Space-Time Coding with Multilevel Protection for Multimedia
Transmission in MIMO Systems
Jian-Jia Weng, Chung-Hsuan Wang, and Li-Der Jeng
†Department of Communications Engineering, National Chiao Tung University, Hsinchu, Taiwan 30010, R.O.C.
†Department of Electronic Engineering, National Chung Yuan Christian University, Chung Li, Taiwan 32023, R.O.C.
[email protected], [email protected], [email protected]
Abstract—In this paper, space-time coding schemes with full transmit diversity are investigated for unequal error protection (UEP). Effective performance indicies are proposed to measure the intrinsic UEP capability of space-time codes, based on which we demonstrate that space-time trellis codes and super-orthogonal space-time trellis codes can be used for UEP as long as the corresponding encoders are properly designed. In addition, UEP convolutional codes are concatenated with space-time block codes to construct another full-diversity UEP scheme which can provide more choices of UEP levels. Finally, good UEP codes are given by a computer search.
I. INTRODUCTION
In many applications, e.g., broadcast systems and vi-sual/speech communication systems, the source data are of unequal error sensitivities or face different levels of noise corruption. To make the best use of the channel bandwidth in those systems, it is desirable to design an error-correcting code with the capability of unequal error protection (UEP) which can provide different levels of protection against errors. However, literatures about UEP were focused on channel codes without transmitter diversity [1]-[6]. Although good UEP codes have been provided, they inevitably demand the undesired bandwidth expansion. Space-time coding not only inherits the high data rate transmission of multiple-input and multiple-output (MIMO) systems but also provides excellent performance against multipath fading [7]-[8]. Nevertheless, only a few of studies are about space-time coding for UEP. Among those, the UEP capability of space-time codes was observed in terms of the different diversity orders embedded for distinct messages [9]; the authors also presented a construc-tion of two-level UEP codes of short block length. Another diversity-based UEP scheme based on the layered space-time architecture was proposed for multimedia transmission in [10]. In [11][12], space-time trellis codes are incorporated with puncturing to construct a powerful UEP scheme with flexible choices of rates and UEP levels.
In this paper, the intrinsic UEP capability of space-time codes is investigated. Firstly, two types of performance indi-cies: the effective rank & determinant for the low-diversity case and the effective distance for the high-diversity case are proposed for space-time codes. Based on the proposed UEP measurements, we observe that different input streams of the space-time trellis codes (STTC) may experience dif-ferent levels of protection; STTC encoders with the special architecture in [7] can thus be used for UEP as long as the associated generator sequences are properly designed. We also demonstrate that super-orthogonal space-time trellis codes
(SOSTTC) [13] originally designed for full-diversity and high-rate transmission can provide the desired UEP capability. In addition, conventional UEP convolutional codes (CC) are concatenated with space-time block codes (STBC) to construct another full-diversity UEP scheme; compared with the UEP-STTC and UEP-SOUEP-STTC, this scheme can provide larger per-formance gaps between UEP levels. Furthermore, good UEP codes based on the above space-time coding schemes with different UEP levels, bandwidth efficiencies, and memories are provided by a computer search. Finally, an example of multimedia transmission is given to visualize the benefits of the proposed UEP schemes.
The rest of this paper is organized as follows. In Section II, the UEP capability of space-time codes is investigated together with the effective measurements. The UEP schemes based on STTC, SOSTTC, and UEPCC-STBC are described in Section III. In Section IV, simulation results are provided for performance verification. Finally, a summary is drawn in Section V to conclude this work.
II. UEP CAPABILITY OFSPACE-TIMECODES AND THE
EFFECTIVEMEASUREMENTS
Consider a space-time code withNT transmit antennas and
NR receive antennas. Let G be the corresponding encoder
which maps K streams of input information into coded se-quences for transmission. Denote byui,t andxj,tthe message
fed to the ith input of encoder and the modulated symbol transmitted by the jth antenna at time t, respectively, ∀ 1 ≤ i ≤ K and 1 ≤ j ≤ NT. For a data frame of
length-L, let Ui= (ui,1, ui,2, · · · , ui,L) be the information sequence
fed to the ith input of G and U = (UT
1, U2T, · · · , UKT)
T be
the information matrix for encoding, where T denotes the operation of taking transpose. The codeword matrix X for transmission is defined by x1,1 x1,2 · · · x1,L x2,1 x2,2 · · · x2,L .. . ... . .. ... xNT,1 xNT,2 · · · xNT,L
where the(j, t)th entry indicates the data transmitted by the jth antenna at time t.
