Joint Uplink-Downlink Resource Allocation for Multi-User IRS-Assisted Systems

08/25/2021
by   Mahmoud Saad Abouamer, et al.
0

We investigate the joint uplink-downlink design for time-division-duplexing (TDD) and frequency-division-duplexing (FDD) multi-user systems aided by an intelligent reflecting surface (IRS). We formulate and solve a multi-objective optimization problem to maximize uplink and downlink rates as a weighted-sum problem (WSP) that captures the trade-off between achievable uplink and downlink rates. We propose a resource allocation design that optimizes the WSP by jointly optimizing the beamforming vectors, power control and IRS phase shifts where the same IRS configuration is used for assisting uplink and downlink transmissions. In TDD, the proposed IRS design reduces the overhead associated with IRS configuration and the need for quiet periods while updating the IRS. In addition, a joint IRS design is critical for supporting concurrent uplink and downlink transmissions in FDD. We investigate the effect of different user-weighting strategies and different parameters on the performance of the joint IRS design and the resultant uplink-downlink trade-off regions. In all FDD scenarios and some TDD scenarios, the joint design significantly outperforms the heuristic of using the IRS configuration optimized for uplink (respectively, downlink) to assist the downlink (respectively, uplink) transmissions and substantially bridges the gap to the upper bound of allowing different IRS configurations in uplink and downlink.

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I Introduction

The relentless demand for increased wireless connectivity, data rates, and quality-of-service (QoS) requirements necessitates improvements in spectrum efficiency, power efficiency, and reliability. Current technologies focus on designing the transmitter or the receiver, and the configuration of the wireless channel iself remains largely beyond the reach of the designer. Intelligent reflecting surfaces (IRS) fill this gap by enhancing the performance of a communication system by tuning the wireless propagation channel. An IRS encompasses a large array of configurable reflecting elements that can collectively reshape the incident signal in terms of its phase and amplitude [1, 2, 3, 4].

An IRS can be implemented using either (nearly) passive or active elements in order to improve the signal power at the receiver and diminish the interference associated with unintended users. A passive IRS consumes (nearly) no power and introduces negligible noise [1, 2]. On the other hand, at the cost of increased power consumption, noise power and hardware complexity, an active IRS allows the reflected signals (and noise) to be amplified and thus provides the benefit of compensating for the ”double-fading” attenuation of the reflected signals [5, 3]. The idea of achieving a middle ground between active and passive IRS by placing few active IRS elements among the passive elements has been discussed in several papers (for e.g. see [6, 7]).

Implementing an IRS using passive elements opens the door to energy and cost efficient smart radio environments. Such an IRS can be incorporated into a communication system in order to improve its performance by jointly designing the active beamforming at the BS with the passive beamforming at the IRS. The literature spans a wide variety of system design objectives including maximizing the weighted sum-rate (WSR) [8], minimizing the total power subject to SINR constraint [9, 10], simultaneous wireless information and power transfer (SWIPT) [11], physical-layer security [12], and improving capacity for indoor and outdoor settings [13, 14]. It should be noted that the majority of the IRS system design literature focuses on downlink transmission (e.g. [8, 9, 10, 11, 12, 13, 14]) with fewer papers focusing on uplink transmission (e.g. [15, 16, 17]).

Joint IRS design for uplink and downlink transmission is an important problem for practical FDD/TDD systems. For FDD systems, using the same IRS configuration is critical in order to support simultaneous uplink and downlink transmissions as they occur over the same time resources [18]. On the other hand, while it is feasible for TDD systems to support different IRS configurations for uplink and downlink transmissions, a joint IRS design for uplink and downlink transmissions in TDD systems reaps the benefits of at least halving the overhead for sending the IRS configuration to update the IRS compared to when a different IRS configuration is used for each of uplink and downlink transmissions. Moreover, a joint IRS design also reduces the need for quiet periods while updating the IRS between uplink and downlink transmission where such guard periods are needed in order to avoid the time-varying behavior of the channel due to a changing IRS configurations. Further, updating the IRS between uplink and downlink transmissions imposes stricter constraints on time synchronization between the BS and the IRS, which in turn increases overhead, hardware complexity and delays [19]. Fig. 1 shows 3 possible update schedules for TDD operation: in case 1 the conventional approach of updating the IRS before every uplink/downlink transmission; in case 2 the joint uplink/downlink approach where the IRS is only updated before each downlink transmission; and if the channel is varying slowly enough, then a joint design will require no update over several uplink/downlink intervals as shown in case 3 of Fig. 1.

