Multilevel Concatenated-Coded M-DPSK Modulation Schemes for the Shadowed Mobile Satellite Communication Channel
Do Jun Rhee, Member, IEEE, and Shu Lin, Fellow, IEEE
Abstract— This paper presents a bandwidth-efficient multi- level concatenated-coded modulation scheme for reliable data transmission over the shadowed mobile satellite communication (MSAT) channel. In this scheme, bandwidth-efficient block mod- ulation codes are used as the inner codes, and Reed–Solomon codes of various error correcting capabilities are used as the outer codes. The inner and outer codes are concatenated in multiple levels. A general method for constructing multilevel concate- nated modulation codes is presented, and a multistage closest coset decoding for these codes is proposed. Specific multilevel concatenated 8-PSK modulation codes have been constructed.
These codes are designed to lower the bit error rate (BER) of the error floor caused by the large Doppler frequency shift due to the motion of vehicles. Simulation results show that these codes perform very well and achieve large coding gains over the uncoded reference modulation systems.
I. INTRODUCTION
C
ODED modulation in conjunction with concatenation is a powerful technique for achieving high reliability, large coding gain, and high spectral efficiency with reduced decoding complexity. This combination of coded modulation and concatenation is known as concatenated-coded modulation [1]. Single-level concatenated trellis-coded modulation (TCM) for additive white Gaussian noise (AWGN) channel was first introduced by Deng and Costello in 1989 [2], [3]. At the same time, Kasami et al. presented a single-level concatenated block-coded modulation (BCM) scheme for reliable data trans- mission over the AWGN channel [1], [4]. Error performances of the single-level concatenated TCM and BCM schemes for the Rayleigh fading channel were first investigated by Vucetic and Lin in 1991 [5] and then by Vucetic in 1993 [6]. All these studies showed that by properly choosing the inner codes and outer codes, large coding gains, and high spectral efficiency can be achieved with a significant reduction in decoding complexity.Block-coded modulation codes are generally constructed from Hamming distance block component codes by using the multilevel coding method proposed by Imai and Hirakawa [7].
The multilevel coding method is a powerful technique for con- structing BCM (or TCM) codes systematically with arbitrarily
Manuscript received May 30, 1995; revised February 19, 1996. This work was supported by NSF Grants NCR-91-1540 and NCR 94-15374 and NASA Grant NAG 5-931. The work of D. J. Rhee was supported by LSI Logic. This paper was presented in part at ISIT’95, Whistler, Canada.
D. J. Rhee is with LSI Logic Corporation, Milpitas, CA 95035 USA.
S. Lin is with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822 USA.
Publisher Item Identifier S 0018-9545(99)07381-8.
large distance parameters for both AWGN and fading channels [8]–[15]. The multilevel structure of the multilevel BCM codes allows the use of multistage decoding procedures that provide good tradeoff between error performance and decoding complexity [16], [17]. Suboptimum error performance can be achieved with a significant reduction in decoding complexity.
In a single-level concatenated-coded BCM system, a prop- erly chosen bandwidth-efficient BCM code is used as the inner code in concatenation with a powerful outer code, usually a Reed–Solomon (RS) code. This setup achieves very low bit error rate (BER), large coding gain, and high spectral efficiency with reduced decoding complexity for both AWGN and Rayleigh fading channels [1], [6]. However, a major shortcoming of a single-level concatenated-coded system with multilevel block inner modulation code is that the outer code corrects all the output bits of the inner code decoder to the same degree. Since a multilevel modulation code is constructed from component codes with different distance profiles, multistage decoding results in different bit error probabilities for different component codes at the output of the inner code decoder. As a result, the overall error performance of a single-level concatenated-coded modulation system is dominated by the worst bit error probability of the component codes of the inner modulation code. To improve the overall error performance, it is necessary to provide different levels of error protection for different inner component codes in a concatenated-coded modulation system. One approach to this improvement is to use multilevel concatenation with multiple outer codes to provide different levels of error protection for different inner component codes. Multilevel concatenation provides the flexibility of choosing outer codes with different error correcting capabilities and furthermore improves the spectral efficiency over the single-level concatenation scheme.
Research in the mobile satellite communications in both Europe and North America received a major stimulus from the 1979 WARC decision which permitted a satellite mobile service in the 806-890-MHz band. Since 1984, NASA has been conducting the mobile satellite experiment (MSAT-X) to determine the feasibility of a communications network in which private and commercial users send linear predictive coded voice and low-speed data via satellite, using low-cost terminals mounted in vehicles. The MSAT-X channel is subject to multipath fading, shadowing, Doppler frequency shift, and adjacent channel interference (ACI). Shadowing is defined as the effect of foliage attenuation or blockage by buildings.
0018–9545/99$10.00 1999 IEEE
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1635
Fig. 1. MDPSK-coded modulation system model for fast-fading Rician channel.
