Non-Linear Estimation of Convolutionally Encoded Sequences

TL;DR

This paper introduces a non-linear estimation method for convolutionally encoded sequences transmitted over AWGN channels, utilizing a novel probability measure to facilitate MAP decoding.

Contribution

It constructs a new probability measure under which observations become independent Gaussian vectors, enabling recursive calculation of conditional probabilities for MAP decoding.

Findings

Derived a recursive formula for conditional probabilities

Connected the measure change to Girsanov theorem

Enabled MAP decoding via the new probabilistic framework

Abstract

Suppose that a convolutionally encoded sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. In this case, pairs of the signal (i.e., code symbol) and observation are not jointly Gaussian and therefore, a linear estimation method cannot be applied. Hence, in this paper, non-linear estimation of convolutionally encoded sequences is discussed. First a probability measure (denoted Q), whose Radon-Nikodym derivative with respect to the underlying probability measure P is an exponential martingale, is constructed. It is shown that with respect to Q, the observations are mutually independent Gaussian random vectors with zero mean and identity covariance matrix. We see that the relationship between observation noises (with respect to P) and observations (with respect to Q) has a close relation to the Girsanov theorem in continuous case. Next, using the probability measure Q, we calculate the conditional probability of an event related to any encoded symbol conditioned by the observations. Moreover, we transform it into a recursive form. In the process of derivation, the metric associated with an encoded sequence comes out in a natural way. Finally, it is shown that maximum a posteriori probability (MAP) decoding of convolutional codes is realized using the derived conditional probability.

Equations

$$Z_k=X_k{C}_k+W_k,k\ge 1,$$

$$\{X_k^{(i)},W_l^{(j)};1\le i\le s,1\le j\le d,k,l\in N\}$$

$$P_{L^2(\mathcal{B})}X=E(X|\mathcal{B}).$$

$$P_{L^2({\mathcal{B}}_n)}{X}_n^{(i)}=P_{H_n}{X}_n^{(i)}.$$

$$Z_n-E(X_n|{\mathcal{B}}_{n-1})C_n,n\ge 1$$

$$z_j$$

$$=$$

$$Z_k=c{X}_k+W_k.$$

$$Z_k=c{X}_k+W_k,$$

$$W_k$$

$$X_k$$

$$Z_k$$

$$E(X_l^{(i)}|{\mathcal{B}}_n)=\sum _{x\in \{-1,+1\}}xP(X_l^{(i)}=x|{\mathcal{B}}_n).$$

$$E(u(X,Y)|\sigma (Y))=\int u(x,Y){\mu }_X(dx),$$

$${\mathcal{F}}_n\stackrel{△}{=}\sigma (X_k;k\in N)\vee \sigma (W_k;k\le n)({\mathcal{F}}_0=\sigma (X_k;k\in N))$$

$${\alpha }_n\stackrel{△}{=}\exp \left\{-c\sum _{k=1}^n(X_k,W_k)-\frac{1}{2}{c}^2{n}_{0}n\right\}({\alpha }_0=1).$$

$$E(\exp \{-c(X_n,W_n)-\frac{1}{2}{c}^2{n}_0\}|{\mathcal{F}}_{n-1})=1.$$

$$E(\exp \{-c(X_n,W_n)-\frac{1}{2}{c}^2{n}_0\}|{\mathcal{F}}_{n-1})$$

$$E({\alpha }_n|{\mathcal{F}}_{n-1})={\alpha }_{n-1}.$$

$${\alpha }_n=\exp \left\{-c\sum _{k=1}^n(\cdot ,W_k)-\frac{1}{2}{c}^2{n}_{0}n\right\}.$$

$$Q(A)\stackrel{△}{=}{\int }_A{\alpha }_{n_p}dP(A\in {\mathcal{F}}_{n_p}).$$

$$Q(A)={\int }_A{\alpha }_{n}dP(A\in {\mathcal{F}}_n)$$

$$E({\alpha }_{n_p}|{\mathcal{F}}_n)={\alpha }_n$$

$${\int }_{A}E({\alpha }_{n_p}|{\mathcal{F}}_n)dP={\int }_A{\alpha }_{n}dP$$

$${\int }_A{\alpha }_{n_p}dP={\int }_A{\alpha }_{n}dP(A\in {\mathcal{F}}_n)$$

$$Q(A)={\int }_A{\alpha }_{n_p}dP={\int }_A{\alpha }_{n}dP.$$

$$Q(A)={\int }_A{\alpha }_{n}dP(A\in {\mathcal{F}}_n).$$

$${\alpha }_n^{-1}$$

$${\beta }_n=\exp \left\{c\sum _{k=1}^n(X_k,Z_k)-\frac{1}{2}\sum _{k=1}^n(c{X}_k,c{X}_k)\right\}.$$

$$W_k=Z_k-c{X}_k,k\ge 1$$

$$P(A|\mathcal{B})=\frac{E_Q({\alpha }^{-1}{\chi }_A|\mathcal{B})}{E_Q({\alpha }^{-1}|\mathcal{B})}(A\in \mathcal{F}),$$