Figure 1: Three TDD cases showing allocation of time slots in Individual design (case 1) and joint design (cases 2 and 3).

In this paper, we consider maximizing the joint uplink-downlink weighted sum-rate problem for FDD and TDD multi-user (MU) multiple-input single-output (MISO) systems. We adopt a joint uplink-downlink IRS design where the same IRS configuration is used to assist both uplink and downlink transmissions in FDD and TDD systems and where the time/frequency resources dedicated to uplink/downlink can be unequal. A set of weights are introduced in order to capture the relative priority of uplink (UL) vs downlink (DL) while configuring the IRS. The aforementioned formulation provides an increase in flexibility in allocating resources to users in uplink vs downlink which is essential in order to deal with asymmetric uplink/downlink demands. Joint uplink-downlink design for IRS was discussed in [18] in the context of an FDD system in which a single-antenna BS is used to serve a single-antenna user and equal bandwidth was allocated to the uplink and downlink transmissions. Moreover, joint design also arises in full-duplex systems such as [20] and [21] where the IRS is used, in part, to improve the performance of the system by suppressing self-interference. However, employing a joint IRS design in multi-user systems to aid uplink and downlink transmissions in practical FDD as well as TDD systems remains an important undiscussed problem. In addition, the performance loss due to joint IRS configuration, compared to the upper bound of time-sharing individually optimized uplink and downlink configurations, has not been quantified. Furthermore, the impact of different user-weighting strategies, beamforming strategy, and ratio of the number of IRS elements to the number of active antennas at the BS on the performance gains of a joint IRS design in TDD and FDD systems has not been studied.

We summarize the contributions of this paper as follows:

  • We consider a MU-MISO communications system which is assisted by an IRS in order to improve the performance of the corresponding FDD/TDD system by jointly optimizing the IRS configuration, DL and UL beamforming vectors, and power control at the users.

  • We formulate a weighted sum problem (WSP) that corresponds to scalarizing the multi-objective optimization problem of maximizing the uplink weighted sum-rate (WSR) and the downlink WSR where the same IRS configuration is optimized for both uplink and downlink transmissions. The UL and DL WSR capture the relative priority of UL vs DL while configuring the IRS and thus such weights can be adjusted in order to compute the trade-off region between UL and DL rates.

  • The WSP formulation is designed to account for unequal UL/DL resources in FDD and TDD systems. Such flexibility in assigning time and frequency resources expands the capacity to accommodate asymmetrical demands for uplink and downlink transmissions that arise in practical systems. Within the UL/DL users, the relative priority of serving a particular user is captured by the user-weights which are modeled here according to three different strategies: equal weights, proportional fair weights and independent weights.

  • We transform the WSP, which is non-convex due to coupling of the decision variables, into sub-problems that can be solved efficiently. The sub-problem of maximizing DL sum-rate is reformulated as an equivalent WMMSE problem which is known to have an efficient iterative algorithm [22]. The sub-problem of optimizing the receive beamforming vector is addressed using the optimal MMSE receiver, and fractional programming (FP) is used to optimize uplink power control at the user. Finally, manifold optimization is used to optimize the IRS configuration. We use numerical experiments to show that the solution converges to a locally optimal solution reasonably quickly.

  • For an IRS of elements, the performance of the joint design in TDD/FDD systems is compared with the upper bound of optimizing the IRS for UL and DL transmissions individually and then these two IRS configurations are time-shared, referred to as Individual design. Moreover, the performance of the joint design is also compared to three alternative schemes defined in Section IV, namely: Fixed Downlink, Fixed Uplink and Ideal Slicing.

  • We study the effect of the three user-weighting strategies on the performance of the joint design with respect to the Individual design upper bound as well as the three alternative schemes

    . Numerical experiments show that as the user-weights become less uniform, corresponding to more variance in user priority, the relative improvement due to joint design compared with the three

    alternative schemes increases.