Investigation of coded modulation schemes for bandwidth- efficient reliable data transmission over the shadowed MSAT began in 1987 [18]. Since then, several TCM and BCM schemes have been proposed and analyzed [19]–[22]. In this paper, we propose and investigate multilevel concatenated- coded modulation schemes for shadowed MSAT channel.
They are intended to achieve high performance, large coding gains, and high spectral efficiencies with reduced decoding complexity. Particularly, they are designed to lower the BER of the error floor caused by the large Doppler frequency shift due to the motion of vehicles.
The organization of the paper is as followed. Section II briefly reviews a statistical model for the shadowed MSAT communication channel. In Section III, a general multilevel concatenated-coded modulation scheme is presented and a multistage closest coset decoding for multilevel concatenated modulation codes is proposed. In Section IV, several specific multilevel concatenated-coded 8-PSK modulation schemes are devised and their error performances with noncoherent de- tection are simulated. Simulation results show that these schemes achieve large coding gains over the uncoded reference modulation systems with the same spectral efficiencies.
II. MODEL FOR SHADOWED MSAT COMMUNICATION CHANNEL
The first statistical model for the shadowed mobile satellite channel was devised by Loo [23]–[26]. This model has been used for analyzing error performances of coded modulation schemes over the shadowed MSAT channel [18]–[22], [27]. In this model, the line-of-sight (LOS) component of the Rician model is subject to lognormal transformation and the mul- tipath is Rayleigh distributed. The lognormal transformation represents the effect of shadowing. There are three kinds of shadowing: light, average, and heavy that correspond to the shadowing faced in the areas of farms, suburban, and city, respectively.
Due to the multipath fading and large Doppler frequency shift in mobile satellite communications, it is very difficult to track the signal phase and use coherent detection methods.
Therefore, it is desirable to use differential encoding and
detection or other noncoherent methods for data transmission over a shadowed MSAT channel. In this paper, we consider the differentially detected 8-PSK as a modulation technique.
Interleaving technique is used for combating burst errors.
The channel model is depicted in Fig. 1, where is the additive white Gaussian noise (AWGN) with zero mean and two-sided power spectral density and is a normal- ized stationary, circular complex Gaussian noise process which represents the fading characteristic of the channel. The noise processes and are independent. Input data is first encoded and interleaved by a block interleaver. We denote a coded symbol sequence by
(1) where represents the MPSK symbol transmitted during the
th signal interval. In polar representation, is given by (2) where the phase is assigned by the signal mapper.
Before transmission, the sequence is differentially en-
coded into a sequence . For ,
the th and th components of , , and , are given by (in phasor notation)
(3) where is the energy per M-DPSK symbol. If a coded system has the spectral efficiency , then the energy per bit joules/bit and the signal-to-noise power ratio per bit (in decibels) is
(4) Let
(5)
be the received signal sequence at the output of the M-DPSK demodulator. For , the th component of
is given by
noise terms (6)
where
1) denotes the conjugate operation;
2) is the sample of the random process at the end of the th signal interval and is called the complex channel gain;
3) is the sample of the AWGN at the end of the th signal interval which is a Gaussian random variable with zero mean and variance .
The complex channel gain is given by
(7) where we have the following.
1) where is a Gaussian random variable with mean and variance . Therefore, is the log normal and is the LOS component of . The parameter and are the mean value and the standard deviation due to shadowing;
2) is a complex Gaussian random variable with zero mean and variance . is the multipath component of and is the multipath power.
In phasor form
(8) The probability density function of is given by
(9) Values of , and for various types of shadowing are given in Table I [19]. For the shadowed MSAT communication channel, Loo et al. [26] showed that the fading process which was created by using third-order Butterworth shaping filter of 3-dB bandwidth has almost the same statistical properties as the experimental data obtained from the shadowed MSAT channel. In Loo’s model, is equal to the maximum Doppler frequency where is the speed of vehicle and is the wavelength of a signal with the carrier frequency . Let be defined as the normalized fading bandwidth where is the symbol interval. For symbol rate
symbols/s, carrier frequency MHz, and
mi/h (or 59.581 Km/h), the normalized fading bandwidth is 0.02. For mi/h (or 148.95 Km/h), is 0.05.
Using Loo’s channel model, the three types of shadowing, light, average, and heavy, correspond to the Rician factors, 6.16, 5.46, and 19.33 dB, respectively.
For the heavy-shadowed MSAT channel (e.g., Rician factor dB) which could be worse than frequency selective
TABLE I
SYSTEMPARAMETERS FOR THESHADOWEDRICIANCHANNEL
Rayleigh fading channel, a coded modulation system suffers very severe distortion due to randomly changing phases and the multipath fading [18]. Especially, if the Doppler frequency shift is large due to the motion of vehicle, a coded modulation system with noncoherent detection (or differential detection) achieves a nonzero error probability regardless of how large we increase the signal-to-noise ratio. Such a phenomenon is referred to as error floor.
