Control theory aspects of power control in UMTS

Control Engineering Practice 11 (2003) 1113–1125

Control theory aspects of power control in UMTS$ Fredrik Gunnarsson*, Fredrik Gustafsson Division of Control & Communication, Department of Electrical Engineering, Linkopings Universitet, SE-581 83 Linkoping, Sweden . . Accepted 12 March 2003

Abstract The global communication systems critically rely on control algorithms of various kinds. In universal mobile telephony system (UMTS)—the third generation mobile telephony system just being launched—power control algorithms play an important role for efficient resource utilization. This survey article describes and discusses relevant aspects of UMTS power control with emphasis on practical issues, using an automatic control framework. Generally, power control of each connection is distributedly implemented as cascade control, with an inner loop to compensate for fast variations and an outer loop focusing on longer term statistics. These control loops are interrelated via complex connections, which affect important issues such as capacity, load and stability. Therefore, both local and global properties are important. The concepts and algorithms are illustrated by simple examples and simulations. r 2003 Elsevier Ltd. All rights reserved. Keywords: Power control; Wireless networks; Distributed control; Stability; Time delays; Smith predictor; Disturbance rejection

1. Introduction The use of control theory applied to communication systems is increasingly popular. More complex networks are being deployed and the critical resource management constitutes numerous control problems. Wireless networks are for example pointed out as a new vistas for systems and control in Kumar (2001). This paper describes power control as standardized for UMTS and provides control theory aspects of the same with an extensive list of citations. It gives an overview from a control perspective of achievements in the area to date with some illuminating examples and pointers to interesting open issues. The power of each transmitter in a wireless network is related to the resource usage of the link. Since the links typically occupies the same frequency spectrum for efficiency reasons, they mutually interfere with each other. Proper resource management is thus needed to utilize the radio resource efficiently. Several aspects $ This work was supported by the Swedish Agency for Innovation Systems (VINNOVA) and in cooperation with Ericsson Research within the competence center ISIS, which all are acknowledged. *Corresponding author. Tel.: +46-13-284-028; fax: +46-13282-622. E-mail addresses: [email protected] (F. Gunnarsson), [email protected] (F. Gustafsson).

0967-0661/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0967-0661(03)00062-5

discussed here are generally applicable, but the focus is on UMTS and the radio interface WCDMA (wideband code division multiple access). More on Radio Resource Management in general can for example be found in Holma and Toskala (2000) and Zander (1997), and with a power control focus in Rosberg and Zander (1998), Hanly and Tse (1999), Gunnarsson (2000) and Gunnarsson and Gustafsson (2002). The radio access network in UMTS is based on DSCDMA (Direct Sequence Code Division Multiple Access), where the links in the system share the same frequency spectrum and are assigned codes to allow the receivers to recover the desired signals. A simplified radio link model is typically adopted to emphasize the network dynamics of power control. The transmitter is using the power pðtÞ % to transmit data scrambled by a user-specific code. The channel is characterized by the power gain gðtÞ (o1), where the ‘‘bar’’ notation % indicates linear scale, while gðtÞ ¼ 10 log10 ðgðtÞÞ is in % logarithmic scale (dB). Correlating the received signal with the code, the receiver extracts the desired signal, % ¼ pðtÞ which has the power CðtÞ and is also subject % gðtÞ; % % from other connections. to an interfering power IðtÞ The perceived quality is related to the signal-to-inter% ference ratio1 (SIR) g% ðtÞ ¼ pðtÞ IðtÞ: For error-free % gðtÞ= % 1

In dB: gðtÞ ¼ pðtÞ þ gðtÞ  IðtÞ:

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transmission (and if the interference can be assumed Gaussian), the achievable data rate RðtÞ is limited from above by (Shannon, 1956) RðtÞ ¼ W log2 ð1 þ g% ðtÞÞ ðbits=sÞ;

g% 1 ðtÞ ¼

where W is the bandwidth in Hz. From a link perspective, power control can be seen as means to compensate for channel variations in gðtÞ: The link % objective with power control can for example be *

*

*

to maintain constant SIR and thereby constant data rate; to use constant power and variable coding to adapt the data rate to the channel variations; to employ scheduling to transmit only when the channel conditions are favorable.

This also depends on the data rate requirements from the service in question. 1.1. Example: two users in a single cell Power control objectives are rather different when considering networks and not only links. Furthermore, the situation is also different in the uplink (mobile to base station) and in the downlink (base station to mobile). Consider the simplistic downlink situation with two mobiles and a single base station in Fig. 1. Apart from information dedicated to each mobile, the base station also broadcast information to all users in the cell. This common information is transmitted with the power P% C ðtÞ; while the dedicated information to the two mobiles is transmitted using the power P% D ðtÞ ¼ p% 1 ðtÞ þ p% 2 ðtÞ: The downlink power is limited to P% max : Hence P% C ðtÞ þ P% D ðtÞ ¼ P% C ðtÞ þ p% 1 ðtÞ þ p% 2 ðtÞpP% max :

gonality is not fully maintained, and the fraction a% 1 ; ð0pa% 1 p1Þ still remains. The SIR at mobile 1 is given by

ð1Þ

All the signals from the base station to a specific mobile have passed through the same channel. Therefore, it can be motivated to use orthogonal channelization codes to reduce the mutual interference between the signals. Due to non-ideal receivers and channel effects, this ortho-

Fig. 1. Simplistic uplink and downlink situation with two mobiles connected to one base station to illustrate fundamental network limitations and objectives.

p% 1 ðtÞg% B1 ðtÞ ; a% 1 ðP% C ðtÞ þ p% 2 ðtÞÞg% B1 ðtÞ þ I%inter;i ðtÞ þ n% 1 ðtÞ

