Data Detection and Soft-Kalman Filter Based Semi ...

Data Detection and Soft-Kalman Filter based Semi-Blind Channel Estimation Algorithms for MIMO-OFDM Systems Kyeong Jin Kim Tony Reid

Ronald A. Iltis

Nokia Research Center 6000 Connection Drive, Irving, TX 75039 Email: {Kyeong.J.Kim,Tony.Reid}@nokia.com

Abstract— In MIMO systems, where multiple antennas are used at both transmitter and receiver to achieve high spectral efficiency, channel impulse responses are often assumed to be constant over a block or packet. This assumption of block stationarity on channels is valid for most fixed wireless scenarios. However, for communications in a high mobility environment, the assumption will result in considerable performance degradation. In this paper, we focus on channel estimation for a MIMO system with OFDM transmission technique. In our system, pilots are placed on subcarriers for a novel channel estimation at the receiver with Kalman filters. With the channels estimated by a Kalman filter, we apply the OFDM MIMO soft data detector with a reasonable computational cost. The soft outputs of soft data detection are fed back to another soft Kalman filter for an improved channel estimation. By alternatively and iteratively using these two Kalman filters, a better overall performance can be obtained.

I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is one of the most competitive candidates for 4G networks. One important advantage of the OFDM transmission technique is that the intersymbol interference (ISI) can be removed if the channel delay spread is less than the inserted guard interval. Thus, a MIMO-OFDM system can achieve high data rates while providing better system performance by using both antenna diversity and frequency diversity, which makes it attractive for high-data-rate wireless applications. In MIMO systems, channel impulse responses are often assumed to be constant over a block or packet. This assumption of block stationarity on channels is valid for most fixed wireless scenarios. However, for communications in a high mobility environment, the assumption will result in considerable performance degradation since the channels vary fast and severely. For this kind of environment, many approaches have been proposed in [16]. Specifically, in [4], a robust interpolator for the channels to the corresponding unknown data symbols are used. In [5], the Kalman filter employing pilots is developed for a channel tracking. When we consider a joint structure, as is now widely recognized, the use of the soft-information for the data can lead to better channel estimators [7]. Thus, replacing a hard data detector by a soft data detector and coupling its soft

Department of Electrical and Computer Engineering University of California Santa Barbara, California 93106 Email: [email protected]

information in the channel estimator should greatly enhance the overall performance of the joint structure. Here, we will propose a new soft-Kalman filter and then combine it with the recently proposed MIMO-OFDM soft data detector [8] to develop a new semi-blind channel estimation and data detection algorithm for MIMO-OFDM systems over high mobility, where we only assume the channel stationarity on OFDM symbol level. The MIMO-OFDM soft data detector can produce reasonable performance results while maintaining a computational complexity comparable to the sum-product algorithm (SPA) based optimal detector [9-10]. The outline of the paper is as follows. Signal and channel models for MIMOOFDM systems are given in Section II. In Section III, the softKalman filter is developed. In Section IV, a new joint semiblind structure is proposed. Finally, some simulation results are presented in Section V, and conclusions are made in Section VI. II. S IGNAL AND C HANNEL M ODELS FOR MIMO-OFDM S YSTEMS In this paper, we consider a baseband model for a received MIMO OFDM signal over a multipath fading channel. The notation used for the MIMO-OFDM system includes the following: • Nf , Nt , Nr : number of multipaths and antennas in transmitter and receiver. • K, N − 1 : number of subcarriers and OFDM data symbols in one packet.
• Tg , Td = KTs , Ts : guard time interval, OFDM data symbol interval, and sampling time. • Tt : training interval, Tt = Mt Ts . g g
• Td : OFDM symbol interval, Td = Tg + Td . • A, a, (A)l,m , (a)k : a matrix, a vector, the (l, m) element of the matrix A, and the k-th element of the vector a. The symbols p, q,k,n are used as indices for the transmit antenna, the receiver antenna, the subcarrier, and the OFDM data symbol respectively, with 1 ≤ p ≤ Nt , 1 ≤ q ≤ Nr , 1 ≤ k ≤ K, 0 ≤ n ≤ N − 1. One packet is composed of