$$P(A\cap B)=E({\chi }_A{\chi }_B)$$

Full text

I Introduction

Consider an error control coding scheme. A decoder estimates a transmitted message based on the received noisy data. Hence, it is natural to think that error control coding has a close connection with estimation of stochastic processes. In this paper, we consider convolutional coding/decoding from the viewpoint of the filtering (or smoothing) theory for discrete-time stochastic processes (cf. [24]). To begin with, we state some basic notions needed in this paper. In the following, the underlying probability space $(\Omega,\mathcal{F},P)$ is implicitly assumed. Here, $\mathcal{F}$ is a $\sigma$-field of subsets of $\Omega$, and $P$ is a probability measure defined on $\mathcal{F}$. Let $X$ be a real-valued random variable defined on $\Omega$ (the set of real numbers is denoted by $R$). In this paper, random variables are expressed in capital letters in principle. Denote by $\sigma(X)$ the $\sigma$-field generated by $X$. Let $\{X_{n},~{}n\in N\}$ ($N$ denotes the set of natural numbers) be a family of real random variables. Then the smallest $\sigma$-field which contains $\cup_{n\in N}\sigma(X_{n})$ is denoted by $\sigma(X_{n},~{}n\in N)$ (or $\vee_{n\in N}\sigma(X_{n})$). Let $\mathcal{B}$ be a sub-$\sigma$-field of $\mathcal{F}$. The conditional expectation of $X$ with respect to $\mathcal{B}$ is denoted by $E(X|\mathcal{B})$ ($E(\cdot)$ is the expectation).

First assume the following:

$W_{k}~{}(k\in N)$ are mutually independent Gaussian random vectors of dimension $d$ with mean [math] and covariance matrix $I_{d}$ ($I_{d}$ is the identity matrix of size $d\times d$). ($W_{k}$ represents an observation noise.)

• 2)

$X_{k}~{}(k\in N)$ are Gaussian random vectors of dimension $s$, and they are independent of $W_{k}~{}(k\in N)$. ($X_{k}$ represents a signal.)

• 3)

Observations $Z_{k}$ are given by

[TABLE]

where $C_{k}$ is an $s\times d$ matrix.

Under these conditions, we see that

[TABLE]

forms a system of Gaussian random variables (i.e., every linear combination of variables contained in the set has a Gaussian distribution). Moreover, we see that $(X_{k},Z_{k})$ are “jointly Gaussian”. (Hence, a “linear” estimation method can be applied.) Denote by $\mathcal{B}_{n}=\sigma(Z_{1},\cdots,Z_{n})$ the $\sigma$-field which represents the observations obtained up to time $n$. Also, Let $H_{n}$ be the Gaussian space [11, 12, 20] generated by $Z_{1},\cdots,Z_{n}$. Then the best estimate (i.e., the least-squares estimate) for $X_{n}^{(i)}~{}(1\leq i\leq s)$ based on $\mathcal{B}_{n}$ (denoted by $\hat{X}_{n}^{(i)}$) is given by the conditional expectation $E(X_{n}^{(i)}|\mathcal{B}_{n})$. Note that under above conditions, we have $E(X_{n}^{(i)}|\mathcal{B}_{n})=P_{H_{n}}X_{n}^{(i)}$ [11, 12, 20, 23], where $P_{H_{n}}X_{n}^{(i)}$ is the orthogonal projection of $X_{n}^{(i)}$ onto the space $H_{n}$.

Remark: Let $\mathcal{B}(\subset\mathcal{F})$ be a sub-$\sigma$-field. Also, let $X\in L^{2}$ be $\mathcal{F}$-measurable, where $L^{2}$ is the set of random variables such that $E(|X|^{2})<\infty$. Let us define as $L^{2}(\mathcal{B})\stackrel{{\scriptstyle\triangle}}{{=}}\{Y\in L^{2};Y~{}\mbox{is}~{}\mathcal{B}-\mbox{measurable}\}$. Denote by $P_{L^{2}(\mathcal{B})}$ the orthogonal projection from $L^{2}$ onto the sub-space $L^{2}(\mathcal{B})$. Then [12, 20, 22, 23, 25] we have

[TABLE]

Since $P_{L^{2}(\mathcal{B}_{n})}X_{n}^{(i)}=E(X_{n}^{(i)}|\mathcal{B}_{n})$ holds (see the above remark), it follows that

[TABLE]

Note that the space $H_{n}$ is smaller than the space $L^{2}(\mathcal{B}_{n})$ [11]. Hence, this is a remarkable feature of a system of Gaussian random variables. Furthermore, we see that $X_{n}^{(i)}-E(X_{n}^{(i)}|\mathcal{B}_{n-1})$ is contained in $H_{n}$ and is orthogonal to $H_{n-1}$. Another important feature is a close connection with the notion of innovations [15, 18, 20, 23, 27]. In fact, it is shown that

[TABLE]

are innovations associated with the observations $Z_{n}$ [15, 20]. Using these properties, the well-known Kalman-Bucy filter [1, 13, 15, 20, 23, 26] is derived. The detailed derivation along the above argument is found in [20, Kunita] and [23, Øksendal] (In the latter, a continuous-time case is dealt with).

Next, consider convolutional coding/decoding. In order to state the problem more precisely, we introduce some additional notions needed for this paper. We always assume that the underlying field is $\mbox{GF}(2)$. Let $G(D)$ be a generator matrix for an $(n_{0},k_{0})$ convolutional code, where $G(D)$ is assumed to be canonical [14] (i.e., minimal [4]). Denote by \mbox{\boldmathi}=\{\mbox{\boldmathi}_{k}\} and \mbox{\boldmathy}=\{\mbox{\boldmathy}_{k}\} an information sequence and the corresponding encoded sequence, respectively, where \mbox{\boldmathi}_{k}=(i_{k}^{(1)},\cdots,i_{k}^{(k_{0})}) is the information block at $t=k$ and \mbox{\boldmathy}_{k}=(y_{k}^{(1)},\cdots,y_{k}^{(n_{0})}) is the encoded block at $t=k$. In this paper, it is assumed that an encoded sequence $y$ is transmitted symbol by symbol over a memoryless additive white Gaussian noise (AWGN) channel using binary phase shift keying (BPSK) modulation [9]. Let \mbox{\boldmathz}=\{\mbox{\boldmathz}_{k}\} be a received sequence, where \mbox{\boldmathz}_{k}=(z_{k}^{(1)},\cdots,z_{k}^{(n_{0})}) is the received block at $t=k$. Each component $z_{j}$ of $z$ is modeled as

[TABLE]

Here, $x_{j}$ takes $\pm 1$ depending on whether the encoded symbol $y_{j}$ is [math] or $1$. $E_{s}$ and $N_{0}$ denote the energy per channel symbol and the single-sided noise spectral density, respectively. Also, $w_{j}$ is a zero-mean unit variance Gaussian random variable. Each $w_{j}$ is independent of all others.