III. MULTILEVELCONCATENATED BCM SCHEME
Concatenated coding is powerful technique to construct long powerful codes from short component codes. Single- level concatenated coding scheme was first devised by Forney in 1966 [28], and later it was generalized to multilevel by Zinoviev [29]. Concatenation allows the use of multistage decoding procedures to attain very low bit error-rate (BER) with significant reduction of decoding complexity.
In this section, we combine concatenation and coded mod- ulation and present a general multilevel concatenated-coded modulation scheme. In this scheme, multiple pairs of outer and inner codes are used. Reed–Solomon (RS) codes will be used as the outer codes and coset codes constructed from a block modulation code and its subcodes will be used as the inner codes.
Let be a positive integer greater than one. In a - level concatenated-coded modulation system, pairs of outer, and inner codes are used. The encoding and decoding are accomplished in stages, respectively.
A. Outer Codes
RS (or shortened RS) codes are used as the outer codes.
For simplicity, we require that all the outer codes have the same length and code symbols from the same Galois field GF . For , let denote the th-level RS outer
code which is an RS code over GF with
length , dimension , and minimum Hamming distance . The th-level RS outer code is interleaved to a depth (or degree) for coset mapping. For , let denote the th copy of the RS outer code . Each code symbol from GF is represented by a binary -tuple (or a -bit symbol). At the th-level of outer code encoding, a message of information bits, regarded as symbols from GF , is encoded into a codeword of symbols in . Each code symbol of this codeword is represented by a -bit symbol. Therefore, in binary form, this codeword is a sequence of bits.
B. Inner Codes
Let be a block modulation code over a certain elementary signal set (e.g., an 8-PSK signal set as shown in Fig. 2)
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1637
Fig. 2. The partitioning and labeling process of an 8-PSK constellation.
with length , dimension , and minimum-squared Euclidean distance . We require that
(10) From , we form a sequence of subcodes (or subspace), , , where consists of the all-zero code- word, i.e., (the sequence of signals of zero phase).
These subcodes satisfy the following conditions: for , is a linear subcode of with dimension
(11) and minimum-squared Euclidean distance . From (10) and (11), we have
...
(12)
Note that . Since consists of only
the all-zero codeword, .
Next, we construct coset codes from , . These coset codes will be used as the inner codes in the proposed -level concatenated-coded modulation scheme. To illustrate the method of coset code construction, we use an example with , , and 8-PSK signal set. The base code is constructed from three binary code using multilevel
coding method proposed by Imai and Hirakawa [7]. The base inner modulation code is a three-level 8-PSK block modulation code which is constructed from three very simple binary Hamming distance component codes of length 8: 1) is the Reed–Muller (RM) code of length 8, dimension 4, and minimum distance 4; 2)
is the even-parity check code of length 8, dimension 7, and minimum distance 2; and 3) . Note that is a subcode of . has a simple four-state trellis diagram and (also ) has a simple two-state trellis diagram as shown in Fig. 3. Therefore, they can be decoded with soft-decision Viterbi algorithm.
The base inner code is constructed as follows. Let
(13) be three codewords in , and , respectively. Form the following sequence:
(14) For , we take as the label for a signal point in the 8-PSK signal constellation as shown in Fig. 2. Let be the mapping which maps the label into its corresponding signal point , i.e., . Then
(15)
(a)
(b)
Fig. 3. The trellis diagrams of Reed–Muller codes (8, 4, 4) and (8, 7, 2) codes: (a) a four-state trellis for the (8, 4, 4) RM code and (b) a two-state trellis for the (8, 7, 2) RM code.
is a sequence of eight 8-PSK signals. The base inner modula- tion code is given as follows:
(16) is an 8-PSK modulation code of length 8 and dimension
. This code has spectral efficiency
bits per signal, minimum-squared Euclidean distance 2.344, minimum symbol distance 2, and minimum product distance 4.
For , let be a subcode of .
Form a three-level 8-PSK modulation code
(17) This modulation code is a subspace (or subcode) of and has dimension . If we partition based on ,
denoted , we obtain cosets modulo .
To construct the coset inner codes, we form the following subspaces of :
We readily see that . Then the three
inner coset codes are given as following partitions:
The minimum-squared Euclidean distances of , and are 2.344, 4, and 8, respectively [15]. The minimum symbol distances of , and are 4, 2, and 2, re- spectively [15]. The minimum product distances of ,
Fig. 4. Multilevel concatenated modulation code encoder.
and are 0.117, 4, and 16, respectively [15]. , and have very simple four-state, two-state, and one-state trellis diagrams, respectively, and, hence, can be decoded with Viterbi algorithm.
General construction of inner coset codes is similar to the above example. First, we construct the base inner code from binary component codes. From , we form a sequence
of subcodes . Partition into cosets
modulo . Let denote the set of cosets of modulo . The minimum-squared Euclidean distance of each coset in is . The minimum-squared distance between two cosets in is . is called the coset code of modulo . Next we partition each coset in into cosets modulo . Let denote the set of cosets of a coset in modulo . It is clear that the minimum- squared Euclidean distance of a coset in is , and the minimum-squared distance among the cosets of a coset in modulo is . We call the coset code of modulo . We continue the above partition process to
form coset codes. For , let be the
coset code of modulo . We partition
each coset in into cosets modulo .
