User:Brian M. Kurkoski/Interference Channel

Revision as of 05:08, 25 December 2009

The title of this document is " Interference Channel".

Introduction

Figure 1: Interference Channel.

The classical communication channel is an input-output probabilistic model in which there are only one message sender and a corresponding receiver. The important quantity to be considered is the maximum rate at which we can transmit information through the channel with essentially no error. Shannon gave the answer for this problem in 1948 (S48). The maximum rate, the capacity, is expressed as the maximum of mutual information \(I(X; Y)\) with respect to the input \(X\) of the channel where \(Y\) is the output of the channel. A modern model of channel in these multi-user communication circumstances is the interference channel where several classical channels are active in parallel, but the data flows of each channel influence each other. Most simple model has two inputs and two outputs constituting two transmission paths interfering each other. Here, we have to consider two transmission rates \(R_1\) and \(R_2\) to be maximized in essentially no error condition. Thus, we want to determine the capacity region \({\mathcal C}_{\rm IC}\) for the interference channel. This problem was also presented by Shannon (S61). To date, this problem is not yet solved in general since the seventies, and is one of famous difficult open problems in information theory. On the other hand, researchers have attacked this problem to restrict to special subclass of interference channels, and sometimes have succeeded to obtain the capacity region for the subclass (refer to (B79),(HK81),(S81),(GC82),(CG87),(AC06),(MYK07),(SLSS08),(LU),(CM09)), or got new outer regions for interference channel (refer to (K04),(ETW08),(XKC09)).

Specifically, let \(\omega(y_1 y_2 | x_1 x_2)\) be the probability of outputs \(y_1, y_2\) when the inputs are \(x_1, x_2\) (see the left of Fig.<ref>ICandcomps</ref>). Two message senders use this channel \(n\) times to transmit their messages \(m_1 \in {\mathcal M}_1, m_2 \in {\mathcal M}_2\) to the corresponding receivers by coding \(m_1, m_2\) into input sequences \({\boldsymbol x}_1(m_1) = x_{11}\dots x_{1n}, {\boldsymbol x}_2(m_2) = x_{21}\dots x_{2n}\), respectively. Here, we assume the memoryless condition on the channel, that is, \(\omega^n({\boldsymbol x}_1, {\boldsymbol x}_2) = \prod_{i=1}^n \omega(y_{1i} y_{2i} | x_{1i} x_{2i})\). Usually, the synchronization of letters is assumed to make the problem simple.

Since each receiver can see only his/her output sequence from which he/she have to decide the message sent, the relevant channels are the marginal distributions \(\omega_1\) and \(\omega_2\) (see the right of Fig. <ref>ICandcomps</ref>): \[ \begin{array}{rcl} \omega_1(y_1 | x_1 x_2) & = & \sum_{y_2} \omega(y_1 y_2 | x_1 x_2) ,\\ \omega_2(y_2 | x_1 x_2) & = & \sum_{y_1} \omega(y_1 y_2 | x_1 x_2) . \end{array} \]

Figure 2: Coding System for Interference Channel.

By giving good codes \({\mathcal C}_1 = \{ {\boldsymbol x}_1 = \varphi_1(m_1) : m_1 \in {\mathcal M}_1 \}, {\mathcal C}_2 = \{ {\boldsymbol x}_2 = \varphi_2(m_2) : m_2 \in {\mathcal M}_2\}\), we want to enlarge the cardinalities \(M_1 = |{\mathcal M}_1|\) and \(M_2 = |{\mathcal M}_2|\) of the message sets \({\mathcal M}_1, {\mathcal M}_2\) as far as messages are transmitted with essentially no error (see Fig. <ref>ICcodingsystem</ref>). More precisely, we call the rates pair \((R_1, R_2) = \lim_{n \rightarrow \infty} ( \frac{1}{n} \log M_1, \frac{1}{n} \log M_2 )\) achivable if there were a sequence of coding systems \(\{ \varphi_1^n, \varphi_2^n ; \psi_1^n, \psi_2^n \}\) such that the average decoding errors goes to zero, that is, \[ \begin{array}{rcl} \frac{1}{M_1 M_2} \sum_{i=1}^{M_1} \sum_{j=1}^{M_2}\left\{1- \omega_1^n((\psi_1^n)^{-1}(i)|\varphi_1^n(i), \varphi_2^n(j)) \right\} \rightarrow 0 , \\ \frac{1}{M_1 M_2} \sum_{i=1}^{M_1} \sum_{j=1}^{M_2}\left\{1- \omega_2^n((\psi_2^n)^{-1}(j)|\varphi_1^n(i), \varphi_2^n(j)) \right\} \rightarrow 0 , \end{array} \] when \(n\) tends to the infinity where \((\psi_1^n)^{-1}(i)\) is the domain of the map \(\psi_1^n\) for \(i\), etc. The set of all achievable rates pair is called the capacity region \({\mathcal C}_{\rm IC}\) of the interference channel \(\omega\).

