Analysis of the CFO Successive Interference Cancellation for the OFDMA Uplink

Abstract

The uplink of orthogonal frequency division multiple access or single-carrier frequency division multiple access suffers multiple access interference when carrier frequency offset (CFO) is not properly estimated and compensated. In particular, multicarrier uplink CFO compensation is highly complex due to the multiuser context. Successive interference cancellation algorithms are effectively employed to compensate for the CFO, where the interference produced by each user is handled sequentially through a series of iterations. The main contribution of this work is the analysis of the CFO compensation performance of efficient successive cancellation algorithms. We study the mean square symbol error, and derive a useful upper-bound of the compensation technique performance at convergence. This result extends the general convergence results for the space-alternating generalized expectation-maximization algorithm in the CFO compensation scenario. Finally, we validate the analysis with numerical simulations.

Appendices

Appendix 1: Proof of Lemma 1

Proof

By replacing ${\hat{\mathbf{s}}}_{q+1}$ by (10), we can rewrite (11) as

$$\begin{aligned} {[}{\mathbf{m}}_{q+1}{]}_n & = {\text {MSE}}\{ [{\hat{\mathbf{s}}}_{q+1}]_n | {\hat{\mathbf{s}}}_{l}: q-K+2\le l \le q \}\\ & = {\text {E} }\left\{ \left| \sum _{l=q-K+2}^{q} [{\varvec{A}}(l,q+1) \left( {\mathbf{s}}_l-{\hat{\mathbf{s}}}_l \right) ]_n + [{\tilde{\mathbf{z}}}]_n \right| ^2\right\} \end{aligned}$$

(21)

Note that ${{\varvec{\Psi }}}^{|q+1|_K}$ is not necessary due to the selection operation $[\cdot ]_n$.

Since the noise is independent of the transmitted data and has zero mean, (21) can be written as in the following expression

$${[}{\mathbf{m}}_{q+1}{]}_n = {\text {E}}\left\{ \left| \sum _{l=q-K+2}^{q} [{\varvec{A}}(l,q+1) \left( {\mathbf{s}}_l-{\hat{\mathbf{s}}}_l \right) ]_n \right| ^2\right\} + \sigma ^2$$

(22)

Using the Cauchy–Schwarz inequality in (22), the equation can be rewritten as

$${[}{\mathbf{m}}_{q+1}{]}_n \le (K-1) \sum _{l=q-K+2}^{q} \text {E}\left\{ \left| [{\varvec{A}}(l,q+1) \left( {\mathbf{s}}_l-{\hat{\mathbf{s}}}_l \right) ]_n \right| ^2\right\} + \sigma ^2$$

(23)

By defining

$$e_l(n)= [(\mathbf{s}_l-{\hat{\mathbf{s}}}_l)]_n, \text { and }$$

(24)

$$\begin{aligned} a_{l,q+1}(n)= & {} \text {IDFT}\{ 1, e^{j2\pi (\xi ^{|l|_K} - \xi ^{|q+1|_K})/N_c}, \ldots , e^{j2\pi (\xi ^{|l|}-\xi ^{|q+1|})(N_c-1)/N_c} \} \end{aligned}$$

(25)

(first column of ${\varvec{A}}(l,q+1)$) and considering n is allocated to user $|q+1|_K$, we can express the circular matrix multiplication of (23) as the following circular convolution

$$\begin{aligned}{[}\mathbf{m}_{q+1}{]}_n\le & {} (K-1) \sum _{l=q-K+2}^{q} \text {E}\left\{ \left| \sum _{p=0}^{N_c-1} a_{l,q+1}(|n-p|_{N_c}) e_l(p) \right| ^2\right\} + \sigma ^2 \end{aligned}$$

(26)

As the energy of the interference is located close to n, we consider only $N_h$ carriers adjacent to n, where $N_h$ is chosen to consider more than 99.9 % of the interference energy. Then, applying again the Cauchy–Schwarz inequality and assuming that the CFO is deterministic and unknown, and n is allocated to user $|q+1|_K$, (26) can be rewritten as

$$\begin{aligned}{[}\mathbf{m}_{q+1}{]}_n\le & {} (K-1)(2N_h+1) \sum _{l=q-K+2}^{q} \sum _{p=-N_h}^{N_h} \left| a_{l,q+1}(|n-p|_{N_c})\right| ^2 [\mathbf{m}_l]_{|n-p|_{N_c}} + \sigma ^2 \end{aligned}$$