By grouping $z_{j}$ together as a branch, we can rewrite the observations as

[TABLE]

Note that $X_{k}~{}(\in\{-1,+1\}^{n_{0}})$ are not Gaussian and accordingly, $Z_{k}$ are not Gaussian random vectors. That is, $(X_{k},Z_{k})$ are “not” jointly Gaussian. Hence, a linear estimation method [1, 3, 15, 23, 26] cannot be applied to our case. As a result, in this paper, we will discuss “non-linear” estimation [6, 13, 20] of convolutionally encoded sequences.

In order to clarify the subsequent argument, we describe the observation model which will be discussed in this paper again. Observations are given as follows:

[TABLE]

where

[TABLE]

Note that the following hold with respect to the triplet $(W_{k},X_{k},Z_{k})~{}(k\in N)$.

$W_{k}~{}(k\in N)$ are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$.

• 2)

$X_{k}~{}(k\in N)$ and $W_{k}~{}(k\in N)$ are mutually independent.

• 3)

$Z_{k}~{}(k\in N)$ are random vectors of dimension $n_{0}$ and have the form $Z_{k}=\phi_{k}(X_{k})+W_{k}$, where $\phi_{k}(\cdot)$ may depend on $X_{j},Z_{j}~{}(j\leq k-1)$ as well. ($Z_{k}$ are not necessarily Gaussian.)

In [20], non-linear filtering of stochastic processes is discussed under the conditions 1), 2), and 3). Then we thought its argument can be used in our case. Hence, we will follow Kunita [20] and repeat the argument there. Also, as in [20], we focus our attention on a conditional probability of the form $P(X_{l}\in B|\mathcal{B}_{n})$, where $\mathcal{B}_{n}=\sigma(Z_{1},\cdots,Z_{n})$. In fact, using a conditional probability $P(X_{l}^{(i)}=x|\mathcal{B}_{n})$, a conditional expectation $E(X_{l}^{(i)}|\mathcal{B}_{n})$ is calculated as

[TABLE]

Now the argument in [20] is not intended to apply to the coding theory. On the other hand, our aim is convolutional coding/decoding. Hence, it is modified to meet our purpose. As a result, although proofs of the results in Section II-A have been given in [20], we will give them again because of our modifications. Subsequently, we will derive a general conditional probability $P(X_{l}\in A|\mathcal{B}_{n})~{}(l\leq n)$. When $l=n$, it is corresponding to filtering of $X_{n}$ based on $\mathcal{B}_{n}$, whereas when $l<n$, it is corresponding to smoothing [13, 16] of $X_{l}$ based on $\mathcal{B}_{n}$. In addition, we transform the obtained conditional probability into a recursive form using the structure of a code trellis. It is shown that the derived result can be used for maximum a posteriori probability (MAP) decoding [2, 21] of convolutional codes.

The rest of the paper is organized as follows. In Section II, a new probability measure $Q$ is constructed from the original probability measure $P$ using an (exponential) martingale. Then it is shown that with respect to $Q$, the (original) observations are mutually independent Gaussian random vectors with zero mean and identity covariance matrix. We see that the relationship between observation noises (with respect to $P$) and observations (with respect to $Q$) has a close connection with the Girsanov theorem [7, 17, 19, 20, 23] in continuous case. Next, in Section III, using the results in Section II, the conditional probability of an event related to any encoded symbol conditioned by the observations is calculated. In the process of derivation, the metric associated with an encoded sequence comes out in a natural way and we find that the argument in this paper has been connected to convolutional coding/decoding. The derived conditional probability is further transformed into a recursive form using the Markov property of state transitions on the associated code trellis. Also, the corresponding computational complexity is evaluated. Moreover, it is shown that MAP decoding [2, 21] of convolutional codes is realized using the obtained result. Finally, conclusions are drawn in Section IV.

II Exponential Martingales and Associated Probability Measures

II-A Martingale $\alpha_{n}$ and the Associated Probability Measure $Q$

The following lemma [12, 20] will be used repeatedly in our discussion.

Lemma II.1

Let $X$ and $Y$ be mutually independent random vectors of dimensions $d_{1}$ and $d_{2}$, respectively. Also, let $u(x,y)~{}(x\in R^{d_{1}},y\in R^{d_{2}})$ be a bounded measurable function. Then we have

[TABLE]

where $\mu_{X}(x)$ denotes the distribution of $X$.

Proof:

See Appendix A. ∎

Remark 1: The above lemma can be extended to a general product space. In Section III, $R^{d_{1}}$ is replaced with $(\{-1,+1\}^{n_{0}})^{n}$, where $n$ is a positive integer.

Define as

[TABLE]

[TABLE]

Here, (\mbox{\boldmatha},\mbox{\boldmathb}) denotes the inner product of vectors $a$ and $b$. (Define |\mbox{\boldmatha}|=(\mbox{\boldmatha},\mbox{\boldmatha})^{\frac{1}{2}}.)