Then is the coset code of
modulo . The minimum-squared Euclidean distance of a
coset in is , and the minimum-squared
distance among the cosets of a coset in modulo is . Note that each coset in
consists of codewords in . Since , each coset in consists of only one codeword in . Hence, the minimum-squared Euclidean distance of each coset is . The minimum-squared Euclidean distance among
the cosets of a coset in modulo is .
The above partition process results in a sequence of coset
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1639
Fig. 5. The ith-level outer code encoder.
codes
...
(18) These coset codes are used as inner codes in the proposed -level concatenated-coded modulation scheme. This -level
concatenated modulation code with as
outer codes and as inner codes is denoted (19) C. Encoding
The overall encoding is accomplished in levels as shown in Fig. 4. Each level consists of two stages, the outer and inner code encodings. Every inner code encoder, except the first- level, has two inputs, one from the output of an outer code encoder and one from the output of the inner code encoder of the preceding level. The output of the th inner code encoder is a sequence of cosets from the th coset inner code . For , the th-level encoding is accomplished in two stages.
1) Outer Code Encoding: At the th stage of the outer code encoding, messages, each of bits long, are encoded into codewords in the th-level RS outer code , respectively, as shown in the Fig. 5. Each codeword is represented by a binary sequence of bits. These code sequences are stored as an by
rectangular array of rows and columns. We call this array, denoted , as the th-level outer code array. Each column of consists of bits, one from each outer code sequence.
2) Inner Code Encoding: Each input coset from the th-level inner encoder is partitioned into cosets
modulo . Each column of bits from the th-level outer code array is encoded (or mapped) into one of the cosets (or a coset leader) in the inner coset code
.
The output of the th-level inner code encoder is a sequence of codewords from the base inner code . Since each codeword in is a sequence of signals from the signal space , the overall output sequence of the overall encoder is a sequence of signals from . The collection of all these signal sequences form a concatenated modulation code of length , dimension
and minimum-squared Euclidean distance
(20) The spectral efficiency and effective rate of are
b/signal (21)
D. Multilevel Closest Coset Decoding
To reduce the decoding complexity, a multilevel concate- nated modulation code is decoded in levels (or stages).
Let
(22) be the transmitted code sequence, where is a codeword in one of the cosets of the coset code with length
. Let
(23) be the received sequence. Decoding is carried out in steps, from the first level to the th-level as shown in Fig. 6.
At the first-level of decoding, each for is decoded into one of the cosets in (this is referred to as coset decoding [16], [17]). Based on the decoded coset, we identify the bits which represents this decoded coset.
Fig. 6. Multilevel concatenated modulation code decoder.
After inner code decodings, an by rectangular array is formed which is an estimate of the first-level transmitted outer code array . Each row is regarded as a sequence of -bit symbol. For , let
(24) denote the th row of the estimated array , where is the th -bit symbol. If there are no errors, is a codeword in . Let
(25) be the decoded codeword. Then the estimated input message of bits is retrieved from this decoded codeword . After outer code decodings, an array is reconstructed. From this array, we reproduce a coset sequence (26) at the output of the first-level decoder, where . Reproducing process is exactly the same as the first-level outer and inner code encodings. This coset sequence is then applied to the second-level inner code decoder.
Now, we perform the second-level decoding. For
, based on the input information , we decode into one of the cosets in . Based on the decoded coset, we identify the -bit symbol which represents the decoded coset. After inner code decodings, an rectangular
array is formed which is an estimate of the second-level transmitted outer code array . Each row is regarded as a sequence of -bit symbols. For , let
(27) denote the th row of , where is the th byte. If there are no errors, is a codeword in the second outer code
. Let
(28) be the decoded codeword in . From this codeword, we retrieve the estimated message of bits. After outer code decodings, an array is reconstructed. From this array, we reproduce a coset sequence
(29) at the output of the second-level decoder, where
. Reproducing process is exactly the same as the second-level outer code and inner code encodings. This coset sequence is then applied to the third-level decoder. Other levels of decoding are carried out in the same manner. Overall multilevel decoder is shown in Fig. 6.
Since decoded information at each level is passed to the next level, decoding at each level depends on decoded information from the preceding level. Therefore, error propagation may occur. To reduce the probability of error propagation, outer codes must be selected by considering the specific channel characteristic. Multilevel concatenated codes are constructed by following rules. For the AWGN channel, strong outer codes must be used for levels where the inner codes have small minimum-squared Euclidean distances. For the Rayleigh fading channel, strong outer codes must used for levels where inner codes have small minimum symbol and product dis- tances.
If have simple trellis diagrams, the coset inner codes, , also have simple trellis dia- grams. If the coset inner codes have simple trellis structure, then we can use Viterbi decoding to decode the coset inner codes. This will decrease decoding complexity of the inner codes drastically.