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(N − 1) OFDM data symbols with one training symbol which is made up of Mt subcarriers. A guard time interval Tg is also included in each data symbol to eliminate ISI. Input symbols {dpk (n)} drive the p-th modulator (a K-point IFFT) which modulates the symbols onto K subcarriers. The symbols dk (n) are chosen from a complex-valued finite alphabet. During the training phase, a fixed symbol is used for each transmit antenna. For convenience, the same signal constellation is employed for all subcarriers and antennas, although the method presented here can be extended to variable-rate constellations. The n-th output of the p-th modulator is sp (t) = spD (t)pD (t − Tt − Td − Tg − Tdg (n − 1)), K−1 g 1
p dk (n)ej2πk(t−Tt −Td −Tg −Td (n−1))/Td . spD (t) = √ K k=0 Here, pD (t) is a pulse with finite support on [0, Td ). The channel between the p-th transmit and q-th receive antenna, {flp,q (n)}, is modeled by a tapped delay line (TDL) [[11],Chap.7], such that the n-th received signal at the q-th antenna is rq (t) =

Nt N
f −1


flp,q (n)spD (t − lTs ) + nq (t).

p=1 l=0

It is assumed in the sequel that • •

•

Multipath delay spread : Nf Ts < Tg . Channel: a set of channels {flp,q (n)} is assumed to be constant over one OFDM symbol duration but varies from symbol to symbol. Receiver: receiver is assumed to be matched to the transmitted pulse.

The additive noise nq (t) is circular white Gaussian with spectral density 2N0 . Having eliminated the guard interval, the k-th demodulator output of the n-th OFDM data symbol (a K-point FFT) is now given by ykq (n)




=

Nt


cTk f p,q (n)dpk (n)

+

zkq (n),

where

ck f p,q (n)

∼ N (zkq (n); 0, 2N0 /Ts ),


= [1, e−j2πkTs /Td , . . . , e−j2πk(Nf −1)Ts /Td ]T ,  T
p,q = f0p,q (n), f1p,q (n), . . . , fN (n) ∈ C Nf(1) . f −1

N (x; mx , σx2 )

Here, denotes a circular Gaussian density with mean mx and variance σx2 . The q-th received signal over all available subcarriers is [12] y (n) =   1 D (n)CT , D2 (n)CT , . . . , DNt (n)CT f q (n) + zq (n), q

yq (n) Dp (n)

 q T q y0 (n), y1q (n), . . . , yK−1 (n) ,  
= diag dp0 (n), dp1 (n), . . . , dpK−1 (n) ∈ C K×K ,


=


[c0 , c1 , . . . , cK−1 ] ∈ C Nf ×K , T 
= f 1,q (n)T , f 2,q (n)T , . . . , f Nt ,q (n)T ∈ C Nt Nf , ∼ N (zq (n); 0, 2N0 /Ts IK ). (2)

C = f q (n) zq (n)

The Jake’s model correlation function (Bessel function) can be well-approximated by the correlation function of an AR process [12-14]. Thus, the channel is assumed to evolve according to flp,q (n)

p,q p,q = αf,l fl (n − 1) + wflp,q (n),

(3)

where wflp,q (n) ∼ N (wflp,q (n); 0, qfp,q ). l III. S OFT-I NFORMATION BASED S OFT-K ALMAN F ILTER For a mapping function g(.), a transmitted symbol dpk (n) is identically represented as a binary Q-tuple, that is, dpk (n) = g(bpk,1 (n), bpk,2 (n), . . . , bpk,Q (n)) : {0, 1}Q → C. (4)  T
For a received signal vector yk (n)= yk1 (n), . . . , ykNr (n) , the log-likelihood ratio (LLR) for bpk,j (n), denoted by  
p b = 1|(yk (n))Nt , . . . , (yk (n))1 , L bpk,j (n) = ln  p b = −1|(yk (n))Nt , . . . , (yk (n))1 b




=

2bpk,j (n) − 1,

is generated by the MIMO OFDM soft data detector [8]. Based on these LLRs, the proposed soft channel estimator first  
computes d¯pk (n)=E [dpk (n)] and E |dpk (n)|2 . For the QPSK modulation, d¯pk (n) =

√ √ tanh(L(bpk,1 (n))/2)/ 2 + j tanh(L(bpk,2 (n))/2)/ 2,

.E[|dpk (n)|2 ] = 1

p=1

zkq (n)

where

(5)

