General random coding bounds: AWGN channels to MIMO fading channels

Abstract

Random coding bounds are obtained for multiple-input multiple-output (MIMO) fading channels. To derive the result in a compact and easy-to-evaluate form, a series of combinatorial codeword enumeration problems are solved for input-constrained MIMO fading channels. The bounds obtained in this paper are shown useful as performance prediction measures for MIMO systems which employ turbo-like block codes as the outer code to derive the space-time inner code. The error exponents for MIMO channels are also derived from the bounds, and then compared with the classical Gallager error exponents as well as the channel capacities. The random coding bounds associated with the maximum likelihood receiver exhibit good match with the extensive system simulation results obtained with a turbo-iterative receiver.

Appendices

Appendix A. Pairwise error probability

In Appendix A and B, we will derive the main theorem. In this section, we start with discussion of pairwise error probability which will be used as the first building block to prove the Theorem.

The PEP from codeword c to codeword c′ is defined as the probability that the receiver, when making an ML decision between a pair of codewords, erroneously decides in preference of c′ when c was actually transmitted. Suppose X and X′ are the two space-time words one-to-one correspondingly mapped from c and c′, respectively.

In case of a Rayleigh MIMO channel, the PEP averaged over the fading channel distribution for the system described by Eq. 1 can be formulated as (see [24] for details)

$$ P({\mathbf{c}} \to {\mathbf{c}}\prime {\text{)}} \leqslant \prod\limits_{t = 1}^T {{{\left( {1 + \frac{1}{{4{N_o}}}{{\left| {{{\mathbf{x}}_t} - {\mathbf{x}}_t^\prime } \right|}^2}} \right)}^{ - N}}} $$

(A20)

where, T is the block length of the space-time word and |·| denotes the L₂ norm of the complex vector. Also, x _t and x_t′ are the tth columns of space-time words X and X′, respectively. Further note that x _t , x _t ′∈{s₀,···,s_J-1}.

One of the key steps involved in the derivation of the union bound is to determine the partition of a codebook into a number of smaller sets so that the PEP in Eq. A20 is to render an identical result within a set. The determination of this set and the calculation of its cardinality are thus the critical steps for deriving our result. To proceed, we introduce two metrics given in the form of definition for easy reference.

Recall that a codeword of length L is segmented into T binary strings of length MK _b and there are $J = {2^{{\text{M}}{{\text{K}}_b}}}$ distinct strings. For each string, we keep track of the number of occurrence of the string within a codeword. Under a particular constellation map, each string is mapped to one of the J channel-symbol vectors in the vector constellation. From a straightforward tracking of the one-to-one correspondence in this manner, we will be able to resolve all the codeword enumeration problems.

Definition 1

Binary string weight profile. There are J binary strings which can be sequenced from 0 to J-1. Likewise, there are J channel symbol vectors which can be indexed from 0 to J-1. Let b _j denote the jth binary string of length MK _b that is modulated onto the jth symbol vector s _j . We will use δ _j to denote the number of occurrences of the jth string b _j in a codeword. They can be stored in an array, referred to here as the binary string weight profile (BSWP). We use $\hat {\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta }}({\mathbf{c}}) = \left( {{\delta_0}({\mathbf{c}}), \cdots, {\delta_{J - 1}}({\mathbf{c}})} \right)$ to denote a BSWP. Each BSWP must satisfy the following four constraints from its definition:

• 1. δ _j (c)∈{0,···,T}

• 2.
  $$ \sum\limits_{j = 0}^{J - 1} {{\delta_j}\left( {\mathbf{c}} \right)} = T $$

• 3. δ _j (c)∈{0,1,2,...,T} and

• 4.
  $$ \sum\limits_{j = 0}^{J - 1} {{\delta_j}\left( {\mathbf{c}} \right) = T} $$

When there is no ambiguity we will use $\hat {\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta }} = \hat {\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta }}\left( {\mathbf{c}} \right)$.

