Improving Differential Detection of MDPSK by Nonlinear Noise ...
Improving Di erential Detection of MDPSK by Nonlinear Noise Prediction and Sequence Estimation Robert Schober, Student Member, IEEE, Wolfgang H. Gerstacker, Member, IEEE, and Johannes B. Huber, Member, IEEE
Abstract A new technique is proposed to improve the performance of di erential detection of MDPSK (M {ary di erential phase{shift keying) signi cantly, applying sequence estimation. In order to obtain an appropriate representation of the received signal, a nonlinear time{variant FIR or IIR prediction{error lter is used. For both lter structures the optimum coecients are derived, assuming transmission over an additive white Gaussian noise (AWGN) channel. Delayed decision{feedback sequence estimation (DDFSE) is employed to estimate the transmitted symbol sequence. It is shown by simulations that even for decision{feedback equalization (DFE), which is a simple special case of DDFSE, a signi cant performance improvement of conventional di erential detection under AWGN conditions results. In contrast to other noncoherent low complexity receivers proposed in literature this receiver is very robust under at fading (Rayleigh and Ricean) conditions. Index Terms{Di erential detection, M {ary DPSK, nonlinear prediction, delayed decision{ feedback sequence estimation (DDFSE), decision{feedback equalization (DFE).
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1 Introduction It is well known, that di erential detection (DD) of MDPSK (M {ary di erential phase{ shift keying) is very attractive when simplicity and robustness of implementation are more important than optimum power eciency. In addition, tracking of the correct carrier phase, as required by coherent detection (CD), may be dicult or even impossible in a fading environment. On the other hand, for an additive white Gaussian noise (AWGN) environment, DD causes a performance loss compared to CD. In literature, modi ed noncoherent detection schemes with an improved performance for the AWGN channel have been proposed. For example, a maximum{likelihood receiver using multiple{symbol detection (MSD) has been reported by Divsalar et al. [1]. The main disadvantage of MSD is its high computational complexity compared to techniques, which rely on feeding back previously decided symbols [2, 3, 4, 5, 7, 8, 9, 10, 11, 12]. Especially the recursive decision{feedback di erential detection (DF{DD) scheme presented in [5] and [7] (in both papers the same lter structure was used as has been pointed out in [6]) and the nonrecursive DF{DD scheme proposed and analyzed in [2, 3, 4] are very attractive for implementation because, respectively, a one tap IIR lter and an FIR lter are employed prior to DD in order to stabilize the reference phase. Hence, the new detector proposed here will be compared with those structures. Although our detector in its most simple form uses decision feedback, too, this approach is di erent from the previous ones. After conversion to complex baseband representation and sampling, the current received symbol is multiplied by the previous complex conjugated one, like in a conventional di erential detector. The resulting signal is corrupted by a noise process which is not proper [14]. Hence, symbol{by{symbol decisions of conventional di erential detectors are not optimum and sequence estimation has to be applied. By means of nonlinear noise prediction a signal representation suitable for the application of delayed decision{ feedback sequence estimation (DDFSE) [15] and decision{feedback equalization (DFE) [16], which is a special case of DDFSE, is derived. Both schemes perform signi cantly better than conventional DD and especially the proposed DFE scheme has a very low computational complexity. For example, for an AWGN channel and M = 4, our receiver using DFE can improve conventional DD by about 1.2 dB at BER = 10,3 . Moreover, our detector is more robust under at Rayleigh and Ricean fading conditions than recursive DF{DD [5, 7] as will be shown by simulations.
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2 Transmission Model Fig. 1 shows a block diagram of the transmission model under consideration. All signals are represented by their complex{valued baseband equivalents. The MDPSK symbols are denoted by a[k] 2 A = fej2 M j 2 f0; 1; : : : ; M , 1gg and the corresponding di erentially encoded MPSK symbols b[k] are given by
b[k] = a[k]b[k , 1]; k 2 Z:
(1)
Both, transmitter lter Ht(f ) and receiver lter Hr (f ) have square{root Nyquist characteristic. Hence, no intersymbol interference occurs as long as signals are transmitted over AWGN or slow at small{scale fading [20] channels. This also holds when there is a small frequency o set f between receiver and transmitter oscillator, i.e., fT 1, where T denotes the symbol interval. For derivation of our detection scheme an AWGN channel with an arbitrary, constant phase shift is assumed. In Section 6, the performance of the new receiver is also tested for frequency o set and at (Rayleigh and Ricean) fading channels. The complex{valued AWGN is denoted by nc (t) and the one{sided power spectral density of the underlying real passband noise process is N0. The samples r[k] of the received signal r(t) can be written as r[k] = r(kT ) = ej b[k] + n0[k]; (2) applying an appropriate normalization to Hr (f ). Here, n0[k] denotes uncorrelated Gaussian noise with variance 2 = ENS0 , where ES is the (mean) received energy per symbol. Like in a conventional di erential detector, the current received symbol r[k] is multiplied with the previous complex conjugated one:
rd[k] = r[k]r[k , 1] = a[k] + z[k];
(3)
where () denotes complex conjugation and, according to Eqs. (1) and (2), the e ective noise z[k] is given by
z[k] = ej b[k]n0[k , 1] + e,j b[k , 1]n0[k] + n0[k]n0[k , 1]:
(4)
By introducing n[k] =4 e,j n0[k], Eq. (4) can be rewritten to
z[k] = b[k]n[k , 1] + b[k , 1]n[k] + n[k]n[k , 1]: Note, that n[k] is still uncorrelated Gaussian noise with the same variance as n0[k].
(5)
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A conventional di erential detector directly uses rd[k] to make a decision on a[k]. The new receiver, however, processes rd[k] in a nonlinear noise prediction{error lter (NPEF), which is embedded in a DDFSE [15] or DFE [16], delivering estimated symbols a^[k , k0], where k0 is the decision delay.
3 Statistical Properties of z
[]
n[] is a proper equivalent complex baseband uncorrelated Gaussian random process [14], i.e., its autocorrelation sequence (ACS) 'nn [] and pseudo{autocorrelation sequence (PACS) '~nn [] are 'nn[] = Efn[k + ]n[k]g = 2[]; '~nn[] = Efn[k + ]n[k]g = 0; 8 ;
(6) (7)
where Efg denotes expectation and [] is the unit pulse sequence, i.e., [0] = 1, [] = 0, 6= 0. Now, the second{order statistics of the e ective noise z[] are determined for a given data sequence a[]. By using straightforward operations and Eqs. (1), (5){(7), the ACS 'zz [] and PACS '~zz [; k] can be calculated to
'zz [] = Efz[k + ]z[k]g = (22 + 4)[]; '~zz [; k] = Efz[k + ]z[k]g = 2a[k]a[k + ][jj , 1]; 8k:
(8) (9)
Note, that for calculation of 'zz [] and '~zz [; k] the transmitted symbol sequence a[] is assumed to be known and expectation is performed only over the noise process n[]. Since the PACS '~zz [; k] does not vanish identically, z[] is not a proper complex random process [14]. In addition, the PACS depends on k, i.e., z[] is a nonstationary process for a xed sequence a[]. Although the ACS 'zz [] implies uncorrelatedness of z[] (cf. Eq. (8)), the PACS '~zz [; k] indicates that z[] and its complex conjugated version z[] are (mutually) correlated. Consequently z[] is not an independent identically distributed (i.i.d.) sequence. This implies, that the maximum{likelihood criterion, which is optimum in the sense of minimum sequence error probability,
a^[] = argmax fprd (rd[]ja~[])g ; ~a[]
(10)
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cannot be simpli ed to symbol{by{symbol decisions as done in conventional di erential detectors, which therefore are suboptimum. Here, prd (rd[]ja[]) denotes the probability density function (pdf) of sequence rd[] conditioned on the transmitted sequence a[]. In order to enable a recursive optimization of the criterion in Eq. (10), an equivalent representation of rd [k] in the form rd[k] = yd[k] + z[k] (11) is desirable, where yd[k] is uniquely determined by rd[k , ] and a[k , ], 0 < 1, and z[k] is i.i.d. Then prd (yd[] + z[]ja[]) = pz (z[]ja[]) and Eq. (10) is equivalent to ^a[] = argmax fpz (z[]ja~[])g ; a~[]
(12)
where pz (z[]ja[]) is the pdf of z[] conditioned on a[]. It would be advantageous if, in addition, pz (z[]ja[]) was a Gaussian pdf with constant variance because then optimization in Eq. (12) could be done by recursively minimizing a sum of quadratic Euclidean distances. In the sequel, the desired decomposition according to Eq. (11) will be developed using a nonlinear NPEF (cf. Fig. 2).
4 Improved Receiver Using a Nonlinear FIR NPEF 4.1 Derivation of the Optimum FIR Filter In this and the next section, an FIR and IIR lter is applied, respectively, for nonlinear noise prediction. The (mutual) correlatedness of z[] and z[] suggests to employ a lter which exploits not only z[] like conventional prediction{error lters [18], but also z[]. For derivation of the lter perfect knowledge of the transmitted sequence a[] is assumed. If an FIR lter is applied, the prediction error z[k] may be expressed as z[k] = z[k] , = ,
N X =0
N X =1
[k]z[k , ] ,
[k]z[k , ] ,
N X =0
N X =1
[k]z[k , ]
[k]z[k , ];
(13)
where [k] and [k], 1 N , are the complex{valued predictor coecients to be 4 determined and 0[k] =4 ,1, 0[k] = 0, 8k, are de ned for convenience. Dependence of the coecients on time k has been introduced because later it will turn out that the optimum
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coecients indeed are time{variant. In this context it is important to note that complex conjugation is not a linear operation, i.e., it is impossible to obtain z[] from z[] by linear ltering. Therefore, the explicit introduction of the complex conjugated process z[] in Eq. (13) is necessary. By using Eq. (5), Eq. (13) can be expressed as z[k] = b[k , 1]n[k] ,
, ,
N X =0 N X =0
N X =0
( +1 [k]b[k , 2 , ] + [k]b[k , ]) n[k , 1 , ]
( [k]b[k , ] + +1[k]b[k , 2 , ]) n[k , 1 , ]
N
X [k]n[k , ]n[k , 1 , ] , =0
[k]n[k , ]n[k , 1 , ];
(14)
4 4 where for simplicity of notation N +1[k] = 0, N +1[k] = 0, 8k, are introduced. The variance of the prediction error z[k] can be calculated from Eqs. (6), (7), and (14):
2 z
=
Efjz[k]j2g = 2
1+ +
N X =0
N X
=0
j +1[k]b[k , 2 , ] + [k]b[k , ]j2
j [k]b[k , ] + +1
N X 4 +
=0
j [k]j2 +
N X =0
[k]b[k , 2 , ]j2
!
!
j [k]j2 ;
(15)
where knowledge of b[k , ], 0 N + 1, is assumed, i.e., expectation is performed only over the noise process n[]. The optimum predictor coecients are chosen to minimize 2 z . Thus, Eq. (15) is di erentiated with respect to [k] and [k], 1 N , using the method for complex di erentiation described in [18], Appendix B: @2 z = 2(( [k]b[k , 1 , ] + [k]b[k + 1 , ])b[k , 1 , ] ,1 @ [k] +( [k]b[k , ] + +1[k]b[k , 2 , ])b[k , ]) + 4 [k]; (16) 2 @z = 2(( [k]b[k , 2 , ] + [k]b[k , ])b[k , ] +1 @ [k]
+( ,1[k]b[k + 1 , ] + [k]b[k , 1 , ])b[k , 1 , ]) + 4 [k]: (17)
By setting Eqs. (16) and (17) equal to zero and applying Eq. (1), 2N equations for determination of 2N coecients are obtained: (2 + 2) [k] + ,1 [k]u[k + 1 , ] + +1[k]u[k , ] = 0; (2 + 2) [k] + ,1 [k]u[k + 1 , ] + +1 [k]u[k , ] = 0;
1 N;
(18) (19)
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where the notation u[k] =4 a[k]a[k , 1] is introduced. These 2N equations can be separated into 2 independent matrix equations:
Ax1 = y1; AT x2 = y2;
(20) (21)
where the N N matrix A and the N 1 vectors x1, x2, y1 and y2 are de ned as
A =
2 2 + 2 6 6 6 u[k 1] 6 6 6 6 0 6 4 .
u[k , 1] 0 ::: 2 + 2 u[k , 2] 0 u[k , 2] 2 + 2 u[k , 3] ... ... ...
