Subspace based DOA estimation of DS-CDMA signals

Abstract

This paper presents a subspace blind method to estimate the direction of arrival of direct sequence code division multiple access signals in a multipath fading environment. The proposed method is based on signal/noise subspace approach. For enhancing the results, the problem is formulated based on both signal and noise subspaces to exploit structures of desired signal, self interference and multiple access interference simultaneously. The main idea in this paper is based on fitting the extended subspace spanned by the desired signature waveform in all different paths into the estimated extended signal subspace. The proposed method is blind in the sense that it utilizes only the desired user’s code and its corresponding path delays. The performance of the proposed method is compared to the Cramer–Rao lower bound. We also propose a method for estimating relative power of different paths in multipath code division multiple access signals.

6 Appendix

Using (7), $z_k^m(i,j)$ can be written as (55),

$$\begin{aligned} z_k^m(i,j)= & {} \sum \limits _{\hat{i}=0}^{Q-1}\sum \limits _{\hat{k}=1}^{K} A_{\hat{k}}b_{\hat{k}}(\hat{i})\sum \limits _{l=1}^{L}a_{\theta _{\hat{k}l,m}}g_{\hat{k}l} r_{s_{\hat{k}}\psi }((i-\hat{i})T_b\nonumber \\{} & {} +jT_c-(\tau _{\hat{k}l}-\tau _{k1}))\nonumber \\{} & {} + \underbrace{\sigma r_{n_{m}\psi }(iT_b+jT_c+\tau _{k1})}_{n_k^m(i,j)} \end{aligned}$$

(55)

where $a_{\theta _{kl},m}$ is mth element of $\varvec{a}_{\theta _{kl}}$ and $r_{fg}(\tau )$ is the cross correlation function between f and g, which is defined as below:

$$\begin{aligned} r_{fg}(\tau )=\int \limits _{0}^{T_c}f(t+\tau )g^*(t)\textrm{d}t. \end{aligned}$$

(56)

To be more specific, $z_k^m(i,j)$ in (55) can be broken down into 4 terms in (57). The first three terms in (57) is related to kth user and the 4th term is effect of other users in filter output.

$$\begin{aligned}{} & {} z_k^m(i,j){}={} A_kb_k(i-1)\nonumber \\{} & {} \quad \sum _{l=2}^{L}a_{\theta _{kl,m}}g_{kl}\,r_{s_{k}\psi }(T_b+jT_c-(\tau _{kl}-\tau _{k1}))\nonumber \\{} & {} \quad {+}\,A_kb_k(i)\sum _{l=1}^{L}a_{\theta _{kl,m}}g_{kl}\,r_{s_{k}\psi }(jT_c-(\tau _{kl}-\tau _{k1}))\nonumber \\{} & {} \quad {+}\,A_kb_k(i+1)\sum _{l=1}^{L-1}a_{\theta _{kl,m}}g_{kl}\,r_{s_{k}\psi }(-T_b+jT_c-(\tau _{kl}-\tau _{k1}))\nonumber \\{} & {} \quad {+}\,\mathop {\sum \sum }_{(\hat{i},\hat{k})\notin \{(i-1,k),(i,k),(i+1,k)\}}\nonumber \\{} & {} \quad \underbrace{A_{\hat{k}}b_{\hat{k}}(\hat{i})\sum \limits _{l=1}^{L}a_{\theta _{\hat{k}l,m}}g_{\hat{k}l} \,r_{s_{\hat{k}}\psi }((i-\hat{i})T_b+jT_c-(\tau _{\hat{k}l}-\tau _{k1}))}_{f_m(\hat{i},\hat{k},j)}\nonumber \\{} & {} \quad {+}\,n_k^m(i,j) \end{aligned}$$

(57)