This computation is based on the signal constellation defined in [15]. Now the q-th received signal vector is equivalently expressed as   yq (n) = D1 (n)CT , . . . , DNt (n)CT f q (n) + zq (n),   1 ¯ Nt (n)CT f q (n) + ¯ (n)CT , . . . , D = D   ˜ Nt (n)CT f q (n) + zq (n),(6) ˜ 1 (n)CT , . . . , D D where ¯ p (n) D ˜ p (n) D

 
= diag d¯p1 (n), . . . , d¯pK−1 (n) ,  
= diag [dp0 (n) − d¯p0 (n)], . . . , [dpK−1 (n) − d¯pK−1 (n)] .

To develop the soft Kalman filter, we use a different representation for yq (n) as   1 ¯ Nt (n)CT f q (n) + z ¯ (n)CT , . . . , D ˜q (n).(7) yq (n) = D

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Here, a new measurement noise process is defined as


˜q (n) z

=

Nt


˜ p (n)CT f p,q (n) + zq (n). D

(8)

p=1

Nt ˜ p Since p=1 D (n)CT f p,q (n) is a summation of many inde˜q (n) may be pendent vectors and f p,q (n) is Gaussian vector, z approximated as a Gaussian distribution using the central limit theorem [16]. However, in a practical system, the number of Nt is not large enough for the central limit theorem. Although ˜q (n) is non-Gaussian, the Kalman filter is still optimal linear z ˜q (n) is uncorrelated in time. Since estimator for f p,q (n) when z p ˜ the symbol error D (n) are uncorrelated in time, this is true. ˜ q (n) of z ˜q (n) can be computed as The covariance matrix R follows:
˜ q (n) = E[˜ zq (n)˜ zq (n)H ] = R

2N0 /Ts I + Nt K−1 

Sk+1 f p,q (n) V (dpk (n))ek+1 eTk+1 ,

(9)

p=1 k=0

where V (dpk (n)) ek+1 Sl (f p,q (n))




= E|dpk (n)|2 − |d¯pk (n)|2 ,


=

[01×k , 1, 01×(K−k−1) ]T ,   01×l−1  
=  (CT E f p,q (n)f p,q (n)H C∗ )(l, :)  . 01×K−l

Note that this is only for the known channels f (n). Its derivation is given in Appendix A. Similarly, for a channel
estimate ˆf q (n|n)=E [f p,q (n)|yq (n), . . . , yq (1)], it is given by p,q

˜ q (n) ≈ 2N0 /Ts I+ R Nt K−1

Sk+1 (ˆf p,q (n|n))V (dpk (n))ek+1 eTk+1 . (10) p=1 k=0

Using previous development, the system and observation equations are of the form   1 ¯ Nt (n)CT f q (n) + z ¯ (n)CT , . . . , D ˜q (n), yq (n) = D f q (n) = Fq f q (n − 1) + wf (n).

(11)

Here, F is a block diagonal system matrix and wf (n) is complex Gaussian with wf (n) ∼ N (wf (n); 0, Qf ). Also, ˜ q (n). With this develop˜q (n) is R the covariance matrix of z ment the soft-Kalman filter can communicate the soft data information with the soft data detector. q

IV. J OINT S EMI -B LIND C HANNEL E STIMATION AND DATA D ETECTION Applying the approach proposed in [6], we can combine both the soft-Kalman filter and the MIMO OFDM soft data detector as shown in Figure 1. For a joint channel estimation and data detection, usually either of approaches is employed, decision-aided (DA) or non-decision- aided (NDA). In our