,
..
y1 y2 x1 x2
::: ::: ... ...
[u[k] 0 : : : 0]T ; [0 0 : : : 0]T ; [ 1[k] 2[k] 3[k] 4[k] : : :]T ; [ 1[k] 2[k] 3[k] 4[k] : : :]T :
= = = =
3
0 7 7 0 777 . . . 77 ; 7 ... 5
(22)
(23) (24) (25) (26)
Here, []T denotes the transpose of a matrix (vector). The elements of x1 and x2, which are calculated in Appendix A, are given by
[k] = [k] =
8 > > x~1 a[k]a[k > > < > > > > : 8 > > > >
> > > : x ~1 a[k]a[k
, ]; even; odd; even;
, ]; odd;
1 N;
(27)
1 N;
(28)
where x~1 is de ned in Eq. (74). Fig. 3a) shows the magnitude of x~1 vs. coecient number for di erent lter orders N and Eb =N0 ratios for a QDPSK constellation (M = 4), i.e., 2 = 2NE0b , where Eb is the (mean) energy per bit. At very high Eb=N0 ratios the magnitude of x~1 decreases linearly with increasing coecient number, whereas it decreases exponentially at low Eb=N0 ratios.
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4.2 ACS and PACS of z [ ]
By combining Eqs. (13), (27) and (28), we get for the prediction error z[k] the expression z[k] = z[k] ,
N X a[k] x~1 (a[k =1
, ]z[k , ])();
(29)
where ()() means that complex conjugation is applied times (e.g. ()(2) = (()) = ()). In Appendix B, it is shown that the ACS 'zz [; k] and the PACS '~zz [; k] can be calculated to [k ]g 'zz [; k] = Ef z [ k + ] z 8 > 2a [k ]a[k + ] PN +1 (~x1 + x~1 )(~x1 > ~1+jj) > =0 ,1 +jj,1 + x > < 2 PN x~1 x~1 = > + even; (30) =0 +jj ; > > > : 0; odd;
and
'~zz [; k] = Ef z[k + ]z[k]g 8 > > 0; even; > > < P = > 2a[k]a[k + ] N=0+1(~x1,1 + x~1 )(~x1+jj,1 + x~1+jj) (31) > > P > : +2 N=0 x~1 x~1+jj ; odd; where for simplicity of notation, the de nitions x~1 =4 0, < 0 or > N , and x~10 =4 ,1 are introduced. Eqs. (30) and (31) show that, in general, z[] is not uncorrelated and z[] and z[] are (mutually) correlated, too. However, as can be seen from Fig. 3b), which again is valid for a QDPSK (M = 4) constellation, the variance 2 z = 'zz [0; k] of the prediction error is lower than that of z[] (corresponding to \no lter"). For nite lter order N a trade{o between prediction gain and complexity exists. Obviously, for N ! 1 and Eb =N0 ! 1, 2 z approaches 2. Thus, in this case the prediction error z[k] has the same variance as the process n[].
4.3 DDFSE and DFE So far it has been assumed, that the transmitted sequence a[] is known. In practice, however, this sequence has to be estimated at the receiver. Applying the previous results, the received
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signal rd[k] may be written according to Eq. (11), where yd[k] can be determined from Eqs. (3) and (29), N X yd[k] = a[k] + a[k] x~1 (a[k , ]z[k , ])(): (32) =1
Although, in general, z[k] is not exactly i.i.d. (cf. Eq. (30), Eq. (31)) and the conditional pdf pz (z[]ja[]) is only approximately Gaussian, simulations justify the application of the Euclidean metric
mk+1 =
k X =,1
jrd[ ] , y~d[ ]j2 =
k X =,1
j~z[ ]j2 = mk + j~z[k]j2:
(33)
In Section 5.3 it will be shown that in the limit N ! 1, Eb =N0 ! 1, the Euclidean metric of Eq. (33) indeed is optimum. y~d[k] is given by
y~d[k] =4 rd[k] + ~z[k];
(34)
and ~z[k] is de ned as 4 ~z[k] = z~[k] , ~a[k]
where the de nition
N X x~1 (~a[k
=1
, ]~z[k , ])();
z~[k] =4 rd[k] , ~a[k]
(35) (36)
is used and a~[k , ] 2 A, 0 N , are equalizer trial symbols. Minimization of Eq. (33) may be done e ectively by the Viterbi algorithm [17]. In order to reduce complexity, DDFSE [15] can be employed. Here, ~z[k] has to be de ned according to 4
~z[k] = z~[k] ,
0 X a~[k] x~1 (~a[k
,a~[k] where the notation
=1
N X
=0 +1
, ]~z[k , ])()
x~1 (^a[k , ]^z[k , ])();
z^[k] =4 rd[k] , ^a[k]
(37) (38)
is used. A comparison of Eqs. (35) and (37) shows that in addition to trial symbols a~[k , ] 2 A, 0 0, 0 0 N , also decision{feedback symbols a^[k , ] 2 A, 0 + 1 N , have been introduced in Eq. (37) instead of the unknown transmitted symbols a[k , ], 0 N . The underlying trellis [15] has Z = M 0 states and T = M 0 +1 transitions.
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Note, that for 0 = N a full{state Viterbi algorithm results. The parameter 0 determines the number of states and the system performance. A trade{o between performance and complexity exists. For small 0, complexity is moderate, however, performance is a ected by error propagation for low to moderate Eb=N0 ratios (cf. Section 6). The extreme case 0 = 0 corresponds to DFE. In the sequel a simple DFE structure is derived. Here, j~z[k]j2 can be directly used to make a decision on a~[k], i.e., according to Eqs. (37) and (36) the estimated symbol a^[k] is determined by n
o
n
^a[k] = argmin j~zj2 = argmin jrd[k]j2 + ja~[k]j2j1 + a~[k]
a~[k]
N X x~1 (^a[k
=1
N X 1 + x~1 (^a[k =1
o
,2Re fa~[k]dFIR[k]g ;
where dFIR[k] is de ned as
dFIR[k] =
, ]^z[k , ])()j2
, ]^z[k , ])((,1))
!
rd [k]:
(39) (40)
Since the rst two terms in Eq. (39) are common to all M possible a~[k], they can be omitted. Hence, the minimum is attained when the real part of a~[k]dFIR[k] is maximum, i.e., the phase of ~a[k] is next to the phase of dFIR[k]. This means that the complex plane can be divided into M sectors and a^[k] is determined by the sector into which dFIR[k] falls. This corresponds to the decision rule usually used for conventional DD, with the di erence that for DD rd[k] is used instead of dFIR [k]. Fig. 4 shows the structure of this simple DFE receiver.
5 Improved Receiver Using a Nonlinear IIR NPEF 5.1 Derivation of the Optimum IIR Filter In this section, the optimum IIR NPEF is derived. Here, we start from the optimum FIR lter of in nite order (N ! 1). In this case, the coecients x~1 can be obtained from Eq. (74) if j 2j < 1 for 2 > 0 is taken into account: ,2(N +1, ) , 1
2 1 +1 , = ,(, 2) ; 1 < 1: (41) x~ = Nlim (,1) 2 ,2(N +1) !1
2 ,1 The corresponding nonlinear time{variant IIR prediction{error lter for z[] can be obtained by substituting x~1 according to Eq. (41) in Eq. (29): z[k] = z[k] + a[k]
1 X
=1
(, 2) (a[k , ]z[k , ])()
(42)
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= z[k] , 2a[k]a[k , 1]z[k , 1] + 22a[k]a[k , 2]z[k , 2] , 23a[k]a[k , 3]z[k , 3] + : : : = z[k] , [k]z[k , 1] + [k] [k , 1]z[k , 2] , [k] [k , 1] [k , 2]z[k , 3] + : : : = z[k] , [k] (z[k , 1] , [k , 1]z[k , 2] + [k , 1] [k , 2]z[k , 3] , : : :) = z[k] , [k]z[k , 1]; (43) where the de nition [k] =4 2a[k]a[k , 1] is used. Eq. (43) corresponds to a nonlinear time{variant one tap IIR lter.
5.2 ACS and PACS of z [ ]
Since we have already calculated the ACS 'zz [; k] and the PACS '~zz [; k] for an FIR NPEF (cf. Eqs. (30), (31)), we use these results for the IIR case. This is possible because Eqs. (27), (28), and (41) describe a nonrecursive lter of in nite order which is equivalent to the optimum IIR lter. Therefore we substitute x~1 according to Eq. (41) in Eqs. (30) 4 and (31). Taking the de nitions x~10 = ,1 and x~1 =4 0, < 0, into account Eq. (30) ( even) yields 1 X 'zz [; k] = 2a[k]a[k + ] (~x1,1 + x~10)(~x1jj,1 + x~1jj) + (~x1 + x~1+1)
(~x1
+jj
=0
+ x~1
+jj+1
1 X 2 ) + x~1 x~1 =0
= 2a[k]a[k + ] ,(~x1jj,1 + x~1jj) + (, 2)jj +
1 2X
(, 2)2
!!
=0
!
+jj
1 X
=0
(, 2)2 (1 , 2)2 !!
1 , 2 + 2 = jj,1 jj) + (, 2 1 + 2 1 , 22 : (44) Now, two cases have to be distinguished: For = 0 the coecient x~1jj,1 = x~1,1 is zero, i.e., 2 4 2 z = 'zz [0; k] = 1 2+ + 1 , 2 : (45) 2 2 Applying some further calculations, where Eq. (72) has to be used, Eq. (45) can be transformed into 2 2 z = 'zz [0; k] = ; (46)
2a[k]a[k + ]
,(~x1
+ x~1
)jj
2
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Note, that Eq. (46) is valid only for optimum 2 according to Eq. (72), whereas Eq. (45) holds for any choice of 2. On the other hand, for 6= 0, the coecient x~1jj,1 equals ,(, 2)jj,1, and Eq. (44) yields 2 ! 1 ,
1 2 2 j j 'zz [; k] = a [k]a[k + ](, 2) , + 1 + 1 + + 1 , 2 2 2 2 ! j j 2 = ,2a[k]a[k + ] (1,+ 2 ) 1 , 2 , 1 , ; even; 6= 0: (47) 2 2 2 Substituting 2 according to Eqs. (72), it is straightforward to show that 'zz [; k] = 0, even, 6= 0. Because of Eq. (30), 'zz [; k] is zero for odd, too, resulting in
'zz [; k] = 2 z []:
(48)
A comparison of Eqs. (30) and (31) shows immediately that '~zz [; k] vanishes identically if x~1 and 2 are chosen according to Eqs. (41) and (72), respectively, i.e.,
'~zz [; k] = 0; 8:
(49)
This means, in this special case z[] is a proper complex random process [14]. This con rms that the derived nonlinear time{variant IIR lter is the optimum NPEF for the process z[] since the prediction error z[k] has minimum variance and there is no correlation left which could be further exploited, i.e., z[] is uncorrelated and z[] and z[] are (mutually) uncorrelated. The optimality of the derived IIR lter is con rmed by Fig. 3b), where the variance according to Eq. (46) is a lower bound for the variances corresponding to the FIR NPEFs.