ð2Þ

where n% 1 ðtÞ is the thermal noise at mobile 1, and I%inter;i ðtÞ the interfering signal power from other cells (inter-cell interference). If the downlink is fully utilized (i.e. all base station power is used to provide services and the hardware investment is fully utilized) user 1 is interfered by the power P% max  p% 1 ðtÞ; which is independent of the power of the other user. The interesting question is thus how the dedicated power P% D ðtÞ; and the resulting service quality and data rate, should be shared between the users. Therefore, the main downlink resources are power and number of codes. When all cells are fully loaded, the interfering signal power from other base stations is fairly constant, and the sharing of resources can thus be managed per base station. In the uplink, the two signals from the mobiles pass through independent channels to the base station. Therefore, it is not meaningful to use orthogonal codes, but rather codes with good correlation properties. Assume that the mobiles are using the powers p% 1 ðtÞ and p% 2 ðtÞ; respectively. The SIR of MS1 is given by g% 1 ðtÞ ¼

p% 1 ðtÞg% 1B ðtÞ ; p% 2 ðtÞg% 2B ðtÞ þ I%inter;B ðtÞ þ n% B ðtÞ

ð3Þ

where n% B ðtÞ represents thermal noise power at base station B; and I%inter;B ðtÞ the interfering power from mobiles connected to other base stations (same for all mobiles connected to this base station). Hence, the connections are mutually interfering, and this fact restricts the achievable SIRs. With only these two mobiles active in the system, SIR is limited by (see Theorem 5) g% 1 g% 2 o1:

ð4Þ

Limited transmission powers and interfering mobiles in other cells might further restrict the achievable SIRs. This stresses the importance of coordinating the resource management between cells in the uplink. Furthermore, the main uplink resource is therefore total received power at the base stations. In practice, the links are subject to variations which act on different time scales. Therefore, fast power control aims at tracking the rapid channel variations, while SIR reference control, rate control and scheduling acts on a slower time scale. A simplification is to model the slower control loop as to regularly re-assign a SIR reference, gti ðtÞ (note the switch to values in dB). Power control is used to maintain this SIR based on feedback of the control error ei ðtÞ ¼ gti ðtÞ  gi ðtÞ: The feedback communication uses valuable bandwidth, and should be kept at a minimum. Let f ðei ðtÞÞ denote the feedback

F. Gunnarsson, F. Gustafsson / Control Engineering Practice 11 (2003) 1113–1125

communication (essentially quantization). With pure integrating control, this yields pi ðt þ 1Þ ¼ pi ðtÞ þ bf ðei ðtÞÞ :

ð5Þ

The default power control algorithm in WCDMA uses single-bit quantization, and hence f ðei ðtÞÞ ¼ signðei ðtÞÞ:

*

1115

Capacity and system load. As indicated by the example above, the available radio resource is limited and have to be shared among the users. An important distinction in Section 5.1 is therefore whether the network can accommodate all the users with associated quality requirements.

1.2. Aspects of power control and paper outline

2. System model

Being subjective, the list below with pointers forward to related sections constitutes important aspects of power control:

Most quantities will be expressed both in linear and logarithmic scale (dB). Linear scale is indicated by the bar notation gðtÞ: %

*

*

*

*

*

*

*

Objectives. It is vital to clarify the aim of power control. As indicated in the example above, throughput maximation leads to different control strategies compared to fair objectives where all users experience roughly the same quality of service. Centralized/decentralized control. Centralized power control is not practically tractable. As discussed in Section 5, it mainly serves as theoretical performance bounds to the decentralized algorithms in Section 4. However, due to the distributed architecture of power control, some information is not available where it is used, which means that some semi-centralized processing takes place in controlling nodes in the network as further explored in Section 3.4. Feedback bandwidth. The feedback bandwidth should be stated as the number of available bits per second for feedback communication. Then, this becomes a trade-off between error representation accuracy and feedback rate as discussed in Section 4.1. Power constraints. The transmission powers are constrained due to hardware limitations such as quantization and saturation, which are in focus in Section 4.2. Time delays. Measuring and control signaling take time, resulting in time delays in the distributed feedback loops. The time delays are typically fixed due to standardized signaling protocols, and are further treated in Section 4.2. Disturbance rejection. The controller’s ability to mitigate time varying power gains and measurement noise is an important performance indicator further discussed in Section 4.2. Soft handover. One important coverage improving feature in UMTS systems is that a mobile can be connected to a multitude of base stations. This puts some specific requirements on power control which are briefly touched upon in Section 3.3. Stability and convergence. Studying stability and oscillatory behavior of the distributed control loops as in Section 4.4 is necessary, but not sufficient. The cross-couplings between the loops also have to be considered. This is addressed in Section 5.4.

2.1. Power gain By neglecting data symbol level effects, the communication channel can be seen as a time varying power gain made up of three components gðtÞ ¼ gp ðtÞ þ gs ðtÞ þ gm ðtÞ as illustrated by Fig. 2. The signal power decreases with distance d% to the transmitter, and the path loss is % (Okamura, Ohmori, modeled as gp ¼ Kp  ap log10 ðd) Kawano, & Fukuda, 1968). Terrain variations cause diffraction phenomenons and this shadow fading gs is modeled as ARMAðna ; nb Þ-filtered Gaussian white noise (na is typically 1–2, nb ¼ na  1; S^rensen, 1998). The multipath model considers scattering of radio waves, yielding a rapidly varying gain gm (Sklar, 1997). The simulations in this paper considers mobiles at 5 km=h in a fading environment sampled at 1500 Hz: 2.2. Wireless networks Consider a general network with m transmitters using the powers p% i ðtÞ and m connected receivers. For generality, the base stations are seen as multiple transmitters (downlink) and multiple receivers (uplink). The signal between transmitter i and receiver j is attenuated by the power gain g% ij : Thus the receiver

-80 Power Gain [dB]

*

gs gp

-85 -90 -95 gm

-100 0

5 10 15 Travelled distance [m]

20

Fig. 2. The power gain gðtÞ is modeled as the sum of three components: path loss gp ðtÞ; shadow fading gs ðtÞ and multipath fading gm ðtÞ: Here this is illustrated when moving from a reference point and away from the transmitter.

