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Trellis-Coded Quadrature Amplitude Modulation with 2N-Dimensional Constellations for Mobile Radio Channels Corneliu Eugen D. Sterian, Senior Member, IEEE, Frank Laue, and Matthias Pätzold, Senior Member, IEEE

Abstract —– Using a modified Wei method, originally designed for AWGN channels, we have constructed four- and six-dimensional trellis codes with rectangular signal constellations for frequency-nonselective mobile radio channels. Applying a novel way of partitioning the 2-D constituent constellations, both into subsets with enlarged minimum Euclidean distance and subrings including equal energy signal points, we have obtained partitions of the 2N-D signal sets into subsets with a Hamming distance between signal points which equals N. This is fundamental for constructing good trellis codes to transmit data over flat fading channels. Index Terms — Constellation shaping, fading channel models, frequency-nonselective mobile radio channels, multidimensional trellis coded modulation, quadrature amplitude modulation, set partitioning, shell mapping.

I. INTRODUCTION Mobile radio channels exhibit a time varying behavior in the received signal envelope, which is called fading. This is caused if the receiving antenna, used in mobile radio links, picks up multipath reflections. While there are other degradations like additive white Gaussian noise (AWGN), the fading is by far the main impairment encountered on this type of channel. Trellis-coded modulation (TCM) is a standard technique used to improve the performance of a digital transmission system. Originally, TCM has been introduced by Ungerboeck in a seminal paper [1] for AWGN channels. The scheme proposed by Ungerboeck uses an expanded one- or two-dimensional constellation with 2b+1 signals to transmit b information bits per signaling interval without increasing the bandwidth or the transmitted power. The constellation is partitioned into 2m+1 subsets with enlarged intrasubset minimum Euclidean distance. Of the b information bits that arrive in each signaling interval, m enter a rate-m/m+1 convolutional encoder, and the resulting m+1 coded bits specify which subset is to be used. The remaining b-m information bits specify which point from the selected subset is to be transmitted. In the receiver, a soft-decision maximum likelihood decoder attempts to recover the original information from the channel output. The first important application of TCM has been a two-dimensional eight-state nonlinear trellis code with 4-dB coding gain designed by Wei [2] which was adopted in the Recommendations V.32, V.32 bis and V.33 of ITU-T (formerly CCITT) for data transmission over voice-band telephone channels. Three four-dimensional trellis codes have been adopted in the Recommendation V.34 of ITU-T for 33.6 kbit/s transmission over the switched telephone network [3]. These codes have been designed using a method invented by Wei [4]. A 2N-dimensional signal point is the concatenation of N two-dimensional points. In Wei’s method, the 2-D constituent constellation has 2b inner points which is the size of the constellation used by the reference system, and 2b/N outer points which provide the redundancy necessary for the error control. Since the number 2b/N of the outer points must be a positive integer, N must be a power of two. This limitation has been removed in [5] which allows us to construct 6-D trellis codes. The beginning of TCM for mobile radio channels is related to the pioneering work of Divsalar and Simon [6], [7]. Many years, only TCM with constant-amplitude modulation schemes, like Phase Shift Keying (PSK) and Continuous Phase Modulation (CPM), have been considered [8], [9].

2

The reason for this is that, for efficiency, the High Power Amplifier (HPA) of the transmitter antenna was operated in a very nonlinear region. However, a number of important papers appeared recently in which Quadrature Amplitude Modulation (QAM) is used [10] - [13]. Since QAM is more bandwidth-efficient than PSK, for which also the data rate is limited to b = 3 bit/signaling interval, we consider it in this paper. In contradistinction to TCM for AWGN channels, where the primary objective is to maximize the Euclidean distance between symbol sequences, in designing TCM for fading channels, the main task is to maximize the smallest Hamming distance of the trellis code. Remember that the Hamming distance between two sequences of symbols is defined as the number of positions where the symbols are different. A secondary objective is to maximize the product distance, defined as the product of the non-zero squared Euclidean distances between the symbols in the same position of two sequences having the same beginning and the same end [21], [22]. The concept of time diversity plays a crucial role in the performance of coded modulation for fading channels. Independent fading in the different symbols is established by means of interleaving. Full interleaving can greatly reduce the required transmit power on fading channels. A block interleaver can be regarded as a buffer with d rows which represent the depth of interleaving, and s columns which represent the span of interleaving. In this paper, we do not address the problem of interleaver design. However, we suppose that the transmission chain includes an interleaver/deinterleaver. Our work has been motivated by the need of digital mobile communication systems having higher transmission rates by keeping the bandwidth as low as possible. One way to achieve this goal is to use trellis-coded QAM instead of constant-amplitude modulation schemes. When using QAM, it is a well established fact that larger Hamming distances and lower average energy of the signal constellations can be obtained going from 2-D to a higher dimension (e.g. 4-D or more). In this paper, we propose a novel way of partitioning the 2-D constituent QAM constellations, both in subsets with enlarged minimum Euclidean distance and subrings including equal energy points. We have thus obtained partitions of the 2N-D signal sets into subsets with a Hamming distance between points which equals N. According to Divsalar and Simon [7], this is fundamental for constructing good trellis codes to transmit data over flat fading channels. The paper is organized as follows. In Section II, we consider 4-D rectangular signal constellations which are also used for AWGN channels, but partition them in a novel way in order to maximize the Hamming distance dH between the points of the same subset. We then design TCM schemes to transmit b = 3 and 4 information bits per signaling interval using QAM. While for b = 3, there exist schemes for both PSK and QAM, the data rate b = 4 bits/signaling interval can only be obtained with QAM. Section III describes the transmission chain including the interleaver/deinterleaver and some considerations are given to the decoding strategy. In this section, we also present an efficient computer-based technique to simulate realistic mobile radio channel scenarios. The performance of the proposed trellis-coded QAM system is then investigated in Section IV. Finally, Section V concludes our paper.