  • We investigate the effect of the number of IRS elements and the number of base station (BS) antennas on the performance of the joint design. Simulation results show that increasing the ratio of the number of IRS elements to the number of BS antennas improves the performance enhancement due to joint design.

  • We investigate the effect of beamforming strategy on the joint IRS performance by comparing the performance of WMMSE and Zero-forcing (ZF). Simulation results show that when ZF is used, the relative performance gains due to joint design are about the same as for WMMSE.

  • In FDD, we find that the proposed joint IRS design performs substantially better than the 3 alternative schemes. In TDD, provided that the ratio of the number of IRS elements to the number of BS elements is large and/or when the user weights are independent, we find that the joint design substantially improves the performance, compared with the alternative schemes. Otherwise, the heuristics of fixed uplink/downlink perform almost as well as the individual design bound.

Organization

In Section II, the system model and the performance metrics are provided. In Section III, the WSP of jointly maximizing the DL-WSR and UL-WSR is formulated, solved and analyzed. Numerical experiments that show the efficacy of the proposed joint IRS design compared to upper bounds and lower bounds are discussed in Section IV. Conclusions are presented in Section V.

(a) System diagram
(b) Top-view layout
Figure 2: (a) System diagram and (b)top-view layout for MU-MISO IRS assisted system.

Notation

The sets of real and complex numbers are denoted by and . Moreover, and are used to represent the real and the imaginary parts of a complex variable, respectively. The transpose, conjugate and conjugate transpose operator are denoted using the superscripts , and , respectively. Vectors are denoted using lower case bold symbols (e.g ) and matrices are denoted using upper case bold symbols (e.g ). For a vector , is the diagonal matrix whose diagonal entries are filled with the elements of

. The expectation of a random variable is denoted by

and is used to denote a circularly symmetric complex Gaussian RV with a mean of 0 and variance .

Ii System Model and Performance metrics

Ii-a Model

In this paper, a single-cell communications system is considered where an -antenna BS is assisted by an -element IRS in order to serve single-antenna users. The users are indexed by . The system diagram and layout are shown in Fig. 2.

In the DL, the precoded signal transmitted by the BS is expressed as where and are the downlink transmit beamforming vector and the information symbol in the downlink associated with user , respectively. The information symbols are zero-mean and satisfy , i.e., are i.i.d. with unit power.

In the UL, the signal transmitted by user can be expressed as where and are the uplink power transmitted by user and the information symbol in the uplink associated with user , respectively. The UL information symbols satisfy . Additionally, is the unit-norm receive combining vector at the BS for user .

The elements of the IRS are indexed by with the vector containing the IRS phase shifts expressed as . For each IRS element , the reflection coefficient is represented as where is the phase shift associated with element . The phase-shift matrix associated with the IRS can then be expressed as .

All channels are quasi-static flat and it is assumed that channel state information (CSI) is perfectly known. Let denote the baseband equivalent direct channel between the BS and user , denote the channel between the IRS and user , and denote the direct channel between the BS and IRS, where indicates whether the uplink channel or the downlink channel is considered. In TDD systems, the downlink and uplink transmissions occur over the same frequency bands but at different time slots. Moreover, channel reciprocity in TDD, as experimentally established for IRS-assisted systems, implies that the uplink and downlink channels are the same [23, 24]. In FDD, the downlink and uplink transmissions occur concurrently over different frequency bands, thus the channels are generally different for uplink and downlink transmissions.

The effective channels between the BS and user in DL and UL are expressed as

(1a)
(1b)

It is important to note that in general TDD systems, the IRS configuration can be different for UL and DL transmissions, whereas for FDD the same IRS configuration is used to support simultaneous UL-DL transmissions. In this work, a joint IRS configuration is used for UL and DL transmissions. However, the case where the IRS configurations can be different for UL and DL transmissions is also considered to obtain an upper bound on system performance. We emphasize that such an upper bound is feasible for TDD systems, with increased overhead and quiet periods, but is not feasible for FDD systems.

Ii-B Linear beamforming and performance metrics

The signal received by user in the DL is

(2)

where is circularly symmetric complex additive white Gaussian noise (AWGN) at user . Similarly, the signal received by the BS in UL is expressed as

(3)

If is the combining beamforming vector at the BS for user , the combined signal is given by

(4)

where is the AWGN at the BS.