experiments, the DA approach uses previous decision to estimate current channels, which results in a symbol delay. The NDA approach can track varying channels well but pilot-aided techniques must be used for online channel estimation. It is observed that the NDA approach obtains a better bit error rate (BER) performance than the DA approach [4], but the expense for the better BER performance is a loss of bandwidth on pilots. Therefore, a trade-off between the amount of bandwidth on pilots and the final system performance has to be made. To get a better performance, we employ the NDA, where Np pilots are placed on of K subcarriers. It was proved in [17] that the MMSE estimate of channels occurs if Np pilots are equal-spaced when the noise is AWGN. With the observation of the demodulation outputs, we can easily divide original subcarriers into a set of data subcarriers, ks ∈ {s1 , . . . , sNs } and a set of pilot subcarriers, kp ∈ {p1 , p2 , . . . , pNp }, that is,   q
y (n) yq (n)= q y• (n)  1  q   T t D (n)CT , . . . , DN z (n) q  (n)C = ¯1 , ¯ Nt (n)CT f (n) + z ˜q• (n) D• (n)CT• , . . . , D • •  p 
¯ p (n)=diag {d¯ks (n)} ∈ C Ns ×Ns , D • 
 C• = {cks } ∈ C Nf ×Ns , T
 q y•q (n)= {yks (n)} ∈ C Ns , T
 zkqs (n)} ˜q• (n)= {˜ z ∈ C Ns . (12) In a real OFDM system, there are NNULL subcarriers for an anti-aliasing reconstruction filter, that is, K = Ns + Np + NNULL . Similarly we can define Dp (n), C , yq (n), and zq (n) in terms of the pilot subcarriers index. Note that yq (n) and y•q (n) share the common channel state vector. With a linear state-space model, a Kalman filter (KF) can provide optimal linear least mean- squares estimate (LLMS) of the state variable given prior observations of pilots, but this LLMS estimate is not optimal for signals since the prior observations of signals are not counted. In the proposed scheme, we first estimate the channel ˆf q (n|n) with pilots employing a regular Kalman filter based on the following equations,  q  1 T t D (n)CT , . . . , DN f (n) + zq (n), yq (n) =  (n)C f q (n)

= Fq f q (n − 1) + wf (n).

(13)

Now the MIMO OFDM soft detector, called the soft-QRD-M, generates the soft data information based on channel estimates {ˆf q (n|n)} obtained from a regular Kalman filter. It is given by ˆ k (n)soft-QRD-M = d s arg

min

dks (n)∈|S|Nt

ˆ k (n|n)dk (n)||2 , ||yks (n) − F s s

 T
t (n) , dks (n)= d1ks (n), d2ks (n), . . . , dN ks 
 d¯pks (n)=E dpks (n) ,  T
yks (n)= yk1s (n), yk2s (n), . . . , ykNsr (n) ,

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ˆ k (n|n)dk (n) + zk (n), =F s s  s
ˆ k (n|n))i,j ∈ C Nr ×Nt , ˆ k (n|n)= ( F F s s 
ˆ k (n|n))i,j =ck ˆf i (n|n) = ck ˆf i,j (n|n). (F s s s j

R EFERENCES

(14)

Having obtained a set of soft data information     p q ˆ ¯ and the channel estimate f (n|n) , the dks (n) soft Kalman filter first builds a estimated observation
t ¯ p (n)CT ˆf p,q (n|n) and then corrects a ˆ q (n)= N y D p=1 previous channel estimate. The one step channel prediction {ˆf q (n + 1|n)} is also coming out from the soft Kalman filter for the next iteration. V. S IMULATION R ESULTS In simulations, we evaluate the performance of the MIMOOFDM system over time-varying channel with the QPSK subcarrier modulation, and assume the channel is constant only one symbol interval. The values of some parameters are •

• •

K = 64, kp ∈ {8, 22, 44, 58},ks ∈ {(2 : 7), (9 : 21), (23 : 27), (39 : 43), (45 : 57), (59 : 64)}, and NNULL = 12. Nt = Nr = 4 and N = 10. Fading channel powers, Nf = 5, {0.5610, 0.2520, 0.1132, 0.0509, 0.0229}, ∀p, q.

Figures 2 and 3 respectively shows the bit error rate (BER) and packet error rate (PER) performance for the QPSK subcarrier modulation at a fading condition fd Td = 0.016. To simplify our approaches, the fading speed is assumed to be identical to p,q , ∀p, q, l, and approximately all antennas and paths, αf = αf,l determined by the zero-order Bessel function. This plot shows that applying two Kalman filters alternatively and iteratively we can significantly improve the overall performance of the MIMO-OFDM system compared to the originally proposed scheme in [6]. This gain is purely obtained from a new combining of the soft-data detection algorithm and the softKalman filter. Particularly, even one iteration in the proposed scheme improves the performance relatively. To have a similar performance at the third iteration, the original semi-blind structure requires more iterations than the new scheme. It is very interesting that at a 10−4 BER, we can achieve up to 1 [dB] SNR difference with three iterations compared to the optimal data detection [9-10] under an ideal channel condition at the receiver. Figure 4 is the corresponding mean squared error (MSE) of the channel estimation. VI. C ONCLUSIONS The pilot-aided channel estimation using hard/soft-Kalman filter is proposed and investigated in an iterative structure. The simulation results show the new joint semi-blind channel estimation and data detection approach is feasible for high mobility environments.