E. Choice of Outer Codes
In the multilevel closest coset decoding process, decoded estimates from the present level are passed to the next level decoding. In the shadowed MSAT channel, when fading is severe, error propagation from one level to the next is likely to occur. Outer codes must be chosen properly to minimize this error propagation. Multilevel concatenation provides the flexibility to choose the error correcting capabilities of outer codes to reduce the effect of the error propagation. In the fol- lowing, a method to determine the error correcting capabilities of the outer code is discussed.
To find the best combination of RS outer codes for a multilevel concatenated-coded modulation system, we must determine the conditional -bit symbol error probability at the
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1641
Fig. 7. BER of the three-level 8-DPSK block modulation code 30over a heavy-shadowed MSAT channel.
output of each inner code decoder. At the th-level inner closest coset decoding, we assume that all the inner code and outer code decodings of previous levels are correct. Based on this assumption, we determine the conditional -bit symbol error probability after the th-level inner code decoding. Once the -bit symbol error probabilities for all levels of inner code decoding are determined, we can determine the error correcting capabilities of the outer codes. Suppose the error correcting capabilities of the outer codes are chosen to achieve bit error rate (BER) 10 to 10 at each level for desired channel SNR. For bit error rates in the range of 10 to 10 , the error propagation becomes negligible. Therefore, we can achieve reliable and error floor free data transmission at the BER 10 over the shadowed MSAT channel for desired channel SNR.
The above method can be used for constructing multilevel concatenated-coded modulation system with any modulation signal set .
IV. SPECIFICMULTILEVELCONCATENATED-CODED
8-DPSK SCHEMES FOR THESHADOWEDMSAT CHANNEL
In this section, we present three specific multilevel concatenated-coded 8-DPSK modulation schemes for the shadowed MSAT channel. These schemes are devised to achieve large coding gains over the uncoded 4-DPSK reference system and particularly to achieve error floor free communications at the BER of 10 .
For all the three schemes, the same base inner modulation code is used. This base inner modulation code is a three- level 8-PSK block modulation code described in Section III-B.
is an 8-PSK modulation code of length 8 and dimension . This code has spectral efficiency bits per signal, minimum-squared Euclidean distance 2.344, minimum symbol distance 2, and minimum product distance 4. The bit error performance of this code using three-stage decoding
with noncoherent detection for the shadowed MSAT channel is shown in Fig. 7. We see that the error floor occurs at the BER of 4.744 10 .
A. Scheme 1—A Three-Level Concatenated-Coded 8-DPSK System
For a three-level concatenated-coded modulation system, three pairs of outer and inner codes are needed. Inner coset codes , and are given in Section III-B. Three RS codes with symbols from GF are used as outer codes. Error correcting capability of the outer code at each level is chosen based on the conditional symbol error probability at the output of the inner code decoder at that level. To minimize the effect of error propagation to the next level, the error correcting capability of the outer code at a given level is chosen to achieve a BER less than 10 . Based on these design criteria, following three RS outer codes are chosen such that the overall coded system achieves the best possible BER performance for the heavy-shadowed MSAT channel with close to 2-b/symbol spectral efficiency for BT and
RS code RS code RS code
With the above chosen inner and outer codes, we obtain the following three-level concatenated-coded 8-PSK modulation code
(30) A codeword in contains
information bits and the spectral efficiency of the code is 2.004 b/symbol.
Three-level closest coset decoding is used to decode the code . Each level decoding consists of the inner closest
TABLE II
BITERRORPERFORMANCE OF THEDIFFERENTIALLYDETECTED8-DPSK THREE-LEVEL
CONCATENATEDMODULATIONCODE C(1)OVER THESHADOWEDMSAT CHANNEL
Fig. 8. BER of the three-level 8-DPSK concatenated modulation code over a light-shadowed MSAT channel.
coset decoding and the outer code decoding. At th-level decoding, the inner coset code is decoded by using decoded estimates from first level decoder to th-level decoder.
In the multilevel concatenated scheme, the th-level outer code is designed to reduce the error propagation from
th-level decoder to th-level decoder.
The bit error performances of the three-level concatenated 8-PSK modulation code with differential detection over the light-, average-, and heavy-shadowed MSAT channels with BT and are summarized in Table II and shown in Figs. 8–10, respectively. In the case of the differentially detected 8-DPSK code, we assume that the phase of a received 8-DPSK signal is randomly changing due to the multipath fading. From the table and figures, we see that the three-level concatenated system achieves large coding gains over the uncoded 4-DPSK system for all three types of shadowing.
The error floor phenomenon appears for the uncoded 4-DPSK system for all three types of shadowing for BER’s between
10 to 10 . However, for light- and average-shadowing, the three-level concatenated 8-DPSK system does not show any error floor for BER greater than 10 . For the heavy- shadowed MSAT channel with BT , the three-level concatenated-coded system does not show any error floor for BER greater than 10 , however, the uncoded 4-DPSK system shows an error floor at the BER 3 10 . For the heavy-shadowed MSAT with BT , the uncoded 4-DPSK system shows an error floor at the BER , however, the coded system does not show any error floor until BER smaller than 10 . Therefore, the coded system provides an impressive improvement in error performance over the uncoded 4-DPSK system.