II. 4-D TRELLIS-CODED MODULATION In this section, we will consider 4-D rectangular signal constellations to transmit b = 3 and 4 bits per signaling interval using QAM. The points of the 2-D constituent signal constellation belong to a rectangular grid and have odd integer coordinates. In other words, if Ζ is the set of integers, then the coordinates of the 2-D points belong to the set {2Ζ + 1}2. Then, following Wei [4] and with reference to Fig. 1, we partition this infinite set into four 2-D subsets A, B, C, and D according to A = {4Ζ + 1}2 B = {4Ζ + 3}2

(1a) (1b)

3

C = {4Ζ + 1}{4Ζ + 3} D = {Ζ + 3}{4Ζ + 1}

(1c) (1d)

If we denote the minimum squared Euclidean distance (MSED) of the set {2Z + 1}2 as δ02, then the MSED of every subset A, B, C, and D is 4δ02.

Im B2

3

C1

2 A1 -3 C2

D0 -2

1

-1 B0

A0 1

-1

D2 2

C0

3

Re

B1

-2 D1

-3

A2

Fig. 1. 12-point 2-D constellation partitioned into four subsets (A, B, C, D) and into three rings (R0, R1, R2).

Sixteen 4-D types may then be defined, each corresponding to a concatenation of two 2-D subsets, and denoted as (A,A), (A,B),…, and (D,D). The 16 4-D types are grouped into four subsets with Hamming distance between types dH = 2 in two different ways. Partition I: The first partition is performed as follows SS0 = (A,A)∪(B,B)∪(C,C)∪(D,D) SS1 = (A,C)∪(B,D)∪(C,B)∪(D,A) SS2 = (A,B)∪(B,A)∪(C,D)∪(D,C) SS3 = (A,D)∪(B,C)∪(C,A)∪(D,B).

(2a) (2b) (2c) (2d)

Note with reference to Fig. 1 that these four subsets are invariant under 900, 1800, and 2700 rotation. Partition II: The even-indexed subsets are the same as before, but the odd-indexed ones are replaced by SS1’= (A,C)∪(B,D)∪(C,A)∪(D,B) SS3’ =(A,D)∪(B,C)∪ (C,B)∪ (D,A).

(2b’) (2d’)

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Note that these two subsets are invariant under 1800, but they aren´t under 900 and 2700 rotation. Let us consider two generic 4-D points of coordinates ( x n , y n , x n +1 , y n +1 ) and ( x n’ , y n’ , x n’ +1 , y n’ +1 ). Define the product distance (PD) between these two points as

PD = [( xn − xn’ )2 + ( yn − yn’ )2 ][( xn +1 − xn’ +1 )2 + ( yn +1 − yn’ +1 )2 ].

(3)

It can be verified by looking at Fig. 1 that for the Partition I the intrasubset minimum product distance (MPD) is 16 for four subsets S 0 , S 1 , S 2 , and S3 , the other twelve having a MPD of 64. For the Partition II, only six subsets S 4 , S 6 , S 8 , S 1 0 , S 1 2 , and S14 have a MPD of 64, the other ten subsets having a MPD of 16. Taking into consideration the less good rotational and product distance properties of the partition II, we will not use it. Two coded bits, let us say Z0p and Z1p , where p = n and n + 1, are used to select one out of the four 2-D subsets as shown in Table I.

TABLE I CORRESPONDENCE BETWEEN Z1pZ0p AND THE FOUR 2-D SUBSETS 2-D subset A B C D

Z1p Z0p 0 1 0 1

0 0 1 1

A. Partition of the 4-D Signal Constellation for Transmitting 3 bits per Signaling Interval Let us define the norm or energy of a 2N-D point of coordinates (x1, y1,…, xN, yN) as the squared distance to the origin, i.e.

E=

N

∑ (x

2 i

+ y 2i ).

(4)

i=1

Let us furthermore define a length N frame as the concatenation of N two-dimensional points in the signal constellation. Use now a technique called shell mapping which is applied in the V.34 voiceband high-speed modem [3]. For N an integer power of two, partition the (1 + 1/N)2b point twodimensional constituent constellation of the 2N-D signal set into N inner rings and one outer ring of equal size 2b/N. In our case, where we have N = 2, the first inner ring R0 contains 2b-1 2-D points of least norm, and the second inner ring R1 contains the next 2b-1 points such that, taken together, ring R0 and ring R1 form the 2b point signal constellation which would be used by the uncoded reference system to transmit b bits per signaling interval. The outer ring R2 contains 2b-1 redundant points chosen in ascending order of the norm. The 22b+1 point 4-D constellation is the union of the Cartesian products of the rings (Ri, Rj) such that at most one ring Ri or Rj is the outer ring. Define a 4-D inner point as a 4-D point which belongs to an inner subset (R0,R0), (R0,R1), (R1,R0) or (R1,R1) and a 4-D outer point as one which belongs to the subsets (R0,R2), (R1,R2), (R2,R0) and (R2,R1). The probability of sending a 4-D inner point equals the probability of sending a 4-D outer point and is ½. However, for each constituent 2-D constellation, an inner ring is used three times (generally 2N

5

– 1 times) as often as the outer ring R2. This produces the well-known effect of shaping the constellation and results in a small gain of the signal to noise ratio [3]. Note that, when applying this known method, we have further partitioned each of the (N + 1) rings into 2b-3 four-point subrings such that any 2-D point may be obtained by rotating any other 2D point within a given subring by 900, 1800, or 2700. Two coded bits, let us say Z2p and Z3p, where p = n and n + 1, are used to select one out of three rings as shown in Table II.

TABLE II CORRESPONDENCE BETWEEN Z3pZ2p AND THE THREE RINGS Ring R0 R1 R2

Z3pZ2p 0 0 0 1 1 0

Using these rings, partition the 4-D signal constellation into four subsets called shells as follows SH0 = (R0,R0)∪(R1,R1) SH1 = (R0,R1)∪(R1,R0) SH2 = (R0,R2)∪(R2,R0) SH3 = (R1,R2)∪(R2,R1).

(5)

The partition has been done in such a way that the Hamming distance between the ring types inside a given shell is equal to dH = 2. Combining the four subsets SSi defined by (2.a) - (2.d) with the four shells SHj , we obtain sixteen 4-D subsets Sk (k = 0, 1,..., 15) as shown in Table III. Note that the index k of the subset Sk is given by the relation k = 4j + i.