5.3 DDFSE and DFE Using Eqs. (3) and (43), in the case of an IIR lter, yd[k] according to Eq. (11) can be expressed as yd[k] = a[k] + [k]z[k , 1]; (50) and again, DDFSE can be applied. In order to obtain an appropriate expression for ~z[k], the de nition [k] = 2a[k]a[k , 1] has to be used in Eq. (43): z[k] = z[k] , 2a[k]a[k , 1]z[k , 1]
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= = = =
z[k] , 2a[k]a[k , 1](z[k , 1] , 2a[k , 1]a[k , 2]z[k , 2]) z[k] , 2a[k]a[k , 1]z[k , 1] + 22a[k]a[k , 2]z[k , 2] ::: 0 X z[k] + a[k] (, 2) (a[k , ]z[k , ])() =1 0 +1
+a[k](, 2)
(a[k , 1 , 0]z[k , 1 , 0])((0+1)):
(51)
By replacing the unknown transmitted symbols a[k , ], 0 < 1, in Eq. (51) by trial symbols ~a[k , ], 0 0, 0 0 < 1, and decision{feedback symbols a^[k , ], 0 + 1 < 1, ~z[k] may be de ned as 4
~z[k] = z~[k] + a~[k]
0 X
(, 2) (~a[k , ]~z[k , ])()
=1 0 +1
+~a[k](, 2)
(^a[k , 1 , 0]^z[k , 1 , 0])((0+1));
(52)
where ^z[k , 1 , 0] is given by 4 ^z[k , 1 , 0] = z^[k , 1 , 0] , 2a^[k , 1 , 0]^a[k , 2 , 0]^z[k , 2 , 0]:
(53)
z~[k] and z^[k] are de ned in Eqs. (36) and (38), respectively. A comparison of Eqs. (37) and (52) shows the similarity of both DDFSE schemes, i.e., the same number of states and transitions is required. In the limit Eb=N0 ! 1, the prediction{error variance 2 z approaches 2 (cf. Fig. 3b)) when an IIR or an equivalent nonrecursive lter of in nite order is applied. On the other hand, in this case x~1 tends to ,(,1) , 1 < 1, cf. Eqs. (41), (72), and thus, z[k] ! b[k , 1]n[k] can be obtained from Eq. (77). This implies, pz (z[]ja[]) asymptotically is Gaussian and z[] is i.i.d. Thus, in this limit case the Euclidean metric is optimum. A full{state Viterbi algorithm can not be used in the IIR case because it would require an in nite number of states. However, the DFE solution (0 = 0) yields a very simple structure when an IIR NPEF is used. From Eqs. (36) and (52) we get n
a^[k] = argmin j~zj2 ~a[k]
o
n
o
= argmin jrd[k]j2 + ja~[k]j2 j1 + 2^a[k , 1]^z[k , 1]j2 , 2Refa~[k]dIIR[k]g ; (54) ~a[k]
where dIIR[k] is de ned as
dIIR[k] =4 (1 + 2a^[k , 1]^z[k , 1])rd[k]:
(55)
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
14
Like dFIR [k] in the FIR case, dIIR[k] can be used directly to make a decision on a~[k], i.e., the complex plane is divided into M sectors and a^[k] is determined by the sector in which dIIR[k] falls. If 2 is not chosen according to Eq. (72), but equal to zero, dIIR[k] = rd[k] results, i.e., the DFE scheme is identical with conventional DD. In order to obtain a structure with low complexity, we de ne q[k] =4 1 + 2a^[k , 1]^z[k , 1]; (56) i.e., dIIR[k] = q[k]rd[k]. Applying Eq. (3) and ^z[k , 1] = 1= 2 ^a[k , 1](q[k] , 1) (cf. Eq. (56)) to Eq. (53), the recursion
q[k] = 1 + 2(rd[k , 1]^a[k , 1] , q[k , 1]) (57) is obtained. The corresponding simple structure is shown in Fig. 5. As has been shown above, in the limit Eb=N0 ! 1, z[k] is a Gaussian random variable with variance 2 in the IIR case. Hence, the proposed DFE detector has the same performance as a coherent detector for high Eb=N0 ratios, since in both cases the decision variable has a Gaussian pdf with variance 2. Since DDFSE performs at least as good as DFE but is bounded by CD, the performance of CD is attained by DDFSE for Eb =N0 ! 1, too.
6 Simulation Results In this section, we evaluate the performance of our receiver for a QDPSK (M = 4) constellation by computer simulations. First we assume an AWGN channel and an optimum design of the prediction{error lters (i.e., 2 is chosen according to Eq. (72)). In Fig. 6, conventional DD, CD, and our receiver using an FIR NPEF of order N = 2 are compared. Obviously, our receiver improves conventional DD signi cantly, e.g. at BER = 10,3 power eciency of conventional DD is improved by 1.3 dB, i.e., the remaining gap to CD is only 0.5 dB, when a full{state Viterbi algorithm (corresponding to DDFSE, 0 = 2) is used. The gain compared to DD decreases with decreasing number of states of DDFSE. For 0 = 1 and 0 = 0 (corresponding to DFE) the improvement is 1.2 dB and 0.9 dB, respectively. A genie{ aided DFE, operating with error{free feedback symbols, even outperforms CD of QDPSK at low Eb=N0 ratios. Since the genie avoids error propagation, the BER of genie{aided DFE is lower{bounded by the BER of coherent QPSK and not by that of coherent QDPSK. Fig. 6 shows, that the performance improvement for N = 2 is only moderate if DDFSE is used instead of DFE, whereas complexity increases considerably. Further simulations have
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
15
shown that for N > 2 an even smaller gain is achieved by DDFSE. Therefore, in the sequel we only consider receivers applying DFE. In Fig. 7, a comparison is made between DFE receivers using di erent NPEFs. The receiver applying an IIR lter yields the best result. This was to be expected since this lter causes the lowest prediction{error variance 2 z (cf. Fig. 3b)). Note, that this very simple receiver outperforms conventional DD by 1.2 dB at BER = 10,3 whereas computational complexity is increased only slightly. The power eciency of receivers using FIR lters increases with N but is always inferior to that of the receiver with IIR lter. For N = 4 the di erence between FIR and IIR structure is very small. This behaviour is con rmed by Fig. 3b) because for 10 log10(Eb=N0 ) 10 dB the di erence of both structures in 2 z is only small. At this point, it is interesting to compare the results obtained for the DFE receiver using FIR NPEFs with nonrecursive DF{DD [2, 3, 4]. The complexity of both schemes is similar and for a fair comparison the number of feedback symbols should be the same in both cases. From Fig. 1 of [4] and Fig. 7 of this paper1 it can be seen, that our scheme performs better than DF{DD for small numbers of feedback symbols, e.g. if 1 and 2 feedback symbols are used, our scheme requires an Eb=N0 ratio of 8.5 dB and 8.2 dB to obtain BER = 10,3 , whereas the nonrecursive DF{DD scheme requires 8.65 dB and 8.3 dB, respectively. On the other hand, for large numbers of feedback symbols DF{DD approaches CD for arbitrary Eb=N0 ratios, whereas the scheme proposed here approaches CD only at high Eb=N0 ratios. A comparison of the DFE receiver using IIR NPEFs and the recursive DF{DD scheme proposed in [5, 7] will be made in Fig. 9. So far only optimum NPEFs have been considered, whose coecients depend on the channel noise variance 2 because 2 depends on 2 (cf. Eq. (72)). In practice, xed lters would be preferable. Therefore, the NPEF might be designed for a mean 2 = 2 (i.e.,
2 = 2) and then kept xed for all Eb=N0 ratios. Further simulations showed that xed lters with 2 = 0:126 (corresponding to 10 log10(Eb=N0) = 6 dB) lead only to a loss of less than 0:03 dB in power eciency for 2 dB 10 log10(Eb=N0 ) 9 dB. The design parameter
2 corresponding to this 2 is 2 = 0:7. Figs. 8a) and b) show, respectively, the normalized prediction{error variance 2 z =2 (calculated from Eq. (45)) and BER vs. 2 for the DFE receiver applying an IIR NPEF. The optimum values for 2 in the sense of minimum 2 z (cf. Eq. (72)) are 0.64, 0.73, and 0.80 1
Note, that in [4] the number of feedback symbols is ( , 1) ( is de ned in [4]), whereas here it is . L
L
N
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
16
for 10 log10(Eb=N0 ) = 4 dB, 7 dB, and 10 dB, respectively. The corresponding values for 2 z =2 and BER are denoted by triangles. It is con rmed that these optimum values for 2 which yield minimum 2 z also yield minimum BER. However, it can be seen from Fig. 8b) that the BERs corresponding to 2 = 0:7 are only slightly higher than the minimum BERs for all Eb =N0 ratios considered here. Although we cannot provide an analytical expression for BER as a function of 2 z , Figs. 8a) and b) show that BER depends monotonically on 2 z . For derivation of our receiver a constant phase has been assumed. Now, the performance of the receiver is investigated for the case of phase variations. Fig. 9 shows the in uence of a frequency o set f on the receiver performance at 10 log10(Eb=N0 ) = 8 dB, assuming the symbol duration of IS{136 digital cellular standard (T = 41:2 s). In the FIR case, xed lters with 2 = 2 = 0:7 are chosen. It can be observed that the sensitivity to frequency o set increases with FIR lter order N . For example, when f exceeds 430, 340 and 230 Hz, the receiver with lter order 1, 2 and 4, respectively, performs even worse than conventional DD. In the IIR case a similar result is obtained. Here, the sensitivity to frequency o set depends on the magnitude of 2. For large j 2j, the magnitude of the taps of an equivalent nonrecursive lter decays slowly and a high sensitivity results. This is con rmed by Fig. 9: If f exceeds 120 Hz, 2 = 2 = 0:3 yields better results than 2 = 2 = 0:7. It can be concluded from Fig. 9 that the proposed detection scheme requires an e ective frequency o set compensation technique. However, this is not a major problem because such techniques are well{known for MDPSK, e.g. [7, 13]. Although our receiver has been designed for AWGN channels it performs also well under
at fading conditions as is shown in Appendix C. In Fig. 10, we compare our receiver with IIR NPEF ( 2 = 2 = 0:7) with the recursive DF{DD scheme proposed in [5, 7] for AWGN, at Rayleigh and Ricean fading channels. The complexity of both schemes is similar and in all fading simulations Jakes model [21] has been used. The normalized Doppler frequency fdT has been chosen to 0:0075 (corresponding to worst case in IS{136 digital cellular standard) and 0:03 (corresponding to worst case in [7]) for Rayleigh and Ricean fading, respectively. The Ricean factor R [20] for the Ricean fading channel was 7 dB. The IIR lter coecient de ned in [7] has been chosen to 0:6 in order to get approximately the same performance for both schemes under AWGN conditions. Fig. 10 shows that in this case both techniques perform better than DD. In many applications (for example terrestrial
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
17
and satellite mobile communications) the receiver has to cope with both, AWGN and at fading channels. According to Fig. 10, for fading conditions, our detector has the same power eciency as DD2 as was to be expected from Appendix C and outperforms the detection scheme proposed in [5, 7] which causes an error oor nearly twice as high. Note, that the lter coecients of both schemes were constant for all channel models. It should be mentioned, that > 0:6 would improve performance of the scheme of [5, 7] for the AWGN channel (for ! 1 CD is approached [7]), but would cause a signi cant degradation for fading channels. In general, a Ricean factor estimation circuit proposed in [7] would reduce the error oor for the technique there, however, computational complexity would increase considerably. On the other hand, the predictor coecients derived here are optimized for the AWGN channel and are suboptimum for fading channels. If the coecients of the nonlinear NPEFs were optimized for fading channels it might also be possible to lower the error oor of conventional DD. However, this is beyond the scope of this paper.
7 Conclusions In this paper, a new technique has been proposed in order to improve the performance of DD of MDPSK for AWGN channels. It has been shown that the noise process contained in the decision variable for conventional DD is not proper, which means sequence estimation has to be applied in an optimum receiver instead of symbol{by{symbol decisions. Suitable algorithms can be derived using nonlinear noise prediction. Optimum coecients for FIR and IIR prediction{error lters have been calculated. In order to reduce complexity, DDFSE or DFE may be used to estimate the transmitted symbol sequence. Simulations show for both cases that the power eciency of a conventional di erential detector is improved considerably. Although the optimum FIR and IIR lter coecients depend on the channel noise variance 2, it turned out that for practical interesting Eb =N0 ratios performance degrades only slightly when xed lters are employed, which are optimized for a mean variance 2 = 2 corresponding to 2 = 2 = 0:7. Especially the DFE structure using an IIR noise prediction{error lter with xed coecients is well suited for implementation because of its The BERs for DD shown in Fig. 8 of [7] for Ricean fading ( d = 0 03, = 7 dB) are slightly higher than these in Fig. 10 of this paper. Presumably this is due to the fast acquisition clock recovery used there, whereas perfect synchronisation is assumed here. 2
f T
:
R
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
18
very low computational complexity. This simple receiver improves DD by 1:2 dB at BER = 10,3 for AWGN conditions while no performance degradation compared to DD is observed for Rayleigh and Ricean fading channels and Doppler frequencies considered here. In contrast to that, the low complexity detector presented in [5, 7] degrades considerably under fading conditions as has been demonstrated by simulations.
Acknowledgement The authors would like to thank TCMC (Technology Centre for Mobile Communication), Philips Semiconductors, Germany, for supporting this work.