For comparison purposes, we construct a single-level concatenated-coded 8-PSK system, denoted , with as the inner code and the (255, 227, 29) RS code over GF as the outer code. The outer code is interleaved to a depth of 18. This
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1643
Fig. 9. BER of the three-level 8-DPSK concatenated modulation code over an average-shadowed MSAT channel.
Fig. 10. BER of the three-level 8-DPSK concatenated modulation code over a heavy-shadowed MSAT channel.
single-level concatenated-coded system has spectral efficiency 2.003 b/symbol which is about the same as the three-level concatenated-coded system . In decoding of , the inner code is decoded in three stages [17], code
first, code next, and code
last. The decoded estimates from the inner code decoder are passed to the outer code decoder at the same time.
Therefore, all the output information bits of the inner code are protected with the same degree by the single (255, 227, 29) RS outer code. However, the BER of the four information bits of the first-level component code , BER of the seven information bits of the second-level component code , and BER of the seven information bits of the third-level component code are all different as
shown in Fig. 7. Consequently, the overall performance of the single-level concatenated-coded system is dominated by the worst BER of the output bits of the inner code decoder.
In this case, the decoding of gives the worst BER at the output of the inner code decoder because has smallest symbol and product distances which dominate the error performance for a fading channel. This is why in the three-level concatenated-coded system , we choose three different outer codes which provide different degrees of protection for the three groups of output bits of inner code decoder.
The bit error performances of the single-level concatenated- coded system with differential detection over various types of shadowed channels are shown in Figs. 8–10. At the
TABLE III
BITERRORPERFORMANCE OF THE DIFFERENTIALLYDETECTED8-DPSK SIX-LEVELCONCATENATED
MODULATIONCODESC(2)OVER THESHADOWEDMSAT CHANNEL
BER with BT , the code achieves 0.711, 1.109, and 1.124 dB coding gains over the code for the light-, average-, and heavy-shadowed MSAT, respectively,
and at the BER with BT , the code
achieves 0.631, 1.108 coding gains over the code for the light- and average-shadowed MSAT, respectively. For the heavy-shadowed MSAT with BT , the single-level concatenated system shows an error floor at the BER
, however, for the three-level concatenated-coded system , the error floor occurs for BER smaller than 10 . For BER , achieves a 7.3-dB coding gain over . B. Scheme 2—A Six-Level Concatenated-Coded
8-DPSK System
In the three-level concatenated-coded system , three levels of error protection are provided for the output bits of three inner coset codes to improve the overall error per- formance. Error performance can be further improved if we provide bit-level error protection for the output of the inner code decoder using six-level concatenation with a little more additional decoding complexity.
Consider the inner code where
for , is a linear binary block
code of length , dimension , minimum Hamming distance
. For , let be a linear subcode code
of of dimension . Let denote the coset representatives of the cosets in the partition . Then
can be decomposed as follows:
(31) where denotes the set of coset representatives of partition [30]. The minimum symbol distance of is the same as but less than that of . Therefore, the error performance of will be dominated by that of . In the three-level concatenated-coded system , the same th-level RS outer code is used to protect information bits for and information bits for . If we choose the error correcting capability of th-level RS outer code based on the error performance of th , we waste spectral efficiency because we protect information bits for with too much error
correcting capability. To attain better error performance and higher spectral efficiency, we decode first and use a proper RS outer code to produce better estimates of the information bits for . Then use these estimates to decode and use a proper RS outer code to prevent error propagation to the next level decoding. By doing this at each level, we have a six-level concatenated-coded system and better error performance and higher spectral efficiency can be achieved with little more additional decoding complexity.
For the second proposed multilevel concatenated system, we use the same base inner code as in the Scheme 1,
. Form the following sequence of subcodes of :
Then, the six inner coset codes are given as follows:
Based on the above six inner coset codes, the following six RS outer codes are chosen such that the overall coded system achieves the best possible BER performance for the heavy- shadowed MSAT channel with close to 2-b/symbol spectral efficiency for BT and :
RS code RS code
RS code
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1645
Fig. 11. BER of the six-level 8-DPSK concatenated modulation code over a light-shadowed MSAT channel.
RS code RS code RS code
Using the above inner and outer codes, we obtain the following six-level concatenated 8-DPSK system:
(32)
A codeword in contains
information bits and the spectral efficiency of the code is 2.002 94 b/signal.
The error performance of with differential detection over the light-, average-, and heavy-shadowed MSAT channels with BT and are summarized in Table III and shown in Figs. 11–13, respectively. We see that there is no error floor for the light- and average-shadowed MSAT channels for BER greater than 10 . For the heavy-shadowed MSAT channel, there is no error floor for BER greater than 10 . At the BER 10 with BT , the code achieves 0.56-, 1.944-, and 2.088-dB coding gains over the code for the light-, average-, and heavy-shadowed MSAT, respectively.