(6)

The decimal values of the indices i and j are given by (see Fig.2) i = 2 I1n + Y 0 n

(7)

j = 2 I 3n + I 2 n

(8)

and

respectively. The sixteen subsets Sk are numbered from 0 to 15. Group the sixteen subsets into two families F0 and F1 as follows F0 =

U Sk

(9a)

US .

(9b)

k =even

F1 =

k = odd

k

6

TABLE III PARTITION OF THE 128-POINT 4-D SIGNAL SET INTO 16 SUBSETS Sk (k = 0, 1,...,15) 4-D 4-D I1n+1 Family Subset F0 S0 0 1 S2 0 1 S4 0 1 S6 0 1 S8 0 1 S10 0 1 S12 0 1 S14 0 1 F1 S1 0 1 S3 0 1 S5 0 1 S7 0 1 S9 0 1 S11 0 1 S13 0 1 S15 0 1

I3n

I2n

I1n

Y0n

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

0

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

0

1

1

1

0

1

1

1

1

1

4-D Type (A0,A0), (B0,B0), (C0,C0), (D0,D0) (A1,A1), (B1,B1), (C1,C1), (D1,D1) (A0,B0), (B0,A0), (C0,D0), (D0,C0) (A1,B1), (B1, A1), (C1,D1), (D1,C1) (A0,A1), (B0,B1), (C0,C1), (D0,D1) (A1,A0), (B1,B0), (C1,C0), (D1,D0) (A0,B1), (B0,A1), (C0,D1), (D0,C1) (A1,B0), (B1, A0), (C1,D0), (D1,C0) (A0,A2), (B0,B2), (C0,C2), (D0,D2) (A2,A0), (B2,B0), (C2,C0), (D2,D0) (A0,B2), (B0,A2), (C0,D2), (D0,C2) (A2,B0), (B2, A0), (C2,D0), (D2,C0) (A1,A2), (B1,B2), (C1,C2), (D1,D2) (A2,A1), (B2,B1), (C2,C1), (D2,D1) (A1,B2), (B1,A2), (C1,D2), (D1,C2) (A2,B1), (B2, A1), (C2,D1), (D2,C1) (A0,C0), (B0,D0), (C0,B0), (D0,A0) (A1,C1), (B1,D1), (C1,B1), (D1,A1) (A0,D0), (B0,C0), (C0,A0), (D0,B0) (A1,D1), (B1,C1), (C1,A1), (D1,B1) (A0,C1), (B0,D1), (C0,B1), (D0,A1) (A1,C0), (B1,D0), (C1,B0), (D1,A0) (A0,D1), (B0,C1), (C0,A1), (D0,B1) (A1,D0), (B1,C0), (C1,A0), (D1,B0) (A0,C2), (B0,D2), (C0,B2), (D0,A2) (A2,C0), (B2,D0), (C2,B0), (D2,A0) (A0,D2), (B0,C2), (C0,A2), (D0,B2) (A2,D0), (B2,C0), (C2,A0), (D2,B0) (A1,C2), (B1,D2), (C1,B2), (D1,A2) (A2,C1), (B2,D1), (C2,B1), (D2,A1) (A1,D2), (B1,C2), (C1,A2), (D1,B2) (A2,D1), (B2,C1), (C2,A1), (D2,B1)

As it may be seen from Table III, every one of the sixteen subsets contains eight 4-D points which are different of any other one in both the first and the second 2-D component of it. Use now this partition for TCM with b = 3.

B. Design of 4-D Trellis Codes for Transmitting 3 bits per Signaling Interval The 12-point 2-D constituent constellation of the 4-D signal set is shown in Fig. 1. For each point, the capital letter indicates the subset A, B, C, or D and the number refers to the ring 0, 1 or 2. To send b = 3 information bits per signaling interval using a rate 3/4 trellis code with a 4-D constellation partitioned into 24 = 16 subsets, three of the six information bits (I1n, I2n, I3n) arriving

7

in each block of two signaling intervals enter the trellis encoder, and the resulting 4 coded bits specify which 4-D subset is to be used. The remaining 3 information bits (I1n+1, I2n+1, I3n+1) specify which point from the selected 4-D subset is to be transmitted. Denote the six information bits gathered at the input of the trellis-coded modulator in two successive signaling intervals n and n+1 as I1n , I2n , I3n , I1n+1, I2n+1, and I3n+1. As shown in Fig. 2, the first three bits enter a rate ¾ systematic convolutional encoder which outputs the coded bit Y0n . In order to make the scheme transparent to all the phase ambiguities of the constellation, we choose the bit pair I3n+1I2n+1 and differentially encode it in such a way that if we translate a sequence of this bit pair by the same number of positions, one, two, or three, in a circular sequence, 00, 01, 10, 11, then the sequence of 2-D points produced by the 4-D constellation mapping procedure will be rotated by 90 0, 1800, and 2700 clockwise, respectively. Therefore, a differential encoder of the form

I 3’n I 2’n = ( I 3’n − 2 I 2’n − 2 + I 3n I 2 n ) mod 100 base 2

(10)

shown in Fig. 2 and a corresponding differential decoder of the form

I 3n I 2 n = ( I 3’n I 2’n − I 3’n − 2 I 2’n − 2 ) mod 100 base 2

(11)

at the output of the trellis decoder will remove all the phase ambiguities of the constellation [2], [4], [5]. A bit converter (see Fig. 2) converts the four bits Y0n, I1n, I 2’n +1 and I 3’n +1 into two pairs of selection bits Z0nZ1n and Z0n+1Z1n+1, which are used to select the pair of 2-D subsets corresponding to the 4-D type. With the correspondence between the bit pair Z0pZ1p and the 2-D subsets A, B, C, and D as shown in Table I, the operation of the bit converter for the Partition I is as shown in Table IV.

I3’n+1

I3n+1

DIFFERENTIAL I2n+1 I1n+1 I3n I2n I1n

ENCODER

I2’n+1 Z3n+1 Z2n+1 Z3n Z2n

4-D BLOCK ENCODER BIT CONVOLUTIONAL

Y0n

CONVERTER

Z1n+1 Z0n+1 Z1n Z0n

ENCODER Fig. 2. General structure for rotationally invariant trellis-coded modulation with 4-D QAM to send b = 3 bits per signaling interval.