Under a specific constellation map, each binary string b _j maps to a corresponding channel symbol vector s _j ; likewise each codeword, a sequence of T binary strings, maps to a space-time word, a sequence of channel symbols s _j . Making use of definition 1, we note that there are δ _j (c) number of channel symbol vectors s _j in X. Likewise, we can find the numbers of other channel-symbol vectors in X.

Now the following definition will help us identify those pairwise error events which lead to an identical PEP under the input/output relationship given in Eq. 1. For this, we momentarily assume that a codeword c is selected and have it held fixed. Relating to definition 1, its BSWP$\hat {\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta }}$ is fixed as well.

Definition 2

Pairwise distance profile. In a pair of codewords c and c′, there are a total of T binary string pairs. Likewise, in the corresponding pair of space-time words X and X′, there are a total of T channel symbol pairs (x _t , x _t ′) for t∈{1,···,T}. We use δ _j,k to denote the number of time indices that a particular channel symbol pair (x _t  = s _j , x _t ′ = s _k ) appears in a pair of ST words. Note that all different combinations of j,k∈{0,···,J-1} are possible. The collection of all δ _j,k can be stored into an array of size T ². The array is referred to as the pairwise distance profile (PDP) between X and X′ (or between c and c′). Let’s use $\underline \delta : = \left( {{{\underline \delta }_0},\,{{\underline \delta }_1},\,...,\,{{\underline \delta }_{J - 1}}} \right)$ to denote the collection and each δ _j is further defined as δ _j : = (δ _j,0, δ _j,1, …, δ _j,J-1) for each j∈{0,···,J-1}.

Using the definition of pairwise distance profile δ, one can succinctly represent a set of erroneous words c′ each of which leads to an identical PEP. Namely, for a fixed c, the group of words c′ satisfy the following two constraints

$$ \begin{gathered} 0 \leqslant {\delta_{j,k}} \leqslant {\delta_j}\left( {\mathbf{c}} \right),\,{\text{and}} \hfill \\ \sum\limits_{k = 0}^{J - 1} {{\delta_{j,k}} = {\delta_j}\left( {\mathbf{c}} \right)} \hfill \\ \end{gathered} $$

(A21)

for each j∈{0,···, J-1}. We note then that the sum of all elements in the profile should be equal to T because it is the total number of symbol vector pairs in any pair of space-time words.

In summary, we have the following defined:

• 1. δ _j,k is the count of occurrences of a channel-symbol pair in a pair of sequence for j, k = 0, 1, …, J-1.

• 2. A pairwise distance profile δ(c) = (δ ₀, δ ₁, …, δ _J-1) is an array of collection of all δ _j,k . This distance profile is between the two words in a pair. One is the test word c (or its corresponding space-time word x(c)). The other is an erroneous word c′ (or its corresponding x′(c′)). A single profile is sufficient to represent a group of erroneous binary words (c′) (or x′(c′)) whose PEPs (Eq. A20) are the same.

• 3. The collection of all words with a same PDP δ(c) and its size: for any fixed word c, we may want to count all the words which share the same PDP δ(c). Any erroneous error pattern belonging to this collection will generate the same PEP. The cardinality of this group of words is of interest and it can be written as the following,

$$ \prod\limits_{j = 0}^{J - 1} {\left( {\begin{array}{*{20}{c}} {{\delta_j}\left( {\mathbf{c}} \right)} \\ {{\delta_{j,0}},...,{\delta_{j,J - 1}}} \\ \end{array} } \right)}, $$

(A22)

where $\left( {\begin{array}{*{20}{c}} {{\delta _j}\left( c \right)} \\ {{\delta _{j,0}}, \ldots {\delta _{j,J - 1}}} \\ \end{array} } \right)$ is the multinomial coefficient.

The following remarks show the usefulness of PDP.

Remark 2

The same PDP leads to the same PEP; but not vice versa.

• (a) The set of candidate codewords c′ can be partitioned with respect to distinct PDP such that each partition contains codewords with an identical PDP.

