The Chernoff Bounding Parameter For A Multilevel Modulation Scheme Using Psk-signaling

The Cherno Bounding Parameter for a Multilevel Modulation Scheme Using PSK-signaling 1

Karin Engdahl and Kamil Sh. Zigangirov Department of Information Technology Telecommunication Theory Group Lund University, Box 118, S-221 00 Lund, Sweden Phone +46 46 222 3450, Fax +46 46 222 4714, e-mail: [email protected] Abstract{ We analyze a PSK version of the multilevel modulation scheme with multi-

stage decoding intruduced by Imai and Hirakawa [5], when transmission takes place over a Gaussian channel. Each stage in the decoder uses a suboptimal metric. The results of our analysis can be used not only for the case of multilevel PSK, but for any binary modulation using PSK-signaling and multiple representation of bits. We argue that the conventional approximation of the Cherno bounding parameter Z (used as argument in the code generating function when calculating error probabilities) is not adequate, and new values of Z that tightens the error bounds without causing them to lose their validity are given. The capacity for this scheme is also calculated, and the result hereof is that the use of the suboptimal metric in multistage decoding causes very little degradation in capacity compared to when optimal metric is used in each stage of the multistage decoder. Keywords{ Multilevel modulation, multistage decoding, PSK-signalling, set parti-

tioning. 1 This work was supported in part by Swedish Research Council for Engineering Sciences under Grant 95-164

1

I. INTRODUCTION The principle of trellis-coded modulation was described by Ungerbck in 1982 [7], and the concept of multilevel modulation was introduced by Imai and Hirakawa [5]. The multilevel scheme enables the usage of a multistage decoder [5], which performes almost as well as an optimum joint maximum likelihood sequence estimator over all levels [4] but is much less complex. In this paper we will study a multilevel modulation scheme using PSK-signaling, transmitting over a Gaussian channel, and employing a multistage decoder in the receiver. To further reduce complexity we use a suboptimal metric in each decoding stage. This is shown to generate very slight performance loss in terms of capacity. In Section II we present the PSK multilevel modulation scheme. The channel characteristics and the decoding procedure are also given. Upper bounds on block and burst error probabilities (for block and convolutional codes respectively) are given in Section III. The upperbounding of these error probabilities reduces to the calculation of code generating functions with the Cherno bounding parameter Z as argument. This parameter is a function of the intra-set squared Euclidean distance,

k2, on level k, the noise variance 2 and the number of signal points in the PSK signal set on the corresponding level of set partitioning. In [2] and [3] Z was calculated for dierent levels of a multilevel modulation scheme using QAM-signaling. For 2-PSK transmission (the last level of the multilevel PSK scheme) Z = exp



k2 2

8



. This

value is a lower bound on Z for other levels of the scheme, and is often used as an approximation of Z , though the approximation results in that the bounds on error probability lose their validity. The inaccuracy increases with the ratio k . An 2

2

2

accurate upper bound on Z for any level of the PSK scheme is Z = 2 exp



k2 2

8



.

This is a consequence of the \nearest neighbor error events principle" [4], [6], and the fact that each signal point of a M -PSK set (M  4) has 2 nearest neighbors. This value of Z yields, especially for small values of k , loose bounds on the error 2

2

probabilities. In Section IV we calculate a better upper bound on Z that tightens the error bounds without causing them to lose their rigor. This is done using the Cherno bounding method, which gives exponentially tight bounds [10]. Finally, in Section V we use the distribution of the metric described in Section IV to calculate the capacity of this multilevel modulation scheme using PSK-signaling, with this type of suboptimal decoder, and compare to the capacity for the same scheme using optimal metric in each decoding stage.

II. SYSTEM DESCRIPTION The transmitter and receiver described in Figure 1 is closely related to [5] and is the PSK-version of the scheme presented in [2]. A binary information sequence u is partitioned into K binary subsequences u(1) ; u(2) ; : : : ; u(K ), and each subsequence is encoded by an independent binary component code Ck (block or convolutional).

n

o

A set of K bits, v(1) (n) ; v(2) (n) ; . . . ; v(K ) (n) , one bit from each of the code sequences v(1) ; v(2) , . . . ,v(K ), are synchronously mapped onto one of the 2K -PSK signal points, s (n). The mapping is illustrated in Figure 2 for K = 3. The mapping procedure results in that on each level, except the last one, we have multiple representations of the transmitted bits. 3