Let \mbox{\boldmath\mathcal{G}}=\{\mathcal{G}_{n}\} be an increasing family of sub-$\sigma$-fields of $\mathcal{F}$. Let $\{\zeta_{n},~{}n\in N\}$ be a discrete-time stochastic process, where $\zeta_{n}$ is $\mathcal{G}_{n}$-measurable. If $E(\zeta_{n+1}|\mathcal{G}_{n})=\zeta_{n}$ holds for $n\in N$, then $\{\zeta_{n},~{}n\in N\}$ is said to be a $\mathcal{G}$-martingale [3, 11, 12, 18, 20, 22, 25, 26, 27]. We have the following.

Lemma II.2** (Kunita [20])**

$\{\alpha_{n}\}$* is a positive $\mathcal{F}$-martingale with mean $1$, where \mbox{\boldmath\mathcal{F}}=\{\mathcal{F}_{n}\}.*

Proof:

Let us show that

[TABLE]

Note that $W_{n}$ and $\sigma(W_{k};k\leq n-1)$ are mutually independent given $\sigma(X_{k};k\in N)$. Hence, it follows from Lemma 2.1 that

[TABLE]

Multiplying both sides by $\exp\{-c\sum_{k=1}^{n-1}(X_{k},W_{k})-\frac{1}{2}c^{2}n_{0}(n-1)\}$ (this is $\mathcal{F}_{n-1}$-measurable), we have

[TABLE]

$E(\alpha_{n})=1$ follows from $\alpha_{0}=1$ and from the property of a martingale (i.e., it has a constant mean). ∎

Remark 2: From the definition of $\mathcal{F}_{n}=\sigma(X_{k};k\in N)\vee\sigma(W_{k};k\leq n)$, $\mathcal{F}_{n}$ contains $\sigma(X_{k};k\in N)$. Accordingly, given $\sigma(X_{k};k\in N)$, we can rewrite $\alpha_{n}$ as

[TABLE]

Then $\alpha_{n}$ becomes a martingale from the property of $W_{k}$.

Now let $n_{p}$ be a positive integer and define a probability measure $Q$ by

[TABLE]

We have the following.

Lemma II.3** (Kunita [20])**

Suppose that $n\leq n_{p}$ and $A\in\mathcal{F}_{n}$. Then

[TABLE]

holds.

Proof:

Since $\{\alpha_{n}\}$ is an $\mathcal{F}$-martingale,

[TABLE]

holds and we have

[TABLE]

for $A\in\mathcal{F}_{n}$. Since $A\in\mathcal{F}_{n}$, the left-hand side is equal to $\int_{A}\alpha_{n_{p}}dP$ from the definition of a conditional expectation. Hence,

[TABLE]

holds. On the other hand, if $A\in\mathcal{F}_{n}$, then $A\in\mathcal{F}_{n_{p}}$ and we have

[TABLE]

∎

The above means that on the $\sigma$-field $\mathcal{F}_{n}$, $\alpha_{n}$ is the Radon-Nikodym derivative [10, 22, 25, 26] of $Q$ with respect to $P$ (denoted by $dQ/dP=\alpha_{n}$).

When $X_{k},Z_{k}~{}(k\in N)$ are viewed from the probability measure $Q$, we have the following.

Proposition II.1** (Kunita [20])**

$X_{k}$* and $Z_{k}$ have the following properties with respect to $Q$.*

The distribution of $X_{k}~{}(k\in N)$ with respect to $Q$ is identical to the distribution with respect to $P$.

• 2)

$Z_{k}~{}(1\leq k\leq n_{p})$* are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$.*

• 3)

$X_{k}~{}(k\in N)$* and $Z_{k}~{}(1\leq k\leq n_{p})$ are mutually independent.*

Proof:

See Appendix B. ∎

II-B In Relation to the Girsanov Theorem [7]

Note Lemma 2.3 and Proposition 2.1. The probability measure $P$ is the underlying probability measure and the probability measure $Q$ is constructed from $P$ using the exponential martingale $\alpha_{n}$. Here reverse the positions of $P$ and $Q$. This is possible as follows. First we can assume that $Q$ is a given probability measure with respect to which conditions 2) and 3) in Proposition 2.1 are satisfied. Next, let us show that $P$ is absolutely continuous with respect to $Q$. This follows from the relation

[TABLE]

Since $\alpha_{n}>0$, if $Q(A)=0$, then we have $P(A)=0$. That is, $P$ is absolutely continuous with respect to $Q$.

On the other hand, we know that $dQ/dP=\alpha_{n}$. Hence, we have $dP/dQ=\alpha_{n}^{-1}$. $\alpha_{n}^{-1}$ is calculated as follows:

[TABLE]

Thus $dP/dQ=\alpha_{n}^{-1}=\beta_{n}$. Note that $\beta_{n}$ has an alternative expression:

[TABLE]

From the assumption, $Z_{k}$ are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$. Using this property, it is shown that $\{\beta_{n}\}$ is an $\mathcal{F}$-martingale. Furthermore, with respect to $P$ ($dP=\beta_{n}dQ$),

[TABLE]

are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$. In the above expression, the term “$-cX_{k}$” is regarded as a shift due to the change of probability measures. In summary,

$Z_{k}$ are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$ with respect to $Q$.

• 2)

$W_{k}=Z_{k}-cX_{k}$ are mutually independent Gaussian random vectors of dimension $n_{0}$ with mean [math] and covariance matrix $I_{n_{0}}$ with respect to $P$, where $dP=\beta_{n}dQ$.

We remark that $Z_{k}$ and $W_{k}$ are corresponding to the Wiener process in continuous case. Hence, we see that the above is corresponding to a discrete-time version [19] of the Girsanov theorem [7, 17, 20, 23] for continuous processes.