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connected to transmitter i will experience a desired signal power C% i ðtÞ ¼ p% i ðtÞg% ii ðtÞ and an interference from other connections plus noise I%i ðtÞ: The SIR at receiver i can be defined by C% i ðtÞ g% ii ðtÞp% i ðtÞ ; ð6Þ g% i ðtÞ ¼ ¼P g I%i ðtÞ % % j ðtÞ þ n% i ðtÞ jai ij ðtÞp where n% i ðtÞ is thermal noise at receiver i: Depending on the receiver design, propagation conditions and the distance to the transmitter, the receiver is differently successful in utilizing the available desired signal power p% i g% ii : Assume that receiver i can utilize the fraction d% i ðtÞ of the desired signal power. Then the remainder ð1  d% i ðtÞÞp% i g% ii acts as interference, denoted auto-interference (Godlewski & Nuaymi, 1999). We will assume that the receiver efficiency changes slowly, and therefore can be considered constant. Hence, the SIR expression in Eq. (6) transforms to d% i g% ii ðtÞp% i ðtÞ : ð7Þ g% i ðtÞ ¼ P % ij ðtÞp% j ðtÞ þ ð1  d% i Þp% i ðtÞg% ii ðtÞ þ n% i ðtÞ jai g From now on, this quantity will be referred to as SIR. For efficient receivers, d% i ¼ 1; and expressions (6) and (7) are equal. In logarithmic scale, the SIR expression becomes gi ðtÞ ¼ pi ðtÞ þ di þ gii ðtÞ  Ii ðtÞ:

ð8Þ

In the downlink, many of the interfering signals go through the same channels. It is therefore instructive to discuss an alternative model to Eq. (7). Let P% k ðtÞ; k ¼ 1; y; B denote the total powers of the B base stations and ki denote the base station which mobile i is connected to. Furthermore, the interfering signal powers from the same base station are attenuated by a% i due to the effect of the channelization codes, which are orthogonal on the transmission side. Then, the downlink SIR is given by

where g% ki denotes the power gain from base station k to mobile station i: For example, by dimensioning the system based on a certain load assumption in terms of base station powers, one can get a rough idea of the downlink situation in a cell planning phase. 2.3. Power control algorithms We adopt the loglinear power control model in Dietrich, Rao, Chockalingam, and Milstein (1996) and Blom, Gunnarsson, and Gustafsson (1998) to embrace the power control approaches. The cascade control block diagram of a generic distributed SIR-based power control algorithm is depicted in Fig. 3. The receiver computes the error ei as the difference between the reference SIR gti and SIR (measured, subject to measurement noise wi and possible filtered by the device Fi ). In practice, the desired signal power and the interference are typically filtered separately, which is discussed in Section 3.1. The control error is coded into power control commands ui by the device Ri ; affected by command errors xi on the feedback channel and decoded on the transmitter side by Di : The control loop is subject to power update delays of np samples and measurement delays nm samples. Since the controller Ri causes a unit delay, the total round-trip delay is nRT ¼ 1 þ np þ nm : Typically, np ¼ 1 and nm ¼ 0 and hence nRT ¼ 2: An outer loop adjusts the reference SIR to assure that the quality of service is maintained. Outer loop control is typically based on block error rate (BLER), which is the error rate of received and decoded data blocks considering the effects of coding, interleaving etc. Also the bit error rate (BER) provides relevant information if available, especially since there are more bits than blocks, which enable better estimation . & performance (Olofsson, Almgren, Johansson, Ho. ok, Kronestedt, 1997; Wigard & Mogensen, 1996).

g% i ðtÞ ¼

d% i g% ki i ðtÞp% i ðtÞ P ; a% i ðP% ki ðtÞ  d% i p% i ðtÞÞg% ki i ðtÞ þ kaki P% k ðtÞg% ki ðtÞ þ n% i ðtÞ ð9Þ

3. Standardized power control algorithms Several power control algorithms are standardized by 3GPP (1999) to be used in WCDMA [3GPP, TS 25.214].

Fig. 3. Block diagram of the receiver–transmitter pair i when employing a general SIR-based power control algorithm. In operation, the controller result in a closed local loop.

F. Gunnarsson, F. Gustafsson / Control Engineering Practice 11 (2003) 1113–1125

RNC

t –2

Downlink

1117

t –1

t

Transmitter Update Receiver

...

Node B

Uplink

(a)

Est.

PC

t –2

(a)

...

t –1

t

Transmitter

(b) UE

Fig. 4. (a) A simplified picture of the WCDMA radio access network architecture using the notation as in the standard. The mobiles (UE— user equipment) are connected to one or several base stations (Node B), which in turn are managed by a radio network controller (RNC). (b) Location of pilot bits for estimation in the slots of the transport channel. In the downlink, control information is inter-foilated with data, whereas it is sent in parallel with the data in the uplink.

