Power Control for Mobile Phones - Essay Example

Section Number Page Number Section I Introduction Section II Interference Constraints 3 Section III Minimum Power Assignment Method 4 Section IV Synchronous Power Control using Minimum Power Assignment Method Theorem 1 Lemma 1 Lemma 2 Theorem 2 5 5 6 7 8 Section V Asynchronous Power Control using Minimum Power Assignment Method Assumptions Theorem 3 Theorem 4 9 9 10 10 Section VI Conclusion 12 Section I: Introduction: Cellular communication devices, more popularly known as mobile phones, have transformed the face of communication in the 21st century. They have now become a necessity rather than a luxury that was enjoyed by the privileged few. As the number of cellular subscribers increases exponentially, there are several issues that have to be addressed to ensure effective and efficient usage of this technology while providing the users with a satisfying communication experience. This paper addresses the issue of transmission power control in cellular communication devices and is based entirely on the work of Roy D. Yates. In cellular communication systems, the power transmitted by a communication device has to be regulated in such a way as to provide an adequate connection to the user, which, in turn, is accomplished by reducing the interference caused by other such communication devices. There are three main bodies of work addressing this concern, firstly, the user can be assigned a fixed base station, secondly, the user can be iteratively assigned to the base station which has the maximum signal to interference ratio and thirdly, a user's signal is a combination of signals from various base stations. This paper aims to concentrate upon the second method mentioned above, i.e. Minimum Power Assignment, which is essentially an iterative method of obtaining the optimum signal strength by dynamically assigning the user to the base station that provides the best signal to interference ratio. In this model, the problem of uplink power control will be reduced to finding a vector (p) of users' transmitter power such that it satisfies the following condition: p>=I (p) Where constraint number j i.e. pj Ij (p) describes the interference that must be surmounted by user j to obtain an acceptable connection. Section II: Interference Constraints: Assumptions and Symbols Used: 1. Number of users = N 2. Number of base stations = M 3. Transmitted power of user j = pj 4. Gain of user j to base k = hkj 5. Receiver noise at base station = 6. Signal to interference Ratio = SIR The general interference constraints that apply to any system state that at a base station (k), a user receives a signal of power (hKjPj). At the same time, the user experiences an interference that can be denoted by ij hkipi + k. Thus, the SIR of a user j at base station (k) and under the power vector (p) can be denoted as pjkj(p) where, hkj kj(p) = ------ ij hkipi + k Section III: Minimum Power Assignment (MPA) Method: The MPA is an iterative procedure and at each step, the user is assigned to the base station at which its SIR is optimized. We can analyze the MPA for two situations: 1. Continuous power adjustment 2. Discrete power adjustment If we denote target SIR for a user by , then, the SIR constraint of a user following the MPA procedure can be denoted as: j pj I j MPA (p) = min k --- kj(p) According to the MPA iteration, i.e. p (t +1) = I MPA( p(t)), the user is assigned to the base station where the power consumed by the mobile device to attain its target SIR i.e. is minimized. This iteration is applicable with the assumption that other users corresponding to the same base station are currently maintaining a fixed transmission power level. We shall now examine the two cases where MPA is applicable in more detail. Section IV: Synchronous Iterative Power Control using Minimum Power Assignment method: Here, we assume that I(p), i.e., the interference to be overcome in order to attain an acceptable connection is a standard interference function, and based on this assumption, we examine the properties of the standard power control algorithm (p(t +1) = I MPA( p(t)) ). This section is based on the analysis of continuous power adjustment performed by Yates, Huang and Hanly. Assuming that we begin synchronous iterative power control with an initial power vector, say (p), after an arbitrary number of iterations say n of the standard power control algorithm, we obtain a new power vector value: In(p). The following are the theorems related to the convergence of the sequence In(p). Theorem 1: If the standard power control algorithm has a fixed point, then, that fixed point is unique Assumptions: p and p' are distinct fixed points Additional Properties of standard deviation function I(p) used: 1. Monotonicity: If p p', then, I(p) I(p') 2. Positivity: I(p)>0 Proof : Step 1: I(p)>0 for all values of power vector p (by the positivity property) Step 2: From step 1 we can conclude that pj >0 (2.1) And p'j >0 (2.2) Step 3: We can assume without loss of generality that there exists a user j such That pj < p'j. (3.1) Step 4: We can conclude from Step 3 that there must also exist a numeric value >1 such that p p' and thus, for some user j, pj pj' Step 5: According to the monotonicity and scalability properties: pj' = I(p') Ij(p) < Ij(p) = pj But, pj' = pj Hence, we find a contradiction which implies that the fixed point must be unique, thus proving the theorem. Lemma 1: If p is a feasible power vector, then, In(p) is a monotonic decreasing sequence of feasible power vectors that converges to a unique fixed point p*. Proof: Step 1: Assume p(0) = p and p(n) = In(p). Feasibility of p implies that p(0) p(1) Step 2: Suppose that p(n-1) P(n) (as a derivation from step 1) Monotonicity implies that I(p(n-1)) I(p(n)) More specifically, p(n) I(p(n)) = p(n+1) Step 3: Thus, p(n) is a decreasing sequence of feasible power vectors. Since the sequence has a lower bound of 0, we can conclude from Theorem 1 that the sequence must converge to a unique fixed point. We shall denote this point by the symbol p*. Interpretation