• (b) Each codewords c′ in a partition has the same PEP.

A codeword pair can be described by a PDP δ. A PDP is for each and every possible channel-symbol pair, and for each it specifies the total number of times a particular channel-symbol pair appears in a pair of ST words. Once a PDP is given, the PEP in Eq. A20 can be rewritten as,

$$ \begin{array}{*{20}{c}} {P\left( {{\mathbf{c}} \to {\mathbf{c}}\prime } \right)}\begin{gathered} \leqslant \prod\limits_{j = 0}^{J - 1} {\prod\limits_{k = 0}^{J - 1} {{{\left[ {{{\left( {1 + \frac{1}{{4{N_o}}}{{\left| {{s_j} - {s_k}} \right|}^2}} \right)}^{ - N}}} \right]}^{ - {\delta _{j,k}}}}} } \hfill \\ = \prod\limits_{j = 0}^{J - 1} {\prod\limits_{k = 0}^{J - 1} {{\beta _{j,k}}^{{\delta _{j,k}}}} } \hfill \\ \end{gathered} \\ \end{array} $$

(A23)

by grouping the like terms under each power exponent δ _j,k . Here, we made use of the memory-less property of the ergodic channel (Eq. 1). Note that the critical information needed to write (Eq. A23) is stored in the PDP. Note that terms β _j,k are defined as

$$ {\beta_{j,k}}: = {\left( {1 + \frac{1}{{4{N_0}}}{{\left| {{s_j} - {s_k}} \right|}^2}} \right)^{ - N}} $$

(A24)

which are completely determined and held fixed once a vector constellation and power spectral density of the noise are given.

The rationale behind the definition of the PDP δ as a distance metric should be clear now: For a given SNR, the pairwise distance profile δ completely determines the upper bound formulation of the PEP. As will be noted in the subsequent sections, use of the two profiles greatly simplifies the union bound evaluation.

In the union bounds for binary transmission over AWGN channels, for example, the use of distance profiles based on Hamming weights and Hamming distances greatly simplifies the calculation of union bound. The calculation of union bounds becomes quite complex for a J-ary vector constellation. The two profiles play the roles similar to the Hamming weight and Hamming distance in AWGN channels, and simply the union bound evaluation for MIMO channels.

The following lemma is the multinomial expansion. It will prove useful to write it here; while we omit the proof.

Lemma

Consider a set of J utility variables {z _j } _{j=0,1,...,J-1} and the T-th power of the sum of the J utility variables${\left( {\sum\limits_{j = 0}^{J - 1} {{z_j}} } \right)^T}$. Then, for an array of integersv = (v₀, v₁, …, v_J-1), the T-th power of the sum can be expanded as,

$$ {\left( {\sum\limits_{j = 0}^{J - 1} {{z_j}} } \right)^T} = \sum\limits_{\underline v \in \Omega } {\left( {\begin{array}{*{20}{c}} T \\ {{v_0},{v_1}, \cdots {v_{J - 1}}} \\ \end{array} } \right)} \prod\limits_{j = 0}^{J - 1} {z_j^{{v_j}}} $$

(A25)

where, Ω is the collection of arrays v, i.e.,

$$ \Omega : = \left\{ {\underline v \left| {{v_j} \in \left\{ {0,1,...,T} \right\},\sum\limits_{j = 0}^{J - 1} {{v_j} = T} } \right.} \right\}, $$

and

$$ \left( {\begin{array}{*{20}{c}} {\sum {v_i}} \\ {{v_0},{v_1}, \cdots {v_{n - 1}}} \\ \end{array} } \right): = \frac{{\left( {\sum {v_i}} \right)!}}{{\prod {v_i}!}} $$

is the multinomial coefficient.