The feedback channel is implemented as a transmitter power control command bit si ðtÞ transmitted at 1500 Hz: The operation of power control is affected by the network architecture in WCDMA, which is briefly illustrated in Fig. 4a. The control information is used for estimation purposes, and it is transmitted differently in up- and downlink. In the former, the control information is continuous in time and sent with a p=2 phase shift compared to the data. while control and data are separated in time in the latter, see Fig. 4b. The resources are coordinated by the RNC, which thus setup and remove links to base stations throughout the lifetime of a connection. In large networks, several RNCs are used, which means that there is also an interface between the RNCs to coordinate resources and to manage UE’s that uses resources controlled by different RNCs. 3.1. SIR estimation Little is standardized with respect to SIR estimation. It is stated that the uplink SIR estimate should reflect the control information, and it is proposed as relevant to do the same in the mobile on the downlink. The pilot information among the control information is located early in the slot to facilitate fast feedback of power control commands to minimize power control roundtrip delay, as illustrated in Fig. 5. Note that the estimation accuracy is strongly dependent on the number of bits considered in the estimation and therefore the delay. Typically, estimates in practice are biased (Freris, Jeans, & Taaghol, 2001; Kurniawan, Perreau, Choi, & Lever, 2001). The number of available pilot bits is typically different for each service. In practice, SIR is estimated as the ratio between the powers of the extracted desired signal and the interference, which are estimated separately. The average interference is assumed to change rather slowly, and the raw interference estimates are therefore low-pass filtered. To minimize the power control round-trip delay,

Update Receiver

(b)

Est.

PC

Fig. 5. Typical situations in WCDMA describing time of estimation (over pilot bits and possibly some data bits), power command generation and signaling (PC), and power update. The relatively long estimation and processing time to favor estimation accuracy in (a) result in a time delay np ¼ 1: This is avoided in (b) by considering fewer symbols in the estimation.

the desired signal power is not filtered. Therefore, the generic power control model in Fig. 3, is not completely true. 3.2. Inner loop power control 3.2.1. Fixed-step power control The power level is increased or decreased depending on whether the measured SIR is below or above target SIR, and implemented as: Receiver :

ei ðtÞ ¼ gti ðtÞ  gi ðtÞ;

ð10aÞ

si ðtÞ ¼ signðei ðtÞÞ;

ð10bÞ

Transmitter : pTPC;i ðtÞ ¼ Di si ðtÞ; pi ðt þ 1Þ ¼ pi ðtÞ þ pTPC;i ðtÞ:

ð10cÞ ð10dÞ

This is the default choice both in the uplink and the downlink inner closed-loop power control. The uplink situation is slightly modified when the mobile is in soft handover. Then, the mobile receives power control commands from every connected cell. To ensure that the power is adapted to the best cell, the mobile only increases the power if all commands are equal to +1, otherwise the power is decreased. Uplink alternatives. This alternative algorithm is a different command decoding than above and is denoted ULAlt1. It makes it possible to emulate slower update rates, or to turn off uplink power control by transmitting an alternating series of TPC commands. In a 5-slot cycle ð j ¼ 1; y; 5Þ; the power update pTPC;i ðtÞ in (10c) is computed according to: 8 P > ðt mod 5 ¼ 0Þ&ð 4j¼0 si ðt  jÞ ¼ 5Þ Di > < P pTPC;i ðtÞ ¼ Di ðt mod 5 ¼ 0Þ&ð 4j¼0 si ðt  jÞ ¼ 5Þ > > : 0 otherwise: Downlink alternatives. There are two downlink alternatives, both aiming at reducing the risk of using

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excessive powers. In the first one, here denoted by DLAlt1, the control commands are repeated over three consecutive slots. The second one, denoted DLAlt2, reduces the controllers ability to follow deep fades by limiting the power raise. As with the ULAlt1, the commands are decoded differently than in Section 3.2, described as an alternative to (10c): 8 > < Di si ðtÞo0; pTPC;i ðtÞ ¼ Di ðsi ðtÞ > 0Þ&ð psum;i ðtÞ þ Di odsum Þ; > : 0 otherwise; where psum;i ðtÞ is the sum of the previous N power updates and N and dsum are configurable parameters. 3.3. Soft handover One core feature in DS-CDMA systems is soft handover, where the mobile can connect to several base stations simultaneously. For best performance, the mobile controls its power with respect to the signal from the base station with the most favorable propagation conditions. Intuitively, the mobile only increases the power if the TPC commands from all the base stations require it to do so. When command errors occur, this might lead to unwanted effects. The algorithm of the TPC command combination in the mobile is not standardized, but some performance requirements are provided in [3GPP, TS 25.214]. The problem is also addressed by Grandell and Salonaho (2001). In the downlink, the mobile combines the signals from the connected base stations. For power control, all these base stations adjusts its powers according to the received TPC command from the mobile. Thereby, the relations between the base station powers are maintained. However, the TPC commands might be interpreted differently in the base stations due to feedback errors, changing the power level relations. To compensate for this drift, a centralized power balancing is proposed in the standards, see [3GPP, TS 25.433]. It is not fully standardized, and the following should be seen as replacing Eq. (10d): pi ðt þ 1Þ ¼ pi ðtÞ þ pTPC;i ðtÞ þ pbal;i :

ð11Þ

The base station computes the average transmission power over a time frame pave;i and computes the balancing term as pbal;i ¼ ð1  rÞð pref ;i  pave;i Þ ;

ð12Þ

where the adjustment ratio r and the reference power pref ;i (possibly individual for each link) are signaled to the base station by the RNC. The need for the central controlling node RNC in Fig. 4 is evident when considering soft handover. Since the uplink information is available in two different base stations, the actual service quality can only be evaluated