When passed through the discrete memoryless Gaussian channel the complex sequence s = s (1) ; s (2) ; : : : ; s (n) ; : : : (where s (n) = a (n) + jb (n) is the channel input at the nth moment of time, a (n) = sin  (n), b (n) = cos  (n) and

n o  (n) 2 22K m : m = 0; 1; : : : ; M 1 ) is corrupted by the error sequence e = e (1) ; e (2) ; : : : ; e (n) ; : : : such that the complex received sequence is r = s + e. Here

e (n) = e(I ) (n) + je(Q) (n), where e(I ) (n) and e(Q) (n) are independent Gaussian random variables with zero mean and variance 2. The multistage decoder consists of a set of suboptimal decoders matched to the codes used on the corresponding levels of encoding. Each decoding stage consists of calculation of distances (metrics) to the received sequence r from all possible code words on the corresponding level of set partitioning. The side information from the previous decoding stages determines, according to the set partitioning structure (illustrated in Figure 2), the signal set upon which the metrics are calculated. When the decoder calculates the metrics, it uses the following suboptimal principle. Let us suppose that a binary block code (the extension to convolutional codes is straight forward) of length N is used on the kth level of the encoding, and that the decoding in the previous (k 1) decoding stages determines the subsets

S (k 1) (1) ; S (k 1) (2) ; : : : ; S (k 1) (N ), to which the transmitted symbols of the binary codeword v(k) = v(k) (1) ; v(k) (2) ; : : : ; v(k) (N ) belong.

Let S0(k 1) (n) and S1(k 1) (n) be subsets of S (k 1) (n), corresponding to transmis-

sion of v(k) (n) = 0 and v(k) (n) = 1 respectively. Let s(k) (n) 2 S (k 1) (n), n =

1) (1) ; 1; 2; : : : ; N and s(k) = s(k) (1) ; s(k) (2) ; : : : ; s(k) (N ). Finally let S(vkk 1) = Sv(kk (1) ( )

( )

1) (2) ; : : : ; S (k 1) (N ) be the sequence of subsets corresponding to transmission Sv(kk (2) v k (N ) ( )

( )

4

of the code word v(k). Then the distance (metric) between the received sequence

r = r (1) ; r (2) ; : : : ; r (N ) and the codeword v(k) is determined as      r; v(k) = k mink dE r; s(k) ; s(

)

2S(v(k)1)

(1)

where dE (x; y) is the squared Euclidean distance between the N -dimensional vectors

x and y. The decoding consists of choosing the codeword v(k) for which the metric    r; v(k) above is minimal.

III. ERROR PERFORMANCE The performance of a multilevel coded modulation system, which employs a multistage decoder, is commonly estimated by average bit error probability of each component code or by block error probability and burst error probability for block and convolutional coding respectively [1], [4]. When a linear block code is used on the kth signaling level, an upper bound on

  the probability of decoding error P "(k) for that level is [2], [9]   P "(k) < G(k) (D) jD=Z k ; ( )

(2)

and when a convolutional code is used, the upper bound on the burst error proba-

  bility P "(k) and bit error probability Pb(k) are [8], [11]   P "(k) < T (k) (D) jD=Z k

(3)

@ T (k) (D; L; I ) j k Pb(k) < @I D=Z ;L=I =1

(4)

( )

and

( )

where G(k) (D) and T (k) (D) are the generating functions of the codes, and T (k) (D; L; I ) is the re ned path enumerator [11]. 5

Hence, the commonly used upper bounds for bloc, burst and bit error probabilities in the block and convolutional coding cases are both functions of the parameter

Z (k). Calculation of the minimal possible Z (k) yields tight upperbounding of the decoding error probability on each decoding level. As we mentioned earlier exp