References [1] [2] [3] [4] [5] [6] [7] [8]
D. Divsalar, M. K. Simon: Multiple-Symbol Di erential Detection of MPSK, IEEE Trans. Commun., Vol. 38, Mar. 1990, pp. 300 - 308. H. Leib, S. Pasupathy: The Phase of a Vector Perturbed by Gaussian Noise and Di erentially Coherent Receivers, IEEE Trans. on Information Theory, Vol. 34, No. 6, Nov. 1988, pp. 1491 - 1501. F. Edbauer: Bit Error Rate of Binary and Quaternary DPSK Signals with Multiple Di erential Feedback Detection, IEEE Trans. Commun., Vol. 40, Mar. 1992, pp. 457 - 460. F. Adachi, M. Sawahashi: Decision Feedback Multiple{Symbol Di erential Detection for M{ary DPSK, Electronics Letters, Vol. 29, No. 15, Jul. 1993, pp. 1385 1387. H. Leib: Data{Aided Noncoherent Demodulation of DPSK, IEEE Trans. Commun., Vol. 43, Feb.{Apr. 1995, pp. 722 - 725. H. Leib: Comments on \Di erential Detection with IIR Filter for Improving DPSK Detection Performance", IEEE Trans. Commun., Vol. 45, Jun. 1997, p. 637. N. Hamamoto: Di erential Detection with IIR Filter for Improving DPSK Detection Performance, IEEE Trans. Commun., Vol. 44, Aug. 1996, pp. 959 - 966. F. Adachi, M. Sawahashi: Decision Feedback Di erential Phase Detection of M{ ary DPSK Signals, IEEE Trans. Vehic. Tech., Vol. 44, May 1995, pp. 203 - 210.
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
19
F. Adachi: Reduced{state Viterbi di erential detection using a recursively estimated phase reference for M{ary DPSK, IEE Proc.{Commun, Vol. 142, No. 4, Aug. 1995, pp. 263 - 270. R. Y. Wei, M. C. Lin: Di erential Phase Detection Using Recursively Generated Phase References, IEEE Trans. Commun., Vol. 45, Dec. 1997, pp. 1504 - 1507. F. Adachi: Adaptive Di erential Detection Using Linear Prediction for M {ary DPSK, IEEE Trans. Vehic. Tech., Vol. 47, Aug. 1998, pp. 909 - 918. R. Schober, W. H. Gerstacker, J. B. Huber: Decision{Feedback Di erential Detection of MDPSK for Flat Rayleigh Fading Channels, accepted for publication in IEEE Trans. Commun., 1999. N. R. Sollenberger, J. C.{I. Chuang: Low{Overhead Symbol Timing and Carrier Recovery for TDMA Portable Radio Systems, IEEE Trans. Commun., Vol. 38, Oct. 1990, pp. 1886 - 1892. F. D. Neeser, J. L. Massey: Proper Complex Random Processes with Applications to Information Theory, IEEE Trans. Inf. Theory, Vol. 39, Jul. 1993, pp. 1293 1302. A. Duel{Hallen, C. Heegard: Delayed Decision{Feedback Sequence Estimation, IEEE Trans. Commun., Vol. 37, May 1989, pp. 428 - 436. C.A. Bel ore, J.H. Park: Decision{feedback equalization, Proc. IEEE, Vol. 67, Aug. 1979, pp. 1143 - 1156. G. D. Forney, Jr.: The Viterbi Algorithm, Proc. IEEE, Vol. 61, 1973, pp. 268 278. S. Haykin: Adaptive Filter Theory, 3rd edition, Prentice Hall, New Jersey, 1996. J. G. Proakis: Digital Communications, McGraw-Hill, New York, 1995. T. S. Rappaport: Wireless Communications, Prentice Hall, New Jersey, 1996. W. C. Jakes, Jr., Ed.: Microwave Mobile Communications, Wiley, New York, 1974. E. J. Borowski, J. M. Borwein: The Harper Collins Dictionary of Mathematics, Harper Perennial, New York, 1991.
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A Calculation of the Optimum FIR Coecients In the following the optimum FIR lter coecients are calculated from Eqs. (20) and (21). In order to simplify Eq. (20), which depends on the transmitted sequence a[], we introduce the variables x~1 [k] =4 a[k] a[[kk], ] ; even; (58) 1 N: 4 [k ] 1 x~ [k] = a[k]a[k , ] ; odd; (59) Applying Eqs. (58), (59) to Eq. (20) and dividing row no. of the resulting set of linear equations by a[k](a[k , ])((+1)) yields
A~ N x~ 1 = y~ 1;
(60)
where the N N matrix A~ N and the N 1 vectors x~ 1 and y~ 1 are de ned as 2 6 6 6 6 6 6 4 ~ N = 666 6 6 6 6 6 4
A
1 0 ... ... 0
1 1 ... ... :::
0 1 ... ... :::
::: 0 1 ... ... 0
::: ::: ... ... ... 1
y~ 1 =4 [1 0 : : : 0]T ; x~ 1 =4 [~x11 x~12 : : : x~1N ]T :
3
0 7 0 777 ... 77 7 . . . 777 ; 7 7 1 775
(61)
(62) (63)
4 In Eq. (61) the notation = 2 + 2 is used. Note, that a time{independent solution results because A~ N and y~ 1 are independent of k. Therefore, discrete time has been omitted in the elements of vector x~ 1. Now, the determinant d = detA~ of the matrix A~ , 1 N , is introduced. The elements x~1 of vector x~ 1 can be obtained from Eq. (60) by applying Cramer's Rule [22]. Because of the special form of A~ N and y~ 1, x~1 can be written as (64) x~1 = (,1)+1 ddN , ; 1 N; N
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
21
and the determinant of A~ can be calculated recursively according to
d = d,1 , d,2 ; > 0; d0 =4 1; d =4 0; < 0:
(65) (66) (67)
Eqs. (65), (66) and (67) may be rewritten to the di erence equation
d , d,1 + d,2 = [ ]; 0;
(68)
which can be solved using the z{transform. Introducing D(z) = Zfd g, the z{transform of Eq. (68) leads to 2 D(z) = z2 , zz + 1 : (69) Assuming 2 > 0,
D(z) = ,1 z ,z , ,2 z ,z 1 2 1 1 2 2 can be obtained, where the de nitions q
1 =4 12 ( + 2 , 4); q
2 =4 12 ( , 2 , 4)
(70) (71) (72)
have been used. Inverse z-transform of Eq. (70) yields +1
2+1 ; 0: d = 1 , (73) 1 , 2 By combining Eq. (64) and Eq. (73) and using the relation 1 2 = 1, x~1 can be expressed as ,2(N +1, ) x~1 = (,1)+1 2, 2 ,2(N +1) , 1 ; 1 N: (74)
2 ,1 From Eqs. (58), (59), and (74), the optimum FIR coecients contained in vector x1 (cf. Eq. (25)) are obtained. x2 (cf. Eq. (26)) can be determined in a similar way like x1. The result is
x2 = 0:
(75)
22
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
B Calculation of 'zz ; k for the FIR case [
]
In the sequel 'zz [; k] is derived. Using Eq. (29) and the de nition x~10 =4 ,1, z[k] can be written as N X z[k] = ,a[k] x~1 (a[k , ]z[k , ])(): (76) =0
Applying some straightforward calculations to Eq. (76) leads to z[k] = ,a[k] +
N X =0
NX +1 =0
(~x1,1 + x~1 )(b[k , ]n[k , ])()
x~1 (a[k , ]n[k , ]n[k , 1 , ])()
!
;
(77)
where Eqs. (1), (5) and the de nition x~1 =4 0, < 0 or > N , have been used. Eq. (77) may be exploited for calculation of 'zz [; k]:
'zz [; k] = Efz[k + ]z[k]g
+1 NX +1 NX
= a[k]a[k + ]
=0 =0
(~x1,1 + x~1 )(~x1,1 + x~1)(b[k , ])()
(b[k , + ])()Ef(n[k , ])()(n[k , + ])()g
N X N X + x~1 x~1 (a[k =0 =0
, ])()(a[k , + ])()
Ef(n [k , ]n[k , 1 , ])()(n[k , + ]n[k , 1 , + ])()g : (78)
From Eqs. (6) and (7)
Ef(n [k , ])()(n[k ,
8 > < ( ) + ]) = > :
g
2[ , + ]; even; 0; odd;
(79)
and
Ef(n[k , ]n[k , 1 , ])()(n[k , + ]n[k8, 1 , + ])()g > < 4 [ , + ]; even; = > :
can be obtained. Thus,
'zz [; k] = 0; odd;
0;
odd;
(80) (81)
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
23
results. For even, only terms which satisfy the condition , + = 0 contribute to the sums in Eq. (78). Hence, this equation can nally be simpli ed to
'zz [; k] = 2a[k]a[k + ]
NX +1 =0
(~x1,1 + x~1 )(~x1+jj,1 + x~1+jj) N
X +2 x~1 x~1+jj =0
!
; even:
(82)
Using the same method as for calculation of 'zz [; k], it is straightforward to determine the PACS '~zz [; k].
C Receiver Performance Under Flat Fading In this appendix, the performance of the DFE receiver using an IIR NPEF is analyzed. From Eqs. (55) and (56) it can be seen that, in this case, the decision variable is given by
dIIR[k] = rd[k]q[k]:
(83)
Under fading conditions, most errors occur when the fading amplitude is small (i.e., the instantaneous signal{to{noise ratio (SNR) is low). Thus, if the in uence of the channel noise is neglected, rd[k , 1] 1 results and according to Eq. (57), q[k] may be approximated by q[k] 1 , 2q[k , 1]: (84) From Eq. (84), q[k] 1=(1 + 2) can be obtained under steady{state conditions and dIIR[k] 1 +1 rd [k] (85) 2 follows from Eq. (83). Since the real, multiplicative constant 1=(1 + 2) has no in uence on the decision, the same decision variable as for conventional DD results. This means, for small fading amplitudes when most decision errors occur the DFE receiver using an IIR NPEF is almost identical with conventional DD. For large fading amplitudes (i.e., the instantaneous SNR is high), however, both schemes produce relatively few errors. Therefore, it can be expected that the DFE receiver using an IIR NPEF has a similar performance like conventional DD under fading conditions. Similar results can be optained for DFE receivers using FIR NPEFs and DDFSE receivers.
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24
Figure Captions: Figure 1: Block diagram of the transmission model for AWGN channel and constant phase shift .
Figure 2: Nonlinear time{variant noise prediction{error lter (NPEF) used in the proposed receiver.
Figure 3: a) Magnitude of x~1 vs. coecient number for di erent lter orders N and Eb =N0
ratios for a QDPSK constellation (M = 4). b) Normalized variance 2 z vs. 10 log10(Eb =N0) for M = 4 and di erent prediction{error lters.
Figure 4: DFE receiver based on an FIR NPEF. Figure 5: DFE receiver based on an IIR NPEF. Figure 6: BER of QDPSK (M = 4) vs. 10 log10(Eb=N0 ) for DDFSE receivers with di erent
numbers of states Z = M 0 applying an FIR NPEF of order N = 2. For comparison the BERs for conventional DD, CD, and genie aided DFE are also shown.
Figure 7: BER of QDPSK (M = 4) vs. 10 log 10(Eb=N0 ) for conventional DD, CD, and DFE receivers applying di erent NPEFs.
Figure 8: a) 2 z =2 and b) BER vs. 2 for QDPSK (M = 4) and the DFE receiver applying
an IIR NPEF. Those values of 2 z =2 and BER, which correspond to the optimum value for 2 (cf. Eq. (72)) are denoted by a triangle.
Figure 9: BER of QDPSK (M = 4) vs. frequency o set f in kHz for T = 41:2 s for conventional DD and DFE receivers applying di erent suboptimum NPEFs at 10 log10(Eb=N0) = 8 dB.
Figure 10: BER of QDPSK (M = 4) vs. 10 log10(Eb=N0 ) for conventional DD, DFE re-
ceiver applying a suboptimum IIR NPEF ( 2 = 0:7), and the scheme proposed in [4, 5] for Rayleigh fading (fdT = 0:0075) and Ricean fading (R = 7 dB, fd T = 0:03) channels.