in the RNC. Essentially, the RNC selects the data blocks from the base station with the best SIR (selection combining). 3.4. Outer loop design SIR might be well correlated to perceived quality, but it is not possible to set a SIR reference offline, that results in a specific service quality in terms of data rate and block error rate. Furthermore, the SIR estimate could be biased, and it is based on the control information and not the data (cf. Fig. 4b). Therefore, the inner control loop is operated in cascade with an outer loop, which adapts the SIR reference for a specific connection. The exact algorithms are not standardized, but the interfaces and performance requirements are. In the uplink, the outer loop is controlled in the RNC using a BLER reference and the SIR reference is signaled to the connected base stations. In the downlink, both the inner and outer loops are contained in the mobile, and the mobile is essentially only provided with a reference BLER. Note that the power is adapted to the SIR (estimated on control information) variations on the short run, but to the average data BLER on a longer time scale. 4. Distributed power control This section further discusses control theory aspects of power control, primarily with respect to the standardized WCDMA power control algorithms in Section 3, but also relates to more academic achievements. 4.1. Feedback bandwidth The feedback signaling bandwidth is limited in real systems. Typically, the communication is restricted to a fixed number of bits per second k: The evident trade-off is between error representation accuracy and feedback command rate. A single bit error representation allow k feedback commands per second, while me bits error representation allow k=me commands per second. This comparison is further explored in Gunnarsson (2001). Different error representations are proposed, for example: single bit (the sign of the error) (Salmasi & Gilhousen, 1991), k-bit linear quantizer (Sim, Gunawan, Soh, & Soong, 1998) and k-bit logarithmic quantizer (Li, Dubey, & Law, 2001). All these correspond to quantizers and decoders in the devices Ri and Di in Fig. 3. Note also that the error representation is related to the command rate on the feedback channel. Single bit quantization is primarily used in WCDMA as discussed in Section 3.2. In the alternative algorithms ULAlt1 and DLAlt1, aggregated bits are used to provide better error protection and not to increase the granularity of the quantization.

F. Gunnarsson, F. Gustafsson / Control Engineering Practice 11 (2003) 1113–1125

The algorithm proposed in Yates (1995) can be interpreted as the controller above with an arbitrary b: A comparison to Fig. 3 yields the following interpretation considering perfect error representation: b ; ui ¼ pi ; Di fpi ðt  np Þg ¼ pi ðt  np Þ: RðqÞ ¼ q1 ð14Þ This has motivated the alternative linear designs of RðqÞ as PI-controllers and more general linear controllers in Gunnarsson, Gustafsson, and Blom (1999). Perfect error representation results in a linear distributed control loop with closed-loop system Gll ðqÞ and sensitivity SðqÞ given by Gll ðqÞ ¼

RðqÞ ; qnp þnm þ RðqÞ

np þnm

SðqÞ ¼

q : qnp þnm þ RðqÞ

ð15Þ

In the typical delay situation nRT ¼ 2 and with the integrating controller in (13), this yields b ; Gll ðqÞ ¼ 2 q qþb

q2  q SðqÞ ¼ 2 : q qþb

ð16Þ

With the single bit power control command as in Section 3.2, the integrating controller becomes pi ðt þ 1Þ ¼ pi ðtÞ þ b signðei ðtÞÞ:

ð17Þ

The transmitter power is constrained in practice, typically quantized and bounded from above and below. Grandhi, Zander, and Yates (1995) proposes an algorithm to deal with powers bounded from above. In loglinear scale it is given by pi ðt þ 1Þ ¼ minfpmax ; pi ðtÞ þ bei ðtÞg:

ð18Þ

This can be interpreted as one out of many possible anti-reset windup implementations for PI-controllers ( om . (Astr & Wittenmark, 1997), which thus can be employed to more general transmitter power constraints. Time delays are critically limiting the closed-loop performance of any feedback system. These time delays are known and fixed, since the signaling and measurement procedures are standardized, and propagation delays are negligible (except possibly in satellite communications). For example, the typical delay situation np ¼ 1 and nm ¼ 0 yields gi ðtÞ ¼ pi ðt  1Þ þ di þ gii ðtÞ  Ii ðtÞ:

ð19Þ

g* i ðtÞ ¼ gi ðtÞ þ Di

nX p þnm

si ðt  cÞ:

ð21Þ

c¼1

With the Smith predictor and a linear controller RðqÞ; the closed-loop system and the sensitivity becomes Gll ðqÞ ¼ SðqÞ ¼

RðqÞ ; qnp þnm ð1 þ RðqÞÞ

ð22Þ

qnp þnm : þ RðqÞÞ

ð23Þ

qnp þnm ð1

In the typical delay situation nRT ¼ 2 and with the integrating controller in (13), we get Gll ðqÞ ¼

b ; qðq  1 þ bÞ

SðqÞ ¼

qðq  1Þ : qðq  1 þ bÞ

ð24Þ

The local behavior of the controllers above is illustrated in simplistic simulations in Fig. 6a–d. We note that the disturbance rejection is satisfactory with most controllers. Furthermore, the benefits of using the Smith predictor are more emphasized with single-bit error representation, see Fig. 6c and d. Roughly the same effect is obtained with linear design, see Fig. 6b.

ei(t) [dB]

ð13Þ

ð20Þ

The actual power levels pi ðtÞ are not available in the receiver considering the WCDMA algorithm in Section 3.2, but rather the power control commands si : However, since the adjustment is based on the difference between power levels, it can be implemented as

2 0 -2

(a) ei(t) [dB]

pi ðt þ 1Þ ¼ pi ðtÞ þ bei ðtÞ:

g* i ðtÞ ¼ gi ðtÞ þ pi ðtÞ  pi ðt  nm  np Þ:

2 0 -2

(b) ei(t) [dB]

Early work such as Foschini and Miljanic (1993) addresses the problem in linear scale based on iterative methods for eigenvector computations (Fadeev & Fadeeva, 1963). Thereby, it is closely related to the global aspects in Section 5. In logarithmic scale, this is the special case b ¼ 1 of an integrating controller

Since the delays are exactly known, time delays can be ( om . compensated for by using the Smith predictor (Astr & Wittenmark, 1997) as described in Gunnarsson and Gustafsson (2001b). Essentially, it is implemented as a measurement adjustment:

2 0 -2

(c) ei(t) [dB]

4.2. Inner loop power control design

1119

(d)

2 0 -2 time index

Fig. 6. Local performance of the algorithms (13) in (a,b) and (17) in (c,d) subject to the typical delay situation nRT ¼ 2: (a) b ¼ 0:9 (b) b ¼ 1 and the Smith predictor (dashed) and b ¼ 0:34 (solid) (c) b ¼ 1 (d) b ¼ 1 and the Smith predictor.