8

(a) CURRENT STATE

NEXT STATE

4-D SUBSET

W1n

W2 n W3n

0 2 4 6 8 10 12 14

0

0

0

0

0

0

1 3 5 7 9 11 13 15

0

0

1

0

0

1

2 0 6 4 10 8 14 12

0

1

0

0

1

0

3 1 7 5 11 9 15 13

0

1

1

0

1

1

4 6 0 2 12 14 8 10

1

0

0

1

0

0

5 7 1 3 13 15 9 11

1

0

1

1

0

1

6 4 2 0 14 12 10 8

1

1

0

1

1

0

7 5 3 1 15 13 11 9

1

1

1

1

1

1

W1n+2 W2n+2 W3n+2

(a)

4-D SUBSET

CURRENT STATE

NEXT STATE

W2 n W3n

W1n+2 W2n+2 W3n+2 W4n+2

W1n

W4n

0

2

4

6

8 10 12 14

0

0

0

0

0

0

0

0

1

3

5

7

9 11 13 15

0

0

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0

0

1

2

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4 10

8 14 12

0

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9 15 13

0

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8 10

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(b)

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Fig. 3. (a) Trellis diagram of 8-state code of Figs. 2 and 4(a); (b) trellis diagram of 16-state code of Figs. 2 and 4(b).

I3n

Y3n

I2n

Y2n Y1n

I1n W1n+2

2T

W1n

W2n+2

2T

W2n

W3n+2

2T

Y0n

W3n

(a) I3n

Y3n

I2n

Y2n

I1n

Y1n

W1n+2

2T

W1n

W2n+2

2T

W2n

W3n+2

2T

W3n

W4n+2

2T

Y0n

W4n

(b) Fig. 4. Trellis encoders of Fig. 2: (a) 8-state; (b) 16-state.

TABLE IV PARTITION OF 4-D 128 POINT RECTANGULAR CONSTELLATION INTO 16 TYPES

I 3’n +1

I 2’n +1

I1n

Y0n

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0

0

0

1

1

0

1

1

4-D Types A,A C,C B,B D,D A,C C,B B,D D,A A,B C,D B,A D,C A,D C,A B,C D,B

Z1n

Z0n

Z1n+1

Z0n+1

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1

0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0

10

A 4-D block encoder then takes three input information bits I2n , I3n , and I1n+1 and generates two pairs of selection bits, Z2nZ3n and Z2n+1Z3n+1 , in accordance with Table V. Each of the bit pairs can assume any of the values given in Table II, but they cannot both assume the value 10 which corresponds to the outer ring R2. TABLE V 4-D BLOCK ENCODER for b = 3 I3n 0 0 0 0 1 1 1 1

I2n 0 0 1 1 0 0 1 1

I1n+1 0 1 0 1 0 1 0 1

4-D Shell R0,R0 R1,R1 R0,R1 R1,R0 R0,R2 R2,R0 R1,R2 R2,R1

Z3n 0 0 0 0 0 1 0 1

Z2n 0 1 0 1 0 0 1 0

Z3n+1 0 0 0 0 1 0 1 0

Z2n+1 0 1 1 0 0 0 0 1

We will design now two convolutional encoders which fit in the general diagram shown in Fig. 2. Note that every one of the sixteen 4-D subsets contains eight 4-D points which are different from each other in both the first and the second 2-D component. Therefore, the intraset Hamming distance of the 16 subsets is maximized to dH = 2. However, the interset Hamming distance only equals 1. Recall that our aim is not to maximize the Euclidean distance between allowed sequences, but the Hamming distance. 8-state convolutional encoder Denote the current and the next states of the trellis encoder as W1pW2pW3p , p = n and n+2. Let us number the states from 0 to 7 by using the relation Wp = 4W1p + 2W2p + W3p.

(12)

The trellis diagram is as shown in Fig. 3(a). It is fully connected and we may express the mapping Wn → Wn + 2 in algebraic form as {0, 1, 2, 3, 4, 5, 6, 7} → {0, 1, 2, 3, 4, 5, 6, 7}. The association of 4-D subsets with the state transitions satisfies the following requirement. Rule 1: The 4-D subsets associated with the transitions originating from a state are different from each other and belong to the same 4-D family F0 or F1; the 4-D subsets associated with the transitions leading to a state are different from each other but may belong to both families F0 and F1. The logic diagram of the 8-state convolutional encoder is given in Fig. 4(a). The shortest error event path is given by parallel paths between successive states of the convolutional encoder. Indeed, although drawn as a single one, there are eight parallel transitions between two successive states in the trellis diagram shown in Fig. 3. The Hamming distance between these parallel transitions is dH = 2. However, the two transitions error event paths also have dH = 2 and the same multiplicity as single transition error event paths. This is since the trellis diagram in Fig. 3 has full connectivity. To improve the performance, a convolutional encoder with a larger number of states must be used in the general structure shown in Fig. 2.

11

16-state convolutional encoder Denote the current and the next states of the trellis encoder as W1pW2pW3pW4p , p = n and n+2. Number the states from 0 to 15 using the relation Wp = 8W1p+4W2p+2W3p+W4p

(13)

The trellis diagram is as shown in Fig. 3(b) and may be also expressed in algebraic form as {0,2,4,6,8,10,12,14} → {0,1,2,3,4,5,6,7} {1,3,5,7,9,11,13,15} → {8,9,10,11,12,13,14,15}.

There are eight parallel transitions between any current state i and a successive state j. Therefore, in this case also, the shortest error event path has a length equal to one transition. However, the multiplicity of two transitions error event paths has been halved. The association of 4-D subsets with the state transitions should satisfy the following requirement. Rule 2: The 4-D subsets associated with the transitions originating from a state are different from each other and belong to the same 4-D family F0 or F1 and likewise for the 4-D subsets associated with the transitions leading to a state. The logic diagram of the 16-state convolutional encoder is given in Fig. 4(b).

C. Partition of the 4-D Signal Constellation and Design of Trellis Codes for Transmitting b = 4 bits per Signaling Interval The transmission rate of b = 4 bits per signaling interval is clearly not possible using TCM with a PSK constellation (32-PSK has unacceptably small MSED). The 24-point 2-D constituent constellation of the 4-D signal set is shown in Fig. 5.