The SINR corresponding to user in DL and UL are given by

(5a)
(5b)

Iii Problem Formulation and Solution

In this section, we formulate the IRS-assisted UL and DL weighted sum-rate problems. Then we scalarize the multi-objective optimization problem by forming a weighted sum objective.

Iii-a Problem Formulation

In DL and UL, the achievable data rates in bit/sec/Hz associated with user are given by

(6a)
(6b)

Asymmetrical demands for UL vs DL transmission arise in practical communications systems. In TDD systems, flexible resource allocation becomes feasible by adjusting the proportion of time resources dedicated for UL vs DL. Rather than assigning equal resources to UL and DL, a normalized weight, , is introduced in order to capture the relative proportion of time resources dedicated to DL vs UL transmissions in TDD systems.

In a TDD system with bandwidth Hz, the rate in UL/DL in bit/sec is then given by

(7a)
(7b)

where is the fraction of time used for DL. Normalizing (7a) and (7b) by , we have

(8a)
(8b)

Likewise, in an FDD system with a bandwidth of dedicated to downlink and dedicated to the uplink, the spectral efficiency is again given by (8a) and (8b) respectively.

The first objective is to maximize the weighted sum-rate in DL by jointly optimizing the phase-shifts at the IRS and the transmit beamforming vectors, , at the BS which are subject to a sum-power constraint. The aforementioned problem is given by

(9a)
subject to (9b)
(9c)

where is the weight associated with user in DL, and captures the relative priority of user in the DL. Moreover, denotes the maximum sum-power at the BS.

In TDD, where is the maximum downlink power spectral density in Watts/Hz. Similarly, the noise power in downlink is expressed as where is the noise power spectral density. In FDD, and .

The second objective is to maximize the weighted sum-rate in UL by jointly optimizing the phase-shifts at the IRS, the unit-norm receive beamforming vectors at the BS defined using and the power control at the users defined using where each is subject to a max-power constraint . Maximizing the WSR in UL is expressed as

(10a)
subject to (10b)
(10c)

where is the weight associated with user in UL that captures the relative priority of user in UL. Moreover, denotes the maximum power that can be transmitted by a user. In TDD, where is the maximum uplink power spectral density in Watts/Hz. Additionally, the noise power in uplink is . In FDD, and .

For a joint IRS design, problems (9) and (10) are coupled through the IRS phase shifts . The multi-objective optimization problem is scalarized by formulating a weighted sum problem (WSP) by introducing a weight that captures the relative priority given to DL vs UL when optimizing the IRS configuration. The WSP problem is then formulated as

(11a)
subject to (11b)
(11c)

Iii-B Solving the WSP optimization problem

To solve the non-convex WSP optimization problem in (11), similar to [8, 25, 26] a BCD algorithm is adopted in order to decompose the coupled optimization variables into blocks of decoupled sub-problems. The algorithm proceeds by iterating over the decoupled blocks and updating the decision variables of a particular block while fixing the variables associated with the other blocks. For the WSP problem in (11), the optimization variables are decomposed into four blocks: , , , and . In particular, the WMMSE is used to optimize the transmit beamforming vectors , an MMSE filter is used to optimize the receive beamforming vectors and fractional programming (FP) is used to update the uplink power . Moreover, manifold optimization is used to update the IRS configuration . The details are in Appendix A.

Iii-C Structure of beamforming vectors and channel reciprocity in IRS systems

The joint IRS problem, solved using the algorithm described in Appendix A, involves optimizing the IRS phase shifts based on the downlink transmit beamforming and the uplink receive beamforming. Consequently, for TDD operation, the joint IRS design would benefit from channel reciprocity and similarity in the structure of the uplink and downlink beamforming vectors. The similarity of the structure can be seen from the uplink beamforming vectors in (17) and the downlink beamforming vectors in (15), as both beamforming vectors arise due to an MMSE filter with different weights and powers (see [27]).

In TDD, channel reciprocity holds for the direct link between the BS and the user as this link is independent of the IRS and corresponds to a classical (non-IRS) system. Recently, experimental results in [28],[29] have confirmed channel reciprocity for the cascaded link through the IRS. Under channel reciprocity, the similarity in the uplink and downlink beamforming vectors becomes more pronounced as both vectors have the same structures with differences only due to different power terms as observed in (17) and (15). Subsequently, the performance gap, due to joint design compared to individual design, is expected to be narrower for TDD compared to FDD.

