Block Coded Modulation Tareq Elhabbash, Yousef Yazji, Mahmoud Amassi

Introduction Encoding means adding redundant bits to the message, this result reduced overall data rate. For Example: Un-coded QPSK sent two bits per symbol. If we encoding this information by encoder with code rate 0.5, it result one bit per symbol.

Introduction Suggested solution: Increase the size of constellation to 16PSK. Problem in this solution: While the constellation size increased the Euclidean distance between the constellation points decreased which increased error performance.

Block Coded Modulation Definition : combining Block Coding and Channel Signal Sets to construct bandwidth efficient codes. The most powerful method for constructing BCM is the multilevel coding technique and multistage decoding of these codes. 4

Distance Concept 5 X(s), Y(s) X(s’’), Y(s’’)

Distance Between Two Codeword 6

7

Set Partitioning of 8-PSK

Multilevel Block Modulation Codes

Properties of code Code Length = n Dimension k = k 0 +k 1 +k 2 (information bit) Spectral efficiency MSE =

3-level 8-PSK BCM Code

Conclusion Minimum Square Euclidean Distance (MSE) = 4 Dimension k = 16 Spectral Efficiency = (16/8) = 2 bit\symbol Coding Gain = 3db over un-coded QPSK

Multistage Decoding Of Multilevel BCM The decoding process is explained by using a 3-level 8-PSK BCM code C= f [ C 0 * C 1 * C 2 ]. Assume an AWGN channel. Let r =( r 0,r 1, ……, r n-1 ) be the received sequence at the output of the demodulator, where for 0 ≤ i ≤ n R i =(x i, y i )  R 2 13

Multistage Decoding Of Multilevel BCM In Multistage soft decision decoding of multilevel BCM code component code are decoded stage by stage as described in the following figure.

Multistage Decoding Of Multilevel BCM

FIRST STAGE DECODING 16

FIRST STAGE DECODING 17

SECOND STAGE DECODING 18

SECOND STAGE DECODING 19

THIRD STAGE DECODING The decoded information at the first and second v’ 0 and v’ 1 is made available to the third decoding stage. For every codeword v 2  C 1 we compute the distance ` 20

THIRD STAGE DECODING 21

Example

Advanced Topic CONCATENATED CODED MODULATION

Concatenated Coding Definition: It’s a powerful technique for constructing long powerful code from short component codes. This technique using: –Outer code: Non-Binary code. –Inner code: Binary code. Generally the outer code is longer than the inner code. 24

Single – Level Concatenated coding: A Single Concatenated code is formed from two codes: –Inner code C 1 (n 1,k 1 ) binary code –Outer code C 2 (n 2,k 2 ) non-binary code with symbols from GF(2 k1 ). The symbols of C 2 are represented by their corresponding bytes of k 1 -tuple.

Single – Level Concatenated coding:

Note that the input of the inner code is the output of the outer code. So, the output from the outer code must be suitable for the inner code. How &Why ?

Encoding of Concatenated Coding:

Second: Inner Encoder: –The result of the first step consider as the input of the second step. –Each n 2 symbol with length of k 1 bits\symbol enter the inner encoder. –The result of this step is n 2 (symbols) with length of n 1 bits\symbol.

Decoding of Concatenated Coding:

Decoding consists from two steps: First: Inner Decoder: –The inner code is usually short and is decoded with soft-decision decoding. Second: Outer Decoder: –The outer code is usually long and is decoded with hard-decision decoding.

Conclusion The Resultant Code is an (n 1 n 2,k 1 k 2 ) binary linear code. The minimum distance of the resultant code d min =d min1 *d min2 where: –Minimum distance C 1 is d min1 –Minimum distance C 2 is d min2

Conclusion Note: Its possible to construct a single concatenated code from: –Single outer code –Multiple inner code

Example: Consider the concatenation code consists from: –RS code (15,11) with symbols from GF(2 4 ) as an outer code (d min =5) –Hamming code (7,4) as an inner code (d min =3) What is the resultant concatenated code? The Resultant concatenated code is (105,44) with d min = 15

Interleaved Concatenated coding system Let the outer code C 2 be an (n 2,k 2 ) linear block code with symbols from GF(2 m ). Let the inner code C 1 be an (n 1,k 1 ) binary linear code with k 1 =λ. m where λ is a positive integer.

Interleaved Concatenated coding system A message of k 2 m-bit bytes is first encoded into an n 2 -byte codeword in C 2. The codeword is then temporarily stored in a buffer as a row in an array. After λ outer codewords have been formed, the buffer stores a λ*n 2 array.