12 Im D5

5

A4

4 C3

B2

3

C1

B3

A0

D2

2 D4 -5 B5

A1 -4

-3 C2

D0 -2

1

-1 B0

1

-1

2

3

C0

B1

A2

D3

A5 4

5

Re

C4

-2 A3

D1

-3 -4

B4

-5

C5

Fig. 5. 24-point 2-D constellation partitioned into four subsets (A, B, C, D) and into six rings (R0, R1, R2, R3, R4, R5). As for b = 3, the 2-D constellation is partitioned into four subsets A, B, C, and D, but the number of subrings is six, numbered from 0 to 5 in ascending order of the norm. Three bits, let us say Z2p, Z3p and Z4p, where p = n and n+1, are used to select one out of these six subrings as shown in Table VI.

TABLE VI CORRESPONDENCE BETWEEN Z4pZ3pZ2p AND THE SIX SUBRINGS Subring R0 R1 R2 R3 R4 R5

Z4pZ3pZ2p 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1

Using these subrings, partition the 4-D signal constellation into eight subsets called shells as follows: SH0 = (R0,R0)∪(R1,R1)∪(R2,R2)∪(R3,R3) SH1 = (R0,R1)∪(R1,R0)∪(R2,R3)∪(R3,R2) SH2 = (R0,R2)∪(R2,R0)∪(R1,R3)∪(R3,R1) SH3 = (R0,R3)∪(R3,R0)∪(R1,R2)∪(R2,R1)

13

SH4 = (R0,R4)∪(R4,R0)∪(R1,R5)∪(R5,R1) SH5 = (R1,R4)∪(R4,R1)∪(R0,R5)∪(R5,R0) SH6 = (R2,R4)∪(R4,R2)∪(R3,R5)∪(R5,R3) SH7 = (R3,R4)∪(R4,R3)∪(R2,R5)∪(R5,R2)

(14)

The partition has been done in such a way that the Hamming distance between the subring types inside a given shells is equal to dH = 2. We combine the four subsets SSi defined as before with the eight shells SHj defined by (14) to obtain 32 4-D subsets such that the relation (6) still holds, but in this case j goes from 0 to 7. For instance, the subset S0 = SS0×SH0 contains sixteen 4-D points as follows: (An,An), (Bn,Bn), (Cn,Cn), and (Dn,Dn), where n = 0,...,3. The 32 subsets are numbered from 0 to 31 and grouped into two families F0 and F1 as in (9). Denote the eight bits gathered at the input of the trellis-coded modulator in two successive signaling intervals n and n+1 as I1n , I2n , I3n , I4n , I1n+1, I2n+1 , I3n+1 and I4n+1. As shown in Fig. 6, the first four bits enter a rate 4/5 systematic convolutional encoder which outputs the coded bit Y0n. In order to make the scheme rotationally invariant, differentially encode the bit pair I3n+1I2n+1 as for the case b = 3. A bit converter converts the four bits Y0n , I1n , I 2’n +1 and I 3’n +1 into two pairs of selection bits as for the case b = 3. A 4-D block encoder then takes five input information bits I2n , I3n , I4n , I1n+1 , and I4n+1 and generates two groups of selection bits, Z2nZ3nZ4n and Z2n+1Z3n+1Z4n+1 in accordance with Table VII. Each of the bit groups can assume any of the values given in Table VI, but they cannot both assume the values 100 and 101 which correspond to the outer subrings R4 and R5, respectively.

TABLE VII 4-D BLOCK ENCODER FOR b = 4 I4n

I3n

I2n

0

0

0

0

0

1

0

1

0

0

1

1

1

0

0

I4n+1 I1n+1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

4-D Shell Type R0,R0 R1,R1 R2,R2 R3,R3 R0,R1 R1,R0 R2,R3 R3,R2 R0,R2 R1,R3 R2,R0 R3,R1 R0,R3 R1,R2 R2,R1 R3,R0 R0,R4 R1,R5 R4,R0 R5,R1

Z4n

Z3n

Z2n

Z4n+1

Z3n+1

Z2n+1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0

0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1

14

1

0

1

1

1

0

1

1

1

0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1

R0,R5 R1,R4 R4,R1 R5,R0 R2,R4 R3,R5 R4,R2 R5,R3 R2,R5 R3,R4 R4,R3 R5,R2

0 0 1 1 0 0 1 1 0 0 1 1

0 0 0 0 1 1 0 0 1 1 0 0

0 1 0 1 0 1 0 1 0 1 0 1

1 1 0 0 1 1 0 0 1 1 0 0

0 0 0 0 0 0 1 1 0 0 1 1

1 0 1 0 0 1 0 1 1 0 1 0

The general diagram of the TCM scheme for b = 4 is shown in Fig. 6. The convolutional encoder should have at least 16 states.

I3n+1 I2n+1 I4n+1 I1n+1 I4n I3n I2n

I3’n+1

DIFFERENTIAL ENCODER

I2’n+1 Z4n+1 Z3n+1 Z2n+1 Z4n Z3n Z2n

4-D BLOCK ENCODER

I1n

BIT CONVOLUTIONAL

Y0n

CONVERTER

Z1n+1 Z0n+1 Z1n Z0n

ENCODER

Fig. 6. General structure for rotationally invariant trellis-coded modulation with 4-D QAM to send b = 4 bits per signaling interval.

16-state convolutional encoder Denote the current and the next states of the trellis encoder as in the case of the 16-state convolutional encoder in Fig. 4(b). However, the trellis has full connectivity, i.e., from any of the 16 originating states, 16 groups of transitions lead to any of the 16 next states. The association of 4D subsets with the state transitions satisfies the Rule 1 as given for the case b = 3. The logical diagram of the convolutional encoder which is part of the general structure in Fig. 6 is shown in Fig. 7(a). The shortest error event path is given by parallel transitions between successive states of the convolutional encoder. The Hamming distance between the 16 parallel transitions is dH = 2. Note that the two transitions error event paths also have dH = 2 and the same multiplicity as single transitions error event paths.