In conventional, space-time codes are used for equal error protection; in such applications, the diversity order defined as the minimum rank of the codeword distance matrix is unarguably an effective parameter for performance evaluation [7]. However, space-time codes with the encoders of multiple
inputs may possess the UEP capability. For example, consider a space-time trellis code with generator sequences:
g1= [(0, 1), (1, 0)], g2= [(2, 2), (0, 2)] (1)
which is equipped with the quaternary phase-shift keying (QPSK) modulation of symbol energy Es for transmission
over a Rayleigh fading channel with the additive white Gaus-sian noise of two-sided power spectral density N0/2. For
binary inputs (u1,t, u2,t), the corresponding encoder outputs
(v1,t, v2,t) are obtained by
(v1,t,v2,t) = u1,t·(0,1)⊕4u1,t−1·(1,0)⊕4u2,t·(2,0)⊕4u2,t−1·(2,2)
where ⊕4 stands for the modulo 4 addition; v1,t and v2,t
are then mapped to the following modulated symbols for transmission:
xi,t=
p Esexp(
√
−1 · vi,t· π/2) for i = 1 and 2.
Suppose the receiver is equipped with 4 antennas. Observed from the performance plots in Fig. 1, u1,t’s and u2,t’s both
experience the same diversity order (since the corresponding BER curves have the same slope asymptotically) but u2,t’s
receive a better protection thanu1,t’s with a coding gain about
2 dB at BER 10−5. This code can hence be used for
two-level UEP as long as the data of distinct BER requirements are properly fed into the encoder. In addition, the embedded diversity order originally addressed in [9] to characterize the UEP capability seems unable to reflect the UEP behavior in this case. Besides the diversity order, new measurements to provide a more precise evaluation of the UEP capability of space-time codes are described below.
To evaluate the BER of the lth input-stream of G for some l ∈ {1, 2, · · · , K}, consider two codeword matrices X and ˆX withUl6= ˆUl. Define the codeword distance matrixA(X, ˆX)
between X and ˆX by
A(X, ˆX) = (X − ˆX) · (X − ˆX)H
whereH stands for the operation of taking Hermitian. Denote by r(X, ˆX) and {λi(X, ˆX), ∀ 1 ≤ i ≤ NT} the rank and
eigenvalues of A(X, ˆX), respectively. Assume λi(X, ˆX) ≥
λi+1(X, ˆX), ∀ i, without loss of generality. For Rayleigh
fad-ing channels, the pairwise error probability that the maximum likelihood decoder decides in favor of ˆX than X to make an error decision on the lth input-stream can be upper bounded by r(X, ˆX) Y i=1 λi(X, ˆX) Es −NR Es 4N0 −r(X, ˆX)NR (2) at high signal-to-noise ratios (SNR), based on the analysis of codeword error probability for ordinary space-time codes in [7]. For the case of r(X, ˆX)NR≥ 4, the upper bound of the
pairwise error probability can be further expressed as 1 4exp − NR 4 · Es N0 · NT X i=1 λi(X, ˆX) ! (3) with a better approximation than (2) by generalizing the derivation in [8]. Letrmin,l(G) denote the minimum effective
rank associated with the lth input of G which is defined as min
{∀X6= ˆX∈C|Ul6= ˆUl}
r(X, ˆX).
According to the value of rmin,l(G)NR, i.e., the effective
diversity gain for the lth input sequence, two types of UEP measurement are given as follows.
A. The Low-Diversity Case: rmin,l(G)NR< 4
Define the minimum effective determinant associated with thelth input of G by
detmin,l(G) = min
{∀X6= ˆX∈C|Ul6= ˆUl} rmin,l(G) Y i=1 λi(X, ˆX) Es . By (2), the BER of the lth input-stream of G, denoted by Pl(G), can be dominated by the following term:
detmin,l(G)−NR·
Es
4N0
−rmin,l(G)NR
at high SNRs. Large values ofrmin,l(G) and detmin,l(G) can
thus imply a small Pl(G).