[1] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. on Veh. Technol., vol. 40, pp. 686–693, Nov. 1991. [2] J. K. Cavers, “Pilot symbol assisted modulation and differential detection in fading and delay spread,” IEEE Trans. on Commun., vol. 43, pp. 2206–2212, July 1995. [3] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. New York, N.Y.: John Wiley and Sons, 1998. [4] G. Li, “Pilot-symbol-aided channel estimation for OFDM in wireless systems,” IEEE Trans. on Veh. Technol., vol. 49, pp. 1207–1215, July 2000. [5] J. K. Tsatsanis, “Pilot symbol assisted modulation in frequency selective fading wireless channels,” IEEE Trans. on Signal Processing, vol. 48, pp. 2353–2365, Aug. 2000. [6] J. Yue, K. J. Kim, T. Reid, and J. D. Gibson, “Joint semi-blind channel estimation and data detection for MIMO-OFDM systems,” in Proceedings of the 6th CAS Symp. on Emerging Technologies: Mobile and Wireless Comm., (Shanghai, China), pp. 709–712, June 2004. [7] M. Tuchler, A. C. Singer, and R. Koetter, “Minimu mean squared error equalization using a priori information,” IEEE Trans. on Signal Processing, vol. 50, pp. 673–683, March 2002. [8] K. J. Kim, T. Reid, and R. A. Iltis, “Soft data detection algorithm for an iterative Turbo coded MIMO OFDM systems.” To appear in the proceedings of Thirty-Eighth Asilomar Conf. on Signals, Systems and Computers, 2004. [9] F. R. Kschischang, B. J. Frey, and H. A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. on Inform. Theory, vol. 47, pp. 498–519, 2001. [10] S. M. Aji and R. J. McEliece, “The generalized distributed law,” IEEE Trans. on Inform. Theory, vol. 46, pp. 325–343, 2000. [11] J. G. Proakis, Digital Communications. New York, N.Y.: McGraw-Hill, 1989. [12] K. J. Kim, Y. Yue, R. A. Iltis, and J. D. Gibson, “A QRD-M/Kalman Filter-based detection and channel estimation algorithm for MIMOOFDM systems.” To appear in the IEEE Trans. on Wireless Commun., 2004. [13] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wesel, “Multi-input multi-output fading channel tracking and equalization using Kalman estimation,” IEEE Trans. on Signal Processing, vol. 50, pp. 1065–1076, May 2002. [14] K. J. Kim, T. Reid, and R. A. Iltis, “A sequential Monte-Carlo Kalman Filter based delay and channel estimation method in the MIMO-OFDM system.” To appear in the proceedings of VTC, 2004. [15] IEEE P802.11a/D7.0, July 1999. [16] C. W. Helstrom, Probability and Stochastic Processes for Engineers. New York, N.Y.: Macmillan Publishing Company, 1984. [17] R. Negi and J. Cioffi, “Pilot tone selection for channel estimation in a mobile ofdm system,” IEEE Trans. on Consumer Electronics,, vol. 44, pp. 1122–1128, 1998.

A. C OMPUTATION OF E QUATION (9) Recall that ˜q (n) z




=

Nt


˜ p (n)CT f p,q (n) + zq (n). D

(A.1)

p=1

Hence   q ˜ (n)˜ E z zq (n)H = 2N0 /Ts I+ Nt  
˜ p (n)H (. A.2) ˜ p (n)CT f p,q (n)f p,q (n)H C∗ D E D p=1

  Note that only in a special case of E fp,q (n)f p,q (n)H = σf2 I, the matrix E CT f p,q (n)f p,q (n)H C∗ becomes Toeplitz. Now