III Non-Linear Estimation of Convolutionally Encoded Sequences

III-A Expectation Operator $E_{Q}$

We have the following [20, 23].

Lemma III.1

Let $P$ and $Q$ be the two probability measures defined on $\mathcal{F}$. Suppose that $Q(P)$ is absolutely continuous with respect to $P(Q)$ and let $dQ/dP=\alpha$ be the corresponding Radon-Nikodym derivative. Also, let $\mathcal{B}$ be a sub-$\sigma$-field of $\mathcal{F}$. Denote by $E_{Q}(\cdot)$ the expectation with respect to $Q$. Then we have

[TABLE]

where $\chi_{A}$ is the indicator function of a set $A$.

Proof:

Let $A\in\mathcal{F}$ and $B\in\mathcal{B}$. Then

[TABLE]

holds. On the other hand, since $dQ=\alpha dP$, we have $dP=\alpha^{-1}dQ$. Hence, the right-hand is equal to $E_{Q}(\alpha^{-1}\chi_{A}\chi_{B})$. Using the properties of a conditional expectation [22, 25, 26], this is modified as follows:

[TABLE]

Since $B\in\mathcal{B}$ is arbitrary, the above equality implies that

[TABLE]

Here note the definition of a conditional probability:

[TABLE]

By replacing $\chi_{A}$ by the above expression, we have $P(A|\mathcal{B})=E(E(\alpha|\mathcal{B})E_{Q}(\alpha^{-1}\chi_{A}|\mathcal{B})|\mathcal{B})$. Since both $E(\alpha|\mathcal{B})$ and $E_{Q}(\alpha^{-1}\chi_{A}|\mathcal{B})$ are $\mathcal{B}$-measurable, these terms are put out of the expectation and we have

[TABLE]

In particular, letting $A=\Omega$,

[TABLE]

is obtained. Then by substituting $\frac{1}{E_{Q}(\alpha^{-1}|\mathcal{B})}$ for $E(\alpha|\mathcal{B})$, we have

[TABLE]

∎

III-B Calculation of a Conditional Probability

Denote by $\mathcal{B}_{n}=\sigma(Z_{1},\cdots,Z_{n})$ the observations obtained up to time $n$. Let $\mathcal{F}=\mathcal{F}_{n}$ and $\mathcal{B}=\mathcal{B}_{n}$. Then by Lemma 3.1, we have

[TABLE]

In the previous section, $\alpha_{n}^{-1}$ has been calculated as

[TABLE]

Note that $\beta_{n}$ is rewritten as

[TABLE]

Here consider the probability density function of $z_{j}$ conditioned by $x_{j}$. This is given by

[TABLE]

Hence, we have

[TABLE]

where $K$ is a constant which depends only on $z_{j}$ and $(x_{j})^{2}=1$ has been used. Accordingly,

[TABLE]

is just the log-likelihood function (i.e., metric) [21] associated with the code sequence $\{X_{k}^{(i)};1\leq i\leq n_{0},1\leq k\leq n\}$. ($cX_{k}^{(i)}Z_{k}^{(i)}-\frac{1}{2}c^{2}$ is the symbol metric associated with the code symbol $X_{k}^{(i)}$ and $c(X_{k},Z_{k})-\frac{1}{2}c^{2}n_{0}$ is the branch metric.) Note that an exponential function having such a quantity as a power exponent can be equally regarded as a metric. Hence, $\beta_{n}$ is regarded as the metric associated with the code sequence $\{X_{k}^{(i)};1\leq i\leq n_{0},1\leq k\leq n\}$ as well. In the following, it is denoted by $\mbox{pm}(X)$. (Here a series of our arguments has been connected to convolutional coding/decoding.)

Taking into consideration Property 3) of Proposition 2.1, it follows from Lemma 2.1 that

[TABLE]

where $P_{X}(\cdot)$ is the distribution of $X_{1},\cdots,X_{n}$. We remark that $P_{X}(\cdot)$ denotes the distribution with respect to the probability measure $Q$. However, from Property 1) of Proposition 2.1, this is identical to the distribution with respect to the original probability measure $P$. We also remark that $Z_{1},\cdots,Z_{n}$ are mutually independent Gaussian random vectors with mean [math] and covariance matrix $I_{n_{0}}$ from Property 2) of Proposition 2.1. This means the following. Let $\{z_{k},~{}1\leq k\leq n\}$ be a set of outcomes generated from the random vectors $Z_{k}=cX_{k}+W_{k}~{}(1\leq k\leq n)$ under the probability measure $P$. Then we can equally regard $\{z_{k},~{}1\leq k\leq n\}$ as the outcomes of mutually independent Gaussian random vectors $Z_{k}~{}(1\leq k\leq n)$ with mean [math] and covariance matrix $I_{n_{0}}$ under the probability measure $Q$. That is, a set of outcomes can be seen in two different ways depending on the probability measures $P$ and $Q$.

In the following, it is assumed that $k_{0}=1$ for simplicity. Let $C$ be a convolutional code generated by $G(D)$. Also, suppose that the corresponding code trellis is terminated in the all-zero state at depth $n$, without loss of generality. Hence, the number of effective information bits is $n-\nu$ ($\nu$ denotes the constraint length). Moreover, it is assumed that the information bits are equally likely. Under these assumptions, the occurrence probability of each code sequence $\{x_{k},~{}1\leq k\leq n\}$ is given by $\frac{1}{2^{n-\nu}}$. (Note that an actual encoded sequence is \{\mbox{\boldmathy}_{k},~{}1\leq k\leq n\}. However, there is a one-to-one correspondence between encoded sequences \{\mbox{\boldmathy}_{k},~{}1\leq k\leq n\} and the corresponding code sequences $\{x_{k},~{}1\leq k\leq n\}~{}(x_{k}\in\{-1,+1\}^{n_{0}})$. Hence, we identify the latter with the former.)