Remark 3

Setting all utility variables to be equal to 1, we note that

$$ \sum\limits_{\underline v \in \Omega } {\left( {\begin{array}{*{20}{c}} T \\ {{v_0},{v_1}, \cdots {v_{J - 1}}} \\ \end{array} } \right) = {J^T}} $$

(A26)

Appendix B. Proof of theorem

We now discuss the proof of Theorem. The sketch of proof goes as follows:

• 1. The random block code is not linear. Unlike linear codes, the all-zero codeword alone is not enough to be selected as the test codeword c. For a given codebook, one must take the average over all randomly selectable test codeword c. This is a difficult task.

• 2. The obstacle is circumvented by taking the ensemble average over all equally probable selection of random codebooks. The two profiles defined in Appendix A are useful to simplify the union bound.

Proof

Consider the ensemble of randomly selectable (L,K) block codes. First, let us consider a code C in the ensemble and the calculation of probability of maximum-likelihood decoding error. A union bound to this error probability is given as follows

$$ \begin{array}{*{20}{c}} {{P_e}\left( C \right) = {E_{c \in \mathcal{C}}}\left[ {{P_{e|c}}} \right]} \\ { \leqslant {E_{c \in C}}\left[ {\sum\limits_{{\mathbf{c}}\prime \in C,{\mathbf{c}}\prime \ne c} {\Pr \left( {{\mathbf{c}} \to {\mathbf{c}}\prime } \right)} } \right]} \\ { = \frac{1}{{{2^K}}}\sum\limits_{_{{\mathbf{c}}\prime \in C,{\mathbf{c}}\prime \ne {\mathbf{c}}}^{\quad \,{\mathbf{c}} \in C}} {\Pr \left( {{\mathbf{c}} \to {\mathbf{c}}\prime } \right)} ,} \\ \end{array} $$

(A27)

where, E _c [•] is the expectation over the choice of a test codeword out of 2^K equi-probably selectable codewords in the code C; P _e|c denotes the error probability conditioned on the transmission of a test codeword c; and the inequality is due to the usual union bound argument.

Then, the average probability of decoding error over the ensemble of codes can be formulated according to Eq. A27 as follows,

$$ \begin{array}{*{20}{c}} {\overline {{P_e}} = {E_{C \in \mathbb{C}}}\left[ {{P_e}\left( C \right)} \right]\frac{1}{{\left| \mathbb{C} \right|}}\left[ {\sum\limits_{C \in \mathbb{C}} {{P_e}} \left( C \right)} \right]} \\ { \leqslant \frac{1}{{{2^K}\left| \mathbb{C} \right|}}\sum\limits_{C \in \mathbb{C}} {\sum\limits_{\mathop {{C \in \mathbb{C}}}\limits_{c\prime \in C,c\prime \ne c} } {\Pr \left( {{\mathbf{c}} \to {\mathbf{c}}\prime } \right)} } } \\ \end{array} $$

(A28)

where we make use of the assumption that each code in the ensemble is selected with equal probability.

In Eq. A28, note that (1) the inner summation shall be conducted over all codeword pairs c and c′ (c ≠ c′) where both should be element codewords in the same codebook C; that (2) the outer summation is implied for each and every code in the ensemble $\mathbb{C}$.

It should be noted that the pilot codeword c should be selected out of 2^K valid codewords within each codebook (the inner summation); but looking at it from the perspective of considering all codebooks in the ensemble, each and every possible 2^L distinct binary string of length L should be considered as the test codeword at least once.

Making use of this observation and changing the order of the summations, Eq. A28 can be rewritten as,

$$ {\overline P _e} \leqslant \frac{1}{{{2^K}\left| \mathbb{C} \right|}}\sum\limits_{\text{c}} {\sum\limits_{\mathop {{c\prime :c\prime \ne c,}}\limits_{c,c\prime \in C,C \in \mathbb{C}} } {\Pr \left( {{\mathbf{c}} \to {\mathbf{c}}\prime } \right)} } $$

(A29)

where the outer sum is now over all 2^L distinct binary string c of length L. The inner sum is to count in all codeword c′ which are different from c, but must coexist with c in the same codebook. Of course, there are only a finite number of codebooks that possess both as its element codewords. Shortly later, this quantity will be obtained explicitly under the assumption of random coding argument (see Eq. A35).