Therefore, $\varvec{z}_k^m(i)$ can be expressed as

$$\begin{aligned} \varvec{z}_k^m(i)= & {} A_kb_k(i-1)\sum _{l=2}^{L}a_{\theta _{kl,m}}g_{kl}\varvec{c}_l^{k,1}\nonumber \\{} & {} {+}\,A_kb_k(i)\sum _{l=1}^{L}a_{\theta _{kl,m}}g_{kl}\varvec{c}_l^{k,2}\nonumber \\{} & {} {+}\,A_kb_k(i+1)\sum _{l=1}^{L-1}a_{\theta _{kl,m}}g_{kl}\varvec{c}_l^{k,3}\nonumber \\{} & {} {+}\,\mathop {\sum \sum }_{(\hat{i},\hat{k})\notin \{(i-1,k),(i,k),(i+1,k)\}}\varvec{f}_m(\hat{i},\hat{k})\nonumber \\{} & {} {+}\,\varvec{n}_k^m(i) \end{aligned}$$

(58)

where $\varvec{f}_m(\hat{i},\hat{k})=[f_m(\hat{i},\hat{k},0),...,f_m(\hat{i},\hat{k},N_{\text {ex}}-1)]^T$, $\varvec{n}_k^m(i)=[n_k^m(i,0),...,n_k^m(i,N-1)]^T$ and $\varvec{c}_l^{k,1}$, $\varvec{c}_l^{k,2}$ and $\varvec{c}_l^{k,3}$ are column vectors, which can be interpreted as augmented version of signature waveforms to produce pattern of the whole frame defined as (59)-(61).

where $\gamma = \dfrac{\tau _{kl}-\tau _{k1}}{T_c}$. Therefore it can be seen that $\varvec{z}_k(i)$ defined in (12) can be written as

$$\begin{aligned} \varvec{z}_k(i)= & {} A_kb_k(i-1)\sum _{l=2}^{L}g_{kl}\varvec{a}_{\theta _{kl}}\otimes \varvec{c}_l^{k,1}\nonumber \\{} & {} {+}\,A_kb_k(i)\sum _{l=1}^{L}g_{kl}\varvec{a}_{\theta _{kl}}\otimes \varvec{c}_l^{k,2}\nonumber \\{} & {} {+}\,A_kb_k(i+1)\sum _{l=1}^{L-1}g_{kl}\varvec{a}_{\theta _{kl}}\otimes \varvec{c}_l^{k,3}\nonumber \\{} & {} {+}\,\mathop {\sum \sum }_{(\hat{i},\hat{k})\notin \{(i-1,k),(i,k),(i+1,k)\}}\varvec{f}(\hat{i},\hat{k})\nonumber \\{} & {} {+}\,\varvec{n}_k(i) \end{aligned}$$

(62)

where $\otimes$ is the Kronecker product and $\varvec{f}(\hat{i},\hat{k})=[\varvec{f}_1(\hat{i},\hat{k}),...,\varvec{f}_M(\hat{i},\hat{k})]^T$ and is interference of other users in $\varvec{z}_k(i)$ (MAI) and $\varvec{n}_k(i)=[\varvec{n}_k^1(i),...,\varvec{n}_k^M(i)]^T$. As aforementioned above, the first three terms in (62) is the contribution of kth user in $\varvec{z}_k(i)$. Therefore mth element of $\varvec{z}_k(i)$ in (62), can be written as (63). From (63), the extended vector $\varvec{z}_k(i)$ can be written as (64).