Iv Simulation results

Iv-a System setup and parameters

The IRS-assisted communications system is comprised of an -antenna BS, an -element IRS, and single-antenna users placed in a 3D coordinate system. Following Fig. 2, the BS is located at and the IRS is placed at . Moreover, the users are placed at a height of while being distributed within an plane disc which is centered at with radius .

A uniform rectangular array (URA) is used at the BS with elements such that where and correspond to the number of elements aligned with the -axis and -axis. Similarly, a URA is used at the IRS with elements such that where and correspond to the number of elements aligned with the -axis and -axis. The default case involves a -antenna base station serving users and an IRS with

For TDD, the uplink/downlink carrier frequency is GHz. In FDD, the uplink carrier frequency is GHz and the downlink carrier frequency is GHz. In both TDD and FDD, the antennas at the BS and the IRS are spaced by where is the speed of light in free space. In TDD, Watts and Watts. In FDD, Watts and Watts. Moreover, dBm/Hz. For TDD, the bandwidth is MHz. In FDD, the total bandwidth is MHz, shared between uplink and downlink transmissions. 3GPP path loss models are used [30]. In particular, the line-of-sight (LOS) and non-line-of-sight (NLOS) models are given by

(12a)
(12b)

where is the 3D distance between two points and is the carrier frequency.

For each user , the IRS-user reflected channel and the BS-IRS channel are modeled using a Rician fading model for . In particular,

(13a)
(13b)

where is as given (12a). Also, denotes the 3D distance between the BS and the IRS and denotes the 3D distance between the IRS and user . The Rician factor captures the relative strength of the LOS component compared to the NLOS components. For the numerical experiments, the Rician factor was . The matrix contains the phases for the LOS components which were computed using the underlying geometry. In particular, for the element on the BS and the element at the IRS, separated by 3D distance of , the corresponding phase shift is given by

where is the wavelength associated with the corresponding carrier frequency. Similarly, the vector captures the phase shifts associated with LOS component between user and the IRS. On the other hand, and capture the NLOS components associated with BS-IRS and IRS-user propagation channels, respectively. Moreover, the Rayleigh fading components and are sampled with each entry i.i.d from .

For each user , the BS-user channel is modeled using Rayleigh fading. Particularly,

(14)

where is as given in (12b) and the components of are sampled from .

Iv-B Alternate schemes and UL/DL user weighting strategies

For an IRS with elements, the performance of the joint design in TDD/FDD systems is compared with the upper bound of optimizing the IRS for UL and DL transmissions individually and then the two IRS configurations are time-shared, which we refer to as Individual design. Such an upper bound is achievable for TDD design but at the cost of increased overhead and complexity as explained before. However, this upper bound is unrealistic for FDD communications as both transmissions need to occur simultaneously. Moreover, the performance of the joint design is compared to three other schemes, namely: Fixed Downlink, Fixed Uplink and Ideal Slicing. In fixed downlink, the IRS is optimized for downlink transmission only and then the same IRS configuration is used to assist uplink transmission. With this fixed IRS configuration, the receive beamforming vectors and the power control at the users are then optimized for uplink. In fixed uplink, the IRS is optimized in order to assist the uplink transmission. Then the same IRS configuration is used for downlink transmission and for this fixed IRS configuration the transmit beamforming vectors at the BS are optimized. In ideal slicing, IRS elements are dedicated (sliced) to each of the UL and DL transmissions where the elements dedicated to UL/DL don’t interfere with the DL/UL signals that impinge on them. While ideal slicing is unrealistic since the interference terms cannot be ignored in practice, it nevertheless provides a benchmark against which we compare the joint design. Also note that ideal slicing an IRS of elements is equivalent to individual design with an IRS of elements. The use of the IRS configured for uplink to assist downlink transmission was suggested for the (interference-free) case when a single-antenna BS serves a single-antenna user under TDD scenario [28].

The weighting strategy of the UL/DL user is an important design parameter that can be chosen to provide uniform service or prioritize users in order to achieve proportional-fairness or provide more flexibility in accommodating the requests for DL or UL transmissions. Three weighting strategies are considered: equal, proportional-fair and independent, and the weights are always normalized such that and .

