An Analytical Model of Iterative Interference Cancellation Receiver for Orthogonal/Orthogonal Overloaded DS-CDMA System

Abstract

One of the efficient schemes to overload a CDMA system is to use two sets of orthogonal signal waveforms (O/O) Sari, Vanhaverbeke and Moeneclaey, IEEE Communication Magazine, 56–59, 2000. In this paper, we have presented the analysis of hard decision interference cancellation (HDIC) receiver for O/O overloaded DS-CDMA system. A closed form expression for the average bit error rate (BER) is obtained, for each stage of HDIC receiver, without considering the correlation between the multiple access interference and Gaussian noise. It is observed that at lower overloadings, analytical result matches closely to that obtained from simulation. But there is a difference between two at higher overloadings. The reason for such a difference is the inaccuracy in variance calculation due to the assumption of zero correlation. By using the Price’s theorem, we have obtained the correlation between tentative data decision and Gaussian noise samples. The overloading performance of a specific Quasi-synchronous sequence (QOS) O/O scheme is also evaluated using interference cancellation receivers.

Appendix 1

1.1 Calculation of Correlated Parameters

We need to refine the variance calculation of the decision variable by considering the correlation between the multiple access interference and Gaussian noise. This approach needs the evaluation of the covariance that involves the non-linear hard decisions on detected bits made in the previous stages. We will deal with the nonlinear problem using Price’s Theorem as Price’s Theorem provides us a tool to cope with the nonlinear relation encountered in Eq. 10.

1.2 Price Theorem

Given two jointly normal random variables x and y, we form the mean

$$ I = E\left\{ {g({\text{x,y}})} \right\} = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {g(x,y)} } f(x,y)dxdy $$

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of some function g(x,y) of (x, y). The above integral is a function of ${\text{I(}}\mu )$ of the covariance $\mu$of the random variables x and y and four parameters specifying the joint density f(x, y) of x and y. If $f(x,y) \to 0$as $( {\text{x,y)}} \to \infty$, then

$$ {\frac{{\partial^{\text{n}} I(\mu )}}{{\partial \mu^{n} }}} = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {{\frac{{\partial^{2n} g(x,y)}}{{\partial x^{n} \partial y^{n} }}}} } .f(x,y)dxdy = E\left[ {{\frac{{\partial^{2n} g({\mathbf{x,y}})}}{{\partial {\mathbf{x}}^{n} \partial {\mathbf{y}}^{n} }}}} \right]. $$

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In the following section, we evaluate the correlation between the tentative decisions and Gaussian noise as well as correlation between tentative decisions for different users.

1.2.1 Evaluation of ${\text{E}}\left\{ {\hat{b}_{2,k}^{(i - 1)} z\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\}$

We define

$$ I_{1} (\mu_{1} ) = E\left\{ {\hat{b}_{2,k}^{(i - 1)} z\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\} $$

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Here $\mu_{1}$=$E\left\{ {z \, (y_{2,k}^{(i - 1)} - m_{{2,k/b_{2} }}^{(i - 1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\}$, is the covariance and $m_{{2,k/{\mathbf{b}}_{{\mathbf{2}}} }}^{(i - 1)}$is the mean of ${\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}}$ conditioned on${\mathbf{b}}_{{\mathbf{2}}}$. By applying Price’s Theorem, it follows that

$$ \begin{aligned} {\frac{{\partial I_{1} (\mu_{1} )}}{{\partial\mu_{1} }}} &= E\left\{ {{\frac{{\partial z\partial \left[{\text{sgn} ({\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}})} \right]\left| {{\mathbf{b}}_{{\mathbf{2}}} }\right.}}{{\partial z\partial {\text{y}}_{{ 2 , {\text{k}}}}^{{ ({\text{i - 1)}}}} }}}} \right\} \\ &= 2E\left\{ {\delta({\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}} )\left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\} =2f_{{{\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.}} \left( 0 \right)\\\end{aligned} $$

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Where

$$ f_{{{\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} ({\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}} ) = N(m_{{2,k\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}}^{(i - 1)} ,\sigma_{{{\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} ) $$