(

) ( ) k2  Z (k) < 2 exp k2 8 2 8 2

(5)

in the PSK case. The bounds (2), (3) and (4) are often abused through the use of the lower bound exp



k2 2

8



as an approximation to Z (k) [1]. Thereby the upper

bounds (2), (3) and (4) lose their validity. An example of this is given in [4], where it can be seen that the actual bit error probability exceeds the bound (4) when Z (k) = exp



k2 2

8



is used. This is equivalent to using the minimum squared

Euclidean distance between code words as a design rule, which is not appropriate when multilevel systems are considered. On the other hand the estimation Z (k) = 2 exp



k2 2

8



sometimes gives bounds that are loose. In the following section we

calculate the values Z (k) that give exponentially tight union bounds, so that (2), (3) and (4) keep their validity. This is done by using the Cherno bounding method. The upper bounds (2), (3) and (4) ignores the fact that errors can propagate in the multistage decoder, and hence for multilevel modulation they are only rigorous for the rst stage in the decoder. The problem of error propagation was addressed in [6] and the bounds presented there will also be improved if our value of Z (k) is used instead of 2 exp



k2 2

8



. If non-multilevel binary modulation using PSK-signaling

and multiple representation of bits is considered, the bounds given are of course also useful. 6

IV. DERIVATION OF THE CHERNOFF BOUNDING PARAMETER Without loss of generality we suppose that v0(k) (the all-zero codeword) is transmitted. That is, at each time instant one of the signal points from the reference set is transmitted. We suppose in addition that vl(k) is a codeword of Hamming

weight wl(k). To simplify the analysis, we also change the order of the transmitted





symbols, such that the rst wl(k) symbols of vl(k) are ones. Comparing  r; v0(k) and

   r; vl(k) to decide in favor of one of the codewords, using the Cherno bounding technique and the union bound yields [2]

Z (k) = min '(k) (s) s0

(6)

where '(k) (s) is the generating function of the suboptimal metric

   

l(k) (n) = dE r (n) ; v0(k) (n) dE r (n) ; vl(k) (n)

(7)

and where

  dE r (n) ; vl(k) (n) =

s(k) n

mink

( )2Sv( (k)1)(n) (n) l

  dE r (n) ; s(k) (n) :

(8)

On the last decoding level the subsets S0(K 1) and S1(K 1) consists of one point each, and the squared Euclidean distance between the subsets is K2 . It is well known





2 that in this case Z (K ) = Z2 = exp 8K2 , i.e. it coincides with the lower bound in (5). The parameter Z (K ) is a function of  = 2K , and since all the expressions of Z (k) in the further evaluation of the error probability on the kth level will also

be functions of , we will keep this notation. Thus, without loss of generality, we consider, on the kth level of set partitioning, the signal set with the energy per 7

transmitted symbol 2 and additive Gaussian noise with the variance 2 equal to 1. Numerical values of the Cherno bounding parameter Z2 (Z for 2-PSK) are shown in Table 1 for a number of dierent . Consider now the decoding on the kth level, when the subsets S0(k 1) and S1(k 1) are 2(K k)-PSK signal sets (Figure 3). On each level, the relation between the symbol energy 2 and the normalized inter-set Euclidean distance

k 

is

k = 2 sin  ;  2

(9)

where  = 2M and M = 2(K k). To study the distribution of (k) (n), we introduce the system of coordinates (x; y), shown in Figure 3. The M sectors de ne regions,

in which a received point has the same nearest point from each set S0(k 1) and S1(k 1). The conditional probability that the received point is in the same region as

in Figure 3, given that v0(k) was transmitted, is equal to 1=M . In this region the

squared Euclidean distance from the received point to the nearest point of S0(k 1) is

  dE r (n) ; v0(k) (n) = 0 = (x (n) )2 + (y (n))2

(10)

and the squared Euclidean distance from the received point to the nearest point of

S1(k 1) is

  dE r (n) ; vl(k) (n) = 1 = (x (n)  cos )2 + (y (n)  sin )2 :

(11)

So the dierence between the squared Euclidean distances becomes

   

(k) (n) = dE r (n) ; v0(k) (n) = 0 dE r (n) ; vl(k) (n) = 1 =

!    2 ( x (n) + x (n) cos  + y (n) sin ) = 4 sin 2 y (n) cos 2 x (n) sin 2 : (12) 8

Since the distribution of (k) (n) does not depend on n, for n = 1; 2; : : : ; wl(k) we will drop the argument n. The random variables X and Y are independent. The con-

ditional joint probability density function of (X; Y ) given that v0(k) was transmitted and given that the received signal is in the sector between the point in the reference set having the smallest phase angle and the point from the opposite set with the smallest phase angle (Figure 3) is (0) (x; y ) = 1 fX;Y