of Lemma 1: Lemma 1 implies that p p* for any feasible power vector. This further implies that p* is the solution of p I(p) and corresponds to the minimum total tranmitted power corresponding to a user under power vector p. This implication is desirable in the case of cellular devices as they mostly work on portable batteries with a fixed amount of stored power and Lemma 1 helps to attain maximum power efficiency for these mobile communication devices. Lemma 2: If I(p) is feasible, then starting from the all zero vector i.e. z, the standard power control algorithm produces a monotonic increasing sequence of power vectors In (z) which converges at p*. Proof: Step 1: Assume that z(n) = In (z). From our initial assumption as stated in lemma 2, it is known that z(0) < p* and that z(1) = I(z) z. Step 2: We can thus suppose that p* z(n) . z(1) z. Step 3: The property of monotonicity implies that: p* = I(p*) I(z(n)) I(z(n-1)) = z(n) i.e. p* z(n+1) z(n) Step 4: Hence, it is found that the sequence corresponding to z(n) is non decreasing, i.e. monotonically increasing and has an upper bound of the value of p*. Theorem 1 also implies that the sequence must converge to p* thus proving lemma2. Theorem 2: If I(p) is feasible, then for any initial power vector p, the standard power control algorithm converges to a unique fixed point p*. Proof: Step 1: Feasibility of I(p) implies the existence of the unique fixed point p*( conclusion from theorem 1). Step 2: We know from lemma 1 and lemma 2 that p*j >0 for all j. Thus, for any initial power vector p, a numeric value can be found such that p* p. Step 3: Extending the result of step 2 and applying the scalability property to it, we can find that p* must be feasible. Step 4: We know that z p p*. The monotonicity property implies that In(z) In(p) In(p*) Step 5: Lemmas 1 and 2 imply that Limn In(p*) = limn In(z) = p* Hence, the theorem stands proven. Section V: Asynchronous Power Control using Minimum Power Assignment: The main issue faced while using the synchronous form of MPA is that users having communication devices with slower processing capabilities may not be able to continuously update the value of the optimal power vector and this may adversely affect the overall power consumption of users under a base station. To counter this problem, an alternative method i.e. asynchronous power control algorithm has been developed. This algorithm utilizes the Totally Asynchronous Model. It allows some users to perform a greater number of power control iterations at a faster rate than other users. Assumptions: 1. Power transmission of user j at time t = pj(t) 2. Power vector at time t = p(t) = (p1(t), , pn(t)) 3. Its assumed that user j may not have access to the most recent values of the components of the power vector at time t. (when user j has out dated information about the received power at a certain base) 4. Most recent time for which pi is known to user j = ij(t). (0 ij(t) t) 5. User j adjusts its transmitter power at time t using the power vector p(j (t)) = (p1(1j (t)), p2(2j (t)), , pn((nj (t))) 6. We also assume a set of times = {0, 1, 2, } at which one or more components pj(t) of p(t) are updated. 7. j is the set of times at which pj(t) is updated Given the above assumptions and the sets 1, 2,, n, the Totally Asynchronous Standard Power Control theorem states that pj(t+1) = Ij(p(j(t)) if t is a subset of T pj(t+1) = pj(t) in other cases Theorem 3 (Asynchronous Convergence Theorem): If there is a sequence of non-empty sets {X(n)} with X(n+1) being a subset of X(n) for all values of n satisfying the following two conditions: 1. Synchronous Convergence Condition: For all n and x X(n), f(x) X(n+1). If {yn} is a sequence such that yn X(n) for all n, then every limit point of {yn} is a fixed point of f. 2. Box Condition: For every n, there exists set Xi(n) X, such that X(n)= X1(n) x X2(n) x .x Xn(n), And the initial solution estimate x(0) belongs to the set X(0), then every limit point of {x(t)} is a fixed point in f. Theorem 4: If I(p) is feasible, then, from any initial power vector p, the asynchronous standard power control algorithm converges to p*. Proof: Step 1: let z denote the all zero vector. Feasibility implies the existence of the fixed point p*. Step 2: Given an initial power vector p, we can choose 1 such that p* p. Step 3: We define X(n) = { p| In(z) p In(p*)} For all n, the set satisfies the box condition. Step 4: Lemma 1 and lemma 2 imply that X(n+1) is a subset of X(n) for all n and limn In(p*) = limn In(z) = p* . Thus, any sequence {p(n)} such that p(n) X(n) for all n must converge to p*. Hence, the theorem stands proven. Section VI: Conclusion: The aim of the minimum power assignment method is to provide users of cellular communication services with an acceptable connection with minimum usage of power on the side of the user so as to elongate battery discharge time, i.e. the amount of time that it takes for the battery of the cellular device to get discharged. The use of the synchronous and asynchronous standard power control algorithms guarantee to find the minimal power level required for establishing an acceptable connection given that the interference from other users that needs to be over come to do so, characterized by the value I(p) is feasible. Asynchronous power control also indicates the robustness of the standard power control algorithm in the face of slower updating of transmission power levels by individual users. The power controlled systems and methods that are described in this report can be implemented by each individual user knowing only its own uplink gains and the total power received at each base station, i.e. a user need not be concerned about the power transmissions of other users corresponding to the same base station, thereby making the algorithm much faster and much less complex. Further development in the field of power control for mobile devices holds the potential to decrease the need for repeated recharging of cellular communication devices and of elongating the standby and talk time simultaneously. Read More