The operational efficiency is measured by the rates pair \((R_1, R_2)\). We want to achieve the best efficiency, that is, to realize the maximal rates pair. There is a tradeoff between the rates \(R_1\) and \(R_2\) in the best operational condition. If we would increase the rate \(R_1\) for user 1, then we should decrease the rate \(R_2\) in sacrificing the efficiency for user 2 in order to keep the achievability.

Strong Interference Channel

One of the early epoch-making results of the interference channel is the realization of the fact that "strong interference can be less harmful than weak interference", or "very intense interference can be as innocuous as no interference at all" (from (C75)). This interesting phenomenon that is today called as the strong or very strong interference, came from the study of Gaussian interference channel on which we will argue in the section 4. Here, we give the discrete version ((CG87)) of this result:

Theorem If the interference channel \(\omega\) satisfies the strong interference conditions, that is, \( I(Y_1; X_1 | X_2) \leq I(Y_2; X_1 | X_2)\) and \( I(Y_2; X_2 | X_2) \leq I(Y_1; X_2 | X_1)\) for any independent input random variables \(X_1, X_2\), then the capacity region of the channel is the closure of union of the regions defined by

<math CGregion>

\left\{ \begin{array}{rcl} 0 & \leq & R_1 \ \leq I( Y_1 ; X_1 | X_2 Q) , \\ 0 & \leq & R_2 \ \leq I( Y_2 ; X_2 | X_1 Q) , \\ 0 & \leq & R_1 + R_2 \ \leq \min \{ I( Y_1 ; X_1 X_2 | Q), I( Y_2 ; X_1 X_2 | Q) \} , \end{array} \right. </math> where \(X_1, X_2\) is independent given \(Q\).

Han-Kobayashi Achievable Region and its Simple Expression

Figure 3: Test Interference Channel.

The largest inner bound for the capacity of interference channel, called the Han-Kobayashi achievable region, is expressed as follows. At first, let us introduce the test channel \(Z = Q U_1 W_1 U_2 W_2 X_1 X_2 Y_1 Y_2\) (see Fig. <ref>{testChannel</ref>). Here, the auxiliary random variables \(U_1, W_1, U_2, W_2\) are independent each other given the superposition random variable \(Q\). The inputs \(X_1\) and \(X_2\) of the channel \(\omega\) are deterministic functions \(f_1(U_1, W_1, Q)\), \(f_2(U_2, W_2, Q)\) of \(U_1, W_1, Q\) and \(U_2, W_2, Q\), respectively. \(Y_1\) and \(Y_2\) are the outputs of the channel \(\omega\). Denote the whole set of test channels by \({\mathcal P}\).

Figure 4: A typical profile of the region \({\mathcal R}_{\rm CMGE}(Z)\).

Then, we can define conditional mutual informations among these random variables defining the test channel:

\[ \left\{ \begin{array}{ccl} a_1 & = & I(Y_1 ; U_1 | W_1 W_2 Q), \\ b_1 & = & I(Y_1 ; W_1 | U_1 W_2 Q), \\ c_1 & = & I(Y_1 ; W_2 | U_1 W_1 Q), \\ d_1 & = & I(Y_1 ; U_1 W_1 | W_2 Q), \\ e_1 & = & I(Y_1 ; U_1 W_2 | W_1 Q), \\ f_1 & = & I(Y_1 ; W_1 W_2 | U_1 Q), \\ g_1 & = & I(Y_1 ; U_1 W_1 W_2 | Q), \end{array} \right. \]

\[ \left\{ \begin{array}{ccl} a_2 & = & I(Y_2 ; U_2 | W_2 W_1 Q), \\ b_2 & = & I(Y_2 ; W_2 | U_2 W_1 Q), \\ c_2 & = & I(Y_2 ; W_1 | U_2 W_2 Q), \\ d_2 & = & I(Y_2 ; U_2 W_2 | W_1 Q), \\ e_2 & = & I(Y_2 ; U_2 W_1 | W_2 Q), \\ f_2 & = & I(Y_2 ; W_2 W_1 | U_2 Q), \\ g_2 & = & I(Y_2 ; U_2 W_2 W_1 | Q). \end{array} \right. \]