M 1 2 X

  1 2 1 (y  sin 2n)2 : ( x  cos 2 n ) 2 2

(13)  n=0 Keeping in mind the multiple representation of v0(k) , the explanation to this expression is as follows. First take the sum of all the two dimensional Gaussian distribuexp

tions centered around each of the signal points of the reference set. All points from the reference set are assumed to be equally likely. Then condition this sum on that the received signal is in the above mensioned sector, this will give a scale factor of 2 times the original sum. If the received signal is in any other region, we get formulas for the representation of (k) analogous to (12) and for the density function of (X; Y ) analogous to (13). The generating function of (k) is

Z 1 Z x tan  n o (0) ( k) exp s (k) fX;Y (x; y) dydx; ' (s) = x=0 y=0

(14)

In polar coordinates, x = r cos  and y = r sin , this becomes

'(k) (s) =

Z Z1

=0 r=0



n o exp s (k) fR;(0) (r; ) drd;

(15)

where and

(k) = 2r ( cos  + cos ( ))

(16)

 1  X1 1 (0) 2 2 fR; (r; ) =  r exp 2 r +  2r cos (2n ) : n=0

(17)

M 2

9

Thus, since we get

n o exp s (k) fR;(0) (r; ) =

1 X1 r exp  1 r2 2r ( cos (2n ) + 2s cos ( ) 2s cos ) + 2  =  n=0 2 M 2

  1   1 X1 1 2 2 2 r exp 2  (A ()) exp 2 (r A ()) = n=0

(18)

A () =  cos (2n ) + 2s cos ( ) 2s cos ;

(19)

M 2

where and since

( 2 ) p  1  2 r exp 2 (r A ()) dr = exp A 2() + 2A () Q ( A ()) ; (20) r=0 n o R where Q (x) = p1 1 exp t dt, the generating function of (k) becomes Z1

2

2

x

2

'(k) (s) =

M ! ( ) 1 X1 Z  exp 2 + p2A () Q ( A ()) exp  1 2 A2 () d =  n=0 =0 2 2 2

( 2 ) 0 s MX1 Z  ( 2 ) 1 = exp  B 1+ 2 A () Q ( A ()) exp A () dC @ A: 2  2 =0 2

n

=0

(21)

This completes the proof of the following theorem.

Theorem 1 For any level of suboptimal decoding of a multilevel coded PSK signal set, the Cherno bounding parameter has the value

Z (k) = ZM = min '(k) (s) ; s0 where

(22)

( 2 ) 1 ( 2 ) 0 s MX1 Z  A () Q ( A ()) exp A 2() dC '(k) (s) = exp 2 B @1 + 2 A (23) n=0 =0 2

10

and

A () =  cos (2n ) + 2s cos ( ) 2s cos :

(24)

Numerical values of ZM (Z for M -PSK), M = 4; 8; 16; 32; 64, are shown in Table 1 for a number of dierent . In Table 2, ZM is compared to its lower bound Z2 (often used as an approximation of ZM ) for dierent values of
, where
= 2 sin( 2 ) is the normalized minimal Euclidean distance in the M -PSK set, shown in Figure 3. It can be seen that the lower bound is rater close to ZM at small
. As
increases, ZM approaches its upper bound 2Z2 due to the \nearest neighbor error events principle". In Figure 4 we show by two examples how the bounds on error probability are improved by the use of ZM instead of 2Z2.

V CAPACITY OF MULTISTAGE DECODING USING SUBOPTIMAL METRIC Analogously to [4] we consider the transmission on each level of modulation as transmission over an individual channel. But in contrast to [4], where the capacities of these individual channels for optimal decoding on each level of modulation was calculated, we calculate the capacities when suboptimal decoding, described in Section II, is used. As an output of the channel we consider the sequence of statistics

(k) (n). According to the de nition of capacity, the error probability can be made arbitrarily small on each decoding stage if the code length approaches in nity [4]. In this section we study the asymptotical behavior of the system, and therefore we do not consider the problem of error propagation due to erroneous decisions in earlier decoding stages. The capacity for the kth level of the given multilevel PSK system 11

using multistage decoding with suboptimal metric (8) is

Z 1 (0) Z 1 (1) C (k) = Haposteriori + Hapriori = 12 f k log f (0)k d (k) + 12 f k log f (1)k d (k) 1 1 Z 1  1 (0) 1 (1)   1 (0) 1 (1)  (25) f k + 2 f k log 2 f k + 2 f k d (k); 1 2 where by symmetry f (1)k = f (0) k . All signal points are assumed to be equally likely. The capacity and computational cuto rate Rc = 1 log2(1+ ZM ) of a level using M ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