And, at the BER 10 with BT , the code achieves 1.081 and 1.638 coding gains over the code for the light- and average-shadowed MSAT, respectively. For the three-level code , each level of inner code is protected by a single RS code. However, for the six-level code , each level of inner code is protected by two different RS codes with proper error correction capabilities. As a result, code outperforms code for light, average, and heavy-shadowed MSAT channel with BT and .
C. Scheme 3—A Two-Level Concatenated-Coded 8-DPSK System
From Figs. 7 and 12, we see that for the heavy-shadowed MSAT channel with large Doppler frequency shift (e.g., BT ), the three-level and six-level concatenated-coded 8- DPSK systems proposed above do not have the waterfall like bit error performance. To achieve better error performance, we need to use longer RS outer codes or more powerful inner coset codes. If the total number of information bits of the inner coset codes is too large, decoding complexities for three-level and six-level concatenated-coded systems with longer outer codes will grow exponentially. To achieve better error performance with complexity less than three-level and six-level concatenated-coded systems, we devise a two-level concatenated-coded 8-DPSK system.
Again, the base modulation inner code is
. To form two inner coset codes, we select the following subcode of :
(33) As a results, we have the following two inner coset codes:
The minimum-squared Euclidean distances of and are 2.344 and 4.688, respectively. The minimum symbol distances of and are 2 and 4, respectively. The minimum product distances of and are 0.1179 and 16, respectively.
The inner coset codes are decoded in multiple stages. The decoding of is carried out as follows.
1) Decode the first-stage coset representative based on the (8, 1, 8) code and obtain the decoded estimates.
2) Decode the second-stage coset representative based on the (8, 4, 4) code
Fig. 12. BER of the six-level 8-DPSK concatenated modulation code over an average-shadowed MSAT channel.
Fig. 13. BER of the six-level 8-DPSK concatenated modulation code over a heavy-shadowed MSAT channel.
by using the decoded estimates from the first decoding stage.
3) Decode the third-stage coset representative based on the (8, 4, 4) code and the second-stage decoded estimates.
4) After the above decodings, decode the first-level RS outer code and reproduce the first-, second-, and third- stage decoded estimates of . These decoded estimates will be used to decode the second-level inner code . Decoding of the second-level inner coset code is carried out as follows.
1) Decode the first-level component code (8, 1, 8) of using the first-stage decoded estimates of .
2) Decode the second-level component code (8, 4, 4) of using the first- and second-stage decoded estimates of and decoded estimates of first-level component code of .
3) Decode the third-level component code (8, 4, 4) of using the first-, second-, and third-stage decoded estimates of and decoded estimates of first- and second-level component codes of .
4) After the above decodings, decode the second-level outer code.
Since we are using RS outer codes over GF , we can provides almost error-free decoded estimates to the second- level inner coset code decoding. As a result, the proposed
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1647
TABLE IV
BITERRORPERFORMANCE OF THEDIFFERENTIALLYDETECTED8-DPSK TWO-LEVELCONCATENATED
MODULATIONCODE C(3)OVER THESHADOWED MSAT CHANNEL
Fig. 14. BER of the two-level 8-DPSK concatenated modulation code over a light-shadowed MSAT channel.
two-level concatenated-coded system can achieve better error performance than the three-level and six-level concatenated systems.
Based on above two inner coset codes, the following two RS outer codes are chosen such that the overall coded system achieves the best possible BER performance for the heavy- shadowed MSAT channel with close to 2-b/symbol spectral efficiency for BT and
RS code RS code
The overall two-level concatenated-coded 8-DPSK system is given as follows:
The spectral efficiency of system is 2.003 424 b/symbol. Its error performances with differential detection over the light-, average-, and heavy-shadowed MSAT channels
are summarized in Table IV and shown in Figs. 14–16, respectively. We see that the performance curves drop like waterfall There is no error floor for BER greater than 10 , even for the heavy-shadowed MSAT channel. System outperforms and . Very large coding gains are achieved over the uncoded 4-DPSK system without bandwidth expansion (in fact, with small bandwidth reduction).
For comparison purpose, we construct a single-level concatenated-coded 8-DPSK system, denoted , with the same spectral efficiency as and using the base code as the inner code and the RS (511, 455, 57) code over GF as the outer code. The performance of this system with differential detection over various types of shadowed MSAT channel are also shown in Figs. 14–16. At the BER with BT , the code achieves 1.06-, 2.049-, and 2.015-dB coding gains over the code for the light-, average-, and heavy-shadowed MSAT, respectively. And, at the BER with BT ,
Fig. 15. BER of the two-level 8-DPSK concatenated modulation code over an average-shadowed MSAT channel.