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This is in line with the results in Kristiansson and Lennartsson (1999). The Smith predictor might compensate for some dynamical effects, but the controllers still show delayed reactions to changes in the power gains. One approach to improve the reactions is to predict the power gain using a model-based approach. Power gain predictions are further studied based on linear model structures (Ericsson & Millnert, 1996; Choel, Chulajata, Kwon, Koh, & Hong, 1999; Ekman, Ahle! n, & Sternad, 2002) and nonlinear model structures (Ekman & Kubin, 1999; Tanskanen, Mattila, Hall, Korhonen, & Ovaska, 1998; Zhang & Li, 1997). It is hard to give a general answer to whether linear or nonlinear models are most appropriate, or whether it is most suitable to predict gðtÞ or gðtÞ; since it depends on the fading situation and the % controller objectives. 4.3. Outer loop Reliable communication can be seen as low block error rate (BLER) requirements, which in turn means that it is very hard to accurately estimate BLER. The errors appear rather seldom and it takes long time before the BLER estimate is stable. One approach increases the SIR reference significantly when an erroneous block is discovered, and decreases the SIR reference somewhat when an error-free block is received. Niida, Suzuki, and Takeuchi (2000) provides experimental results of outer loop power control using this method. One block comprises many bits. Therefore, it is easier to obtain a good BER estimate. Then the relation between BER and BLER can be utilized to predict BLER based on BER measurements (Kawai, Suda, & Adachi, 1999). 4.4. Inner loop power control analysis Local stability analysis is straightforward when the error representation is ideal, since the local control loop is all linear. For example, root locus of the poles to Gll ðqÞ in (15) and (22) can be used to address local stability (Blom et al., 1998). It is easy to see that time delays make the choice of b in the I-controller crucial to ensure local stability. For example in the typical delay situation in (16), the local control loop is stable for bo1: With the Smith predictor, local stability is improved (24) and b ¼ 1 yields a dead-beat controller. See also Fig. 6a and b. The single-bit error representation can be seen as relay feedback in a linear system. This explains the observed oscillatory behavior, which can be approximated by using discrete-time describing functions (Gunnarsson, Gustafsson, & Blom, 2001) as outlined below:

Assume that there is an oscillation in the power control error by making the N-periodic hypothesis   2p t ; eðtÞ ¼ E sinðOe tÞ ¼ E sin N where E is the amplitude of the oscillation and Oe is the normalized angular frequency. Discrete-time describing function analysis yields the equation of the oscillations as Yf ðE; NÞGðei2p=N Þ ¼ 1;

ð25Þ

where the discrete-time describing function Yf ðE; NÞ of the relay is obtained as 4 eiðp=Nde 2p=NÞ ; de Að0; 1Þ Yf ðE; N; de Þ ¼ NE sinðp=NÞ ð26Þ and the linear part of the closed inner loop system is given by Di : GðqÞ ¼ n 1 q RT ðq  1Þ On the unit circle GðqÞ is equal to Di eiðp=2þp=Nþð2p=NÞðnRT 1ÞÞ : GðqÞjq¼e2pi=N ¼ 2 sinðp=NÞ

ð27Þ

Defining that only oscillations with even period N are possible (odd N are seen as transitions between evenperiod oscillations), these expressions lead to the conclusion that the oscillation period in samples is N ¼ 4nRT  2; and the amplitude E ¼ Nb=4: Hence, the longer the delay, the more emphasized oscillatory behavior. Since the Smith predictor reduces the roundtrip delay to a minimum, such oscillations are therefore reduced. This is illustrated in Fig. 6c and d. Further details are provided in Gunnarsson et al. (2001). The relay feedback controller is also locally analyzed in Song, Mandayam, and Gajic (2001), where the relay is approximated by a constant and an additive disturbance yielding the same relay output variance. 4.5. Downlink issues As indicated by the introductory example, downlink power control objectives can be rather different and specific. The power control objectives depend on the policies of the network operator, and can be differentiated in terms of service requirements, resource utilization, subscription conditions, etc. Revisit the introductory example in Fig. 1. Some natural policies are: * * *

Aim at equal data rate ðg% 1 ðtÞEg% 2 ðtÞÞ: Aim at equal power usage ð p% 1 ðtÞEp% 2 ðtÞÞ: Use all power to transmit to the user with highest power gain (i.e. mobile 1 as indicated in the figure) to maximize throughput.

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First use power to meet the quality of service requirements of users with more expensive subscriptions. Use the remainder to low-fare subscription users.

Various downlink power control and resource sharing issues are brought up in Song and Holtzman (1998), Lu and Brodersen (1999), De Bernardi, Imbeni, Vignali, and Karlsson (2000) and Vignali (2001). Another problem is uneven traffic distributions that are not supported by the cell layout. In such a scenario, some cells might be overloaded, while others are under-utilized. The users select base stations based on measured pilot powers from the different base stations. By controlling the pilot powers, the cells can be made larger or smaller. This is often referred to as cell breathing (Hwang, Kim, Oh, Kang, & Son, 1997).

