sion, the differential orthogonal frequency division multiplexing (OFDM) systems with diversity reception are discussed. It is well known that there are two types of differential OFDM systems, namely, the time domain differential OFDM (TD-OFDM) and the
frequency domain differential OFDM (FD-OFDM). In this paper,
the NSD and its special cases are incorporated to the differential OFDM systems. Furthermore, we provide a simple closed-form bit-error-rate (BER) expression for the differential OFDM sys-tems utilizing the noncoherent one-shot detector with diversity reception in the time-varying multipath Rayleigh fading channels. Numerical results have revealed that, with multi-antenna diversity reception, the performance of the noncoherent one-shot detector is improved significantly. However, when only one or two receive antennas are available, the implementation of the linearly pre-dictive DF detector or the linearly prepre-dictive Viterbi receiver is necessary for achieving better and satisfactory performance.
Index Terms—Decision-feedback differential detection,
dif-ferential orthogonal frequency division multiplexing (OFDM), estimator-detector, noncoherent maximum-likelihood sequence detection (NSD), noncoherent one-shot detection, Viterbi algo-rithm.
I. INTRODUCTION
F
OR the single-carrier M-ary differential phase-shift keying (MDPSK) [1], [2], Divsalar et al. [3] and Ho et al. [4] proposed the multiple-symbol differential detector, or the non-coherent maximum-likelihood (ML) sequence detector, to im-prove the conventional product detector [1], or the noncoherent one-shot detector. Since the noncoherent one-shot detector is built assuming that the channel is constant over two consecu-tive symbol durations, its bit-error-rate (BER) performance ex-hibits an error floor in the time-selective fading channel. It is proved theoretically that the noncoherent ML sequence detector (NSD) can lower this error floor significantly by detecting a se-quence of symbols jointly [4]. Although the NSD is optimal, its complexity increases exponentially with the number of symbolsManuscript received May 27, 2004; revised August 9, 2005. This work was supported by the National Science Council, Republic of China, under Grant NSC 93-2213-E027-035.
D.-B. Lin is with the Department of Electronic Engineering, National Taipei University of Technology, Taipei 106, Taiwan, R.O.C. (e-mail: dblin@ en.ntut.edu.tw).
P.-H. Chiang and H.-J. Li are with the Graduate Institute of Communica-tion Engineering, NaCommunica-tional Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: d1942011@ee.ntu.edu.tw; hjli@ew.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TBC.2005.857604
an attractive estimator-detector structure.
Orthogonal frequency-division multiplexing (OFDM) is an excellent technique to reduce the effect of frequency-selective fading by dividing the transmission bandwidth into many narrow-band subcarriers, each of which exhibits an approxi-mately flat fading [9], [10]. For this multicarrier scheme, the differential modulation can be applied on each subcarrier, and the corresponding differential encoding and detection can be performed in either the time domain or frequency domain. According to the direction of the differential encoding and detection (see Fig. 1), the differential OFDM systems are classified into two categories: 1) the time domain differential OFDM (TD-OFDM) [10]–[15]; 2) the frequency domain dif-ferential OFDM (FD-OFDM) [10], [13]. The former has been standardized in the terrestrial digital audio broadcasting (DAB) system [16].
Nevertheless, only the noncoherent one-shot detector was considered in the above cited papers regarding the differential OFDM systems. In this paper, the NSD and its three special cases, namely, the noncoherent one-shot detector, linearly predictive DF detector, and linearly predictive Viterbi receiver are incorporated to the differential OFDM systems with diver-sity reception. Furthermore, we provide a simple closed-form bit-error-rate (BER) expression for the differential OFDM systems utilizing the noncoherent one-shot detector with diver-sity reception in the time-varying multipath Rayleigh fading channels.
The remaining of the paper is organized as follows. In Sec-tion II, a hierarchical interpretaSec-tion of the NSD and its three special cases is presented. In Section III, the preliminaries of the differential OFDM systems with diversity reception are described. In Section IV, the receiver designs of the differential OFDM systems according to the NSD and its three special cases are illustrated. Also, a simple BER expression for the dif-ferential OFDM systems employing the noncoherent one-shot detector is given. Numerical results are shown in Section V, whereas the conclusions are drawn in Section VI.
II. NONCOHERENTML SEQUENCEDETECTOR
In this section, the NSD with diversity reception is reviewed. Then, three special cases of the NSD are introduced, namely, the noncoherent one-shot detector, linearly predictive DF de-tector, and linearly predictive Viterbi receiver. At the end of this 0018-9316/$20.00 © 2005 IEEE
Fig. 1. Comparison between the TD- and FD-OFDM.
section, a hierarchical interpretation of the NSD and its special cases is also given.