Under the equal weight strategy, which corresponds to providing uniform priority to the users. In order to provide further flexibility in assigning resources while maintaining fairness, the weights in (11) can also be chosen based on a proportional-fair (PF) strategy. In particular, memory is incorporated into the system in order to capture the history of the data transmissions provided for each user in the past. Let be the data rate associated with user at time slot in the DL or the UL indicated by . The objective associated with PF at time-slot is [31] [32]

Then for , approximating and by ignoring (which is a constant for each slot), the weights in (11) are given by where is a normalization constant chosen such that .

(a) Convergence for FDD system
(b) Convergence for TDD system
Figure 3: Convergence for FDD and TDD systems

Finally, we also consider a scenario where the weights are uneven, and uncoupled from user channel quality and position. Specifically, we consider independent user-weights where the weights are randomly chosen independently of all user/network parameters and then normalized. This can correspond to a scenario where the demand for downlink and uplink transmissions vary substantially across the users and the variance of the weights reflect the flexibility needed to accommodate the requests for transmissions.

Iv-C Convergence of the BCD algorithm

The performance of the BCD algorithm in Section III-B is evaluated for the aforementioned system setup and parameters in Section IV-A. The numerical results were obtained by averaging the results over 100 independent channel realizations.

We first consider the convergence of the BCD algorithm in Section III-B for FDD and TDD systems. With , and hence , Fig. 3 shows the weighted sum-rate from (11) vs the number of outer-loop iterations for different number of IRS elements , under independent weights. It is observed that the solution converges reasonably quickly, and in all cases the solution reaches of the final value in less than 20 iterations. Similarly, Fig. 3 shows the convergence of the BCD solution for a TDD system with an IRS of -elements.

Iv-D Performance of the BCD algorithm

For a given and , the joint IRS design is obtained with the uplink WSR weighted by and the downlink WSR weighted by . With again, the weighted sum objective as a function of the number of IRS elements was obtained under FDD and TDD scenarios as shown in the joint design curves of Figs. 4 and 4, respectively.

(a) FDD
(b) TDD
Figure 4: Weighted sum-rate for and independent weights.

From Figs. 4 and 4, the improvement in performance due to joint design, compared with the fixed downlink/uplink lower bounds and ideal slicing, is significantly larger for FDD compared to TDD. Similarly, the loss in performance due to the joint design compared to the individual design upper bound is lower in TDD compared to FDD. This occurs as the joint design in TDD benefits from channel reciprocity along with the structure of the beamforming vectors as explained in Section III-C.

It is observed that the fixed downlink bound in Figs. 4 and 4 performs better than its fixed uplink design counterpart in terms of the weighted sum of UL-WSR and DL-WSR. This follows since the power used by the BS in downlink transmissions is substantially larger than the maximum power that can be transmitted by the users in UL and hence the relative contribution of the downlink rates to the weighted sum in (11) is larger than that of the uplink rates. Thus, the overall performance is more sensitive to the IRS phases that passively beamform the downlink signals compared to the uplink signals. However, the fixed uplink design provides better uplink rates than the fixed downlink design. The trade-off between uplink rates and downlink rates for different schemes is investigated in the next section.

Iv-E Uplink-Downlink Trade-off Regions

For fixed time/frequency resource allocation defined by a fixed , the relative priority of uplink vs downlink is defined by the weight . In order to first explore the rate region for a fixed , the optimization problem in (11) is solved for different values of from 0 to 1 under the independent weighting strategy. Figs. 5 and 5 show the trade-off between uplink weighted rates, given by (10a) and (8b), versus downlink weighted rates, given by (9a) and (8a), for and both FDD and TDD systems.

(a) FDD
(b) TDD
Figure 5: Uplink-Downlink tradeoff region for , where for a fixed , , with and under independent weights.