Fig. 16. BER of the two-level 8-DPSK concatenated modulation code over a heavy-shadowed MSAT channel.
the code achieves 1.06, 1.711, and 11.671 coding gains over the code for the light- and average-shadowed MSAT, respectively. We see that outperforms . For the heavy-shadowed MSAT channel with BT , the code faces the error floor around BER .
V. CONCLUSION
In this paper, a general method for constructing multilevel concatenated-coded modulation systems has been presented.
Several specific multilevel concatenated-coded 8-DPSK sys- tems have been constructed for various shadowed MSAT channels. Simulation results showed that if the inner codes, outer codes, partitions of inner codes, and the level of con- catenation are properly chosen, very good error performance
can be achieved with high spectral efficiency and large coding gain. As shown in the example of six-level concatenated code, it is possible to achieve better BER performance than the three-level concatenated code with the same inner block modulation codes and the same spectral efficiency because of good partition of inner code and proper selection of outer code for each level of partition.
For the channel with severe multipath fading and large Doppler frequency shift, the proposed multilevel concatenated- coded schemes achieve very impressive real coding gains over the corresponding single-level concatenated-coded schemes with the same base inner code. Even though single-level concatenated-coded codes faces an error floor before reaching the BER 10 , some of proposed multilevel schemes achieve
RHEE AND LIN: CODED M-DPSK MODULATION SCHEMES FOR SATELLITE COMMUNICATION CHANNEL 1649
an error floor free communication at the BER 10 . Especially, the proposed two-level scheme achieves very impressive error performances for various shadowed MSAT channels.
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Do Jun Rhee (S’85–M’96) received the B.S.E.E.
degree from Hanyang University, Seoul, Korea, in 1983 and the M.S. and P.E. degrees in electrical engineering from George Washington University, Washington, DC, in 1987 and 1989, respectively.
From July 1995 to July 1996, he was a Senior Design Engineer of the Channel Coding Group in the Consumer Product Division of LSI LOGIC, Mil- pitas, CA. From July 1996 to October 1997, he was a Senior Design Engineer in the Communication Product Division of LSI LOGIC, Milpitas. Since November 1997, he has been a Senior Design Engineer of the Channel Coding Group in the Consumer Product Division of LSI LOGIC, Milpitas.
Since 1995, he has been working as a member of the wireless ad hoc group in physical layer of digital audiovisual council (DAVIC) and has participated in making specifications MMDS, LMDS, unidirectional satellite broadcasting, and cable modems. He has published several technical papers in different IEEE TRANSACTIONS. His current research areas include algebraic coding theory, coded modulation for the mobile satellite communication, and VLSI implementation of channel coding decoder for the MMDS, LMDS, cable modems, digital TC for the United States and Japan, and fully digital BPSK/QPSK/8PSK demodulators for broadcasting satellite receivers.
Dr. Rhee is a member of the IEEE Information Theory Society and IEEE Communication Society. He is also a member of the Digital Visual Council.
Shu Lin (S’62–M’65–SM’78–F’80) received the B.S.E.E. degree from the National Taiwan Univer- sity, Taipei, Taiwan, R.O.C., in 1959 and the M.S.
and Ph.D. degrees in electrical engineering from Rice University, Houston, TX, in 1964 and 1965, respectively.
In 1965, he joined the faculty at the University of Hawaii, Honolulu, as an Assistant Professor of Electrical Engineering. He was promoted to Asso- ciate Professor in 1969 and to Professor in 1973.
In 1986, he joined Texas A&M University, College Station, as the Irma Runyon Chair Professor of Electrical Engineering. In 1987, he returned to the University of Hawaii, where he is now the Chairman of the Department of Electrical Engineering. He spent 1978–1979 as a Visiting Scientist at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, where he worked on error control protocols for data communication systems. He spent the academic year 1996–1997 as a Visiting Professor at the Technical University of Munich, Munich, Germany. He has published numerous technical papers in different IEEE TRANSACTIONSand other refereed journals. He is the author of An Introduction to Error-Correcting Codes (Englewood Cliffs, NJ: Prentice-Hall, 1970). He also coauthored with D. J.
Costello Error Control Coding: Fundamentals and Applications (Englewood Cliffs, NJ: Prentice-Hall, 1982) and with T. Kasami, T. Fujiwara, and M. Fossorier, Trellis and Trellis-Based Decoding Algorithms (Norwell, MA:
Kluwer, 1998). His current research areas include algebraic coding theory, coded modulation, error control systems, and satellite communications. He has served as the Principal Investigator on 23 research grants.
Dr. Lin is a member of the IEEE Information Theory Society and the IEEE Communication Society. He served as the Associate Editor for Algebraic Coding Theory for the IEEE TRANSACTIONS ONINFORMATIONTHEORYfrom 1976 to 1978. He was the Program Cochairman of the IEEE International Symposium on Information Theory held in Kode, Japan, in June 1988. He was the President of the IEEE Information Theory Society in 1991. He was a recipient of the Alexander von Humbolt Research Prize for U.S. Senior Scientists in 1996.