PSK is shown in Figure 5 for M = 2; 4; 8; 16; 32; 64. In this gure we also show the capacity for the same system using optimal metric at each decoding stage calculated in [4] (M = 2; 4; 8) and conclude that the capacity for our system using suboptimal metric and multistage decoding is close to the capacity for a system using optimal metric and multistage decoding. This indicates that it is not necessary to use the optimal metric in multistage decoding of multilevel PSK. That is, a complexity reduction by use of the suboptimal metric can be accieved at very small loss in capacity. In Figure 6 we show the total capacity of a multilevel modulation scheme using PSK-signaling with K = 6 levels.

VI CONCLUSIONS The decoding error probability for multilevel PSK with suboptimal multistage decoding has been analyzed. Rigorous union bounds for error probabilities of component codes have been presented, and a formula for the parameter Z , derived using the Cherno bounding method, has been derived. This formula yields a Z that gives tighter bounds than the commonly used \nearest neighbor error events principle", but the bounds do not lose their validity as is the case when the often used approximation Z = exp

n

2 2

8

o

is applied. Comparison to traditional bound12

ing techniques for the probability of decoding error has been made for M -PSK with

M = 2; 4; 8; 16; 32; 64. Calculation of capacity was made, and from this it was concluded that the capacity for suboptimal decoding of M -PSK is close to the capacity for optimal decoding, hence in terms of capacity the need for using optimal metric is small when multilevel PSK with multistage decoding is considered.

References [1] E. Biglieri, D. Divsalar, P. J. McLane and M. K. Simon, Introduction to TrellisCoded Modulation with Applications, Macmillan, 1991.

[2] K. Engdahl and K. Sh. Zigangirov, \On the Calculation of the Error Probability for a Multilevel Modulation Scheme Using QAMsignalling," submitted to IEEE Trans. Information Theory, (available from http://www.it.lth.se/~karin/artiklar.html). [3] K. Engdahl and K. Sh. Zigangirov, \On the Calculation of the Error Probability for a Multilevel Modulation Scheme Using QAM-signaling," Proceedings ISIT -97, p.390.

[4] J. Huber, \Multilevel Codes: Distance Pro les and Channel Capacity," in ITGFachbereicht 130, pp. 305-319, Oct. 1994. Conference Record.

[5] H. Imai and S. Hirakawa, \A New Multilevel Coding Method Using ErrorCorrecting Codes," IEEE Trans. Information Theory, vol. IT-23, pp. 371-377, May 1977. 13

[6] Y. Kofman, E. Zehavi and S. Shamai, \Performance Analysis of a Multilevel Coded Modulation System" IEEE Trans. Communications, vol. COM-42, pp. 299-312, Feb./Mar./Apr. 1994. [7] G. Ungerbck, \Channel Coding with Multilevel/Phase Signals," IEEE Trans. Information Theory, vol. IT-28, pp. 55-67, Jan. 1982.

[8] A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding, McGraw Hill, 1979.

[9] S. G. Wilson, Digital Modulation and Coding, Prentice Hall, 1996. [10] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering, Wiley, 1965. [11] K. Sh. Zigangirov and R. Johannesson, Fundamentals of Convolutional Codes, to be published by IEEE Press 1998.

14

LIST OF FIGURE CAPTIONS Figure 1: System description of the K level modulation scheme using PSK-signaling. Figure 2: Set partitioning of PSK signals when K = 3. Figure 3: The signal constellation on the rst level for K = 3. Table 1: Numerical values of Z for dierent values of . Table 2: Comparison of Z to the conventional approximation. The normalized intra-set Euclidean distance is
= 2 sin( 2 ).

Figure 4: Two examples of the error bound (3) for the rst level of a system with K = 2. The gure shows the bound when the lower and upper bounds in (5) D = Z2 (dashed) and D = 2Z2 (dash-dotted) are used, and when D = Z4 (solid) is applied. The relation between
and the energy per channel use over

N0 is NEs = p
2 . (a) T (D) = 1 D2D , (b) T (D) = D10 5

0

D12

D8 4D6 +5D4 4D2 +3 +4D10 4D8 +2D6 2D4 2D2 +1 .