Let us now consider the inner sum over all binary string c′ with respect to a binary string c that have a binary string weigh profile (δ ₀(c), δ ₁(c), …, δ _J-1(c)). Recall that the pairwise error probability from c to c′ in Eq. A23 is completely determined by their pairwise distance profile δ. The summation over c′ thus can be re-arranged with respect to the PDP. We collect a single representative string per each group of strings c′ which possess the same PDP, and call it c″. That is, a string c″ represents all binary strings c′ each of which has the same pairwise distance profile, δ(c), from the test word c.

Now Eq. A29 can be rewritten as

$$ {\overline P _e} \leqslant \frac{1}{{{2^K}\left| \mathbb{C} \right|}}\sum\limits_{\mathbf{c}} {\sum\limits_{\begin{array}{*{20}{c}} {{\mathbf{c}}\prime \prime :\underline {\:\delta } \in \Omega \left( {\mathbf{c}} \right)} \\ {\underline {\:\delta } \ne {{\underline {\:\delta } }^*}} \\ \end{array} } {{S_{\underline \delta }}\left( c \right)\Pr \left( {{\mathbf{c}} \to {\mathbf{c}}\prime \prime } \right)} } = \frac{1}{{{2^K}\left| \mathbb{C} \right|}}\sum\limits_{\mathbf{c}} {\sum\limits_{\begin{array}{*{20}{c}} {{\mathbf{c}}\prime \prime :\underline {\:\delta } \in \Omega \left( {\mathbf{c}} \right)} \\ {\underline {\:\delta } \ne {{\underline {\:\delta } }^*}} \\ \end{array} } {{S_{\underline \delta }}\left( c \right)\prod\limits_{j,k = 0}^{J - 1} {\beta _{j,k}^{{\delta _{j,k}}}} } } . $$

(A30)

A few explanations are in order per Eq. A30. First, Ω(c) denotes the set of all possible pairwise distance profile δ(c) : = (δ ₀, δ ₁, …, δ _J-1) anchored at the test word c. Making use of our definition in Eq. A21, we have,

$$ \Omega \left( {\mathbf{c}} \right): = \left\{ {\underline \delta \left| {{{\underline \delta }_j} \in {\Omega_j}\left( {\mathbf{c}} \right),{\text{ for }}j{\text{ }} = {\text{ }}0,1,2,{\text{ }}...,J - 1} \right.} \right\}, $$

(A31)

where,

$$ {\Omega_j}\left( {\mathbf{c}} \right): = \left\{ {{{\underline \delta }_j}\left| {{\delta_{j,k}} \in \left\{ {0,1,...,{\delta_j}\left( {\mathbf{c}} \right)} \right\},\sum\limits_{k = 0}^{J - 1} {{\delta_{j,k}} = {\delta_j}\left( {\mathbf{c}} \right)} } \right.} \right\}. $$

(A32)

Since the inner summation is taken over all the distinct strings c″, each representing a group of equivalent strings with the same PDP, the size of the group should be calculable and it is the multinomial coefficient $\prod\limits_{j = 0}^{J - 1} {\left( {\begin{array}{*{20}{c}} {{\delta_j}\left( {\mathbf{c}} \right)} \\ {{\delta_{j,0}},...,{\delta_{j,J - 1}}} \\ \end{array} } \right)}$. It is the number of ways to come up with the equivalent strings which possess the given PDP. We will incorporate this factor into the parameter S _δ (c); see Eq. A35.

Second, δ ^* denotes the unique PDP of a word anchored at itself, and thus δ  ≠  δ ^* is equivalent to c′ ≠  c. Notice that for δ ^* the entries are given as δ _j,k  = δ _j (c) for j = k and δ _jk  = 0 otherwise.