B. The High-Diversity Case:rmin,l(G)NR≥ 4
Let d2(X, ˆX) be the effective distance between X and ˆX
defined by L X t=1 NT X j=1 |xj,t− ˆxj,t|2 and denote by d2
min,l(G) the minimum effective distance
corresponding to thelth input of G, i.e., d2
min,l(G) = min
{∀X6= ˆX∈C|Ul6= ˆUl}
d2(X, ˆX).
Based on the observation that
NT X i=1 λi(X, ˆX) = L X t=1 NT X j=1 |xj,t− ˆxj,t|2 a large value of d2
min,l(G) then implies a better BER for the
lth input-stream of G, since Pl(G) is now dominated by
exp −N4R· ENs 0 · d 2 min,l(G) at high SNRs by (3).
Define the effective rank vector R(G), the effective deter-minant vector ∆(G), and the effective distance vector Ω(G) of G by
R(G) = (rmin,1(G), rmin,2(G), · · · , rmin,K(G))
∆(G) = (detmin,1(G), detmin,2(G), · · · , detmin,K(G))
Ω(G) = d2
min,1(G), d2min,2(G), · · · , d2min,K(G) .
(4) The UEP capability of G can then be characterized by (R(G), ∆(G)) for the low-diversity case and Ω(G) for the high-diversity case, respectively. In general, the larger (R(G), ∆(G)) or Ω(G) is, the better the UEP capability is; the number of distinct components in R(G), ∆(G), Ω(G) also corresponds to the available levels for UEP. Recall the space-time encoder of generator sequences in (1); in this case, K = 2, NT = 2, and NR = 4. By (4), we have
R(G) = (2, 2), ∆(G) = (4, 4), and Ω(G) = (4, 12). Since
rmin,l(G)NR ≥ 4 for l=1,2, the unequal effective distances
inΩ(G) thus successfully reflect the two-level UEP for u1,t’s
andu2,t’s revealed in Fig. 1.
III. FULL-DIVERSITYUEP SCHEMESBASED ONSTTC, SOSTTC,ANDUEPCC-STBC
In conventional approaches [9][10], different diversity or-ders were designed for distinct messages to achieve UEP. The consequent UEP schemes thus can not obtain full transmit diversity and suffer from a poor performance of average BER. However, we observe that some space-time architectures with full diversity are inherently with the UEP capability in terms of the proposed effective measurements, although such a desirable capability is totally ignored by the previous researches. Designs based on those architectures can then accomplish UEP without any loss of transmit diversity. In the following, we first demonstrate the feasibility of STTC and SOSTTC for UEP. A concatenation of UEPCC and STBC which can assure full diversity and provide more choices of UEP levels is presented as well.
A. STTC for UEP
STTC with the special encoder architecture in [7], for which total memory elements are uniformly distributed in the shift-registers for all input streams, have been shown to achieve full-diversity transmission and provide excellent performance against multipath fading [7][8]. Conventional designs of STTC are focused on maximizing the average diversity and coding gains of the space-time coded system. Therefore, most of the optimal codes constructed previously, no matter for the low-diversity or high-low-diversity cases, can provide only a single level of protection. For example, consider an optimal code ˆC obtained for the high-diversity case with the encoder ˆG of generator sequences [8]
ˆ
g1= [(0, 2), (2, 0)] and ˆg2= [(0, 1), (1, 0)] (5)
which is equipped with the QPSK modulation; in this case, K = 2 and NT = 2. By the definitions in Section II, we
have R( ˆG) = (rmin,1( ˆG), rmin,2( ˆG)) = (2,2) and Ω( ˆG) = (d2
min,1( ˆG), d2min,2( ˆG)) = (4,4). ˆC achieves the full diversity
as expected since rmin,1( ˆG) = rmin,2( ˆG) = NT. However,
the observation ofd2
min,1( ˆG) = d2min,2( ˆG) implies that only a
single level of protection is available although K = 2. To provide UEP by STTC, we still employ the same encoder architecture to guarantee full transmit diversity but the cor-responding generator sequences are now searched according to the proposed UEP measurements, instead of the average diversity and coding gains. For the low diversity case, an optimum choice of the generator sequences should result in (R(G), ∆(G)) as large as possible. The generator sequences are then optimized to maximize Ω(G) for the high diversity case. By an exhaustive computer search, families of full-diversity STTC with good UEP capability for both of the low-diversity and high-low-diversity cases have been obtained; only the family with two transmit antennas and QPSK are given in Table I for illustration due to the length limitation. Note that, compared with the optimal codes with the same rates and
memories in [7], UEP codes obtained for the low-diversity case have the same or larger(R(G), ∆(G))’s; the new constructed UEP codes can hence provide better protection for all input streams.