Using these quantities, let us define the \({\mathcal R}_{\rm CMGE}(Z)\) as the whole set of rates pair \((R_1, R_2)\) satisfying the inequalities (see Fig. <ref>regionCMGE</ref>):

<math CMGEregion>

\left\{ \begin{array}{rcl} R_1 & \leq & d_1 ,\\ R_2 & \leq & d_2 ,\\ R_1 + R_2 & \leq & a_1 + g_2 ,\\ R_1 + R_2 & \leq & a_2 + g_1 ,\\ R_1 + R_2 & \leq & e_1 + e_2 ,\\ 2 R_1 + R_2 & \leq & a_1 + g_1 + e_2 ,\\ R_1 + 2 R_2 & \leq & a_2 + g_2 + e_1 ,\\ R_1 & \geq & 0 ,\\ R_2 & \geq & 0 . \end{array} \right. </math>

Temporary Subsection

Then, the Han-Kobayashi region \({\mathcal R}_{\rm HK}\) is the closure of \(\displaystyle \bigcup_{Z \in {\mathcal P}} {\mathcal R}_{\rm CMGE}(Z)\), and it holds

Theorem Any rates pair \((R_1, R_2)\) contained in \({\mathcal R}_{\rm HK}\) is achievable, that is, \[ \begin{array}{rcl} {\mathcal C}_{\rm IC} &\supset& {\mathcal R}_{\rm HK}. \end{array} \]

This formula using (<ref>CMGEregion</ref>) of the HK region introduced by the paper (CMGE08) is the simplest to date. The original proof on achievability of rates pair \((R_1, R_2)\) contained in \({\mathcal R}_{\rm HK}\) was started by introducing an intermediate region \({\mathcal S}_{\rm HK}(Z)\) of rates quadruple \((S_1, T_1, S_2, T_2)\) satisfying two sets of the inequalities induced by the test channel \(Z \in {\mathcal P}\):

<math s1t1t2>

\left\{ \begin{array}{ccl} S_1 & \leq & a_1 ,\\ T_1 & \leq & b_1 ,\\ T_2 & \leq & c_1 ,\\ S_1 + T_1 & \leq & d_1 ,\\ S_1 + T_2 & \leq & e_1 ,\\ T_1 + T_2 & \leq & f_1 ,\\ S_1 + T_1 + T_2 & \leq & g_1 ,\\ S_1 & \geq & 0 ,\\ T_1 & \geq & 0 , \end{array} \right. </math> and

<math s2t2t1>

\left\{ \begin{array}{ccl} S_2 & \leq & a_2 ,\\ T_2 & \leq & b_2 ,\\ T_1 & \leq & c_2 ,\\ S_2 + T_2 & \leq & d_2 ,\\ S_2 + T_1 & \leq & e_2 ,\\ T_2 + T_1 & \leq & f_2 ,\\ S_2 + T_2 + T_1 & \leq & g_2 , \\ S_2 & \geq & 0 ,\\ T_2 & \geq & 0 . \end{array} \right. </math>

Here, \(S_1, S_2\) are the rates of private messages intended to be transmitted to only the corresponding receivers from senders, and \(T_1, T_2\) are the rates of common messages to both of receivers. Any rates quadruple \((S_1, T_1, S_2, T_2)\) contained in \({\mathcal S}_{\rm HK}(Z)\) is proved to be achievable. Then, using the Fourier-Motzkin method, we can project the four-dimensional region \({\mathcal S}_{\rm HK}(Z)\) to two-dimensional region \({\mathcal R}_{\rm HK}(Z)\) by equating \(R_1 = S_1 + T_1, R_2 = S_2 + T_2\) through subtle discussions removing redundant inequalities. The two-dimensional region \({\mathcal R}_{\rm HK}(Z)\) is defined by the same set of inequalities (<ref>CMGEregion</ref>) plus the extra two inequalities:

<math extraineqs>

\left\{ \begin{array}{rcl} R_1 & \leq & a_1 + c_2 ,\\ R_2 & \leq & a_2 + c_1 . \end{array} \right. </math>