1121

definitions are needed. Introduce the matrices

g % t 9diagðg% t1 ; y; g% tm Þ; Z% ¼ ½z%ij 9 % ij ; C g% ii % D9diagð d% 1 ; y; d% m Þ and vectors p% 9½p% i ;



n% i g% ¼ ½Z% i 9 : g% ii

The network itself puts restrictions on the achievable SIRs and there exists an upper limit on the balanced SIR (same SIR to every connection). This is disclosed in the following theorem, neglecting auto-interference and assuming that the noise can be considered zero. Theorem 1 (Zander, 1992). With probability one, there exists a unique maximum achievable SIR in the noiseless case: g% n ¼ maxfg% 0 j ( p% X0: g% i Xg% 0 8ig:

5. Global analysis For practical reasons, power control algorithms in cellular radio systems are implemented in a distributed fashion. However, the local loops are interconnected via the interference between the loops, which affects the global dynamics as well as the capacity of the system. An important global issue is whether it is possible to accommodate all users with their service requirements. The power gains reflect the situation from the transmitters to the receivers, and the results are therefore applicable to both the uplink and the downlink. To illustrate the sometimes abstract concepts, the two-mobile example in the uplink from Fig. 1 will be used throughout the section with the SIR references g% t1 ¼ g% t2 ¼ 0:20: (This corresponds to a rather high data rate, especially when considering the uplink, but it is chosen to considerably load the system.) Sufficient conditions on global stability are derived, including the effect of the system load and time delays. 5.1. Performance upper bounds and feasibility The individual reference SIRs and the power gains are considered constant in the global level analysis, where the latter is motivated by an assumption that the inner loops perfectly meet the provided SIR reference, and thereby mitigate the fast channel variations: d% i g% ii p% i P ¼ g% ti 8i: ð28Þ % ij p% j þ ð1  d% i Þp% i g% ii þ n% i jai g Note that values in linear scale are used in this section. The aim is to characterize the system load, and a few

Furthermore, the maximum is given by 1 ; g% n ¼ n l%  1 % Note that where l% n is the largest real eigenvalue of Z: l% n > 1 implies that g% n > 0: Moreover, the optimal power vector p% n is the eigenvector of l% n (i.e. kp% n for any kARþ constitutes an optimal power vector). Considering the noise, the following can be concluded: Theorem 2 (Zander, 1993). In the noisy case and with no power limitations, there exist power levels that meet the balanced SIR target g% t0 if and only if g% t0 og% n : In the example, we have ! 1 1 Z% ¼ ; g% n ¼ 1: 1 1

ð29Þ

Since the SIR target g% ti ¼ 0:20og% n it is possible to find power levels that meet the requirements of both users. This is generalized to multiple services below. The effects of auto-interference are considered in Godlewski and Nuaymi (1999) and Gunnarsson (2000). The requirements in Eq. (28) can be vectorized to % 1 Z%  EÞp% þ D% 1 g% Þ; p% ¼ C% t ððD ð30Þ where E is the identity matrix. Solvability of the equation above is related to feasibility of the related power control problem, defined as: % t is said Definition 3 (Feasibility). A set of target SIRs C % to be feasible with respect to a network described by Z; % and g% ; if it is possible to assign transmitter powers p% so D that the requirements in Eq. (30) are met. Analogously, % C % g% ; D; % t Þ is said to be the power control problem ðZ;

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1122

feasible under the same condition. Otherwise, the target SIRs and the power control problem are said to be infeasible. The degree of feasibility is described by the feasibility margin, which is defined below. The concept has been adopted from Herdtner and Chong (2000), where similar proofs of similar and additional theorems covering related situations also are provided. Herdtner and Chong used the term feasibility index RI and omitted auto-interference. Definition 4 (Feasibility margin). Given a power con% C% t Þ; the feasibility margin G % g% ; D; % m ARþ trol problem ðZ; is defined by % t is feasibleg: % m ¼ supfxAR: G x% C % A motivation for introducing the name feasibility margin is to stress the similarity to the stability margin of feedback loops. While Theorems 1 and 2 address the case when all users have the same service, the feasibility margin describes the situation with multiple services. The following theorem captures the essentials regarding feasibility margins. Theorem 5 (Gunnarsson, 2000). Given a power control % C % g% ; D; % t Þ; the feasibility margin is obtained as problem ðZ; % m ¼ 1=m% n ; G where m% n is m% n ¼ max eigfC% t ðD% 1 Z%  EÞg: Moreover, if G% m > 1; the power control problem is feasible, and there exists an optimal power assignment, given by % 1 Z%  EÞÞ % 1 g% : % 1 C % t ðD % tD p% ¼ ðE  C The power assignment above can of course be seen as a centralized strategy. However, since full information about the network is required to compile Z% it is not plausible in practice. The result mainly serves as a performance bound. The feasibility margin can also be related to the load of the system. When the feasibility margin is one, the system clearly is fully loaded (only possible when unlimited transmission powers are available). Conversely, when the feasibility margin is large, the load is low compared to a fully loaded system. Thus the following load definition is plausible. Definition 6 (Relative load). The relative load L% r of a system is defined by L% r ¼

1 ð¼ m% n in Theorem 5Þ: %m G

Feasibility of the power control problem is thus equal to a system load less than unity. For a more detailed capacity discussion, see Hanly (1999) and Zhang and Chong (2000), where receiver and code sequence effects also are considered. In the example we have qffiffiffiffiffiffiffiffiffi ð31Þ L% r ¼ g% t1 g% 1t ¼ 0:2o1: Again, the power control problem in the example is feasible. It is interesting to note that the unbalanced situation g% 1t ¼ 1:5 and g% 2t ¼ 0:2 also corresponds to a feasible problem ðL% r ¼ 0:55Þ despite the fact that g% 1t > g% n : 5.2. Approximative downlink capacity As mentioned in the introduction, some crude planning at network deployment can be employed to approximate the downlink capacity. This discussion is partly adopted, but reformulated, from Hiltunen and De Bernardi (2000). Reconsider the SIR expression in Eq. (9), assume equally ideal receivers ðd% i ¼ d% 8iÞ and the same orthogonality properties at every receiver ða% i ¼ a% 8iÞ; and introduce the inter–intra interference power ratio F%i as P % % ki ðtÞ kaki Pk ðtÞg % Fi ¼ : ð32Þ P% ki ðtÞg% ki i ðtÞ Then, the downlink SIR at receiver i is given by d% g% ki i ðtÞp% i ðtÞ : g% i ðtÞ ¼ ða% þ F%i ÞP% ki ðtÞ  a% d% i p% i ðtÞÞg% ki i ðtÞ þ n% i ðtÞ