B. SOSTTC for UEP
SOSTTC were first introduced in [13] to take the advantages of STBC and STTC. By employing the trellis structure similar to STTC but choosing a set of parameterized class of STBC as the space-time signal points for transmission, SOSTTC have been verified to provide a good trade-off between the rate and diversity of space-time coded systems. Conventional designs on SOSTTC are focused on maximizing the minimun coding gain distance [13] of all codeword matrices only; the intrinsic UEP capability of SOSTTC is thereby neglected unwittingly. However, we observe that SOSTTC can be used for UEP as long as the codes are now designed by optimizing the proposed effective measurements rather than the original performance index. Consider an example of SOSTTC with K=4, NT=2,
and the QPSK modulation which chooses the following two types of STBC for transmission:
Type1 STBC: s1 s2 −s∗2 s∗1 , Type2 STBC:−s1 s2 s∗2 s∗1
wheres1ands2are mapped from the input vector(u1,t,u2,t,
u3,t, u4,t) by the encoder G as depicted in Fig. 2, where
the associated trellis module has two states and at each time instant the type of STBC used for transmission is determined by the value of current state. By the definitions in Section II, we have R(G) = (2, 2, 2, 2), ∆(G) = (4, 16, 16, 64), and Ω(G) = (4, 8, 8, 16). This full-diversity code can thus provide three-level UEP for both of the low-diversity and high-diversity cases since the components in∆(G) and Ω(G) are now of three distinct values respectively.
C. UEP Scheme Based on UEPCC-STBC
STBC with the orthogonal design have been verified to achieve full transmit diversity but with the drawback of no coding gain [14]. On the other hand, UEP convolutional codes, originally designed for the system of single transmit antenna, can provide rich of coding gain for different input streams, but the direct use of them in fading channels usually suffers from a performance degradation. To take both of the advantages of STBC and UEPCC, a space-time UEP scheme which concatenates the both schemes as indicated in Fig. 3 is proposed here. In this concatenated scheme, the information bits are first encoded by an(N, K, V ) UEP convolutional code with proper UEP capability to attain the desirable coding gain, whose encoder has K inputs, N outputs, and memory V . Let (c1,t, · · · , cN,t) denote the vector of coded bits at time
t. (c1,t, · · · , cN,t) is then processed by a specially designed
Ns×N mapping matrix Q to generate Nsmodulated symbols
(y1,t, · · · , yNs,t) by (y1,t, · · · , yNs,t)
T
= Q · (c1,t, · · · , cN,t)T.
Finally, (y1,t, · · · , yNs,t) is transformed into a space-time codeword matrix with NT transmit antennas by a STBC
encoder to achieve the full transmit diversity. By an exhaustive computer search, families of full-diversity codes with good
UEP capability for both of the low-diversity and high-diversity cases have been obtained. Only the family with two transmit antennas and the Alamouti STBC [14] with QPSK are given in Table II, where the generator matrices of UEPCC are chosen to be canonical to avoid the catastrophic encoders and the corresponding entries are expressed in octal form, for illustration due to the length limitation.