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
Figures: Figure 1:
25
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Figure 2:
26
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Figure 3:
27
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Figure 4:
28
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Figure 5:
29
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Figure 6:
30
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Figure 7:
31
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Figure 8:
32
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Figure 9:
33
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Figure 10:
34
Abstract A new technique is proposed to improve the performance of di erential detection of MDPSK (M {ary di erential phase{shift keying) signi cantly, applying sequence estimation. In order to obtain an appropriate representation of the received signal, a nonlinear time{variant FIR or IIR prediction{error lter is used. For both lter structures the optimum coecients are derived, assuming transmission over an additive white Gaussian noise (AWGN) channel. Delayed decision{feedback sequence estimation (DDFSE) is employed to estimate the transmitted symbol sequence. It is shown by simulations that even for decision{feedback equalization (DFE), which is a simple special case of DDFSE, a signi cant performance improvement of conventional di erential detection under AWGN conditions results. In contrast to other noncoherent low complexity receivers proposed in literature this receiver is very robust under at fading (Rayleigh and Ricean) conditions. Index Terms{Di erential detection, M {ary DPSK, nonlinear prediction, delayed decision{ feedback sequence estimation (DDFSE), decision{feedback equalization (DFE).
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2
1 Introduction It is well known, that di erential detection (DD) of MDPSK (M {ary di erential phase{ shift keying) is very attractive when simplicity and robustness of implementation are more important than optimum power eciency. In addition, tracking of the correct carrier phase, as required by coherent detection (CD), may be dicult or even impossible in a fading environment. On the other hand, for an additive white Gaussian noise (AWGN) environment, DD causes a performance loss compared to CD. In literature, modi ed noncoherent detection schemes with an improved performance for the AWGN channel have been proposed. For example, a maximum{likelihood receiver using multiple{symbol detection (MSD) has been reported by Divsalar et al. [1]. The main disadvantage of MSD is its high computational complexity compared to techniques, which rely on feeding back previously decided symbols [2, 3, 4, 5, 7, 8, 9, 10, 11, 12]. Especially the recursive decision{feedback di erential detection (DF{DD) scheme presented in [5] and [7] (in both papers the same lter structure was used as has been pointed out in [6]) and the nonrecursive DF{DD scheme proposed and analyzed in [2, 3, 4] are very attractive for implementation because, respectively, a one tap IIR lter and an FIR lter are employed prior to DD in order to stabilize the reference phase. Hence, the new detector proposed here will be compared with those structures. Although our detector in its most simple form uses decision feedback, too, this approach is di erent from the previous ones. After conversion to complex baseband representation and sampling, the current received symbol is multiplied by the previous complex conjugated one, like in a conventional di erential detector. The resulting signal is corrupted by a noise process which is not proper [14]. Hence, symbol{by{symbol decisions of conventional di erential detectors are not optimum and sequence estimation has to be applied. By means of nonlinear noise prediction a signal representation suitable for the application of delayed decision{ feedback sequence estimation (DDFSE) [15] and decision{feedback equalization (DFE) [16], which is a special case of DDFSE, is derived. Both schemes perform signi cantly better than conventional DD and especially the proposed DFE scheme has a very low computational complexity. For example, for an AWGN channel and M = 4, our receiver using DFE can improve conventional DD by about 1.2 dB at BER = 10,3 . Moreover, our detector is more robust under at Rayleigh and Ricean fading conditions than recursive DF{DD [5, 7] as will be shown by simulations.
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2 Transmission Model Fig. 1 shows a block diagram of the transmission model under consideration. All signals are represented by their complex{valued baseband equivalents. The MDPSK symbols are denoted by a[k] 2 A = fej2 M j 2 f0; 1; : : : ; M , 1gg and the corresponding di erentially encoded MPSK symbols b[k] are given by
b[k] = a[k]b[k , 1]; k 2 Z:
(1)
Both, transmitter lter Ht(f ) and receiver lter Hr (f ) have square{root Nyquist characteristic. Hence, no intersymbol interference occurs as long as signals are transmitted over AWGN or slow at small{scale fading [20] channels. This also holds when there is a small frequency o set f between receiver and transmitter oscillator, i.e., fT 1, where T denotes the symbol interval. For derivation of our detection scheme an AWGN channel with an arbitrary, constant phase shift is assumed. In Section 6, the performance of the new receiver is also tested for frequency o set and at (Rayleigh and Ricean) fading channels. The complex{valued AWGN is denoted by nc (t) and the one{sided power spectral density of the underlying real passband noise process is N0. The samples r[k] of the received signal r(t) can be written as r[k] = r(kT ) = ej b[k] + n0[k]; (2) applying an appropriate normalization to Hr (f ). Here, n0[k] denotes uncorrelated Gaussian noise with variance 2 = ENS0 , where ES is the (mean) received energy per symbol. Like in a conventional di erential detector, the current received symbol r[k] is multiplied with the previous complex conjugated one:
rd[k] = r[k]r[k , 1] = a[k] + z[k];
(3)
where () denotes complex conjugation and, according to Eqs. (1) and (2), the e ective noise z[k] is given by
z[k] = ej b[k]n0[k , 1] + e,j b[k , 1]n0[k] + n0[k]n0[k , 1]:
(4)
By introducing n[k] =4 e,j n0[k], Eq. (4) can be rewritten to
z[k] = b[k]n[k , 1] + b[k , 1]n[k] + n[k]n[k , 1]: Note, that n[k] is still uncorrelated Gaussian noise with the same variance as n0[k].
(5)
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A conventional di erential detector directly uses rd[k] to make a decision on a[k]. The new receiver, however, processes rd[k] in a nonlinear noise prediction{error lter (NPEF), which is embedded in a DDFSE [15] or DFE [16], delivering estimated symbols a^[k , k0], where k0 is the decision delay.
3 Statistical Properties of z
[]
n[] is a proper equivalent complex baseband uncorrelated Gaussian random process [14], i.e., its autocorrelation sequence (ACS) 'nn [] and pseudo{autocorrelation sequence (PACS) '~nn [] are 'nn[] = Efn[k + ]n[k]g = 2[]; '~nn[] = Efn[k + ]n[k]g = 0; 8 ;
(6) (7)
where Efg denotes expectation and [] is the unit pulse sequence, i.e., [0] = 1, [] = 0, 6= 0. Now, the second{order statistics of the e ective noise z[] are determined for a given data sequence a[]. By using straightforward operations and Eqs. (1), (5){(7), the ACS 'zz [] and PACS '~zz [; k] can be calculated to
'zz [] = Efz[k + ]z[k]g = (22 + 4)[]; '~zz [; k] = Efz[k + ]z[k]g = 2a[k]a[k + ][jj , 1]; 8k:
(8) (9)
Note, that for calculation of 'zz [] and '~zz [; k] the transmitted symbol sequence a[] is assumed to be known and expectation is performed only over the noise process n[]. Since the PACS '~zz [; k] does not vanish identically, z[] is not a proper complex random process [14]. In addition, the PACS depends on k, i.e., z[] is a nonstationary process for a xed sequence a[]. Although the ACS 'zz [] implies uncorrelatedness of z[] (cf. Eq. (8)), the PACS '~zz [; k] indicates that z[] and its complex conjugated version z[] are (mutually) correlated. Consequently z[] is not an independent identically distributed (i.i.d.) sequence. This implies, that the maximum{likelihood criterion, which is optimum in the sense of minimum sequence error probability,
a^[] = argmax fprd (rd[]ja~[])g ; ~a[]
(10)
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cannot be simpli ed to symbol{by{symbol decisions as done in conventional di erential detectors, which therefore are suboptimum. Here, prd (rd[]ja[]) denotes the probability density function (pdf) of sequence rd[] conditioned on the transmitted sequence a[]. In order to enable a recursive optimization of the criterion in Eq. (10), an equivalent representation of rd [k] in the form rd[k] = yd[k] + z[k] (11) is desirable, where yd[k] is uniquely determined by rd[k , ] and a[k , ], 0 < 1, and z[k] is i.i.d. Then prd (yd[] + z[]ja[]) = pz (z[]ja[]) and Eq. (10) is equivalent to ^a[] = argmax fpz (z[]ja~[])g ; a~[]
(12)
where pz (z[]ja[]) is the pdf of z[] conditioned on a[]. It would be advantageous if, in addition, pz (z[]ja[]) was a Gaussian pdf with constant variance because then optimization in Eq. (12) could be done by recursively minimizing a sum of quadratic Euclidean distances. In the sequel, the desired decomposition according to Eq. (11) will be developed using a nonlinear NPEF (cf. Fig. 2).
4 Improved Receiver Using a Nonlinear FIR NPEF 4.1 Derivation of the Optimum FIR Filter In this and the next section, an FIR and IIR lter is applied, respectively, for nonlinear noise prediction. The (mutual) correlatedness of z[] and z[] suggests to employ a lter which exploits not only z[] like conventional prediction{error lters [18], but also z[]. For derivation of the lter perfect knowledge of the transmitted sequence a[] is assumed. If an FIR lter is applied, the prediction error z[k] may be expressed as z[k] = z[k] , = ,
N X =0
N X =1
[k]z[k , ] ,
[k]z[k , ] ,
N X =0
N X =1
[k]z[k , ]
[k]z[k , ];
(13)
where [k] and [k], 1 N , are the complex{valued predictor coecients to be 4 determined and 0[k] =4 ,1, 0[k] = 0, 8k, are de ned for convenience. Dependence of the coecients on time k has been introduced because later it will turn out that the optimum
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coecients indeed are time{variant. In this context it is important to note that complex conjugation is not a linear operation, i.e., it is impossible to obtain z[] from z[] by linear ltering. Therefore, the explicit introduction of the complex conjugated process z[] in Eq. (13) is necessary. By using Eq. (5), Eq. (13) can be expressed as z[k] = b[k , 1]n[k] ,
, ,
N X =0 N X =0
N X =0
( +1 [k]b[k , 2 , ] + [k]b[k , ]) n[k , 1 , ]
( [k]b[k , ] + +1[k]b[k , 2 , ]) n[k , 1 , ]
N
X [k]n[k , ]n[k , 1 , ] , =0
[k]n[k , ]n[k , 1 , ];
(14)
4 4 where for simplicity of notation N +1[k] = 0, N +1[k] = 0, 8k, are introduced. The variance of the prediction error z[k] can be calculated from Eqs. (6), (7), and (14):
2 z
=
Efjz[k]j2g = 2
1+ +
N X =0
N X
=0
j +1[k]b[k , 2 , ] + [k]b[k , ]j2
j [k]b[k , ] + +1
N X 4 +
=0
j [k]j2 +
N X =0
[k]b[k , 2 , ]j2
!
!
j [k]j2 ;
(15)
where knowledge of b[k , ], 0 N + 1, is assumed, i.e., expectation is performed only over the noise process n[]. The optimum predictor coecients are chosen to minimize 2 z . Thus, Eq. (15) is di erentiated with respect to [k] and [k], 1 N , using the method for complex di erentiation described in [18], Appendix B: @2 z = 2(( [k]b[k , 1 , ] + [k]b[k + 1 , ])b[k , 1 , ] ,1 @ [k] +( [k]b[k , ] + +1[k]b[k , 2 , ])b[k , ]) + 4 [k]; (16) 2 @z = 2(( [k]b[k , 2 , ] + [k]b[k , ])b[k , ] +1 @ [k]
+( ,1[k]b[k + 1 , ] + [k]b[k , 1 , ])b[k , 1 , ]) + 4 [k]: (17)
By setting Eqs. (16) and (17) equal to zero and applying Eq. (1), 2N equations for determination of 2N coecients are obtained: (2 + 2) [k] + ,1 [k]u[k + 1 , ] + +1[k]u[k , ] = 0; (2 + 2) [k] + ,1 [k]u[k + 1 , ] + +1 [k]u[k , ] = 0;
1 N;
(18) (19)
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where the notation u[k] =4 a[k]a[k , 1] is introduced. These 2N equations can be separated into 2 independent matrix equations:
Ax1 = y1; AT x2 = y2;
(20) (21)
where the N N matrix A and the N 1 vectors x1, x2, y1 and y2 are de ned as
A =
2 2 + 2 6 6 6 u[k 1] 6 6 6 6 0 6 4 .
u[k , 1] 0 ::: 2 + 2 u[k , 2] 0 u[k , 2] 2 + 2 u[k , 3] ... ... ...
,
..
y1 y2 x1 x2
::: ::: ... ...