ð33Þ

Suppress the time index t for clarity, solve for pi ; assume that mobiles i ¼ 1; y; m1 are connected to cell 1 and all have the same inter–intra interference power ratio F%i ¼ % and compute the total base station power of cell 1 F; Pm1 P% 1 ¼ PC i¼1 pi to obtain 1 þ  X m1 a% g% i C P% 1 ¼ P% 1 þ P% 1 þ N% 1 ; þ F% %d 1  a% g% i i¼1

P% c1 þ N% 1 Pm 1 P% 1 ¼ ; % 1  ða% =d% þ FÞ % i =ð1  a% g% i Þ i¼1 g

ð34Þ

where N% 1 is a term related to the thermal noises. If we assume that unlimited base station power is available, the capacity requirement is obtained as m1 X g% i d% : ð35Þ o 1  a% g% i a% þ d% F% i¼1

If the base station power is limited by P% max ; and if we neglect the noise term N% 1 ; we can rewrite (34) as   m1 X g% i d% P% c1 1 p : ð36Þ 1  a% g% i a% þ d% F% P% max i¼1

This limits the maximal number of users in the multiple service scenario. With one single service available corresponding to the SIR g% ; it is thus possible to

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nRT ¼ 1; 2; y) converges to a region where the SIR error for every connection is bounded (in dB) by

maximally accommodate the following number of users:   P% c1 1  a% g% d% 1 mmax;1 ¼ : ð37Þ g% a% þ d% F% P% max

Without Smith: jgti  gi ðtÞjp2nRT b;

5.3. Approximative uplink capacity

With Smith: jgti  gi ðtÞjpðnRT þ 1Þb

As for the downlink, it is relevant to derive approximate capacity expressions. The difference is that the capacity is related to the total received signal power I%tot;i at the base station. For clarity, essentially the same notation and situation as for the downlink above is adopted. Solve for p% i in the SIR expression in (7): g% i I%tot;ki d% i g% iki p% i : ð38Þ 3 p% i ¼ g% i ¼ %di g% iki ð1 þ g% i Þ %Itot;ki  d% i g% iki p% i

and b is the step size. The results also hold when subject to auto-interference.

In the uplink, the inter–intra interference power ratio F% is the same for all mobiles in the same cell. The total received power at cell 1 can thus be expressed as m1 X % I%tot;1 ¼ ð1 þ FÞ g% i1 p% i þ n% 1 i¼1

¼ I%tot;1

m1 1 þ F% X g% i þ n% 1 : 1 þ g% i d% i¼1

ð39Þ

Solve for I%tot;1 relative the background noise n% i (popularly referred to as the noise rise): I%tot;1 1 P 1 ¼ : ð40Þ % d% Þ m n% 1 1  ð1 þ F= % i =1 þ g% i i¼1 g Given an acceptable noise rise ½I%tot;1 =n% i max ; and a single service with SIR g% ; the number of users in the multiple service scenario must satisfy   m1 X 1 g% ti d% p : ð41Þ 1  1 þ g% ti 1 þ F% ½I%tot;1 =n% i max i¼1

In the single service scenario with SIR g% t ; the number of users possible to serve is   1 d% 1 þ g% t m1 ¼ 1  : ð42Þ ½I%tot;1 =n% 1 max 1 þ F% g% t Uplink load estimation is addressed in more detail in Geijer-Lundin, Gunnarsson, and Gustafsson (2003).

Proof. The result without the Smith predictor is provided in Herdtner and Chong (2000), while the result with the Smith predictor is from Gunnarsson (2000). Note that the error bound is tighter when using the Smith predictor, and also when using smaller b; which can be interpreted as the step-size. There is thus a tradeoff between small tracking errors and fast responses to changes. Furthermore, the importance of well functioning admission and congestion control mechanisms is evident. Power control stability critically rely on their ability to preserve feasibility despite a time-varying network.

6. Conclusions and future work Power control in the WCDMA standard is surveyed, and the power control algorithms are put into a control theory context to relate to the control nomenclature. With this common framework, it is natural to address critical properties such as stability and convergence. The ambition with the extensive citation list is to provide, yet subjective, an overview of central proposals, and pointers to interesting open problems. Still many problems remains to be solved, and in short, a subjective list of interesting problems include: *

5.4. Convergence and global stability The WCDMA algorithm described in Section 3.2 with single-bit error representation never converges to a fixed point. As disclosed in Section 4.4, the relay feedback and the delays cause an oscillatory behavior around the reference signal. Instead, it converges to a region characterized by the following theorem. Theorem 7. If the power control problem is feasible, the algorithm without and with the Smith predictor (subject to a round-trip delay of totally nRT ¼ 1 þ np þ nm samples,

*

*

Quality estimation and prediction. Quality of service is a subjective matter, which is relevant to map onto more objective quantities. Signal-to-interferences and power gains are commonly used, and much remains to be done to provide accurate and reliable estimates and predictions thereof. Soft handover. Since soft handover is a central part in third generations systems, power control algorithms have to consider all aspects and situations. Downlink power control. The objectives and limitations are very different between the downlink and the uplink. The operators are eager to fully utilize the investments to provide services, and therefore complex trade-offs between fairness, throughput, efficiency, policies, services and pricing are prevalent.

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