Just for a check, we take the sum of all coefficients and find:

$$ \sum\limits_{{\mathbf{c}} \in GF{{(2)}^L}} {\sum\limits_{{\mathbf{c}}\prime \prime :\;\underline \delta \in \Omega \left( {\mathbf{c}} \right),\underline \delta \ne {{\underline \delta }^*}} {\prod\limits_{j = 0}^{J - 1} {\left( {\begin{array}{*{20}{c}} {{\delta _j}({\mathbf{c}})} \\ {{\delta _{j,0}},...,{\delta _{j,J - 1}}} \\ \end{array} } \right)} } = } {2^L}\left( {{2^L} - 1} \right). $$

(A33)

Third, we use S _δ (c) to subsume the rest of the factors. It should point to the number of all erroneous codewords c′ which have the pairwise distance profile δ from c, counted for all valid codebooks. Note that a test word c exists only in a certain number of codebooks. Such occasions should be counted properly in the parameter S _δ (c). Thus, taking the summation of S _δ (c) over all PDP and all codeword pairs shall give a number equal to the product of 2^K(2^K-1) and the cardinality of the (L,K) code ensemble, i.e.,

$$ \sum\limits_{{\mathbf{c}} \in GF{{\left( 2 \right)}^L}} {\sum\limits_{{\mathbf{c}}\prime \prime :\;\underline \delta \in \Omega \left( {\mathbf{c}} \right),\underline \delta \ne {{\underline \delta }^*}} {{S_{\underline \delta }}\left( {\mathbf{c}} \right)} = } \left| \mathbb{C} \right|{2^K}\left( {{2^K} - 1} \right). $$

(A34)

The value of S _δ (c) can be calculated using the usual combinatorial methods:

$$ {S_{\underline \delta }}\left( {\mathbf{c}} \right) = \left[ {\frac{{{2^K}}}{{{2^L}}}\left| \mathbb{C} \right|} \right] \cdot \left[ {\frac{{{2^K} - 1}}{{{2^L} - 1}}} \right] \cdot \left[ {\prod\limits_{j = 0}^{J - 1} {\left( {\begin{array}{*{20}{c}} {{\delta_j}\left( {\mathbf{c}} \right)} \\ {{\delta_{j,0}},...,{\delta_{j,J - 1}}} \\ \end{array} } \right)} } \right], $$

(A35)

for δ ≠ δ*. The first term is the number of codes in the ensemble that include a word c as a codeword. Only a certain fraction of these codes also include c″ as its element codeword, which is the second term. There are 2^K–1 binary strings out of 2^L– 1 available to be selected as the erroneous codeword. Therefore, the number of codebooks which contain both c and c″ simultaneously is the product of the first two terms in Eq. A35. The third term is the tally of all possible ways of having the binary strings for an erroneous codeword which possess a PDP δ anchored at the test word c and thus satisfying all the constraints due in Eq. A31 and Eq. A32.

Substituting Eq. A35 into Eq. A30, we have

$$ {\overline P _e} \leqslant \frac{1}{{{2^L}}}\frac{{{2^K} - 1}}{{{2^L} - 1}}\sum\limits_{{\mathbf{c}} \in GF{{\left( 2 \right)}^L}} {\sum\limits_{_{\quad \underline \delta \ne {{\underline \delta }^*}}^{{\mathbf{c}}\prime \prime :\underline \delta \in \Omega \left( {\mathbf{c}} \right)}} {\prod\limits_{j = 0}^{J - 1} {\left[ {\left( {\begin{array}{*{20}{c}} {{\delta _j}\left( {\mathbf{c}} \right)} \\ {{\delta _{j,0}},...,{\delta _{j,J - 1}}} \\ \end{array} } \right)\prod\limits_{k = 0}^{J - 1} {\beta _{j,k}^{{\delta _{j,k}}}} } \right]} } } . $$

(A36)