$$\begin{aligned}{} & {} \varvec{z}_k^m(i){}={}b_k(i)A_k \lbrace g_{k1} a_{\theta _{k1,m}} \varvec{c}^{k,2}_1 + g_{k2} a_{\theta _{k2,m}} \varvec{c}^{k,2}_2 +... \nonumber \\{} & {} \quad + g_{kL} a_{\theta _{kL,m}} \varvec{c}^{k,2}_L \rbrace \nonumber \\{} & {} \quad {+}\,b_k(i-1)A_k \lbrace g_{k2} a_{\theta _{k2,m}} \varvec{c}^{k,1}_2 + g_{k3} a_{\theta _{k3,m}} \varvec{c}^{k,1}_3 +... \nonumber \\{} & {} \quad + g_{kL} a_{\theta _{kL,m}} \varvec{c}^{k,1}_L \rbrace \nonumber \\{} & {} \quad {+}\,b_k(i+1)A_k \lbrace g_{k1} a_{\theta _{k1,m}} \varvec{c}^{k,3}_1 \nonumber \\{} & {} \quad + g_{k2} a_{\theta _{k2,m}} \varvec{c}^{k,3}_2 +... + g_{k(L-1)} a_{\theta _{k(L-1),m}} \varvec{c}^{k,3}_{L-1} \rbrace \nonumber \\{} & {} \quad {+}\,\mathop {\sum \sum }_{(\hat{i},\hat{k}) \notin \{(i-1,k),(i,k),(i+1,k)\}}\varvec{f}_m(\hat{i},\hat{k})\nonumber \\{} & {} \quad {+}\,\varvec{n}_k^m(i) \end{aligned}$$

(63)

(64)

Finally from 64, the decomposition of (62) can be reformulated as following form:

$$\begin{aligned} \varvec{z}_k(i){}={}b_k(i)\varvec{u}_k+\underbrace{\ldots }_{\text {the rest of terms}} \end{aligned}$$

(65)

where $\varvec{u}_k$ is expressed as below:

$$\begin{aligned} \varvec{u}_k&{}={}&B_k\varvec{\alpha } \end{aligned}$$

(66)

where $B_k$ and $\varvec{\alpha }$ is defined as (15) and (16) respectively.

Therefore (65) can be written as (67).

$$\begin{aligned} \varvec{z}_k(i)= & {} b_k(i)\varvec{u}_k(i)+b_k(i-1)\varvec{u}_k(i-1)+b_k(i+1)\varvec{u}_k(i+1)\nonumber \\{} & {} {+}\,\mathop {\sum \sum }_{(\hat{i},\hat{k})\notin \{(i-1,k),(i,k),(i+1,k)\}}\varvec{f}(\hat{i},\hat{k})\nonumber \\{} & {} {+}\,\varvec{n}_k(i) \end{aligned}$$

(67)

Using the whiteness assumption (8), the covariance matrix of noise can be calculated as following:

$$\begin{aligned} \text {cov}(\varvec{n}_k(i),\varvec{n}_{k^\prime }(i)){}={}E\{\varvec{n}_k(i)\varvec{n}_{k^\prime }(i)^H\}\nonumber \\ {=}\, {\left\{ \begin{array}{ll} I_M \otimes F_k^{k^{\prime }}, \, \text {if} \, |\tau _{k1}-\tau _{k^{\prime }1}|< T_b \\ 0, \text {if} |\tau _{k1}-\tau _{k^{\prime }1}|\ge T_b \end{array}\right. } \end{aligned}$$

(68)

where

$$\begin{aligned} F_k^{k^{\prime }}(s,q)= & {} E\{n_k^m(i,s)n_{k^\prime }^{m*}(i,q)\}\nonumber \\= & {} \sigma ^2\int \limits _{0}^{T_c}\psi ^*(t)\psi (t-(\tau _{k^\prime 1}-\tau _{k1})-(q-s)T_c)\textrm{d}t\nonumber \\= & {} \sigma ^2r_{\psi \psi }((\tau _{k1}-\tau _{k^\prime 1})+(s-q)T_c) \end{aligned}$$

(69)

By these results we can find the autocorrelation of the noise vector $\varvec{n}_k(i)$ as following:

$$\begin{aligned} E\{\varvec{n}_k(i)\varvec{n}_k(i^\prime )^H\}={\left\{ \begin{array}{ll} \dfrac{\sigma ^2}{N}I_{MN_{ex}}, \text {if} i=i^\prime \\ 0, \text {if} i\ne i^\prime . \end{array}\right. } \end{aligned}$$

(70)