Therefore, \({\mathcal R}_{\rm CMGE}(Z) \supset {\mathcal R}_{\rm HK}(Z)\). But, it can be also shown that \({\mathcal R}_{\rm CMGE}(Z) \subset {\mathcal R}_{\rm HK}(Z) \cup {\mathcal R}_{\rm HK}(Z') \cup {\mathcal R}_{\rm HK}(Z'')\) by suitably choosing other test channels \(Z', Z''\). Thus, \(\displaystyle \bigcup_{Z \in {\mathcal P}} {\mathcal R}_{\rm CMGE}(Z) = \bigcup_{Z \in {\mathcal P}} {\mathcal R}_{\rm HK}(Z)\) . Here, we should mention that both sets of inequalities, (<ref>s1t1t2</ref>) and (<ref>s2t2t1</ref>), define polymatroids due to the assumption on the test channel \(Z \in {\mathcal P}\). Thus, it holds that for \(a_i, \dots, g_i, i= 1, 2\), \[ \left\{ \begin{array}{ccl} a_i, b_i \leq & d_i & \leq a_i + b_i, \\ a_i, c_i \leq & e_i & \leq a_i + c_i, \\ b_i, c_i \leq & f_i & \leq b_i + c_i, \\ d_i, e_i, f_i \leq & g_i & \leq c_i + d_i, b_i+e_i, a_i + f_i, \\ a_i + g_i & \leq & d_i + e_i, \\ b_i + g_i & \leq & f_i + d_i, \\ c_i + g_i & \leq & e_i + f_i . \end{array} \right. \] These conditions are used to remove redundant inequalities in applying Fourier-Motzkin method to project \({\mathcal S}_{\rm HK}(Z)\) to \({\mathcal R}_{\rm HK}(Z)\).

Gaussian Interference Channel

Figure 5: Gaussian Interference Channel.

The Gaussian interference channel is shown in Fig. <ref>GIC</ref>, and is mathematically expressed as

<math GICmodel>

\left\{ \begin{array}{rcl} Y_1 & = & a X_1 + b X_2 + n_1^* \\ Y_2 & = & c X_1 + d X_2 + n_2^* , \end{array} \right. </math> where \(Y_1\) is the output for real input \(X_1\) added the interference from unintended input \(X_2\) plus the Gaussian noise \(n_1^*\) with the expectation 0, variance \(N_1\), and \(Y_2\) is the same.

Figure 6: The standard form of Gaussian interference channel.

Usually, setting \(\sqrt{a_{12}} = \frac{b}{d}\,\sqrt{ \frac{N_2}{N_1}}\) and \(\sqrt{a_{21}} = \frac{c}{a}\,\sqrt{ \frac{N_1}{N_2}}\), we discuss the capacity problem with the standard model (see Fig. <ref>standardGIC</ref>) converted from (<ref>GICmodel</ref>):

<math standardCMGmodel>

\left\{ \begin{array}{rcl} y_1 & = & x_1 + \sqrt{a_{12}}x_2 + n_1 \\ y_2 & = & \sqrt{a_{21}}x_1 + x_2 + n_2 , \end{array} \right. </math> where \(n_1, n_2\) are Gaussian noises with expectation \(0\), and variance \(1\), and the inputs \(x_1, x_2\) are subjected to the average power constraints when the channel is used \(n\) times: \[ \begin{array}{c} \frac{1}{n} \sum_{i=1}^n x_{1i}^2 \leq P_1,\ \ \frac{1}{n} \sum_{i=1}^n x_{2i}^2 \leq P_2 . \end{array} \]

For the case of strong interference, we have

Theorem The capacity region of Gaussian interference channel of \(a_{12} \geq 1, a_{21} \geq 1\) is the set of rates pair \((R_1, R_2)\) satisfying the inequalities:

<math stronGIC>

\left\{ \begin{array}{rcccl} 0 & \leq & R_1 & \leq & \frac{1}{2} \log (1 + P_1),\\ 0 & \leq & R_2 & \leq & \frac{1}{2} \log (1 + P_2),\\ 0 & \leq & R_1 + R_2 & \leq & \min \left\{ \frac{1}{2} \log (1 + P_1 + a_{12} P_2), \right.\\ & & & & \left. \ \ \ \ \ \ \ \frac{1}{2} \log (1 + P_2 + a_{21} P_1) \right\} \end{array} \right. </math>

Especially, in case of very strong interference, that is, when \(a_{12} \geq 1+P_1, a_{21} \geq 1+P_2\), the capacity region is identical to the case of no interference:

<math verystronGIC>

\left\{ \begin{array}{rcccl} 0 & \leq & R_1 & \leq & \frac{1}{2} \log (1 + P_1),\\ 0 & \leq & R_2 & \leq & \frac{1}{2} \log (1 + P_2) . \end{array} \right. </math>

Temp Subsection

Figure 7: The HK achievable region of symmetric Gaussian interference channel.