Notice that, for δ  =  δ ^* (i.e., c′  =  c), the multinomial coefficient in Eq. A36 equals 1, and also β _j,k = 1 according to Eq. A24. Thus, Eq. A36 can be rewritten by considering δ  ≠  δ ^* separately,

$$ {\overline P_e} \leqslant \frac{1}{{{2^L}}}\frac{{{2^K} - 1}}{{{2^L} - 1}}\sum\limits_{\mathbf{c}} {\sum\limits_{\underline \delta \in \Omega \left( {\mathbf{c}} \right)} {\prod\limits_{j = 0}^{J - 1} {\left[ {\left( {\begin{array}{*{20}{c}} {{\delta_j}\left( {\mathbf{c}} \right)} \\ {{\delta_{j,0}},...,{\delta_{j,J - 1}}} \\ \end{array} } \right)\prod\limits_{k = 0}^{J - 1} {\beta_{j,k}^{{\delta_{j,k}}}} } \right]} } } - \frac{{{2^K} - 1}}{{{2^L} - 1}}. $$

(A37)

Recalling the definition in Eq. A31 that the J constraints for Ω(c) are not coupled with each other, the sum over δ = (δ ₀, δ ₁, …, δ _J-1) ∈ Ω(c) can be simplified as follows,

$$ \begin{array}{*{20}{c}} {\sum\limits_{\underline \delta \in \Omega (c)} {\prod\limits_{j = 0}^{J - 1} {\left[ {\left( {\begin{array}{*{20}{c}} {{\delta _j}({\mathbf{c}})} \\ {{\delta _{j,0}},...,{\delta _{j,J - 1}}} \\ \end{array} } \right)\prod\limits_{k = 0}^{J - 1} {{\beta _{j,k}}^{{\delta _{j,k}}}} } \right]} } } \\ { = \sum\limits_{{{\underline \delta }_0} \in {\Omega _0}\left( {\mathbf{c}} \right)} \cdots \sum\limits_{{{\underline \delta }_{J - 1}} \in {\Omega _{J - 1}}\left( {\mathbf{c}} \right)} {\prod\limits_{j = 0}^{J - 1} {\left[ {\left( {\begin{array}{*{20}{c}} {{\delta _j}\left( {\mathbf{c}} \right)} \\ {{\delta _{j,0}},...,{\delta _{j,J - 1}}} \\ \end{array} } \right)\prod\limits_{k = 0}^{J - 1} {\beta _{j,k}^{{\delta _{j,k}}}} } \right]} } } \\ { = \prod\limits_{j = 0}^{J - 1} {\sum\limits_{{{\underline \delta }_j} \in {\Omega _j}\left( {\mathbf{c}} \right)} {\left[ {\left( {\begin{array}{*{20}{c}} {{\delta _j}\left( {\mathbf{c}} \right)} \\ {{\delta _{j,0}},...,{\delta _{j,J - 1}}} \\ \end{array} } \right)\prod\limits_{k = 0}^{J - 1} {\beta _{j,k}^{{\delta _{j,k}}}} } \right]} } } \\ { = \prod\limits_{j = 0}^{J - 1} {{{\left( {\sum\limits_{k = 0}^{J - 1} {{\beta _{j,k}}} } \right)}^{{\delta _j}\left( {\mathbf{c}} \right)}}} ,} \\ \end{array} $$

(A38)

where, the last equality follows from the Lemma.

Substituting Eq. A38 into Eq. A36, we have,

$$ {\overline P_e} \leqslant \frac{1}{{{2^L}}} \cdot \frac{{{2^K} - 1}}{{{2^L} - 1}}\sum\limits_{{\mathbf{c}} \in {\text{GF}}{{(2)}^L}} {\prod\limits_{j = 0}^{J - 1} {{{\left( {\sum\limits_{k = 0}^{J - 1} {{\beta_{j,k}}} } \right)}^{{\delta_j}\left( {\mathbf{c}} \right)}}} } - \frac{{{2^K} - 1}}{{{2^L} - 1}}. $$

(A39)

