Symbol Detection of IDMA Systems in the Presence of Carrier Frequency Offsets

Abstract

In this paper we investigate the detection algorithms for interleave division multiple access (IDMA) systems in the presence of carrier frequency offsets (CFOs). The existing IDMA detection algorithm is designed under the zero CFO assumption and its performance will be degraded when the CFOs are present. We first extend the existing algorithm to the nonzero CFO case by utilizing effective channel coefficients which take the CFO effects into account. Then we turn to a more practical scenario with imperfect CFO estimates. We propose an algorithm that can cope with the residual CFO effects by integrating the CFO updating into the iterative receiver. Signal-to-interference-plus-noise ratio analysis and simulations show the feasibility and superiority of our proposed algorithms.

Appendix: Mean and Variance of \(I_{k}^{Re}(n)\)

Denote $\alpha _{k,i}=h_k^*h_i$. It can be shown that

$$\begin{aligned} I_{k}^{Re}(n) =\sum _{i=1,i\ne k}^K \left[ \mathfrak R \{\alpha _{k,i}\}\mathfrak R \{x_i(n)\}- \mathfrak I \{\alpha _{k,i}\}\mathfrak I \{x_i(n)\}\right] +\mathfrak R \{h_k^*z(n)\}, \end{aligned}$$

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which is a function of the Gaussian random variable $z(n)$ and the independent Bernoulli random variables $\mathfrak R \{x_i(n)\}$ and $\mathfrak I \{x_i(n)\}$, $\forall i$. Therefore the mean and variance of $I_{k}^{Re}(n)$ are given by

$$\begin{aligned} \mathrm{E}\{I_{k}^{Re}(n)\}&= \sum _{i=1,i\ne k}^K \left[ \mathfrak R \{{\alpha }_{k,i}\}\mathrm{E}\{\mathfrak{R }\{x_i(n)\} \} -\mathfrak I \{{\alpha }_{k,i}\}\mathrm{E}\{\mathfrak{I }\{x_i(n)\} \}\right] ,\end{aligned}$$

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$$\begin{aligned} \mathrm{var}\{I_{k}^{Re}(n)\}&= \sum _{i=1,i\ne k}^K \left[ \mathfrak R ^2\{{\alpha }_{k,i}\}\mathrm{var}\{\mathfrak{R }\{x_i(n)\} \} +\mathfrak I ^2\{\alpha _{k,i}\}\mathrm{var}\{\mathfrak{I }\{x_i(n)\} \}\right] \nonumber \\&+\frac{\sigma ^2}{2}|h_k|^2, \end{aligned}$$

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where

$$\begin{aligned} \mathrm{E}\left\{ \mathfrak{R }\{x_i(n)\}\right\}&= 1 \cdot P(\mathfrak{R }\{x_i(n)\}=1) +(-1)\cdot P(\mathfrak{R }\{x_i(n)\}=-1)\nonumber \\&= \tanh \left( \frac{L_a\left( \mathfrak{R }\{x_i(n)\}\right) }{2}\right) ,\end{aligned}$$

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$$\begin{aligned} \mathrm{var}\{\mathfrak{R }\{x_i(n)\}\}&= 1-\mathrm{E}^2\{\mathfrak{R }\{x_i(n)\}\}. \end{aligned}$$

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At the first iteration, there is no a priori information on the symbols so $L_a\left( \mathfrak R \{x_i(n)\}\right) =0$ and $\mathrm{E}\left\{ \mathfrak{R }\{x_i(n)\}\right\} =0$.

$\mathrm{E}\{\mathfrak{I }\{x_k(n)\}\}$ and $\mathrm{var}\{\mathfrak{I }\{x_k(n)\}\}$ can be obtained in a similar way.