Remark 1: This paper is concerned with conditional probabilities of the form $P(\cdot|\mathcal{B}_{n})$. Hence, it is natural to think of a maximum a posteriori probability (MAP) decoding algorithm (see Section III-F). In that case, the above metric is not appropriate. In fact, the assumption that the information bits are equally likely does not hold in general (for example, consider an iterative decoding algorithm [8]). Let $S_{l}$ be a state at depth $l$ on the corresponding code trellis. In MAP decoding, the branch metric associated with a state transition $\eta_{l}=(S_{l-1}=s^{\prime},S_{l}=s)$ (denoted by $\gamma_{l}(s^{\prime},s)$) depends on the a priori probability of the associated information bit $i_{l}$. However, if the information bits are equally likely, then $\gamma_{l}(s^{\prime},s)$ is essentially equal to the above (exponential) branch metric (see [21, Section 12.6]).

Remark 2: As stated above, it is assumed that the code trellis is terminated in the all-zero state at depth $n$. On the other hand, we can consider a truncated convolutional code, where all ending states are possible at depth $n$. In this case, the conditional probability $P(X_{n}\in B|\mathcal{B}_{n})$ corresponds to filtering of $X_{n}$ based on $\mathcal{B}_{n}$. Hence, in our situation, $P(X_{l}\in B|\mathcal{B}_{n})~{}(l<n)$ corresponds to smoothing of $X_{l}$ based on $\mathcal{B}_{n}$.

Now we have

[TABLE]

Next, note the relation

[TABLE]

For $1\leq l\leq n$, $A=\{X_{l}\in B\}$ is contained in $\mathcal{F}_{n}$, where $B$ is a set of branch codes between depths $l-1$ and $l$. Then by Lemma 3.1, $E_{Q}(\alpha_{n}^{-1}\chi_{A}|\mathcal{B}_{n})$ is calculated. That is, for $A=\{X_{l}\in B\}$, we obtain

[TABLE]

where $\sum_{x\in C,x_{l}\in B}$ means that summation is carried out for those elements $x\in C$ such that $x_{l}\in B$.

We finally have

[TABLE]

In particular, for $x_{l}=v$ ($v$ is a branch code), we have

[TABLE]

Thus the following has been derived.

Proposition III.1

Let $C$ be a convolutional code generated by $G(D)$. It is assumed that the corresponding code trellis is terminated in the all-zero state at depth $n$. Let $B$ be a set of branch codes between depths $l-1$ and $l$ ($l\leq n$). Then the conditional probability $P(X_{l}\in B|\mathcal{B}_{n})$ based on the observations up to time $n$ is given by

[TABLE]

III-C Conditional Probability in Recursive Form

We know that when a convolutional code is represented using the associated code trellis, a state sequence $\{S_{k}\}$ has the Markov property [5]. Hence, we see that a sequence of state transitions $\eta_{k}=(S_{k-1},S_{K})$ also has the Markov property. On the other hand, each state transition $\eta_{k}=(S_{k-1},S_{K})$ determines the associated branch code. Note that the converse is not true. However, when there is no danger of confusion, we identify a branch code $X_{k}$ with the associated state transition $\eta_{k}$. Noting these facts, the derived conditional probability $P(X_{l}\in B|\mathcal{B}_{n})$ is transformed into a recursive form.

Denote by $P_{k|k-1}(x_{k}|x_{k-1})$ the transition probability associated with a code sequence $\{X_{k}\}$. Also, let

[TABLE]

Then $b_{k}(x_{k},Z_{k})$ represents the metric associated with the branch code $x_{k}$. Using the Markov property of $X_{k}~{}(k\in N)$, we have

[TABLE]

Let us set $C_{s}^{t}\stackrel{{\scriptstyle\triangle}}{{=}}\{(x_{k},s\leq k\leq t);x\in C\}$. $C_{s}^{t}$ represents the set of code sub-sequences $(x_{k},s\leq k\leq t)$, where each code sequence $x$ is restricted for the interval $s\leq k\leq t$. Then the numerator of $P(X_{l}\in B|\mathcal{B}_{n})$ (denoted by $P(X_{l}\in B|\mathcal{B}_{n})_{nu}$) is given by

[TABLE]

Here, we have set

[TABLE]

Accordingly, we have

[TABLE]

Similarly, the denominator of $P(X_{l}\in B|\mathcal{B}_{n})$ (denoted by $P(X_{l}\in B|\mathcal{B}_{n})_{de}$) is given by

[TABLE]

As special cases, we have

[TABLE]

[TABLE]

Thus we have shown the following.

Proposition III.2

Under the same conditions as those for Proposition 3.1, the conditional probability $P(X_{l}\in B|\mathcal{B}_{n})~{}(l\leq n)$ with respect to $\mathcal{B}_{n}$ is given by

[TABLE]

where the quantities on the right-hand side are defined as above.

III-D Details of the Recursions

We describe the recursions in more detail. Our final goal is to calculate the quantity

[TABLE]

($P(X_{l}\in B|\mathcal{B}_{n})_{nu}$ is calculated in a similar way.)

III-D1 Forward Recursion

First consider the forward recursion. Suppose that $\nu+2\leq l$. When $x_{l}~{}(\in C_{l}^{l})$ is fixed, the variable $x_{l-1}$ is restricted according to the transition probability $P_{l|l-1}(x_{l}|x_{l-1})$. Since $k_{0}=1$ is assumed, $P_{l|l-1}(x_{l}|x_{l-1})>0$ holds only for two values of $x_{l-1}$. Let these two values be $v_{l-1}$ and $v_{l-1}^{\prime}$ (see Fig.1). Then we have

[TABLE]

where $x_{1}^{l-2}+v_{l-1}~{}(v_{l-1}^{\prime})$ denotes the concatenation of $x_{1}^{l-2}$ and $v_{l-1}~{}(v_{l-1}^{\prime})$. Also, we have used the relations:

[TABLE]

Hence, we have

[TABLE]

where

[TABLE]

We see that the above equation represents the forward recursion.