(27)

where it is used that $\text {E}\left\{ |e_l(n)|^2\right\} = [\mathbf{m}_l]_{n}$. If we define the $N_c\times N_c$ banded convolution matrix $\mathbf{C}_{l,q+1}$, with elements

$$\begin{aligned}{[}\mathbf{C}_{l,q+1}{]}_{n,p} = \left\{ \begin{array}{ll} |a_{l,q+1}(|n-p|_{N_c})|^2 & \text {if}\; 0\le n\le N_c-1 \quad \text {and}\quad -N_h\le p\le N_h \\ 0 & \text {otherwise} \end{array} \right. \end{aligned}$$

(28)

the MSE as a function of the previous iterations results

$$\begin{aligned}{[}\mathbf{m}_{q+1}{]}_n\le (K-1)(2N_h+1) \sum _{l=q-K+2}^{q} [\mathbf{C}_{l,q+1} \mathbf{m}_l]_n + \sigma ^2 \end{aligned}$$

(29)

$\square$

Appendix 2: Maximum Value of C _sum

The maximum value of $C_{sum}$ is produced by a system with ICAS, where contiguous tiles belong to users with opposed CFO values, i.e. $\xi ^{(m_1)}=-\xi ^{(m_2)}$, if $m_1$ and $m_2$ are contiguous users. On the other hand, the maximum value of CFO that results in full-rank ${\varvec{\varPhi }}$ matrices is $|\xi ^{(m)}|<0.5$ [15].

Considering the worst case $|\xi ^{(m)}|=0.5$, and that $\xi ^{(1)}=-0.5$ (the CFO of the user allocated in the first tile is -0.5),¹ the matrix ${\bar{\mathbf{C}}}$ has the following structure

$$\begin{aligned} {\bar{\mathbf{C}}} = \begin{pmatrix} {\tilde{\mathbf{C}}}_{0,0} &{} {\tilde{\mathbf{C}}}_{0,1} &{} \cdots &{} {\tilde{\mathbf{C}}}_{0,N_t-1} \\ {\tilde{\mathbf{C}}}_{1,0} &{} {\tilde{\mathbf{C}}}_{1,1} &{} \cdots &{} {\tilde{\mathbf{C}}}_{1,N_t-1} \\ \vdots &{} &{} \ddots &{} \vdots \\ \tilde{\mathbf{C}}_{N_t-1,0} &{} \tilde{\mathbf{C}}_{N_t-1,1} &{} \cdots &{} {\tilde{\mathbf{C}}}_{N_t-1,N_t-1} \\ \end{pmatrix} \end{aligned}$$

(30)

where the only non zero matrices are ${\tilde{\mathbf{C}}}_a = {\tilde{\mathbf{C}}}_{1,0} = {\tilde{\mathbf{C}}}_{3,2} = \cdots = {\tilde{\mathbf{C}}}_{N_t-1,N_t-2}$ and ${\tilde{\mathbf{C}}}_b = {\tilde{\mathbf{C}}}_{0,1} = \tilde{\mathbf{C}}_{2,3} = \cdots = {\tilde{\mathbf{C}}}_{N_t-2,N_t-1}$, with dimension $L_t\times L_t$ ; and values

$$\begin{aligned} {\tilde{\mathbf{C}}}_a = \begin{pmatrix} 0 &{} 0 &{} \cdots &{} 1 \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix}\end{aligned}$$

(31)

$$\begin{aligned} \tilde{\mathbf{C}}_b = \begin{pmatrix} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 1 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} \end{aligned}$$

(32)

As in each column or row of (30) there is either one ${\tilde{\mathbf{C}}}_a$ or one ${\tilde{\mathbf{C}}}_b$, it is easy to see that the worst-case value results $C_{sum}=KN_t$.