For the single-carrier transmission, an -symbol se-quence is assumed to be transmitted and received over a single-input-multiple-output (SIMO) time-varying flat Rayleigh fading channel. Let denote the th information symbol, which is
a -PSK symbol, i.e., and
. Each information symbol is differentially encoded as follows [1], [2].
(1) Obviously, the differentially encoded symbol, or channel symbol, still belongs to the -ary constellation . Then, collecting the received signal from all receive antennas gives (2)
where ,
, and . is the
re-ceived signal from the th receive antenna in the th symbol interval. is the path gain with mean zero, variance ,
and autocorrelation , assuming
iden-tical and independent channels. is the AWGN with mean zero and variance . Suppose that the transmission begins at
time and ends at time . Stacking up the
variables involved in the optimum block detection yields the signal model as
(3)
where ,
, , and
. From (1), it is clear that there exists a one-to-one correspondence between the
information symbol vector and the
channel symbol vector .
From (3), it can be shown that the ML estimate of is obtained through
(4)
which is the noncoherent sequence detector [4]. Here and is the covariance matrix of . As a result that its complexity grows exponentially with , the direct implementation of this detector is not recommended. In the following subsections, we introduce three special cases of this NSD, which can be carried out in practice.
A. Linearly Predictive Viterbi Receiver
The NSD given in (4) brings a linear-prediction interpreta-tion of its structure. To avoid that its complexity increases ex-ponentially with , the minimization problem in (4) is solved by using the innovations-based approach [6], which is mathe-matically equivalent to the Cholesky factorization approach [7] [17,Ch. 3.7] as follows. Applying the Cholesky factorization to
the matrix produces
(5)
where is a lower triangular matrix and
. The inverse of is denoted as
..
. ... ... . .. ...
(6) Note that the element and is the th coefficient of the th order one-step forward linear predictor for the fading-plus-noise process and is the corresponding mean square prediction error. Then, substituting (5) and (6) into (4) results in a linearly predictive sequence detector as
(7)
where is the th order prediction of
Each state of the trellis in the th symbol interval is represented
as . There are transitions
emerging from each state and terminating in different states. The branch metric associated with each transition is defined as
(9)
Then the Viterbi algorithm [18,Ch. 19] with a fixed decision delay can be applied to this trellis for solving the minimiza-tion problem in (8).
The algorithmic complexity of the above trellis-based se-quence detector is proportional to the number of states which may be very large. Accordingly, the reduced-state sequence detection [8] can be incorporated to overcome the implementation complexity. A reduced state is de-fined with the most recent channel symbols, namely, , and U K. The calculation
of the branch metric involves observations
and trial channel symbols. Thus, there are still
channel symbols unavailable
in the state . However, one can extract these unavailable channel symbols from the survivor history according to the per-survivor processing (PSP) technique [8]. Thereupon, the branch metric for this reduced-state approach is written as
(10)
where are determined by the
survivor entering the state . Subsequently, this reduced-state trellis-based sequence detector [7] is named the Viterbi receiver (VR).
B. Linearly Predictive Decision-Feedback Detector
Here, we consider an extreme case of for the VR. In this case, only a single path is allowed to survive, then all the channel symbols required by the estimator are found from the
C. Noncoherent One-Shot Detector
Considering the DR with the prediction order , (11) is reduced to
(12)
provided that . (12) is the noncoherent
one-shot detector and is dubbed the conventional receiver (CR), subsequently.
Ignoring the fading correlation in (12) results in a subop-timum detector as
(13) which is the well-known product detector [1]. For this subop-timum detector, Kam derived a closed-form BER expression as follows [2].
(14)
where for differential BPSK (i.e.
and for differential QPSK
(i.e. ). is the average signal-to-noise-ratio (SNR) per receiving branch. Obviously, (14) can be an upper bound for the BER of the CR given by (12).
D. Hierarchy of the NSD and Its Special Cases
Now we interpret the hierarchy of the NSD (4) and its three special cases, namely, the CR (12), DR (11), and VR (10). Firstly, the VR is a suboptimum version of the NSD; secondly, the DR is the one-survivor version of the VR; finally, the CR is the one-order-prediction version of the DR. Moreover, since the NSD and its special cases are inherently of the
esti-mator-detector structure, their performances are dominated by
Fig. 2. Discrete-time baseband equivalent system model for the differential OFDM.
processes. More specifically, the VR employs the PSP tech-nique so that each state in the trellis has its own survivor. Then the estimates contained in the branch metrics are different from state to state or from path to path. Indeed, the VR benefits from the path diversity and thus performs better than its one-survivor version, namely, the DR. Furthermore, with a larger prediction order and hence better estimates, the DR performs better than its one-order-prediction version, i.e. the CR.