15

32-state convolutional encoder Denote the current and the next states of the trellis encoder as W1pW2pW3pW4pW5p , p = n and n+2. Number the states from 0 to 31 using the decimal representation Wp = 16W1p + 8W2p + 4W3p + 2W4p + W5p

(15)

The trellis diagram may be expressed in algebraic form as {0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30} → {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} and

{1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31} → {16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31}.

There are 16 parallel transitions between any current state i and a successive state j. Therefore, in this case also, the shortest event path has a length equal to one transition. However, the multiplicity of two transitions error event path is only half of that for the 16-state convolutional encoder. The association of 4-D subsets with the state transitions satisfies the Rule 2 as given for the case b = 3. The logical diagram of the convolutional encoder is shown in Fig. 7(b). I4n I3n I2n

Y4n Y3n Y2n

I1n

Y1n W1n+2

2T

W2n+2

W1n

2T

W2n

W3n+2

2T

W4n+2

W3n

2T

W4n

Y0n

(a) I4n

Y4n

I3n I2n I1n

Y3n Y2n Y1n

W1n+2

2T

W1n

W2n+2

2T

W2n

W3n+2

2T

W3n

W4n+2

2T

W4n

W5n+2

2T

W5n

Y0n

(b) Fig. 7. Trellis encoders of Fig. 6: (a) 16-state; (b) 32-state.

The authors have also designed 4-D trellis codes for transmitting b = 5 bits per signaling interval and 6-D trellis codes based on signal constellations partitioned such that the Hamming distance between the points within a subset equals 3. These codes have not been included in this paper by considerations of typographical space. The interested readers are kindly invited to contact the authors.

16

III. TRANSMISSION SYSTEM AND CHANNEL MODEL In this section, we describe the transmission system and the channel model by making use of the equivalent complex baseband notation.

A. The Transmission System The transmission system we consider is presented in Fig. 8. A data source generates a random information bit stream that enters the foregoing described trellis encoder. The encoded 2-D output symbols are then interleaved by a block interleaver which can be regarded as a rectangular buffer with d rows and s columns representing the interleaving depth and span, respectively. The M-QAM modulator maps the block interleaved 2-D symbols to the signal points of the signal space diagram for the M-QAM constellation shown in Fig. 1. The transmitted signal is impaired first by a complex multiplicative stochastic process describing the fading behavior of the frequency-nonselective mobile radio channel and second by a complex additive white Gaussian noise (AWGN) process. In the receiver, the received signal is demodulated and deinterleaved before feeding into the trellis decoder. The trellis decoder is based on the maximum likelihood sequence decoding principle by using the classical Viterbi algorithm. It is well-known that the performance of the trellis decoder can be considerably improved if channel state information (CSI) is available. To obtain CSI from the received signal a channel estimator is required in the receiver.

Data Source

Trellis Encoder

Symbol Interleaver

M-QAM Modulator

Fading Process

AWGN

Data Sink

Trellis Decoder

Symbol Deinterleaver

CSI

M-QAM Demodulator

Channel Estimator

Fig. 8. Equivalent complex baseband model of the trellis-coded M-QAM transmission system.

B. The Channel Model An often used statistical model for modeling various types of terrestrial mobile radio channels and especially for land mobile satellite channels is the well-known Suzuki process [14]. Such a

17

process is defined as a product process of a Rayleigh process with uncorrelated underlying in-phase and quadrature components and a lognormal process. Recently, two different modified versions of the classical Suzuki process have been introduced in [15] and [16] which are called extended Suzuki processes of Type I and Type II, respectively. Moreover, it has been shown [17] that both types of extended Suzuki processes can be combined to a joint statistical channel model called generalized Suzuki process. The main advantage of extended and in particular generalized Suzuki processes is that their statistical properties are more flexible than those of the original Suzuki process. Thus, the former processes allow in general a much better fitting of the statistics of the channel model to real-world measurements and that not only with respect to different kinds of measured probability density functions of the received envelope but also with respect to the corresponding higher order statistical properties like the level-crossing rate and the average duration of fades. In this paper, we use for the channel model the extended Suzuki process of Type I which is briefly reviewed in the following. For a detailed description of that process, we are referring the interested reader to [15]. The extended Suzuki process (of Type I), denoted henceforth by η(t), is defined as product process of a Rice process ξ ( t ) with given cross-correlation properties between the underlying in-phase and quadrature components and a lognormal process ζ ( t ) , i.e.,

η( t ) = ξ ( t ) ⋅ ζ ( t ) .

(16)

The Rice process ξ ( t ) is obtained from a zero-mean complex Gaussian noise process

µ ( t ) = µ 1 ( t ) + jµ 2 ( t )

(17)

representing the scattered (diffuse) component and a complex line-of-sight (LOS) component

{(

m( t ) = ρ ⋅ exp j 2πfρ t + θ ρ as follows

)}

ξ ( t ) = µ ( t ) + m( t ) .

(18)

(19)

Thereby, the parameters ρ , fρ , and θ ρ appearing in (18) are the amplitude, Doppler frequency,

and phase of the LOS component, respectively. The Rice process ξ ( t ) is used to model the shortterm fading effects, whereas the lognormal process ζ ( t ) models the long-term fading variations due to shadowing. The lognormal process ζ ( t ) can be derived from a non-linear transformation of a further real Gaussian noise process ν3(t) having zero mean and unit variance according to

ζ ( t ) = exp{σ 3ν 3 ( t ) + m3 }

(20)

where σ3 and m3 are parameters introduced to control the statistics of ζ ( t ) . The second order statistical properties of the extended Suzuki process η(t) are strongly influenced by the Doppler power spectral density (PSD) Sµµ(f) of the complex Gaussian noise process µ(t) introduced by (17). Typical for the extended Suzuki model is that the complex Gaussian noise process µ(t) has cross-correlated in-phase and quadrature components. A crosscorrelation between the generating components can easily be achieved by using an asymmetrical Doppler PSD Sµµ(f), e.g., the left-sided restricted Jakes PSD which is defined by [15]