The initial condition for the forward recursion is related to the initial “transient” sections of the corresponding code trellis. It is given by

[TABLE]

Consider the next step in the forward recursion. We have

[TABLE]

Note that $x_{1}$ is determined (denoted by $v_{1}$) given $x_{2}$. Hence, the right-hand side of the above equation becomes

[TABLE]

Continuing this procedure, we have

[TABLE]

for $1\leq l\leq\nu+1$.

III-D2 Backward Recursion

Next, consider the backward recursion. This recursion is used to determine

[TABLE]

Suppose that $l\leq n-\nu-1$. Then the backward recursion is expressed as

[TABLE]

Here, we have used the relations that $P_{l+1|l}(x_{l+1}=v_{l+1}|x_{l})=\frac{1}{2}$ and $P_{l+1|l}(x_{l+1}=v_{l+1}^{\prime}|x_{l})=\frac{1}{2}$.

As in the forward recursion, the initial condition for the backward recursion is related to the final “transient” sections of the code trellis. It is given by

[TABLE]

where $P_{n|n-1}(x_{n}=v_{n}|x_{n-1})=1$ is used. Note that this value depends on $x_{n-1}$.

Similarly, at the next step in the backward recursion, we have

[TABLE]

where $P_{n-1|n-2}(x_{n-1}=v_{n-1}|x_{n-2})=1$ is used. Note that the value depends on $x_{n-2}$.

Continuing this procedure, we have

[TABLE]

for $n-\nu\leq l\leq n$. This value depends on $x_{l-1}$.

III-E Complexity of Calculating the Conditional Probability

Let us evaluate the complexity required to calculate $P(X_{l}\in B|\mathcal{B}_{n+\nu})_{de}$. (A similar evaluation is possible for $P(X_{l}\in B|\mathcal{B}_{n})_{nu}$.) It suffices to note the expression

[TABLE]

First fix $x_{l}~{}(\in C_{l}^{l})$ arbitrarily and consider the summation

[TABLE]

Since $x_{l}~{}(\in C_{l}^{l})$ is fixed, the number of $x_{1}^{l-1}~{}(\in C_{1}^{l-1})$ is equal to that of code sub-sequences which start in [math] (the all-zero state) at depth [math] and end in state $S_{l-1}$ at depth $l-1$. It is given by $2^{l-1-\nu}$. Since $P_{l|l-1}(x_{l}|x_{l-1})>0$ holds only for two $x_{l-1}$’s given $x_{l}$ ($k_{0}$ is assumed to be $1$), the expression $2^{l-2-\nu}\times 2=2^{l-1-\nu}$ is more accurate (see the previous sub-section). Hence, the complexity of the summation is proportional to $2^{l-1-\nu}$.

Next, consider the summation

[TABLE]

The number of $x_{l+1}^{n}~{}(\in C_{l+1}^{n})$ given $x_{l}$ is equal to that of code sub-sequences which start in state $S_{l}$ at depth $l$ and end in [math] at depth $n$ and is given by $2^{n-\nu-l}$. Hence, the complexity of the summation is proportional to $2^{n-\nu-l}$.

Finally, note that the number of $x_{l}~{}(\in C_{l}^{l})$ is $2^{\nu+1}$.

As a result, the total complexity $Q_{c}$ is given by

[TABLE]

where $K_{c}$ is some constant. We remark that $2^{n-\nu}$ is the number of code sequences on the associated code trellis. Hence, the result is reasonable.

III-F Application to MAP Decoding

Since we have obtained an expression for the conditional probability $P(X_{l}\in B|\mathcal{B}_{n})$, we immediately see that it can be applied to MAP decoding [2, 21] of convolutional codes. Suppose that $k_{0}=1$ as before. Let $x$ and $\hat{x}$ be the transmitted code sequence and the decoded code sequence, respectively. Here, taking into consideration the argument in Section III-B (see the remark in the sub-section), let us assume that the information bits are equally likely. When this assumption holds, the (ML) Viterbi algorithm maximizes P(\mbox{\boldmath\hat{x}}=\mbox{\boldmathx}|\mathcal{B}_{n}). However, this does not guarantee that $P(\hat{i}_{l}=i_{l}|\mathcal{B}_{n})$ is also maximized, where $i_{l}$ is the transmitted information bit and $\hat{i}_{l}$ is the decoded information bit (see [21, Section 12.6]). Hence, although the above assumption restricts the applications, we can still consider MAP decoding.

Let $i_{l}$ be the information bit at $t=l$ and suppose that a state at depth $l$ on the code trellis has the form $S_{l}=(i_{l-\nu+1},\cdots,i_{l-1},i_{l})$. Denote by $B_{0}$ the set of branch codes whose branches enter into the states $S_{l}^{0}=(\cdots,i_{l}=0)$ at depth $l$. Similarly, denote by $B_{1}$ the set of branch codes whose branches enter into the states $S_{l}^{1}=(\cdots,i_{l}=1)$ at depth $l$. Then we have

[TABLE]

Using these equations, the ratio of the a posteriori probability (APP) of $i_{l}$ being [math] to the APP of $i_{l}$ being $1$ is calculated as

[TABLE]

Hence, MAP decoding of convolutional codes can be realized based on the above APP ratio $\Lambda(i_{l})$.

