Unified framework for the analysis and design of linear uplink power control in CDMA systems

Abstract

In this work, it is proposed a unified framework to design and analyze uplink distributed power control schemes over flat-fading channels from a control theory perspective. The effects of linear detectors and round trip delays are explicitly characterized in this study. First, the optimal solution to the power minimization problem under signal to interference-noise ratio (SINR) restrictions is reviewed, where sufficient conditions for its existence are presented that depends on the detection strategy. Four different linear detection schemes are studied in this work: Matched Filter, Decorrelator, MMSE and Projector. Specifically, two special cases are analyzed with respect to the spreading codes properties: uniform cross-correlation and orthogonal codes, and under both conditions an explicit expression for the central solution is obtained. Nevertheless, one drawback of the central solution is its lack of robustness against channel estimation errors, transport delays and noise. Hence, it is proposed closed-loop control laws with linear power assignment which are capable of provide robustness to these channel effects. It is then presented that under certain conditions, stable feedback loops can be obtained considering SINR quantification, transmission and processing delays, and the resulting closed-loop power solutions tend to the central ones. Finally, it is illustrated that the selection of the linear detectors does not affect the resulting closed-loop dynamics, but the uplink transmission power in steady-state. An exhaustive simulation evaluation is included to validate the mathematical analysis presented for open and closed-loop solutions.

Appendix

1.1 Proof of Proposition 1

First, it is rewritten the representation of the estimated data bit vector in (3) as [31, 34]:

$$ \hat{\bf b}[k]=\varvec{\Uplambda}_{d} \tilde{\bf b}[k]+ \varvec{\Uplambda}_{nd}\tilde{\bf b}[k]+{\varvec{\mu}}\left[k\right]. $$

(42)

Then, to measure the SINR, the following matrices are defined:

• $\varvec{\Uplambda}_{d}=diag(\mathbf{X}\mathbf{C})$ represents a diagonal matrix that contains the information about the estimated users’ amplitudes.

• $\varvec{\Uplambda}_{nd}=\mathbf{X}\mathbf{C}-\varvec{\Uplambda}_{d}$ denotes the matrix XC with zeros on its diagonal (MAI component).

• $\varvec{\mu}\left[k\right]$ characterizes the noise samples obtained from $\mathbf{X} \mathbf{n}\left[k\right].$

In specific, for the ith user, the representation of the previous estimated data bit vector is given by

$$ {\hat{\bf b}}_i[k]=\left[\varvec{\Uplambda}_{d}{\tilde{\bf b}}[k]\right] _i+ \left[\varvec{\Uplambda}_{nd}{\tilde{\bf b}}[k]\right]_i+\left[{\mathbf{n}}[k]\right]_i, $$

(43)

where $\left[\cdot\right]_i$ denotes the ith element in the vector. Therefore, the SINR factor at the detector output for ith active user is given by

$$ \gamma_i=\frac{E\{([\varvec{\Uplambda}_{d}{\tilde{\bf b}}(k)]_i)^{2}\}} {E\{ ([\varvec{\Uplambda}_{nd}{\tilde{\bf b}}(k)]_i )^{2}\}+ E\{ ([{\varvec{\mu}}(k)]_i )^{2}\}}. $$

(44)

where $E\{(\cdot)^2\}$ is the expected energy. Thus, each component in (44) can be expressed in terms of the SVD representation of $\mathbf{C}.$ In particular, since $\mathbf{W}_1^* \mathbf{W}_1=\mathbf{I} \Rightarrow \mathbf{C}^T \mathbf{C}= \mathbf{V} \varvec{\Upsigma}_1^2 \mathbf{V}^*,$ and the results in (5)–(7) are obtained.$\square$

1.2 Proof of Proposition 2

First note that

$$ \frac{\partial \gamma_i[k]}{\partial p_i[k]} = \frac{\delta_{ii} \,|h_i[k]|^{2}}{\sum_{j\neq i}{\delta_{ij}p_j[k]|h_j[k]|^{2}}+\chi_i \sigma^2} > 0 $$

(45)

then the optimal solution for (9) is achieved at the boundary of the inequalities condition, i.e. $\gamma_i[k]=\gamma_i^{obj}.$ Note that (4) can be expressed by

$$ \varvec{\Upgamma}_{bit} {\mathbf{H}}[k] {\mathbf{p}}[k]= \varvec{\Uplambda}_{obj} \left \{ \varvec{\Upgamma}_{MAI} {\mathbf{H}}[k] {\mathbf{p}}[k] + {\varvec{\eta}} \right \} $$

(46)

$$ \Rightarrow \left \{ \varvec{\Uplambda}_{obj}^{-1}\varvec{\Upgamma}_{bit} - \varvec{\Upgamma}_{MAI} \right \} {\mathbf{H}}[k] {\mathbf{p}}[k] = {\varvec{\eta}} $$

(47)

Hence, in order to guarantee a unique solution to the previous system of linear equations, it is needed $\Uppsi \,\triangleq\, \left \{ \varvec{\Uplambda}_{obj}^{-1}\varvec{\Upgamma}_{bit} - \varvec{\Upgamma}_{MAI} \right \}$ to be non-singular. A sufficient condition is for $\Uppsi$ to be diagonally dominant [35], i.e.

$$ \frac{\delta_{ii}}{\gamma_i^{obj}} > \sum_{j=1, j\neq i}^U \delta_{ij} \quad i=1,\ldots,U $$

(48)

which is equivalent to (8). Finally, by a direct inversion, the result in (10) is deduced.$\square$