IV. SIMULATIONRESULTS
To verify the validity of the proposed UEP measurements, UEP schemes based on STTC, SOSTTC, and UEPCC-STBC are simulated for transmission over a block fading channel, for which the fading gain is assumed to be constant within each data frame of 130 symbols. In Fig. 1, the BER curves corresponding to different input streams of the high-diversity STTC are plotted. From the simulation results, all curves are observed to have the same slope asymptotically; the code in Table I achieves two-level UEP but the optimal code by the conventional design can provide only a single level of protection. Compared with the observation thatR(G) = R( ˆG) = (2,2), Ω(G) = (4, 12), and Ω( ˆG) = (4,4), the proposed measurements can successfully predict the UEP capability. Performance plots of the SOSTTC illustrated in Section III-B are shown in Fig. 4; the three-level UEP observed in Fig. 4 is also consistent with the prediction of ∆(G) = (4, 16, 16, 64) for the low-diversity case. In addition, consider the UEP scheme based on the (4,2,2) UEP convolutional code in Table II with∆(G) = (4, 144) and Ω(G) = (4, 24). Observed from Fig. 5, the proposed measurements can precisely reflect the UEP behavior for the both cases of NR = 2 and 4. Owing to
the different coding gains introduced by the UEP convolutional code, this scheme can provide apparent performance gaps (up to6dB SNR gain at BER 10−5) between two UEP levels.
To demonstrate the benefit of applying UEP, a gray image is transmitted over Rayleigh fading channels under the protection of the both STTCs with the same rate and memory considered in Fig. 1. At SNR =−1dB, the reconstructed images with and without applying UEP are shown in Fig. 6(a) and Fig. 6(b). It can be seen that the better visual quality is obtained with the usage of UEP. The peak signal-to-noise ratio (PSNR) curves in Fig. 7 also shows the improvement of the reconstructed images with the aid of UEP.
V. CONCLUSION
In this paper, two types of performance indicies: (R(G), ∆(G)) and Ω(G) are proposed to measure the intrinsic UEP capability of space-time codes. Based on those measurements, we demonstrate that STTC and SOSTTC can be used for UEP as long as the corresponding encoders are properly designed. In addition, UEPCC are concatenated with STBC to construct another full-diversity UEP scheme which can provide more choices of UEP levels. Simulation results also verify the effectiveness of the proposed UEP measurements for multimedia transmission. Finally, tables of space-time codes with good UEP capability are given by a computer search.
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TABLE I
GOODUEP STTCWITHQPSK MODULATION ANDTWOTRANSMIT
ANTENNAS
Memory Generator Sequences UEP Measurements