Now, let us move on to the summation over c: it is over all 2^L distinct binary strings of length L, as mentioned before. Similar to the case for c′, this summation can be reorganized with respect to the binary string weigh profile $\underline {\hat \delta } \left( {\mathbf{c}} \right) = \left( {{\delta_0},\,{\delta_1},...,{\delta_{J - 1}}} \right)$ associated with each c. That is,

$$ {\overline P_e} \leqslant \frac{1}{{{2^L}}}\frac{{{2^K} - 1}}{{{2^L} - 1}}\sum\limits_{\underline {\hat \delta } \in \widehat{\Omega }} {{{\widehat{A}}_{\underline {\hat \delta } }}\prod\limits_{j = 0}^{J - 1} {{{\left( {\sum\limits_{k = 0}^{J - 1} {{\beta_{j,k}}} } \right)}^{{\delta_j}}}} } - \frac{{{2^K} - 1}}{{{2^L} - 1}}. $$

(A40)

where, ${\widehat{A}_{\underline {\hat \delta } }}$ is the number of binary strings of length L that have a metric $\underline {\hat \delta }$; i.e., each of these strings can be regarded as a concatenation of a number δ _j of binary sub-string b _j (j = 1, 2, …, J-1). Let $\widehat{\Omega }$ denote the set of all possible metric $\underline {\hat \delta }$ and according to definition 1, we have,

$$ \widehat{\Omega }: = \left\{ {\underline {\hat \delta } \left| {{\delta_j} \in \left\{ {0,1,...,T} \right\},\sum\limits_{j = 0}^{J - 1} {{\delta_j} = T} } \right.} \right\}. $$

(A41)

Similar to S _δ (c), we use the combinatorial analysis to calculate ${\widehat{A}_{\underline {\hat \delta } }},$

$$ {\hat A_{\underline {\hat \delta } }} = \left( {\begin{array}{*{20}{c}} T \\ {{\delta_0},...,{\delta_{J - 1}}} \\ \end{array} } \right), $$

(A42)

which is the number of ways to arrange a number δ _j of binary sub-strings b _j (for j = 0, 1, 2, … J-1). As expected, we can verify that ${\sum_{\underline {\hat \delta } \in \widehat{\Omega }}}{\hat A_{\underline {\hat \delta } }} = {J^T} = {2^{M{K_b}T}} = {2^L}.$

Substituting Eq. A42 into Eq. A40, we have

$$ \begin{array}{*{20}{c}} {{{\overline P }_e} \leqslant \frac{1}{{{2^L}}}\frac{{{2^K} - 1}}{{{2^L} - 1}}\sum\limits_{\underline {\hat \delta } \in \widehat{\Omega }} {\left( {\begin{array}{*{20}{c}} T \\ {{\delta _0},...,{\delta _{J - 1}}} \\ \end{array} } \right)\prod\limits_{j = 0}^{J - 1} {{{\left[ {\sum\limits_{k = 0}^{J - 1} {{\beta _{j,k}}} } \right]}^{{\delta _j}}}} } - \frac{{{2^K} - 1}}{{{2^L} - 1}}} \\ { = \frac{1}{{{2^L}}}\frac{{{2^K} - 1}}{{{2^L} - 1}}{{\left( {\sum\limits_{j,k = 0}^{J - 1} {{\beta _{j,k}}} } \right)}^T} - \frac{{{2^K} - 1}}{{{2^L} - 1}},} \\ \end{array} $$

(A43)

where the equality is obtained by applying the Lemma.

The bound can be further upper-bounded by

$$ {\overline P_e} \leqslant \frac{1}{{{2^L}}}\frac{{{2^K}}}{{{2^L}}}{\left( {\sum\limits_{j,k = 0}^{J - 1} {{\beta_{j,k}}} } \right)^T}. $$

(A44)

Using L = TMK _b and R _c  = K/L and rewriting Eq. A44 in an exponential form, we reach the result of the Theorem 2. □