1.3 Proof of Proposition 3

Recalling that by the SVD decomposition of C, it is satisfied $\mathbf{C}^T \mathbf{C}= \mathbf{V} \varvec{\Upsigma}_1^2 \mathbf{V}^*,$ and consequently

$$ {\tilde{\bf v}}_i^{*} \varvec{\Upsigma}_1^2 {\tilde{\bf v}}_j = \left \{ \begin{array}{ll} 1 & i=j \\ -\epsilon & i\neq j \end{array} \right . $$

( 49)

Hence, components δ _ii  = α, δ _ij  = β and $\chi_i=\zeta$ in (5)–(7) are independent of indexes i and j. Next, by using the SINR definition in (4), it is obtained

$$ \beta \sum_{j=1}^U |h_j[k]|^2 p_j[k]+ \zeta \sigma^2 = \left( \frac{\alpha}{\gamma_i^{obj}}+\beta \right ) |h_i[k]|^2 p_i[k] \quad \forall i $$

(50)

and as a consequence

$$ \left( \frac{\alpha}{\gamma_i^{obj}}+\beta \right ) |h_i[k]|^2 p_i[k] = \left( \frac{\alpha}{\gamma_j^{obj}}+\beta \right ) |h_j[k]|^2 p_j[k] \quad \forall i \neq j $$

(51)

By a direct substitution into (50), it is deduced the central solution

$$ p_i^*[k]=\frac{\zeta \sigma^2 / |h_i[k]|^2}{\left( \frac{\alpha}{\gamma_i^{obj}}+\beta \right ) \left [1- \beta \sum_{j=1}^U \left( \frac{\alpha}{\gamma_i^{obj}}+\beta \right )^{-1} \right ] } $$

(52)

Finally, the expressions for α, β and ζ can be derived from (5)–(7) for each linear detector; and by a substitution of these expressions into (52), the central solution in each case is concluded.$\square$

1.4 Proof of Proposition 4

From Fig. 2, it can be deduced that

$$ {\mathbf{p}}[z] = z^{-n_m-n_p} {\mathbf{C}}(z) \left[ z^{-n_m-n_p} {\mathbf{C}}(z) - {\mathbf{I}} \right ]^{-1} \hat{{\mathbf{p}}}[z]. $$

(53)

where $\mathbf{p}[z]=\mathcal{Z}\{\mathbf{p}[k]\}$ and $\hat{\mathbf{p}}[z]=\mathcal{Z}\{{\hat{\bf p}}[k]\}.$ As a result, if conditions I. and II. are satisfied, then $\mathbf{p}[k] \approx \hat{\mathbf{p}}[k]$ in steady state. Next, by a direct substitution in (25), the convergence to the central solution in (10) can be concluded.$\square$

1.5 Proof of Proposition 5

First, the stability and convergence of the Foschini-Miljanic control law [33] will be derived. The corresponding closed-loop transfer function of the ith active user by considering (33) is

$$ T_i(z) =\frac{ \theta z^{-n_m-n_p-1}}{1-z^{-1}+ \theta z^{-n_m-n_p-1}} $$

(54)

Then, it is satisfied

$$ \lim_{z \rightarrow 1} T_i(z)=1, $$

(55)

which ensures reference tracking for the QoS requirement. Therefore, the closed-loop stability is characterized by the roots of the polynomial

$$ \rho(z)=z^{n_m+n_p+1}- z^{n_m+n_p}+\theta. $$

(56)

According to [38], the necessary conditions for stability are

• ρ(1) > 0,

• $(-1)^{n_m+n_p+1}\rho(-1)>0$

which imply that θ > 0. On the other hand, by re-writing the characteristic equation in root-locus format [38], it is obtained

$$ 1+\frac{k_i}{z^{n_m+n_p}(z-1)}=0 $$

(57)

Therefore, the n _m  + n _p roots of ρ(z) will move away from the origin and one from z = 1, but they will remain inside the unit circle until a limiting gain is reached. In order to perfectly characterize this value, it is then computed the roots of ρ(z) on the unit circle $\left . \rho(z) \right |_{z=e^{j \theta}}=0.$ Consequently, it is deduced

$$ k_i=e^{j \theta n_m+n_p}\left (1-e^{j \theta} \right ), $$

(58)

where by taking the absolute value and argument, at both sides of the previous equation, it is obtained

$$ k_i^2 = (1-\cos \theta)^2+\sin^2\theta $$

(59)

$$ 0 = \theta n_m+n_p+ \arctan \left ( \frac{-\sin \theta}{1 - \cos\theta}\right ). $$

(60)

After some algebraic manipulations, these two equation are equivalent to

$$ k_i^2 = 2(1-\cos \theta) $$

(61)

$$ 0 = \tan (\theta n_m+n_p) \tan (\theta/2)-1, $$

(62)

and then it is concluded θ = π/(2(n _m  + n _p ) + 1), and in consequence, the upper bound in (34) is reached.

Next, the stability and convergence proof of the LQ algorithm is introduced. By a direct substitution of (35) into (30), it is derived ∀ i

$$ T_i(z) = \frac{\Upomega z^{-n_m-n_p-1}}{1-(1-\Upomega)z^{-1}} $$

(63)

Consequently, T _i (z) has a pole at $(1-\Upomega)$ and if $0<\Upomega \leq 1,$ the pole is always inside the unit-circle to guarantee stability. Finally, from (63), it is easy to see that $\hbox{lim}_{z \rightarrow 1}T_i(z)=1.$ Hence, the restrictions in Proposition 4 are fulfilled for both controllers in (33) and (35).$\square$