[u[k] 0 : : : 0]T ; [0 0 : : : 0]T ; [ 1[k] 2[k] 3[k] 4[k] : : :]T ; [ 1[k] 2[k] 3[k] 4[k] : : :]T :
= = = =
3
0 7 7 0 777 . . . 77 ; 7 ... 5
(22)
(23) (24) (25) (26)
Here, []T denotes the transpose of a matrix (vector). The elements of x1 and x2, which are calculated in Appendix A, are given by
[k] = [k] =
8 > > x~1 a[k]a[k > > < > > > > : 8 > > > >
> > > : x ~1 a[k]a[k
, ]; even; odd; even;
, ]; odd;
1 N;
(27)
1 N;
(28)
where x~1 is de ned in Eq. (74). Fig. 3a) shows the magnitude of x~1 vs. coecient number for di erent lter orders N and Eb =N0 ratios for a QDPSK constellation (M = 4), i.e., 2 = 2NE0b , where Eb is the (mean) energy per bit. At very high Eb=N0 ratios the magnitude of x~1 decreases linearly with increasing coecient number, whereas it decreases exponentially at low Eb=N0 ratios.
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4.2 ACS and PACS of z [ ]
By combining Eqs. (13), (27) and (28), we get for the prediction error z[k] the expression z[k] = z[k] ,
N X a[k] x~1 (a[k =1
, ]z[k , ])();
(29)
where ()() means that complex conjugation is applied times (e.g. ()(2) = (()) = ()). In Appendix B, it is shown that the ACS 'zz [; k] and the PACS '~zz [; k] can be calculated to [k ]g 'zz [; k] = Ef z [ k + ] z 8 > 2a [k ]a[k + ] PN +1 (~x1 + x~1 )(~x1 > ~1+jj) > =0 ,1 +jj,1 + x > < 2 PN x~1 x~1 = > + even; (30) =0 +jj ; > > > : 0; odd;
and
'~zz [; k] = Ef z[k + ]z[k]g 8 > > 0; even; > > < P = > 2a[k]a[k + ] N=0+1(~x1,1 + x~1 )(~x1+jj,1 + x~1+jj) (31) > > P > : +2 N=0 x~1 x~1+jj ; odd; where for simplicity of notation, the de nitions x~1 =4 0, < 0 or > N , and x~10 =4 ,1 are introduced. Eqs. (30) and (31) show that, in general, z[] is not uncorrelated and z[] and z[] are (mutually) correlated, too. However, as can be seen from Fig. 3b), which again is valid for a QDPSK (M = 4) constellation, the variance 2 z = 'zz [0; k] of the prediction error is lower than that of z[] (corresponding to \no lter"). For nite lter order N a trade{o between prediction gain and complexity exists. Obviously, for N ! 1 and Eb =N0 ! 1, 2 z approaches 2. Thus, in this case the prediction error z[k] has the same variance as the process n[].
4.3 DDFSE and DFE So far it has been assumed, that the transmitted sequence a[] is known. In practice, however, this sequence has to be estimated at the receiver. Applying the previous results, the received
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signal rd[k] may be written according to Eq. (11), where yd[k] can be determined from Eqs. (3) and (29), N X yd[k] = a[k] + a[k] x~1 (a[k , ]z[k , ])(): (32) =1
Although, in general, z[k] is not exactly i.i.d. (cf. Eq. (30), Eq. (31)) and the conditional pdf pz (z[]ja[]) is only approximately Gaussian, simulations justify the application of the Euclidean metric
mk+1 =
k X =,1
jrd[ ] , y~d[ ]j2 =
k X =,1
j~z[ ]j2 = mk + j~z[k]j2:
(33)
In Section 5.3 it will be shown that in the limit N ! 1, Eb =N0 ! 1, the Euclidean metric of Eq. (33) indeed is optimum. y~d[k] is given by
y~d[k] =4 rd[k] + ~z[k];
(34)
and ~z[k] is de ned as 4 ~z[k] = z~[k] , ~a[k]
where the de nition
N X x~1 (~a[k
=1
, ]~z[k , ])();
z~[k] =4 rd[k] , ~a[k]
(35) (36)
is used and a~[k , ] 2 A, 0 N , are equalizer trial symbols. Minimization of Eq. (33) may be done e ectively by the Viterbi algorithm [17]. In order to reduce complexity, DDFSE [15] can be employed. Here, ~z[k] has to be de ned according to 4
~z[k] = z~[k] ,
0 X a~[k] x~1 (~a[k
,a~[k] where the notation
=1
N X
=0 +1
, ]~z[k , ])()
x~1 (^a[k , ]^z[k , ])();
z^[k] =4 rd[k] , ^a[k]
(37) (38)
is used. A comparison of Eqs. (35) and (37) shows that in addition to trial symbols a~[k , ] 2 A, 0 0, 0 0 N , also decision{feedback symbols a^[k , ] 2 A, 0 + 1 N , have been introduced in Eq. (37) instead of the unknown transmitted symbols a[k , ], 0 N . The underlying trellis [15] has Z = M 0 states and T = M 0 +1 transitions.
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Note, that for 0 = N a full{state Viterbi algorithm results. The parameter 0 determines the number of states and the system performance. A trade{o between performance and complexity exists. For small 0, complexity is moderate, however, performance is a ected by error propagation for low to moderate Eb=N0 ratios (cf. Section 6). The extreme case 0 = 0 corresponds to DFE. In the sequel a simple DFE structure is derived. Here, j~z[k]j2 can be directly used to make a decision on a~[k], i.e., according to Eqs. (37) and (36) the estimated symbol a^[k] is determined by n
o
n
^a[k] = argmin j~zj2 = argmin jrd[k]j2 + ja~[k]j2j1 + a~[k]
a~[k]
N X x~1 (^a[k
=1
N X 1 + x~1 (^a[k =1
o
,2Re fa~[k]dFIR[k]g ;
where dFIR[k] is de ned as
dFIR[k] =
, ]^z[k , ])()j2
, ]^z[k , ])((,1))
!
rd [k]:
(39) (40)
Since the rst two terms in Eq. (39) are common to all M possible a~[k], they can be omitted. Hence, the minimum is attained when the real part of a~[k]dFIR[k] is maximum, i.e., the phase of ~a[k] is next to the phase of dFIR[k]. This means that the complex plane can be divided into M sectors and a^[k] is determined by the sector into which dFIR[k] falls. This corresponds to the decision rule usually used for conventional DD, with the di erence that for DD rd[k] is used instead of dFIR [k]. Fig. 4 shows the structure of this simple DFE receiver.
5 Improved Receiver Using a Nonlinear IIR NPEF 5.1 Derivation of the Optimum IIR Filter In this section, the optimum IIR NPEF is derived. Here, we start from the optimum FIR lter of in nite order (N ! 1). In this case, the coecients x~1 can be obtained from Eq. (74) if j 2j < 1 for 2 > 0 is taken into account: ,2(N +1, ) , 1
2 1 +1 , = ,(, 2) ; 1 < 1: (41) x~ = Nlim (,1) 2 ,2(N +1) !1
2 ,1 The corresponding nonlinear time{variant IIR prediction{error lter for z[] can be obtained by substituting x~1 according to Eq. (41) in Eq. (29): z[k] = z[k] + a[k]
1 X
=1
(, 2) (a[k , ]z[k , ])()
(42)
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= z[k] , 2a[k]a[k , 1]z[k , 1] + 22a[k]a[k , 2]z[k , 2] , 23a[k]a[k , 3]z[k , 3] + : : : = z[k] , [k]z[k , 1] + [k] [k , 1]z[k , 2] , [k] [k , 1] [k , 2]z[k , 3] + : : : = z[k] , [k] (z[k , 1] , [k , 1]z[k , 2] + [k , 1] [k , 2]z[k , 3] , : : :) = z[k] , [k]z[k , 1]; (43) where the de nition [k] =4 2a[k]a[k , 1] is used. Eq. (43) corresponds to a nonlinear time{variant one tap IIR lter.
5.2 ACS and PACS of z [ ]
Since we have already calculated the ACS 'zz [; k] and the PACS '~zz [; k] for an FIR NPEF (cf. Eqs. (30), (31)), we use these results for the IIR case. This is possible because Eqs. (27), (28), and (41) describe a nonrecursive lter of in nite order which is equivalent to the optimum IIR lter. Therefore we substitute x~1 according to Eq. (41) in Eqs. (30) 4 and (31). Taking the de nitions x~10 = ,1 and x~1 =4 0, < 0, into account Eq. (30) ( even) yields 1 X 'zz [; k] = 2a[k]a[k + ] (~x1,1 + x~10)(~x1jj,1 + x~1jj) + (~x1 + x~1+1)
(~x1
+jj
=0
+ x~1
+jj+1
1 X 2 ) + x~1 x~1 =0
= 2a[k]a[k + ] ,(~x1jj,1 + x~1jj) + (, 2)jj +
1 2X
(, 2)2
!!
=0
!
+jj
1 X
=0
(, 2)2 (1 , 2)2 !!
1 , 2 + 2 = jj,1 jj) + (, 2 1 + 2 1 , 22 : (44) Now, two cases have to be distinguished: For = 0 the coecient x~1jj,1 = x~1,1 is zero, i.e., 2 4 2 z = 'zz [0; k] = 1 2+ + 1 , 2 : (45) 2 2 Applying some further calculations, where Eq. (72) has to be used, Eq. (45) can be transformed into 2 2 z = 'zz [0; k] = ; (46)
2a[k]a[k + ]
,(~x1
+ x~1
)jj
2
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Note, that Eq. (46) is valid only for optimum 2 according to Eq. (72), whereas Eq. (45) holds for any choice of 2. On the other hand, for 6= 0, the coecient x~1jj,1 equals ,(, 2)jj,1, and Eq. (44) yields 2 ! 1 ,
1 2 2 j j 'zz [; k] = a [k]a[k + ](, 2) , + 1 + 1 + + 1 , 2 2 2 2 ! j j 2 = ,2a[k]a[k + ] (1,+ 2 ) 1 , 2 , 1 , ; even; 6= 0: (47) 2 2 2 Substituting 2 according to Eqs. (72), it is straightforward to show that 'zz [; k] = 0, even, 6= 0. Because of Eq. (30), 'zz [; k] is zero for odd, too, resulting in
'zz [; k] = 2 z []:
(48)
A comparison of Eqs. (30) and (31) shows immediately that '~zz [; k] vanishes identically if x~1 and 2 are chosen according to Eqs. (41) and (72), respectively, i.e.,
'~zz [; k] = 0; 8:
(49)
This means, in this special case z[] is a proper complex random process [14]. This con rms that the derived nonlinear time{variant IIR lter is the optimum NPEF for the process z[] since the prediction error z[k] has minimum variance and there is no correlation left which could be further exploited, i.e., z[] is uncorrelated and z[] and z[] are (mutually) uncorrelated. The optimality of the derived IIR lter is con rmed by Fig. 3b), where the variance according to Eq. (46) is a lower bound for the variances corresponding to the FIR NPEFs.