Interleaved Concatenated coding system Each column of the array consisted of λ.m bits add is encoded into n 1 -bit codeword in C 1. Each encoded column is transmitted serially.

Multilevel Concatenated Codes In Multilevel concatenated coding system, multiple inner and outer codes are used. Its provides more powerful coding technique, and allows the use of multistage decoding to reduce decoding complexity.

Multilevel Concatenated Codes An m-level concatenated code is formed from a set of m inner codes and a set of m outer codes. The m inner codes are coset codes formed from a binary (n,k) linear block code A 1 and a sequence of m linear subcodes of A 1, denoted by A 2,A 3,…,A m+1 = {0} such that

The concatenated code is the direct-sum of m component concatenated codes: The Min Distance of C: The dimension of C:

Example :

Example Cont.

Concatenated Coded Modulation The combination of two coded techniques is called the concatenated coded modulation CCM. In CCM the inner code in BCM

Example: BCM codes: 1.Repetition (8,1,4) 2.Even Parity Check (8,7,1) 3.Universal code (8,8,1) BCM codes: K = = 16 dmin = 4*1*1 = 4

Example. Cont.

Advanced Topic Multilevel Coded modulation For Unequal Error Protection 51

Introduction Data may consist information of several parts that have different degrees of significance hence require different levels of protection against noise. Codes that are designed to provide different levels of data protection are Known as Unequal Error Protection ( UEP ) First study By ( Mansink & Wolf ) 52

Cont. Let m = (m 0,m 1,……m l-1 ) be a message of k bits, where for 0 ≤ i ≤ l, m i is the i th part of the message and consist of k i information bits. Suppose –m 0 is the most significant part of the message m. –m l-1` is the least significant part of the message m. Hence m 0 require more protection against error than the other parts 53

Properties For 0 ≤ i ≤ l let Ci be an ( n, k i,d i ) binary LBC of length n, dimension k i, and minimum Hamming distance d i, Consider the l-level BCM code f [ C 0 *C 1 *…….*C l-1 ] over a signal space S. Suppose a message m = (m 0,m 1,……m l-1 ) is to be encoded into a signal sequence in f [ C 0 *C 1 *…….*C l-1 ]. First : the i th part m i is encoded into a codeword v i in C i Then : interleaved sequence v 0 * v 1 *…..v l-1 is mapped into a signal sequence f [ v 0 * v 1 * ……. * v l-1 ] in the l-level BCM code f [ C 0 *C 1 *…….*C l-1 ] this sequence is the codeword for the message m 54

UEP

Owing to the large increase in effective error coefficient in the first several decoding stages

UEP Let A (0) d0 denote the number of nearest neighbours of a codeword in C 0 with multistage decoding, the error Coefficient at the first decoding stage becomes 2 d 0. So for Large d 0,, 2 d 0. A (0) d0 becomes very large, For small-medium SNR the error performance determined by the error coefficient, This leads to destroy the UEP capability, m 0 of message is no longer well protected, So, ( poor performance in the first results in poor overall performance in terms of unequal error protection )

UEP Example : Let we have 3-level 8-PSK BCM code with  (64,18,22),(64,57,4),(64,63, 2) MSE distance at three level  (64,18,22)  22*0.586 =  (64,57,4)  4 *2 = 8  (64,63, 2)  2 *4 = 8 Spectral efficiency of the code is u[c] = bit/symbol. This code does provide two-level of error protection,  However for small-medium SNR the increase in error coefficient by factor 2 22 at the first stage  destroy the UEP capability.

Solution Several approaches have been proposed for designing good multilevel BCM codes to provide distinct unequal error protection for various levels. Using nonconventional signal set partition nonconventional signal constellation or both.

UEP In this section we present a nonconventional signal set partition of conventional signal constellation for designing multilevel BCM codes for unequal error protection ( UEP ) Target : to reduce the error coefficient and prevent or minimize error prorogation from the first stage of decoding

conventional 8-PSK signal space

UEP Examples : Consider the conventional 8-PSK signal space. First stage : the signal set is partitioned into two subsets of equal size denoted by Q(0) and Q(1) ; The four points of Q(0) lie in the left-half plane with first labeling (a 0 =0)…. And so on … Let ( x,y ) denote the coordinates of an 8-PSK signal point in the real plane R 2, Labelling prosperities :

Decoder

Example

Simulation

Thanks