III. PRELIMINARIES FORDIFFERENTIALOFDM SYSTEMS WITHDIVERSITYRECEPTION
It is well known that there are two types of differential OFDM systems, namely, the TD- and FD-OFDM. The comparison is il-lustrated in Fig. 1. In this section, the system model and some statistical properties of the differential OFDM systems with di-versity reception are provided. Then, the receiver designs will be discussed later in Section IV.
A. System Model
As shown in Fig. 2, we consider the DFT-based OFDM trans-mission over the SIMO WSSUS Rayleigh fading channel and assume sufficient cyclic prefix (CP) is inserted such that the inter-block-interference (IBI) is eliminated completely [9]. Let denote the -PSK information symbol for the th subcar-rier in the th OFDM block interval. Then, the channel symbol
is produced via the differential encoding as [10], [13]
( - )
( - ). (15)
Let , , and represent the number of subcarriers, guard samples, and taps, respectively. Then, the discrete-time base-band representation of the received signal from the th receive antenna is [9], [19]
for (16)
where is the equivalent AWGN [19] with
mean zero and variance . Here, , ,
and are the multiplicative distortion (MD), inter-carrier interference (ICI), and frequency domain noise, respectively.
B. Statistical Properties
For the ease of demonstrating the receiver designs later in Section IV, the statistical properties of the MD , and ICI are clarified. Considering the exponential power delay pro-file [20] with the constraint , the fading power of the th tap is given by
(17) where and the delay control dominates the normal-ized root-mean-square (RMS) delay spread . Here is the symbol duration (or the reciprocal of the system bandwidth).
1) Correlation of the Multiplicative Distortion: For the
spa-tially independent and identical WSSUS Rayleigh fading chan-nels with the classical Doppler spectrum, let
, for , where is
the correlation of . Then, the correlation of the MD is given as [19]
(18) where is the zero-order Bessel function of the first kind and is the maximum Doppler frequency. Thereupon, the time correlation can characterize the channel time-selectivity, while the frequency correlation can represent the channel frequency-selectivity.
2) Variances of the Multiplicative Distortion and ICI: From
OFDM systems, namely, the TD- and FD-OFDM. The theoret-ical BER’s for both systems employing the CR are also given.
A. TD-OFDM
For the TD-OFDM, since the differential encoding and de-tection are independent and simultaneous on all subcarriers (see Fig. 1), only the detection on the th subcarrier is illustrated subsequently. In light of (2) and (16), the received signal vector for the th subcarrier in the th block interval from all receive antennas is
(20)
where ,
, and . Then,
stacking up the received signal vectors results in the signal model for the NSD as
(21)
where ,
, , and
. Then, from (4), the ML
estimate of the information symbol vector
is given by
(22) where
(23)
and , the covariance matrix of
, is evaluated via (18) with .
1) CR: In light of (12), the noncoherent one-shot detector
obtains the estimate for the information symbol according to
(24) The last step is due to that the fading correlation is real-valued. Thereupon, the BER of the TD-OFDM employing the
of the trellis in the th block interval is denoted as
, and the corresponding branch metric is expressed as
(26)
where the prediction coefficients are the same as the ones for the DR.
B. FD-OFDM
For the FD-OFDM, since the differential encoding and detec-tion are independent and sequential for each OFDM blocks (see Fig. 1), only the detection in the th block interval is described here. The NSD performs optimum block detection for the entire information symbols as a whole. Piling up the received signal vectors given by (20) yields the signal model for the NSD as
(27)
where ,
, , and
. From (4), one can obtain the ML estimate of the information symbol vector
through
(28) where
(29)
and , the covariance matrix of
, is calculated via (18) with .
1) CR: In light of (12), the CR evaluates the estimate for the
information symbol according to
Fig. 3. BER comparisons for LTLF (j j = 0:9993;j j = 0:9988) and Q = 1, 2, 4.
However, based on (13), the suboptimum detector ignores the complex-valued fading correlation and gets a suboptimum estimate of by
(31) The BER of the FD-OFDM using this suboptimum detector is computed via (14) with the substitutions of and
. Apparently, this BER expression is an upper bound on the performance of the FD-OFDM utilizing the CR, given by (30).
2) DR: Based on (11), the DF detector performs
symbol-by-symbol detection through
(32) where the prediction coefficients are computed via (5), (6), and (29).
3) VR: According to (10), the reduced-state trellis-based
se-quence detector with states is defined as follows. Each state of the trellis for the th subcarrier is represented as
, and the corresponding branch metric is expressed as
(33)
where the prediction coefficients are the same as the ones for the DR.