5.3 DDFSE and DFE Using Eqs. (3) and (43), in the case of an IIR lter, yd[k] according to Eq. (11) can be expressed as yd[k] = a[k] + [k]z[k , 1]; (50) and again, DDFSE can be applied. In order to obtain an appropriate expression for ~z[k], the de nition [k] = 2a[k]a[k , 1] has to be used in Eq. (43): z[k] = z[k] , 2a[k]a[k , 1]z[k , 1]
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= = = =
z[k] , 2a[k]a[k , 1](z[k , 1] , 2a[k , 1]a[k , 2]z[k , 2]) z[k] , 2a[k]a[k , 1]z[k , 1] + 22a[k]a[k , 2]z[k , 2] ::: 0 X z[k] + a[k] (, 2) (a[k , ]z[k , ])() =1 0 +1
+a[k](, 2)
(a[k , 1 , 0]z[k , 1 , 0])((0+1)):
(51)
By replacing the unknown transmitted symbols a[k , ], 0 < 1, in Eq. (51) by trial symbols ~a[k , ], 0 0, 0 0 < 1, and decision{feedback symbols a^[k , ], 0 + 1 < 1, ~z[k] may be de ned as 4
~z[k] = z~[k] + a~[k]
0 X
(, 2) (~a[k , ]~z[k , ])()
=1 0 +1
+~a[k](, 2)
(^a[k , 1 , 0]^z[k , 1 , 0])((0+1));
(52)
where ^z[k , 1 , 0] is given by 4 ^z[k , 1 , 0] = z^[k , 1 , 0] , 2a^[k , 1 , 0]^a[k , 2 , 0]^z[k , 2 , 0]:
(53)
z~[k] and z^[k] are de ned in Eqs. (36) and (38), respectively. A comparison of Eqs. (37) and (52) shows the similarity of both DDFSE schemes, i.e., the same number of states and transitions is required. In the limit Eb=N0 ! 1, the prediction{error variance 2 z approaches 2 (cf. Fig. 3b)) when an IIR or an equivalent nonrecursive lter of in nite order is applied. On the other hand, in this case x~1 tends to ,(,1) , 1 < 1, cf. Eqs. (41), (72), and thus, z[k] ! b[k , 1]n[k] can be obtained from Eq. (77). This implies, pz (z[]ja[]) asymptotically is Gaussian and z[] is i.i.d. Thus, in this limit case the Euclidean metric is optimum. A full{state Viterbi algorithm can not be used in the IIR case because it would require an in nite number of states. However, the DFE solution (0 = 0) yields a very simple structure when an IIR NPEF is used. From Eqs. (36) and (52) we get n
a^[k] = argmin j~zj2 ~a[k]
o
n
o
= argmin jrd[k]j2 + ja~[k]j2 j1 + 2^a[k , 1]^z[k , 1]j2 , 2Refa~[k]dIIR[k]g ; (54) ~a[k]
where dIIR[k] is de ned as
dIIR[k] =4 (1 + 2a^[k , 1]^z[k , 1])rd[k]:
(55)
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14
Like dFIR [k] in the FIR case, dIIR[k] can be used directly to make a decision on a~[k], i.e., the complex plane is divided into M sectors and a^[k] is determined by the sector in which dIIR[k] falls. If 2 is not chosen according to Eq. (72), but equal to zero, dIIR[k] = rd[k] results, i.e., the DFE scheme is identical with conventional DD. In order to obtain a structure with low complexity, we de ne q[k] =4 1 + 2a^[k , 1]^z[k , 1]; (56) i.e., dIIR[k] = q[k]rd[k]. Applying Eq. (3) and ^z[k , 1] = 1= 2 ^a[k , 1](q[k] , 1) (cf. Eq. (56)) to Eq. (53), the recursion
q[k] = 1 + 2(rd[k , 1]^a[k , 1] , q[k , 1]) (57) is obtained. The corresponding simple structure is shown in Fig. 5. As has been shown above, in the limit Eb=N0 ! 1, z[k] is a Gaussian random variable with variance 2 in the IIR case. Hence, the proposed DFE detector has the same performance as a coherent detector for high Eb=N0 ratios, since in both cases the decision variable has a Gaussian pdf with variance 2. Since DDFSE performs at least as good as DFE but is bounded by CD, the performance of CD is attained by DDFSE for Eb =N0 ! 1, too.
6 Simulation Results In this section, we evaluate the performance of our receiver for a QDPSK (M = 4) constellation by computer simulations. First we assume an AWGN channel and an optimum design of the prediction{error lters (i.e., 2 is chosen according to Eq. (72)). In Fig. 6, conventional DD, CD, and our receiver using an FIR NPEF of order N = 2 are compared. Obviously, our receiver improves conventional DD signi cantly, e.g. at BER = 10,3 power eciency of conventional DD is improved by 1.3 dB, i.e., the remaining gap to CD is only 0.5 dB, when a full{state Viterbi algorithm (corresponding to DDFSE, 0 = 2) is used. The gain compared to DD decreases with decreasing number of states of DDFSE. For 0 = 1 and 0 = 0 (corresponding to DFE) the improvement is 1.2 dB and 0.9 dB, respectively. A genie{ aided DFE, operating with error{free feedback symbols, even outperforms CD of QDPSK at low Eb=N0 ratios. Since the genie avoids error propagation, the BER of genie{aided DFE is lower{bounded by the BER of coherent QPSK and not by that of coherent QDPSK. Fig. 6 shows, that the performance improvement for N = 2 is only moderate if DDFSE is used instead of DFE, whereas complexity increases considerably. Further simulations have
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
15
shown that for N > 2 an even smaller gain is achieved by DDFSE. Therefore, in the sequel we only consider receivers applying DFE. In Fig. 7, a comparison is made between DFE receivers using di erent NPEFs. The receiver applying an IIR lter yields the best result. This was to be expected since this lter causes the lowest prediction{error variance 2 z (cf. Fig. 3b)). Note, that this very simple receiver outperforms conventional DD by 1.2 dB at BER = 10,3 whereas computational complexity is increased only slightly. The power eciency of receivers using FIR lters increases with N but is always inferior to that of the receiver with IIR lter. For N = 4 the di erence between FIR and IIR structure is very small. This behaviour is con rmed by Fig. 3b) because for 10 log10(Eb=N0 ) 10 dB the di erence of both structures in 2 z is only small. At this point, it is interesting to compare the results obtained for the DFE receiver using FIR NPEFs with nonrecursive DF{DD [2, 3, 4]. The complexity of both schemes is similar and for a fair comparison the number of feedback symbols should be the same in both cases. From Fig. 1 of [4] and Fig. 7 of this paper1 it can be seen, that our scheme performs better than DF{DD for small numbers of feedback symbols, e.g. if 1 and 2 feedback symbols are used, our scheme requires an Eb=N0 ratio of 8.5 dB and 8.2 dB to obtain BER = 10,3 , whereas the nonrecursive DF{DD scheme requires 8.65 dB and 8.3 dB, respectively. On the other hand, for large numbers of feedback symbols DF{DD approaches CD for arbitrary Eb=N0 ratios, whereas the scheme proposed here approaches CD only at high Eb=N0 ratios. A comparison of the DFE receiver using IIR NPEFs and the recursive DF{DD scheme proposed in [5, 7] will be made in Fig. 9. So far only optimum NPEFs have been considered, whose coecients depend on the channel noise variance 2 because 2 depends on 2 (cf. Eq. (72)). In practice, xed lters would be preferable. Therefore, the NPEF might be designed for a mean 2 = 2 (i.e.,
2 = 2) and then kept xed for all Eb=N0 ratios. Further simulations showed that xed lters with 2 = 0:126 (corresponding to 10 log10(Eb=N0) = 6 dB) lead only to a loss of less than 0:03 dB in power eciency for 2 dB 10 log10(Eb=N0 ) 9 dB. The design parameter
2 corresponding to this 2 is 2 = 0:7. Figs. 8a) and b) show, respectively, the normalized prediction{error variance 2 z =2 (calculated from Eq. (45)) and BER vs. 2 for the DFE receiver applying an IIR NPEF. The optimum values for 2 in the sense of minimum 2 z (cf. Eq. (72)) are 0.64, 0.73, and 0.80 1
Note, that in [4] the number of feedback symbols is ( , 1) ( is de ned in [4]), whereas here it is . L
L
N
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16
for 10 log10(Eb=N0 ) = 4 dB, 7 dB, and 10 dB, respectively. The corresponding values for 2 z =2 and BER are denoted by triangles. It is con rmed that these optimum values for 2 which yield minimum 2 z also yield minimum BER. However, it can be seen from Fig. 8b) that the BERs corresponding to 2 = 0:7 are only slightly higher than the minimum BERs for all Eb =N0 ratios considered here. Although we cannot provide an analytical expression for BER as a function of 2 z , Figs. 8a) and b) show that BER depends monotonically on 2 z . For derivation of our receiver a constant phase has been assumed. Now, the performance of the receiver is investigated for the case of phase variations. Fig. 9 shows the in uence of a frequency o set f on the receiver performance at 10 log10(Eb=N0 ) = 8 dB, assuming the symbol duration of IS{136 digital cellular standard (T = 41:2 s). In the FIR case, xed lters with 2 = 2 = 0:7 are chosen. It can be observed that the sensitivity to frequency o set increases with FIR lter order N . For example, when f exceeds 430, 340 and 230 Hz, the receiver with lter order 1, 2 and 4, respectively, performs even worse than conventional DD. In the IIR case a similar result is obtained. Here, the sensitivity to frequency o set depends on the magnitude of 2. For large j 2j, the magnitude of the taps of an equivalent nonrecursive lter decays slowly and a high sensitivity results. This is con rmed by Fig. 9: If f exceeds 120 Hz, 2 = 2 = 0:3 yields better results than 2 = 2 = 0:7. It can be concluded from Fig. 9 that the proposed detection scheme requires an e ective frequency o set compensation technique. However, this is not a major problem because such techniques are well{known for MDPSK, e.g. [7, 13]. Although our receiver has been designed for AWGN channels it performs also well under
at fading conditions as is shown in Appendix C. In Fig. 10, we compare our receiver with IIR NPEF ( 2 = 2 = 0:7) with the recursive DF{DD scheme proposed in [5, 7] for AWGN, at Rayleigh and Ricean fading channels. The complexity of both schemes is similar and in all fading simulations Jakes model [21] has been used. The normalized Doppler frequency fdT has been chosen to 0:0075 (corresponding to worst case in IS{136 digital cellular standard) and 0:03 (corresponding to worst case in [7]) for Rayleigh and Ricean fading, respectively. The Ricean factor R [20] for the Ricean fading channel was 7 dB. The IIR lter coecient de ned in [7] has been chosen to 0:6 in order to get approximately the same performance for both schemes under AWGN conditions. Fig. 10 shows that in this case both techniques perform better than DD. In many applications (for example terrestrial
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
17
and satellite mobile communications) the receiver has to cope with both, AWGN and at fading channels. According to Fig. 10, for fading conditions, our detector has the same power eciency as DD2 as was to be expected from Appendix C and outperforms the detection scheme proposed in [5, 7] which causes an error oor nearly twice as high. Note, that the lter coecients of both schemes were constant for all channel models. It should be mentioned, that > 0:6 would improve performance of the scheme of [5, 7] for the AWGN channel (for ! 1 CD is approached [7]), but would cause a signi cant degradation for fading channels. In general, a Ricean factor estimation circuit proposed in [7] would reduce the error oor for the technique there, however, computational complexity would increase considerably. On the other hand, the predictor coecients derived here are optimized for the AWGN channel and are suboptimum for fading channels. If the coecients of the nonlinear NPEFs were optimized for fading channels it might also be possible to lower the error oor of conventional DD. However, this is beyond the scope of this paper.
7 Conclusions In this paper, a new technique has been proposed in order to improve the performance of DD of MDPSK for AWGN channels. It has been shown that the noise process contained in the decision variable for conventional DD is not proper, which means sequence estimation has to be applied in an optimum receiver instead of symbol{by{symbol decisions. Suitable algorithms can be derived using nonlinear noise prediction. Optimum coecients for FIR and IIR prediction{error lters have been calculated. In order to reduce complexity, DDFSE or DFE may be used to estimate the transmitted symbol sequence. Simulations show for both cases that the power eciency of a conventional di erential detector is improved considerably. Although the optimum FIR and IIR lter coecients depend on the channel noise variance 2, it turned out that for practical interesting Eb =N0 ratios performance degrades only slightly when xed lters are employed, which are optimized for a mean variance 2 = 2 corresponding to 2 = 2 = 0:7. Especially the DFE structure using an IIR noise prediction{error lter with xed coecients is well suited for implementation because of its The BERs for DD shown in Fig. 8 of [7] for Ricean fading ( d = 0 03, = 7 dB) are slightly higher than these in Fig. 10 of this paper. Presumably this is due to the fast acquisition clock recovery used there, whereas perfect synchronisation is assumed here. 2
f T
:
R
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
18
very low computational complexity. This simple receiver improves DD by 1:2 dB at BER = 10,3 for AWGN conditions while no performance degradation compared to DD is observed for Rayleigh and Ricean fading channels and Doppler frequencies considered here. In contrast to that, the low complexity detector presented in [5, 7] degrades considerably under fading conditions as has been demonstrated by simulations.
Acknowledgement The authors would like to thank TCMC (Technology Centre for Mobile Communication), Philips Semiconductors, Germany, for supporting this work.