IV Conclusion

We have considered a standard observation model where a convolutionally encoded sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation and have discussed it from the viewpoint of filtering (smoothing) for discrete-time stochastic processes. In this case, since pairs of the signal and observation are not jointly Gaussian, a linear estimation method cannot be used. Then we have applied a non-linear estimation method to the problem. (We have used the argument given in [20].) More precisely, we have used a discrete-time version of the Girsanov theory, which states a finite-dimensional Gaussian distribution is invariant under appropriate shifts of variables and a transformation of the underlying probability measure. As a result, we have derived the conditional probability of an event related to any encoded symbol conditioned by the observations. We have also transformed it into a recursive form. Moreover, we have shown that the derived conditional probability can be used for MAP decoding of convolutional codes. We remark that the method in this paper can be applied to block codes as well. We think a connection between the coding theory and the estimation theory for stochastic processes has been more clarified through the discussion.

Appendix A Proof of Lemma 2.1

Let $\mathcal{B}_{d_{1}}$ and $\mathcal{B}_{d_{2}}$ be the Borel $\sigma$-fields in $R^{d_{1}}$ and $R^{d_{2}}$, respectively. If $u(x,y)$ has the form $u(x,y)=u_{1}(x)u_{2}(y)$, then the lemma holds. Consider a general case. Let $\{A_{j}\times B_{j},~{}1\leq j\leq j_{0}\}$ be pairwise disjoint, where $A_{j}\in\mathcal{B}_{d_{1}}$ and $B_{j}\in\mathcal{B}_{d_{2}}$. Denote by $\sum_{j=1}^{j_{0}}A_{j}\times B_{j}$ the union of $A_{j}\times B_{j}~{}(1\leq j\leq j_{0})$. In this case, we have

[TABLE]

Note that a family of sets of the form $\sum_{j=1}^{j_{0}}A_{j}\times B_{j}$ generates the $\sigma$-field $\mathcal{B}_{d_{1}}\times\mathcal{B}_{d_{2}}$ in $R^{d_{1}+d_{2}}$ [10, 22]. Using these facts, it is shown that $u(x,y)$ is approximated by a linear combination of functions of the form $u_{1}(x)u_{2}(y)$.

Appendix B Proof of Proposition 2.1

Since $\alpha_{0}=1$, $Q(A)=\int_{A}\alpha_{0}dP=P(A)$ holds for $A\in\mathcal{F}_{0}$. Here note the relation $\mathcal{F}_{0}=\sigma(X_{k};k\in N)$. Thus 1) is proved.

Next, let us show 2). For $n\times n_{0}$ vector $\xi=(\xi_{1},\cdots,\xi_{n})$, we define the following quantity:

[TABLE]

where $i\stackrel{{\scriptstyle\triangle}}{{=}}\sqrt{-1}$. Then we have

[TABLE]

By Lemma 2.1,

[TABLE]

is obtained, where we have set $u_{n}=i\xi_{n}-cX_{n}$. (The last equality is derived by applying the Cauchy integral theorem in complex analysis.)

Using this fact, it is shown that $\{\alpha_{n}\gamma_{n}^{\xi}\}$ is an $\mathcal{F}$-martingale. A proof is similar to that for $\alpha_{n}$. Hence, for $m<n~{}(\leq n_{p})$,

[TABLE]

holds and for $A\in\mathcal{F}_{m}$, we have

[TABLE]

The left-hand side is equal to $\int_{A}\alpha_{n}\gamma_{n}^{\xi}dP$ from the definition of a conditional expectation. Hence,

[TABLE]

is obtained.

On the other hand, if $A\in\mathcal{F}_{m}$, then $A\in\mathcal{F}_{n}$ and by Lemma 2.3, we have

[TABLE]

Hence, both $dQ=\alpha_{n}dP$ and $dQ=\alpha_{m}dP$ hold on the $\sigma$-field $\mathcal{F}_{m}$ and it follows that

[TABLE]

That is, for $A\in\mathcal{F}_{m}$, we have

[TABLE]

This implies that

[TABLE]

or equivalently,

[TABLE]

on the $\sigma$-field $\mathcal{F}_{m}$. Then

[TABLE]

is obtained for $A\in\mathcal{F}_{m}$. Note that this is rewritten as

[TABLE]

Here note the relation $W_{k}+cX_{k}=Z_{k}$. By letting $A=\Omega$ and $m=0$, 2) is proved from the property of a characteristic function [12, 13, 20].

Finally, let us show 3). Again, note the relation

[TABLE]

It is modified as

[TABLE]

where $\chi_{A}$ the indicator function of $A$. Hence, for a simple function [10]

[TABLE]

we have

[TABLE]

Let $U$ be an arbitrary $\mathcal{F}_{m}$-measurable function. In the following, we will show that

[TABLE]

holds.

In the expression

[TABLE]

both $\cos(\xi U)$ and $\sin(\xi U)$ are $\mathcal{F}_{m}$-measurable and can be approximated by simple functions [10, 22] as

[TABLE]

Here note that

[TABLE]

also holds. Hence, we have

[TABLE]

Thus an equality

[TABLE]

has been shown. Since $\exp\{i\xi U\}$ can be approximated by simple functions $\sum_{p=1}^{p_{0}}a_{p}\chi_{A_{p}}+i\sum_{q=1}^{q_{0}}b_{q}\chi_{B_{q}}$ with arbitrary accuracy [10, 22], the above equality means that

[TABLE]

holds. This final expression implies that $Z_{k}~{}(m+1\leq k\leq n)$ and $\mathcal{F}_{m}$ are mutually independent [13]. In particular, letting $m=0$ and $n=n_{p}$, we have 3).