V. NUMERICALRESULTS
The simulation parameters are detailed as follows. 1) the car-rier frequency and the system bandwidth are 1.8 GHz and 800 kHz, respectively, and thus the symbol duration is ; 2) the number of subcarriers and guard samples are
and , respectively, and hence the total OFDM block duration is ; 3) the modulation is differen-tial BPSK, i.e. ; 4) the number of uncorrelated paths is ; 5) the number of receive antennas is , 2, and 4; 6) for both the DR and VR, the prediction order is ; 7) for the VR, , i.e., the number of states is , and the decision delay is .
As shown in Figs. 3–6, the BER’s of differential OFDM systems are compared in four extreme scenarios (see Table I), namely, LTLF, HTLF, LTHF and HTHF, respectively. For instance, LTHF stands for low time-selectivity and high fre-quency-selectivity. In each figure, the performances of the CR, DR, and VR are presented from left to right. The matched filter bounds (MFB’s) shown in all figures are the BER’s of the single-carrier differential BPSK systems employing the noncoherent one-shot detector with one-, two-, and four-branch diversity reception, respectively, in the quasistatic Rayleigh fading channel. These bounds are produced via (14) by setting . Apparently, they are the lower bounds of the BER’s of differential OFDM systems utilizing the CR. How-ever, at low SNR level, they are slightly higher than the BER’s of the systems using the DR or VR in all channel conditions. This is due to that the DR and VR, originated from the NSD,
Fig. 4. BER comparisons for LTHF (j j = 0:9993;j j = 0:9859) and Q = 1, 2, 4.
Fig. 5. BER comparisons for HTLF (j j = 0:9826;j j = 0:9972) and Q = 1, 2, 4.
are inherently of better performances than the noncoherent one-shot detector (see Section II-D).
As shown in Fig. 3 (LTLF), since the channel selectivity is low, both TD-OFDM and FD-OFDM experience BER’s similar to that of the corresponding MFB’s regardless of which receiver structure is employed. On the other hand, as shown in Figs. 4–6,
error floors appear when the channel selectivity is higher, espe-cially for the systems utilizing the CR. The reasons for the pres-ence of these error floors are twofold: 1) the ICI induced when the channel is not constant over an OFDM block duration; 2) the worse estimates of the fading-plus-noise processes resulted from faster channel variations along the differential detection
Fig. 6. BER comparisons for HTHF (j j = 0:9826;j j0:9843) and Q = 1, 2, 4. TABLE I
PARAMETERSETTINGS FORDIFFERENTCHANNELCONDITIONS
direction (see Section II-D). If the TD- and FD-OFDM undergo the same channel time-selectivity, the former results in the same performance degradation. However, the later lifts the error floors further due to the estimation error increases with the channel se-lectivity. Thereupon, they obtain similar performance for both LTLF (Fig. 3) and HTHF (Fig. 6). In addition, the TD-OFDM performs better than the FD-OFDM for LTHF (Fig. 4), whereas the TD-OFDM performs worse than the FD-OFDM for HTLF (Fig. 5).
Moreover, as observed from Fig. 5, it is evident that: 1) the VR and DR perform better than the CR; 2) the performances of the FD-OFDM with different receiver structures are similar. The former is due to that the VR and DR detect signals with the aid of better estimates (see Section II-D). The later is consequent on the low frequency-selectivity. In this case, the qualities of the estimates are good and similar. As seen from (16), the ICI is treated as an equivalent noise term, thus no reduction on the ICI is made in all receiver structures. Thereupon, the presence of these error floors is mainly resulted from the ICI.
Finally, as viewed from Fig. 6, both the VR and DR yield significant performance improvements over the CR when one
or two receive antennas are employed. However, when four re-ceive antennas are available, all rere-ceivers benefit from the larger spatial diversity gain and hence their performances are equally good. In this case, since the performance of the CR is satisfac-tory, the consideration for implementing the DR and VR is not necessary.
VI. CONCLUSIONS
In this paper, we review the NSD and its three special cases, namely, the CR, DR, and VR. Based on the estimator-detector structures, a hierarchical interpretation of the NSD and its spe-cial cases are also presented. Then the NSD and its spespe-cial cases are applied to the differential OFDM systems with diversity re-ception. Moreover, assuming sufficient CP, we provide a simple closed-form BER expression for differential OFDM systems employing the CR with diversity reception in the time-varying multipath Rayleigh fading channels. Numerical results have revealed that, with multi-antenna diversity reception, the per-formance of the CR is improved significantly. However, when few receive antennas are available, the implementation of the DR or VR is necessary for achieving better and satisfactory performance.
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