References [1] [2] [3] [4] [5] [6] [7] [8]
D. Divsalar, M. K. Simon: Multiple-Symbol Di erential Detection of MPSK, IEEE Trans. Commun., Vol. 38, Mar. 1990, pp. 300 - 308. H. Leib, S. Pasupathy: The Phase of a Vector Perturbed by Gaussian Noise and Di erentially Coherent Receivers, IEEE Trans. on Information Theory, Vol. 34, No. 6, Nov. 1988, pp. 1491 - 1501. F. Edbauer: Bit Error Rate of Binary and Quaternary DPSK Signals with Multiple Di erential Feedback Detection, IEEE Trans. Commun., Vol. 40, Mar. 1992, pp. 457 - 460. F. Adachi, M. Sawahashi: Decision Feedback Multiple{Symbol Di erential Detection for M{ary DPSK, Electronics Letters, Vol. 29, No. 15, Jul. 1993, pp. 1385 1387. H. Leib: Data{Aided Noncoherent Demodulation of DPSK, IEEE Trans. Commun., Vol. 43, Feb.{Apr. 1995, pp. 722 - 725. H. Leib: Comments on \Di erential Detection with IIR Filter for Improving DPSK Detection Performance", IEEE Trans. Commun., Vol. 45, Jun. 1997, p. 637. N. Hamamoto: Di erential Detection with IIR Filter for Improving DPSK Detection Performance, IEEE Trans. Commun., Vol. 44, Aug. 1996, pp. 959 - 966. F. Adachi, M. Sawahashi: Decision Feedback Di erential Phase Detection of M{ ary DPSK Signals, IEEE Trans. Vehic. Tech., Vol. 44, May 1995, pp. 203 - 210.
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
19
F. Adachi: Reduced{state Viterbi di erential detection using a recursively estimated phase reference for M{ary DPSK, IEE Proc.{Commun, Vol. 142, No. 4, Aug. 1995, pp. 263 - 270. R. Y. Wei, M. C. Lin: Di erential Phase Detection Using Recursively Generated Phase References, IEEE Trans. Commun., Vol. 45, Dec. 1997, pp. 1504 - 1507. F. Adachi: Adaptive Di erential Detection Using Linear Prediction for M {ary DPSK, IEEE Trans. Vehic. Tech., Vol. 47, Aug. 1998, pp. 909 - 918. R. Schober, W. H. Gerstacker, J. B. Huber: Decision{Feedback Di erential Detection of MDPSK for Flat Rayleigh Fading Channels, accepted for publication in IEEE Trans. Commun., 1999. N. R. Sollenberger, J. C.{I. Chuang: Low{Overhead Symbol Timing and Carrier Recovery for TDMA Portable Radio Systems, IEEE Trans. Commun., Vol. 38, Oct. 1990, pp. 1886 - 1892. F. D. Neeser, J. L. Massey: Proper Complex Random Processes with Applications to Information Theory, IEEE Trans. Inf. Theory, Vol. 39, Jul. 1993, pp. 1293 1302. A. Duel{Hallen, C. Heegard: Delayed Decision{Feedback Sequence Estimation, IEEE Trans. Commun., Vol. 37, May 1989, pp. 428 - 436. C.A. Bel ore, J.H. Park: Decision{feedback equalization, Proc. IEEE, Vol. 67, Aug. 1979, pp. 1143 - 1156. G. D. Forney, Jr.: The Viterbi Algorithm, Proc. IEEE, Vol. 61, 1973, pp. 268 278. S. Haykin: Adaptive Filter Theory, 3rd edition, Prentice Hall, New Jersey, 1996. J. G. Proakis: Digital Communications, McGraw-Hill, New York, 1995. T. S. Rappaport: Wireless Communications, Prentice Hall, New Jersey, 1996. W. C. Jakes, Jr., Ed.: Microwave Mobile Communications, Wiley, New York, 1974. E. J. Borowski, J. M. Borwein: The Harper Collins Dictionary of Mathematics, Harper Perennial, New York, 1991.
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A Calculation of the Optimum FIR Coecients In the following the optimum FIR lter coecients are calculated from Eqs. (20) and (21). In order to simplify Eq. (20), which depends on the transmitted sequence a[], we introduce the variables x~1 [k] =4 a[k] a[[kk], ] ; even; (58) 1 N: 4 [k ] 1 x~ [k] = a[k]a[k , ] ; odd; (59) Applying Eqs. (58), (59) to Eq. (20) and dividing row no. of the resulting set of linear equations by a[k](a[k , ])((+1)) yields
A~ N x~ 1 = y~ 1;
(60)
where the N N matrix A~ N and the N 1 vectors x~ 1 and y~ 1 are de ned as 2 6 6 6 6 6 6 4 ~ N = 666 6 6 6 6 6 4
A
1 0 ... ... 0
1 1 ... ... :::
0 1 ... ... :::
::: 0 1 ... ... 0
::: ::: ... ... ... 1
y~ 1 =4 [1 0 : : : 0]T ; x~ 1 =4 [~x11 x~12 : : : x~1N ]T :
3
0 7 0 777 ... 77 7 . . . 777 ; 7 7 1 775
(61)
(62) (63)
4 In Eq. (61) the notation = 2 + 2 is used. Note, that a time{independent solution results because A~ N and y~ 1 are independent of k. Therefore, discrete time has been omitted in the elements of vector x~ 1. Now, the determinant d = detA~ of the matrix A~ , 1 N , is introduced. The elements x~1 of vector x~ 1 can be obtained from Eq. (60) by applying Cramer's Rule [22]. Because of the special form of A~ N and y~ 1, x~1 can be written as (64) x~1 = (,1)+1 ddN , ; 1 N; N
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
21
and the determinant of A~ can be calculated recursively according to
d = d,1 , d,2 ; > 0; d0 =4 1; d =4 0; < 0:
(65) (66) (67)
Eqs. (65), (66) and (67) may be rewritten to the di erence equation
d , d,1 + d,2 = [ ]; 0;
(68)
which can be solved using the z{transform. Introducing D(z) = Zfd g, the z{transform of Eq. (68) leads to 2 D(z) = z2 , zz + 1 : (69) Assuming 2 > 0,
D(z) = ,1 z ,z , ,2 z ,z 1 2 1 1 2 2 can be obtained, where the de nitions q
1 =4 12 ( + 2 , 4); q
2 =4 12 ( , 2 , 4)
(70) (71) (72)
have been used. Inverse z-transform of Eq. (70) yields +1
2+1 ; 0: d = 1 , (73) 1 , 2 By combining Eq. (64) and Eq. (73) and using the relation 1 2 = 1, x~1 can be expressed as ,2(N +1, ) x~1 = (,1)+1 2, 2 ,2(N +1) , 1 ; 1 N: (74)
2 ,1 From Eqs. (58), (59), and (74), the optimum FIR coecients contained in vector x1 (cf. Eq. (25)) are obtained. x2 (cf. Eq. (26)) can be determined in a similar way like x1. The result is
x2 = 0:
(75)
22
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
B Calculation of 'zz ; k for the FIR case [
]
In the sequel 'zz [; k] is derived. Using Eq. (29) and the de nition x~10 =4 ,1, z[k] can be written as N X z[k] = ,a[k] x~1 (a[k , ]z[k , ])(): (76) =0
Applying some straightforward calculations to Eq. (76) leads to z[k] = ,a[k] +
N X =0
NX +1 =0
(~x1,1 + x~1 )(b[k , ]n[k , ])()
x~1 (a[k , ]n[k , ]n[k , 1 , ])()
!
;
(77)
where Eqs. (1), (5) and the de nition x~1 =4 0, < 0 or > N , have been used. Eq. (77) may be exploited for calculation of 'zz [; k]:
'zz [; k] = Efz[k + ]z[k]g
+1 NX +1 NX
= a[k]a[k + ]
=0 =0
(~x1,1 + x~1 )(~x1,1 + x~1)(b[k , ])()
(b[k , + ])()Ef(n[k , ])()(n[k , + ])()g
N X N X + x~1 x~1 (a[k =0 =0
, ])()(a[k , + ])()
Ef(n [k , ]n[k , 1 , ])()(n[k , + ]n[k , 1 , + ])()g : (78)
From Eqs. (6) and (7)
Ef(n [k , ])()(n[k ,
8 > < ( ) + ]) = > :
g
2[ , + ]; even; 0; odd;
(79)
and
Ef(n[k , ]n[k , 1 , ])()(n[k , + ]n[k8, 1 , + ])()g > < 4 [ , + ]; even; = > :
can be obtained. Thus,
'zz [; k] = 0; odd;
0;
odd;
(80) (81)
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23
results. For even, only terms which satisfy the condition , + = 0 contribute to the sums in Eq. (78). Hence, this equation can nally be simpli ed to
'zz [; k] = 2a[k]a[k + ]
NX +1 =0
(~x1,1 + x~1 )(~x1+jj,1 + x~1+jj) N
X +2 x~1 x~1+jj =0
!
; even:
(82)
Using the same method as for calculation of 'zz [; k], it is straightforward to determine the PACS '~zz [; k].
C Receiver Performance Under Flat Fading In this appendix, the performance of the DFE receiver using an IIR NPEF is analyzed. From Eqs. (55) and (56) it can be seen that, in this case, the decision variable is given by
dIIR[k] = rd[k]q[k]:
(83)
Under fading conditions, most errors occur when the fading amplitude is small (i.e., the instantaneous signal{to{noise ratio (SNR) is low). Thus, if the in uence of the channel noise is neglected, rd[k , 1] 1 results and according to Eq. (57), q[k] may be approximated by q[k] 1 , 2q[k , 1]: (84) From Eq. (84), q[k] 1=(1 + 2) can be obtained under steady{state conditions and dIIR[k] 1 +1 rd [k] (85) 2 follows from Eq. (83). Since the real, multiplicative constant 1=(1 + 2) has no in uence on the decision, the same decision variable as for conventional DD results. This means, for small fading amplitudes when most decision errors occur the DFE receiver using an IIR NPEF is almost identical with conventional DD. For large fading amplitudes (i.e., the instantaneous SNR is high), however, both schemes produce relatively few errors. Therefore, it can be expected that the DFE receiver using an IIR NPEF has a similar performance like conventional DD under fading conditions. Similar results can be optained for DFE receivers using FIR NPEFs and DDFSE receivers.
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Figure Captions: Figure 1: Block diagram of the transmission model for AWGN channel and constant phase shift .
Figure 2: Nonlinear time{variant noise prediction{error lter (NPEF) used in the proposed receiver.
Figure 3: a) Magnitude of x~1 vs. coecient number for di erent lter orders N and Eb =N0
ratios for a QDPSK constellation (M = 4). b) Normalized variance 2 z vs. 10 log10(Eb =N0) for M = 4 and di erent prediction{error lters.
Figure 4: DFE receiver based on an FIR NPEF. Figure 5: DFE receiver based on an IIR NPEF. Figure 6: BER of QDPSK (M = 4) vs. 10 log10(Eb=N0 ) for DDFSE receivers with di erent
numbers of states Z = M 0 applying an FIR NPEF of order N = 2. For comparison the BERs for conventional DD, CD, and genie aided DFE are also shown.
Figure 7: BER of QDPSK (M = 4) vs. 10 log 10(Eb=N0 ) for conventional DD, CD, and DFE receivers applying di erent NPEFs.
Figure 8: a) 2 z =2 and b) BER vs. 2 for QDPSK (M = 4) and the DFE receiver applying
an IIR NPEF. Those values of 2 z =2 and BER, which correspond to the optimum value for 2 (cf. Eq. (72)) are denoted by a triangle.
Figure 9: BER of QDPSK (M = 4) vs. frequency o set f in kHz for T = 41:2 s for conventional DD and DFE receivers applying di erent suboptimum NPEFs at 10 log10(Eb=N0) = 8 dB.
Figure 10: BER of QDPSK (M = 4) vs. 10 log10(Eb=N0 ) for conventional DD, DFE re-
ceiver applying a suboptimum IIR NPEF ( 2 = 0:7), and the scheme proposed in [4, 5] for Rayleigh fading (fdT = 0:0075) and Ricean fading (R = 7 dB, fd T = 0:03) channels.
R. Schober et al.: IMPROVING DIFFERENTIAL DETECTION OF MDPSK
Figures: Figure 1:
25
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Figure 2:
26
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Figure 3:
27
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Figure 4:
28
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Figure 5:
29
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Figure 6:
30
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Figure 7:
31
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Figure 8:
32
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Figure 9:
33
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Figure 10:
34