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fˆ q ,i (n | n − 1)

fˆ q ,i (n | n)

q-th Kalman Filter

fdTd=0.016,αf = 0.9975, M=16, Nt=Nr=4

0

fˆ 1,i (n | n)

y q◊ (n)

10

D1 ( n )

Soft-QRD-M

fˆ N r ,i (n | n)

D p (n) D N t (n) −1

10

MSE

y •q (n) D1 ( n )

One iteration delay

fˆ q ,i +1 (n | n − 1)

q-th D p (n) Soft-Kalman Filter D N (n)

−2

10

h−KF+h−QRDM (loop=0) h−KF+h−QRDM (loop=1) h−KF+h−QRDM (loop=3) h−KF+h−QRDM (loop=4) h−KF+h−QRDM (loop=8) s−KF+s−QRDM (loop=0) s−KF+s−QRDM (loop=1) s−KF+s−QRDM (loop=3) s−KF+s−QRDM (loop=4)

t

Fig. 1. The KF-based iterative channel estimation model in the n-th OFDM symbol interval.

−3

10

4

6

8

10

12 E /N [dB] s

Fig. 4.

14

16

18

20

0

MSE performance of the channel estimation.

f T =0.016,α = 0.9975, M=16, N =N =4 d d

0

f

t

r

10

using the following relationship ˜ p (n)H = ˜ p (n)CT f p,q (n)f p,q (n)H C∗ D D   K 01×l−1
˜ p (n)H ,  (CT f p,q (n)f p,q (n)H C∗ )(l, :)  d˜p (n)D l−1 01×K−l l=1 (A.3)

−1

10

−2

BER

10

−3

10

h−KF+h−QRDM (loop=0) h−KF+h−QRDM (loop=1) h−KF+h−QRDM (loop=3) h−KF+h−QRDM (loop=4) h−KF+h−QRDM (loop=8) s−KF+s−QRDM (loop=0) s−KF+s−QRDM (loop=1) s−KF+s−QRDM (loop=3) s−KF+s−QRDM (loop=4) s−QRDM + Exact Channel Optimum Detection + Exact Channel

−4

10

−5

10

4

Fig. 2.

6

8

10

12 14 E /N [dB] s 0

16

18

20

22

BER performance with the QPSK subcarrier modulation.

we have   ˜ p (n)H = ˜ p (n)CT f p,q (n)f p,q (n)H C∗ D E D   K
   T p,q 01×l−1  E C f (n)f p,q (n)H C∗ (l, :)  01×K−l l=1   ˜ p (n)H , E d˜pl−1 (n)D (A.4) from which the expectation involved with data is    ˜ p (n)H E d˜pk (n)D = E|dpk (n)|2 − |d¯pk (n)|2 ek+1 eTk+1 ,

f T =0.016,α = 0.9975, M=16, N =N =4 d d

0

f

t

ek+1

r

10




=

[01×k , 1, 01×(K−k−1) ]T .

(A.5)

PER

In the computation of (A.5), we use the following fact     2 2 E d˜pk (n)d˜pk
(n)∗ = E |dpk (n)| − d¯pk (n) δk,k
. by Here δk,k
is the Kronecker delta function. Denoting  01×l−1   
  Sl (f p,q (n)) =  CT E f p,q (n)f p,q (n)H C∗ (l, :)  the l01×K−l th channel expectation in (A.4), we have   E Dp (n)CT f p,q (n)f p,q (n)H C∗ Dp (n)H =

−1

10

−2

10

12

h−KF+h−QRDM (loop=0) h−KF+h−QRDM (loop=1) h−KF+h−QRDM (loop=3) h−KF+h−QRDM (loop=4) h−KF+h−QRDM (loop=8) s−KF+s−QRDM (loop=0) s−KF+s−QRDM (loop=1) s−KF+s−QRDM (loop=3) s−KF+s−QRDM (loop=4) s−QRDM + Exact Channel Optimum Detection + Exact Channel

14

K−1


16

18

20

22

  Sk+1 (f p,q (n)) E|dpk (n)|2 − |d¯pk (n)|2 ek+1 eTk+1 .

k=0

(A.6)

E /N [dB] s

Fig. 3.

0

PER performance of Figure 2.

Now substituting (A.6) into (A.2) yields (9), which completes the derivation.

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