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is the probability density function of ${\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}}$ conditioned on ${\mathbf{b}}_{{\mathbf{2}}}$. Integrating Eq. 23, we get

$$ \begin{aligned} I_{1} (\mu_{1} ) = 2f_{{{\text{y}}_{{ 2 ,{\text{k}}}}^{{ ( {\text{i - 1)}}}} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (0).\mu_{1} + I_{1} (0)&= 2f_{{{\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i - 1)}}}}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}}(0).{\text{E}}\left\{ {{\text{z (y}}_{{ 2 , {\text{k}}}}^{{ ({\text{i - 1)}}}} - m_{{2,k/{\mathbf{b}}_{{\mathbf{2}}} }}^{(i -1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\} +I_{1} (0) \\ &= 2f_{{{\text{y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i- 1)}}}} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}}(0).E\left\{ {{\text{z y}}_{{ 2 , {\text{k}}}}^{{ ( {\text{i -1)}}}} } \right\} + I_{1} (0) \\ &= 2f_{{{\text{y}}_{{ 2 ,{\text{k}}}}^{{ ( {\text{i - 1)}}}} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (0).E [ {\text{z y}}_{{ 2, {\text{k}}}}^{{ ( {\text{i - 1)}}}} ] = 2f_{{{\text{y}}_{{ 2 ,{\text{k}}}}^{{ ( {\text{i - 1)}}}} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (0)\rho_{l,k}\sigma_{n}^{2} \\ \end{aligned} $$

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Equation 25 shows the correlation between the tentative data decision and Gaussian noise. It is that when the crosscorrelation between l-th set-1 and k-th set-2 users is high, the correlation between the tentative data decision and Gaussian noise is significant.

1.2.2 Evaluation of $E\left[ {b_{2,m} \hat{b}_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right]$, $E\left[ {b_{2,n} \hat{b}_{2,m}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right]$

Define

$$ I_{2} (\mu_{2} ) = {\text{E(}}b_{2,m} \hat{b}_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.) $$

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$I_{2} (\mu_{2} )$ can be evaluated from its definition, i.e.,

$$ \begin{aligned} I_{2} (\mu_{2} ) &= b_{2,m} \int\limits_{ -\infty }^{\infty } {\text{sgn} (y_{2,n}^{(i - 1)} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.} ).f_{{y_{2,n}^{(i - 1)}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (y_{2,n}^{(i - 1)}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.).dy_{2,n}^{(i - 1)}\\ &= b_{2,m} \left\{ \begin{array}{ll} \int\limits_{ - \infty }^{0}{ - 1.} f_{{y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}}} \right.}} (y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}}} \right.).dy_{2,n}^{(i - 1)}\\ + \int\limits_{0}^{\infty } {1.}f_{{y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} }\right.}} (y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} }\right.).dy_{2,n}^{(i - 1)}\\ \end{array} \right\} \\ &= 2b_{2,m}\left\{ {\int\limits_{0}^{\infty } {1.} f_{{y_{2,n}^{(i - 1)}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (y_{2,n}^{(i - 1)}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.).dy_{2,n}^{(i - 1)}} \right\} \\ &= 2b_{2,m} \int\limits_{0}^{\infty }{{\frac{1}{{\sqrt {2\pi \sigma_{n}^{2} } }}}e^{{ - {\frac{{ -\left( {y_{2,n}^{(i - 1)} - m_{2,k}^{(i - 1)} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right)^{2}}}{{2\sigma_{n}^{2} }}}}} dy_{2,n}^{(i - 1)} } \\ &= 2b_{2,m}\int\limits_{{ - {\frac{{m_{2,k}^{(i - 1)} \left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.}}{{\sigma_{n} }}}}}^{\infty} {{\frac{1}{{\sqrt {2\pi } }}}e}^{{ - {\frac{{z^{2} }}{2}}}} dz\\ &= 2b_{2,m} \left\{ {2Q\left( { - {\frac{{m_{2,k}^{(i - 1)}\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}}{{\sigma_{n} }}}}\right) - 1} \right\}\\ \end{aligned} $$

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Similarly we can find

$$ E[b_{2,n} \hat{b}_{2,m}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.] = 2b_{2,n} \left\{ {2Q\left( { - \, {\frac{{m_{2,k}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}}{{\sigma_{n} }}}} \right) - 1} \right\} $$

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Equation 28 gives the correlation between the tentative decision of user-n and transmitted data of user-m form set-2.