18

 2σ 20 ,   πfmax 1 − ( f fmax )2 Sµµ ( f ) =    0,

− κ o fmax ≤ f ≤ fmax (20)

else

where fmax denotes the maximum Doppler frequency, the parameter κo is within the interval [0,1], and 2σ 20 defines the maximum mean power (variance) of (17) obtained for κ 0 = 1 . Note that for the special case where κ 0 = 1 , the relation (20) results in the classical Jakes PSD [20] which has a symmetrical shape, and, consequently, the in-phase and quadrature components of the complex Gaussian noise process µ(t) are in this case uncorrelated. On the other hand, if κ 0 = 0 , then the (20) results in an asymmetrical right-sided Jakes PSD and thus the in-phase and quadrature components of µ(t) are strongly correlated. It is also important to note that the fading rate of the channel model can easily be reduced (without changing the maximum Doppler frequency fmax ) by reducing the quantity κo. This is important because the fading rate of real-world mobile radio channels is often much lower than the theoretical expected fading rate. The above parameters σ 20 , κo, fmax, ρ, σ3, m3, and fρ are the primary model parameters of the extended Suzuki process. These parameters have been optimized successfully in [15] in such a way that not only the cumulative distribution function of η(t) but also the level-crossing rate and average duration of fades are very close to measured data of a real-word land mobile satellite channel in different (light and heavy) shadowing environments. The optimized primary parameters of the channel model are listed in Table VIII.

TABLE VIII OPTIMIZED PARAMETERS OF THE EXTENDED SUZUKI CHANNEL MODEL [15] Shadowing

σ 20

κo

ρ

σ3

m3

fρ/fmax

Heavy

0.0409

4.4E-11

0.118

0.1175

0.4906

0.6366

Light

0.2022

5.9E-08

0.9856

0.0101

0.0875

0.7326

The above described extended Suzuki process η(t) is an analytical (mathematical) model that cannot be implemented exactly on a computer. In order to enable the simulation of such processes, an efficient simulation model was also derived in [15] (see Fig. 9) by applying the concept of deterministic channel modeling (e.g. [18], [19]). The parameters ρ, σ3, m3, and fρ appearing in Fig. 9 are obtained directly from Table VIII, whereas the remaining parameters of the simulation model ( fi , n , ci , n , θ i , n ) have to be determined according to the procedure described in [15].

19

Rice process with cross-correlated components c 1,1 cos(2 π f 1,1 t + θ 1,1) cos(2 π f 1,N t + θ 1,N ) 1

cos(2 π f 2,N t + θ 2,N )

sin(2 π f 1,N t + θ 1,N )

c 2,1 c 2,N

~ µ ρ(t)

2

c 1,1 c 1,N

1

1

sin(2 π f 2,1 t + θ 2,1) sin(2 π f 2,N t + θ 2,N ) 2

~ µ 1(t)

2

sin(2 π f 1,1 t + θ 1,1)

1

1

1

cos(2 π f 2,1 t + θ 2,1)

2

m (t) = ρ cos(2 π fρ t + θρ ) 1 c 1,N

- c 2,1

.

~ ξ (t)

~ µ 2(t)

- c 2,N2

~ =~ . ~ η(t) ξ (t) ζ (t)

m (t) = ρ sin(2 π fρ t + θρ ) 2

2

Lognormal process cos(2 π f 3,1 t + θ 3,1) cos(2 π f 3,2 t + θ 3,2)

c 3,1 c 3,2

c 3,N cos(2 π f 3,N t + θ 3,N ) 3

3

~ ν3(t)

~ (t) µ 3

exp( .)

~ ζ (t)

3

σ3

m3

Fig. 9. Structure of an efficient simulation model for extended Suzuki processes of Type I.

IV. PERFORMANCE The discussion of the performance of the proposed 2N-D dimensional trellis-coded M-QAM system is restricted for short to the case N = 2 and M = 12. A 4-D trellis code for transmitting b=3 bits per signaling interval was designed according to the procedure described in Section II-B. For the convolutional encoder (see Fig. 2) the 16-states trellis encoder shown in Fig. 4(b) has been selected. For the depth d and span s describing the interleaver (deinterleaver) the moderate values d=16 and s=16 have been chosen. Hence, due to the finite interleaver (deinterleaver) size, the digital channel between the interleaver input and the deinterleaver output is nonideally interleaved (correlated fading). Furthermore, we have assumed that ideal CSI is available within the decoding unit. The trellis decoder operates on optimum (unquantized) soft-decisions made by the 12-QAM demodulator; and finally, the decoding depth L of the Viterbi algorithm is finite (L=16). The simulation of the complete trellis-coded 12-QAM transmission system shown in Fig. 8 has been performed by choosing a symbol rate to sampling rate ratio of fS f A = 1 8 and symbol rate to maximum Doppler frequency ratio of fS fmax = 0.02 . The resulting bit error rate (BER) of the trellis-coded 12-QAM transmission system by using the extended Suzuki channel model with two realistic scenarios (light and heavy shadowing) are presented in Fig. 10. For reasons of comparison, we have also shown in this figure the simulation results for the BER by using an AWGN channel and a Rayleigh channel in conjunction with the classical Jakes PSD. It should be mentioned that the extended Suzuki process includes the Rayleigh process as a special case. The results in Fig. 10 show us that, for a large range of the Eb / N0 ratio,

20

the standard Rayleigh channel model does not represent the worst case condition, but the extended Suzuki process on the heavy shadowing condition does it. On the other side, the BER determined for the light shadowing situation is for a given Eb / N0 ratio always below the corresponding results obtained for the Rayleigh channel, as it was intuitively expected. A profound insight into the performance of the proposed scheme is obtained by comparing the BER of our b=3 4-D TCM system with the BER of an appropriate reference system. As reference system we used an uncoded 8-PSK system which has the same information bit rate as the proposed trellis-coded 12-QAM transmission system with b=3 bits per signaling interval. The resulting BER of the reference system is also depicted in Fig. 10. The presented results show us that the coding gain ranges from 2.0 dB (AWGN) up to 5.4 dB (heavy shadowing) at a BER of 10-3. Note that the proposed TCM scheme has especially been designed for fading channels, what explains the fact that the achieved coding gain is higher for the heavy shadowing condition than for the AWGN channel.