Setting the superposition variable \(Q\) to a constant, the auxiliary variables to Gaussian, and the deterministic functions to the addition, the HK achievable regions \({\mathcal G}_{\rm HK}(Z)\) corresponding to the Han-Kobayashi regions \({\mathcal R}_{\rm HK}(Z)\) are plotted in Fig. <ref>regionSymIC</ref> for a symmetric Gaussian interference channel (\(P=P_1=P_2, a=a_{12} = a_{21}\)) parametrized by the interference coefficient \(a\).

Figure 8: The asymptotic profile of symmetric rate \(R_{\rm HK}^{\rm sym}\) over \(C_{\rm AWGN}\).

The paper (ETW08) shows that simple strategies of Han-Kobayashi scheme can achieve to within a single bit of the capacity of the complex Gaussian interference channel even in cases of weak interference. The simple strategy is that the power allocation should be to balance the interference \(|a|^2 P_u\) due to the private message \(P_u\) equal to the noise power \(N\), where the total restricted power \(P\) is partitioned for the private message and the common message as \(P = P_u + P_w\). By applying the simple scheme, the whole quantities of conditional mutual information for the HK region expression can be expressed by only two quantities, that is, the signal-to-noise ratio \({\rm SNR} = |a|^2 P/N\) and the interference-to-noise ratio \({\rm INR} = |b|^2 P/N\) for the symmetric Gaussian interference channel where \(a\) is the direct coefficient, and \(b\) is the cross-over coefficient. We write as \(R_{\rm HK}^{\rm sym}\) for the maximum rate \(R\) such that \((R,R)\) is achievable by the simple HK scheme. Then, we have the formula for this quantity:

<math symmetricRate>

R_{\rm HK}^{\rm sym} =  \min

\left\{ \begin{array}{l} \frac{1}{2} \log \left( 1 + {\rm SNR} + {\rm INR} \right) + \frac{1}{2} \log \left( 2 + \frac[[:Template:\rm SNR]][[:Template:\rm INR]] \right) - 1 , \\ \log \left( 1 + {\rm INR} + \frac[[:Template:\rm SNR]][[:Template:\rm INR]] \right) - 1 . \end{array} \right. </math>

The first term in "\(\min\)" corresponds to the upper bound of inequalities \(R_1 + R_2 \leq a_1 + g_2 = a_2 + g_1\), and the sencond term to that of inequality \(R_1 + R_2 \leq e_1 + e_2\) in (<ref>CMGEregion</ref>). Comparing \(R_{\rm HK}^{\rm sym}\) with the capacity \(C_{\rm AWGN} = \log (1 + {\rm SNR})\) of standard complex additive Gaussian noise channel, we obtain an interesting two modes behavior depending on the interference level \(0 \leq \alpha = \frac{\log {\rm INR}}{\log {\rm SNR}} \leq 1\) in an asymptotic condition of \({\rm SNR} \gg 1, {\rm INR} \gg 1, {\rm SNR} \gg {\rm INR}\) when the interference is weak (see Fig. <ref>generalDF</ref>):

<math GDF>

\frac{R_{\rm HK}^{\rm sym}}{C_{\rm AWGN}} \approx \min \left\{ \begin{array}{l} 1 - \frac{1}{2} \alpha , \\ \max \left\{ \alpha, 1-\alpha \right\} . \end{array} \right. </math>

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again

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not yet

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None

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\bibitem{LU08} N.Liu and S.Ulukus, ``The capacity region of a class of discrete degraded interference channels, {\em IEEE Trans. Inform. Theory}, vol.54, no.9, pp.4372--4378, Sep. 2008.

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\bibitem{ETW08} R.H.Etkin, D.N.C.Tse and H.Wang, ``Gaussian interference channel capacity to within one bit, {\em IEEE Trans. Inform. Theory}, vol.54, no.12, pp.5534--5562, Dec. 2008.

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\bibitem{CM09} H-F.Chong and M.Motani, ``The capacity region of a class of semideterministic interference channels, {\em IEEE Trans. Inform. Theory}, vol.55, no.2, pp.598--603, Feb. 2009.

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\bibitem{XKC09} X.Shang, G.Kramer and B.Chen, ``A new outer bound and the noisy-interference sum-rate capacity for Gaussian interference channels, {\em IEEE Trans. Inform. Theory}, vol.55, no.2, pp.689--699, Feb. 2009.