFT outputs at sampling rate, followed by MIMO equalizer special case: PTEQ similar to bank of N parallel filters SIMO structure Steffen Trautmann DMT Equalization / 45

34 Other Methods Other Methods (cont d) DFE MIMO Equalizer instead of cyclic prefix, use Decision-Feedback Equalizer (DFE) for full equalization difficult to apply for DMT, since decoding after the FFT efficient structures proposed by Al-Dhahir and Cioffi ( ) Redundant Filterbanks general zero-forcing filterbanks with introduced redudancy channel impulse response must be shorter than P = M + L complexity Transmitter Pre-Coding pre-distort transmit symbol according to channel characteristics to simplify equalization at the receiver side Tomlinson-Harashima Pre-coding (THP) adopted for DMT with insufficient cyclic prefix by Cheong and Cioffi (1998) increased transmit power, increased Crestfactor Steffen Trautmann DMT Equalization / 45

35 Conclusions and Outlook Conclusions and Outlook DFT as the transform base for DMT/OFDM was not the optimal choice in terms of spectral containment of the subchannels Cyclic Prefix was introduced further extensions to the original scheme (TEQ, Windowing,...) necessary to improve spectral selectivity of the subchannels and thus to reduce ISI/ICI extended FEQ methods like GDMT and PTEQ are able to also incorporate other tasks like RFI suppression, Echo cancellation, Windowing, Crestfactor reduction,... joint optimization to reduce the amount of required redundancy Steffen Trautmann DMT Equalization / 45

Infinite Length Results for Channel Shortening Equalizers

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