1.2.3 Evaluation of $E\left[ {\hat{b}_{2,m}^{(i - 1)} \hat{b}_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right]$

Let

$$\begin{aligned} I_{3} (\mu_{3} ) &= E\left\{ {\hat{b}_{2,m}^{(i- 1)} \hat{b}_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}}} \right.} \right\}\\ &= E\left\{ {\text{sgn} \left[ {y_{2,m}^{(i- 1)} } \right]\text{sgn} \left[ {y_{2,n}^{(i - 1)} }\right]\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\}\end{aligned}$$

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As $y_{2,m}^{(i - 1)}$and $y_{2,n}^{(i - 1)}$ are jointly normal when conditioned on${\mathbf{b}}_{{\mathbf{2}}}$, we can apply Price’s Theorem:

$$ \begin{aligned} {\frac{{\partial I_{3} (\mu_{3} )}}{{\partial \mu_{3} }}} &= E\left\{ {{\frac{{\partial^{2} [\text{sgn} (y_{2,m}^{(i - 1)} )\text{sgn} (y_{2,n}^{(i - 1)} )]}}{{\partial y_{2,m}^{(i - 1)} \partial y_{2,n}^{(i - 1)} }}}\left|{{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\} \\ &= E\left\{{4\delta (y_{2,m}^{(i - 1)} )\delta (y_{2,n}^{(i - 1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right\} \\ &= 4f_{{(y_{2,m}^{(i - 1)} ,y_{2,n}^{(i - 1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (0,0) \\ \end{aligned} $$

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Where $f_{{(y_{2,m}^{(i - 1)} ,y_{2,n}^{(i - 1)} )/{\mathbf{b}}_{{\mathbf{2}}} }} (y_{2,m}^{(i - 1)} ,y_{2,n}^{(i - 1)} )$ is the joint probability density function of $y_{2,m}^{(i - 1)}$ and$y_{2,n}^{(i - 1)}$.

By integrating Eq. 30

$$ I_{3} (\mu_{3} ) = 4f_{{(y_{2,m}^{(i - 1)} ,y_{2,n}^{(i - 1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} (0,0)\mu_{3} + I_{3} (0) $$

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$$ \mu_{3} = E\left[ {(y_{2,m}^{(i - 1)} ,y_{2,n}^{(i - 1)} )\left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right] - E\left[ {y_{2,m}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.} \right]E[y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.] $$

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$$ I_{3} (0) = \left[ {2Q\left( { - {\frac{{m_{{y_{2,m}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} }}{{\sigma_{{y_{2,m}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} }}} - 1} \right)} \right]\left[ {2Q\left( { - {\frac{{m_{{y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} }}{{\sigma_{{y_{2,n}^{(i - 1)} \left| {{\mathbf{b}}_{{\mathbf{2}}} } \right.}} }}} - 1} \right)} \right] $$

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Equation 33 shows the correlation between the tentative decision between user-m and user-n from set-2.

1.2.4 Improved BER expression

Denoting the unconditional variance of $y_{1,l}^{i}$ as$\sigma_{{y_{1,l}^{i} }}^{2}$, it follows from Eq. (3.2.6) that

$$ \sigma_{{y_{1,l}^{i} }}^{2} = \sum\limits_{j} {\sigma_{{y_{1,l}^{i} \left| {{\mathbf{b}}_{2j} } \right.}}^{2} p({\mathbf{b}}_{2j} )} $$

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where ${\mathbf{b}}_{2j}$ denotes a specific combination of ${\mathbf{b}}_{2}$ and $p({\mathbf{b}}_{2j} )$ denotes the possibility for a specific vector in the space of ${\mathbf{b}}_{2}$.

When the interference is assumed to be Gaussian, the expression for BER of the set-1, l-th user (l = 1, 2, 3,…,N) is given by

$$ P_{e,(1,l)}^{(i)} = Q\left( {{\frac{1}{{\sigma_{{y_{1,l}^{i} }} }}}} \right) $$

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From the above discussion, we observe that, the tentative data decisions and noise samples become correlated in the process of iterative interference cancellation in multistage detection, in an overloaded environment. These correlation values are non-zero and contribute to the total multiple access interference from other users.