Figure 5: Capacities (solid lines) for M -PSK using suboptimal decoding (M = 2; 4; 8; 16; 32; 64). The capacities for M -PSK with M = 2; 4; 8 using optimal decoding[4], are also shown (stars). Also plotted is the computational cuto rates (dashed lines).

Figure 6: Total capacity (solid line) for a multilevel modulation scheme using 64PSK (K = 6 levels) and the capacities of each level (dashed lines).

15

u(1) C v(1) 1 (2) (2) v u C 2

u Partition u(3) of information

C3

v(3)

p p p

2K -PSK s AWGN r subopti- u^ mal mapper  decoder

u(K ) C v(K ) K Figure 1:

v(1) = 0

v(2) = 0



eue e e eue

eue u u eue

   

uuu u u uuu

HHH

@ v(2) = 1 @@ R e

HHHj

v(2) = 0



uee e e eeu

e e u u eee

Figure 2:

16

S (0)

v(1) = 1

ueu e e ueu

@ v(2) = 1 @@R e e u e e uee

S (1) S (2)

@@ @@h @@


BxB B

BB

Bh

y

6 x



@ @@

@ @@ @@

x

Figure 3:

17





(k) HHH dE r (n) ; vl (n) = 1  received point  HY d r (n) ; v(k) (n) = 0 0  -E x

h

x

@h @

u 2 S0(k 1)

e 2 S1(k 1)

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Z2 0.6065 0.1353 0.0111 0.0003 4e-05 1e-08 2e-11 1e-14 3e-18 2e-22 5e-27

Z4 0.9334 0.5650 0.1889 0.0353 0.0038 0.0002 1e-05 2e-07 3e-09 3e-11 1e-13

Z8 0.9994 0.9639 0.8010 0.5417 0.3005 0.1392 0.0547 0.0184 0.0053 0.0013 0.0003 0.0001

Z16 1.0000 1.0000 0.9976 0.9766 0.9140 0.8077 0.6736 0.5317 0.3986 0.2846 0.1941 0.1267 0.0793 0.0477 0.0275 0.0153 0.0082 0.0042 0.0021 0.0010 0.0005 0.0002 0.0001

Table 1:

18

Z32 1.0000 1.0000 1.0000 1.0000 0.9999 0.9988 0.9938 0.9804 0.9553 0.9176 0.8682 0.8091 0.7430 0.6725 0.6004 0.5288 0.4597 0.3946 0.3347 0.2805 0.2324 0.1904 0.1542 0.1236 0.0981 0.0770 0.0598 0.0460 0.0350 0.0264

Z64 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9991 0.9975 0.9944 0.9892 0.9813 0.9704 0.9564 0.9390 0.9185 0.8951 0.8689 0.8402 0.8095 0.7770 0.7430 0.7080 0.6723 0.6361 0.5998


0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Z4=Z2 1.03 1.11 1.22 1.34 1.46 1.57 1.67 1.76 1.82 1.88 1.92 1.94 1.96 1.98 1.99 1.99 1.99 1.996 1.997 1.997

Z8=Z2 Z16=Z2 Z32=Z2 Z64 =Z2 1.03 1.03 1.03 1.03 1.13 1.13 1.13 1.13 1.28 1.30 1.31 1.31 1.45 1.49 1.50 1.50 1.61 1.65 1.66 1.66 1.73 1.77 1.78 1.78 1.83 1.86 1.87 1.87 1.89 1.92 1.92 1.92 1.94 1.95 1.96 1.96 1.96 1.98 1.98 1.98 1.98 1.99 1.99 1.99 1.99 1.99 1.99 1.99 1.99 1.996 1.996 1.996 1.996 1.997 1.997 1.997 1.997 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 1.998 Table 2:

19

−2

10

−4

10 −3

10

−6

10 −4

10

−8

10 −5

10

−10

10 −6

10

−12

10 −7

10

−14

10 −8

10

3

4




5

6

3

4




5

Figure 4: bits per level and channel use

1

0.8

0.6

0.4

0.2

0 −1 10

0

1

10

10

 Figure 5:

20

2

10

6

bits per channel use

6

5

4

3

2

1

0 −1 10

0

1

10

10

 Figure 6:

21

2

10