2 g1= [(3, 2), (2, 0)] g2= [(1, 0), (0, 3)] R(G) = (2, 2) ∆(G) = (4, 8) 3 g1= [(2, 2), (1, 2)] g2= [(2, 0), (2, 3), (2, 1)] R(G) = (2, 2) ∆(G) = (8, 16) Low Diversity Case 4 g1= [(3, 0), (1, 1), (2, 0)] g2= [(1, 2), (1, 2), (0, 2)] R(G) = (2, 2) ∆(G) = (12, 16) 5 g1= [(0, 3), (1, 1), (0, 2)] g2= [(0, 2), (2, 2), (1, 1), (2, 0)] R(G) = (2, 2) ∆(G) = (16, 32) 6 g1= [(1, 1), (3, 2), (3, 0), (3, 2)] g2= [(2, 0), (1, 0), (3, 2), (3, 3)] R(G) = (2, 2) ∆(G) = (28, 32) 2 g1= [(0, 1), (1, 0)] g2= [(2, 2), (0, 2)] R(G) = (2, 2) Ω(G) = (4, 12) 3 g1= [(0, 2), (0, 3), (1, 2)] g2= [(2, 0), (1, 3)] R(G) = (2, 2) Ω(G) = (8, 12) High Diversity Case 4 g1= [(3, 1), (3, 0), (0, 1)] g2= [(1, 2), (2, 1), (2, 2)] R(G) = (2, 2) Ω(G) = (8, 12) 5 g1= [(0, 3), (1, 1), (0, 3)] g2= [(2, 1), (3, 2), (3, 1), (2, 3)] R(G) = (2, 2) Ω(G) = (8, 14) 6 g1= [(0, 2), (0, 1), (0, 1), (1, 1)] g2= [(2, 0), (3, 1), (2, 1), (3, 1)] R(G) = (2, 2) Ω(G) = (12, 16)
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TABLE II
UEP SPACE-TIMECODESBASED ONUEPCC-STBC
WITHQPSK MODULATION ANDTWOTRANSMITANTENNAS
(N, K, V ) Generator Matrix UEP Measurements
(4, 2, 2) 1 0 0 1 7 7 5 0 ! ∆(G) = (4, 144) Ω(G) = (4, 24) (4, 2, 3) 1 1 0 0 13 0 15 17 ! ∆(G) = (64, 196) Ω(G) = (16, 28) (4, 2, 4) 1 0 0 1 25 33 37 0 ! ∆(G) = (4, 256) Ω(G) = (4, 32) (4, 2, 5) 1 0 7 6 14 13 15 5 ! ∆(G) = (144, 256) Ω(G) = (24, 32) (4, 2, 6) 0 5 7 1 35 11 35 20 ! ∆(G) = (64, 256) Ω(G) = (16, 32) (4, 3, 2) 0 0 1 1 1 0 1 0 5 7 0 0 ∆(G) = (4, 4, 64) Ω(G) = (4, 4, 16) (4, 3, 3) 1 1 1 1 2 3 1 0 5 3 4 0 ∆(G) = (16, 36, 36) Ω(G) = (8, 12, 12) (4, 3, 4) 1 1 1 1 1 2 3 0 16 5 15 0 ∆(G) = (16, 64, 100) Ω(G) = (8, 16, 20) (4, 3, 5) 1 1 1 1 1 2 3 0 36 11 31 0 ∆(G) = (16, 64, 196) Ω(G) = (8, 16, 28) (4, 3, 6) 1 1 1 1 3 1 2 0 57 40 21 0 ∆(G) = (16, 100, 144) Ω(G) = (8, 20, 24) Mapping Matrix Q: 2 0 0 1 0 2 1 0 ! 0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 E b/N0(dB) BER Average BER of U
1 for STTC with g1 & g2
Average BER of U
2 for STTC with g1 & g2
Average BER of U
1 for STTC with g1 & g2
Average BER of U
2 for STTC with g1 & g2
^ ^
^ ^
Fig. 1. Performance plots of STTC with generator sequences: g1 =
[(0, 1), (1, 0)], g2 = [(2, 0), (2, 2)] and ˆg1 = [(0, 1) , (1, 0)], ˆg2 = [(0, 2) , (2, 0)], respectively. 0 Z 1 Z 1 Z 0 Z
Minimum Effective Determinant
4 16 16 64 00 Z0 Z1 Z00 Z01 Z10 Z11 Z000 Z001 Z010 Z011 Z100 Z101 Z110 Z111 22 02 20 11 33 13 31 01 23 10 32 03 21 12 30
QPSK Modulation Indicies for s1 and s2
I1I2 Type-1 STBC Encoder s2 I1I2 0 1 Type-2 STBC Encoder s 1 s1 s2 1 3 2 0 State0 State1 State0 State1 u1,t u2,t u3,t u4,t u1,t u2,t u3.t u4,t
Fig. 2. The encoder structure of SOSTTC.
10 12 14 16 18 20 22 10−4 10−3 10−2 Eb/N0(dB) BER N R=1 Averge BER of U1
NR=1 Averge BER of U2 & U3
N
R=1 Averge BER of U4
Fig. 4. Performance plots of SOSTTC with NT= 2 and NR= 1.
0 2 4 6 8 10 12 14 16 10−6 10−5 10−4 10−3 10−2 10−1 E b/N0(dB) BER
Average BER of U1 for NR=2
Average BER of U2 for NR=2
Average BER of U1 for NR=4
Average BER of U2 for NR=4
Fig. 5. Performance plots of UEPCC-STBC with two transmit antennas.
(a) Without UEP (b) With UEP
Fig. 6. The reconstructed images without and with UEP at SNR = −1dB.
-2 -1 0 1 2 3 4 40 42 44 46 48 50 52 54 56 58 60 Eb/N0(dB) P S N R (d B ) Without UEP With UEP
Fig. 7. PSNR curves of both reconstructed images for coding schemes with/without UEP.