10

0

Rayleigh (without CSI) 10

BER

10

10

10

10

-1

12-QAM (TCM, ideal CSI) 8-PSK (uncoded)

-2

Rayleigh extended Suz. (heavy shadowing)

-3

AWGN extended Suz. (light shadowing)

-4

-5

0

10

20 E¯ b /N 0 (dB)

30

40

Fig. 10. BER of the trellis-coded 12-QAM transmission system with b=3 bits per signaling interval.

V. CONCLUSIONS A novel way of designing TCM codes for radio mobile fading channels using rectangular signal constellations has been demonstrated. In order to obtain large minimum intrasubset Hamming distance, the signal constellation is partitioned both into subsets with enlarged minimum Euclidean

21

distance and into shells. It was inspired by the methods applied in the very performant V.34 modem used to transmit data over the voice-band telephone channel, which of course is not affected by fading. However, our aim in so doing was different. For 4-D, this resulted in rather simple TCM schemes which have the advantage of maximizing the minimum intrasubset Hamming distance without neglecting the Euclidean distance. The obvious disadvantage of these schemes is that the trellis encoders have rather a large number of states. When moving to higher dimensions (the next step is 6-D), the benefits diminish quickly and the complexity grows unacceptably high.

ACKNOWLEDGMENT The first author would like to thank the German Service for Academic Exchanges DAAD (Deutscher Akademischer Austauschdienst) for the scholarship which allowed him to spend a fruitful month in the Department of Digital Communications Systems of the Technical University Hamburg-Harburg, working in the research group of Prof. U. Killat. This international collaboration has been kindly encouraged by Mr. Sorin Pantis, Minister, and Mr. Dan Chirondojan, Secretary of State, both with the Ministry of Communications of Romania.

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G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, no. 1, pp. 55-67, Jan. 1982. [2] L.-F. Wei, “Rotationally invariant convolutional channel coding with expanded signal space – Part I: 180 degrees and Part II: Nonlinear codes,” IEEE J. Select. Areas Commun., vol. SAC-2, pp. 659-686, Sept. 1984. [3] ITU-T, “V.34 – A modem operating at data signalling rates of up to 33600 bit/s for use on the general switched telephone network and on leased point-to-point 2-wire telephone-type circuits,” Sept. 1994. [4] L.-F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory, vol. IT-33, no. 4, pp. 483-501, July 1987. [5] C.E.D. Sterian, “Wei-type trellis-coded modulation with 2N-dimensional rectangular constellation for N not a power of two,” IEEE Trans. Inform. Theory, vol. IT-43, no. 2, pp. 750-758, March 1997. [6] D. Divsalar and M.K. Simon, “Trellis coded modulation for 4800-9600 bits/s transmission over a fading mobile satellite channel,” IEEE J. Select. Areas Commun., vol. SAC-5, no. 2, pp. 162-175, Feb. 1987. [7] D. Divsalar, and M.K. Simon, “Multiple trellis coded modulation (MTCM),” IEEE Trans. Commun., vol. 36, pp. 410-419, Apr. 1988. [8] C-E. W. Sundberg and N. Seshadri, “Coded modulations for fading channels: an overview,” European Trans. Telecommun. (ETT), vol. 4, pp. 309-324, May-June 1993. [9] C. Schlegel, “Trellis coded modulation on time-selective fading channels,” IEEE Trans. Commun., vol. 42, pp. 1617-1627, Feb./March/April 1994. [10] D. Subasinghe-Dias and K. Feher, “A coded 16 QAM scheme for fast fading mobile radio channels,” IEEE Trans. Commun., vol. 43, pp. 1906-1916, May 1995. [11] X. Giraud and J.C. Belfiore, “Constellations matched to the Rayleigh fading channels,” IEEE Trans. Inform. Theory, vol. IT-42, no. 1, pp. 106-115, Jan. 1996.

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[12]

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[23]

J. Boutos, E. Viterbo, C. Rastello, and J.C. Belfiore, “Good lattice constellations for both Rayleigh and Gaussian channels,” IEEE Trans. Inform. Theory, vol. IT-42, no. 2, pp. 502518, Mar. 1996. X. Giraud and E. Boutillon, “Algebraic tools to build modulation schemes for fading channels,” IEEE Trans. Inform. Theory, vol. IT-43, no. 3, pp. 938-952, May 1997. H. Suzuki, “A statistical model for urban radio propagation,“ IEEE Trans. Commun., vol. 25, no. 7, pp. 673 - 680, July 1977. M. Pätzold, U. Killat, and F. Laue, “An extended Suzuki model for land mobile satellite channels and its statistical properties,“ IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 254269, Feb. 1998. M. Pätzold, U. Killat, Y. Li, and F. Laue, “Modeling, analysis, and simulation of nonfrequency-selective mobile radio channels with asymmetrical Doppler power spectral density shapes,“ IEEE Trans. Veh. Technol., vol. 46, no. 2, pp. 494-507, May 1997. M. Pätzold and F. Laue, “Generalized Rice processes and generalized Suzuki processes for modeling frequency-nonselective mobile radio channels,“ submitted for publication in IEEE Trans. Veh. Technol. M. Pätzold, U. Killat, and F. Laue, “A deterministic digital simulation model for Suzuki processes with application to a shadowed Rayleigh land mobile radio channel,” IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 318-331, May 1996. M. Pätzold, U. Killat, F. Laue, and Y. Li, “On the statistical properties of deterministic simulation model for mobile fading channels,“ IEEE Trans. Veh. Technol., vol. 47, no. 2, 617-630, May 1998. W. C. Jakes, Ed., Microwave Mobile Communications. New Jersey: IEEE Press, 1993. D. Divsalar and M.K. Simon, “The design of trellis coded MPSK for fading channels: performance criteria,” IEEE Trans. Commun., vol. 36, pp. 1004-1012, Sept. 1988. D. Divsalar, and M.K. Simon, “The design of trellis coded MPSK for fading channels: set partitioning for optimum code design,” IEEE Trans. Commun., vol. 36, pp. 1004-1012, Sept. 1988. L.-F. Wei, “Coded M-DPSK with built-in time diversity for fading channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 1820-1839, November 1993.

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