By optimizing the fraction of time or band spent on uplink vs downlink, i.e. optimizing , the entire trade-off between UL-rate and DL-rate is found and is given by the envelope of the curves in Figs. 5 and 5, and shown in solid black. This envelope represents the trade-off between UL weighted rates and DL weighted rates under joint design and UL/DL resource optimization. The performance of the joint design envelope with respect to the individual design upper bound and the fixed downlink/uplink lower bounds are shown in Figs. 6-8 for the three user weighting schemes. We devise 4 metrics to compare the performances: maximum DL gain of joint design over fixed uplink, maximum UL gain of joint design over fixed downlink, and maximum DL and maximum UL loss of joint design compared to individual design. The two maximum gain metrics correspond to the maximum improvement yielded by employing a joint design vs employing a fixed downlink/uplink design. Moreover, the loss of performance due to joint design compared with the individual design upper bound is quantified using the maximum DL/UL losses.

(a) FDD,
(b) TDD,
Figure 6: Uplink-Downlink Rate region for and equal weights.

Figs. 6 and 6 show the trade-off curves for equal weighting strategies for FDD and TDD systems with along with the 4 performance metrics described above. Similarly, Figs. 7 and 7 as well as Figs. 8 and 8 show the trade-off curves for PF weighting and independent weighting, respectively, again for . Under the FDD scenario, the maximum UL gain due to joint design is and the maximum DL gain is when equal weights are used, shown in Fig. 6. This is compared to and when PF weights are used (Fig. 7) and compared to and when independent weights are used (Fig. 8). The same pattern applies for TDD where the maximum UL and DL are and when independent weights are used. This is in contrast with max-UL and max-DL gains of and when PF weights are used and and when equal weight are used as shown in Figs. 7 and 6, respectively. It follows that the improvement due to joint design compared with the fixed downlink/uplink designs becomes more significant as we deviate from equal weights and as more flexibility in resource allocation is needed.

The max-UL loss and max-DL loss are lower for TDD compared to FDD. Under TDD, the max-UL loss is , , and for equal, PF, and independent weights, respectively. This is compared to , , and for FDD. Similarly, for TDD the max-DL loss is , and for equal, PF, and independent weights, respectively. This is contrast with the higher losses in FDD of , , and . The diminished losses for TDD compared to FDD are attributed to how TDD benefits from channel reciprocity as outlined in Section III-C.

(a) FDD,
(b) TDD,
Figure 7: Uplink-Downlink Rate region for and PF weights.
(a) FDD,
(b) TDD,
(c) FDD,
(d) TDD,
Figure 8: Uplink-Downlink region for and , under independent weighting.

As shown in Figs. 8 and 8 compared to Figs. 8 and 8, the joint design provides greater benefits for larger . The max-DL and max-UL gains increase from and for to and for in the case of FDD configuration. We attribute this to the fact that as increases, the performance improvement due to the IRS increases and hence the performance becomes more dependent on properly optimizing the phase shifts at the IRS and hence joint IRS design becomes more beneficial compared to fixed downlink/uplink designs.

(a) FDD
(b) TDD
Figure 9: Uplink-Downlink Rate region for different and independent weights.

Fig. 9 shows, for independent weights, the trade-off between the uplink rate and downlink rate achieved by joint and individual designs for different values of under both FDD and TDD. The figure reveals that the joint design outperforms the ideal slicing benchmark. This follows since for all values of in Fig. 9, the joint design curve lies above the performance of individual design with elements and thus the rate region of the joint design using elements strictly contains the region of dedicating elements for each of UL and DL. The same results follow for TDD design where the joint design of an -element IRS outperforms the -element design of optimizing the IRS for UL/DL individually.

(a) FDD,
(b) FDD,
(c) TDD,
(d) TDD,
Figure 10: Uplink-Downlink Trade-off region for under FDD/TDD for different with independent weights.
(a) FDD
(b) TDD
Figure 11: Uplink-Downlink Rate region for under ZF beamforming and independent weighting.

For a fixed number of IRS elements , the number of antennas at the BS can impact the performance improvement due to joint IRS design. Fig. 10 shows the impact of the number of antennas at the BS on the max-UL gain, the max-DL gain, the max-UL loss and the max-DL loss metrics under independent weights. Figs. 10 and 10 show the uplink-downlink trade-off regions for and under FDD system with an IRS of elements. Comparing Figs. 10 and 10 with the default case of in Fig. 8, the max-DL gain decreases from to to as the number of antennas at the BS increases from to to . In a similar manner, the max-UL gain decreases from to to as the number of antennas at the BS increases from to to . The same behavior was observed in TDD as demonstrated by Figs. 10 and 10 compared with Fig. 8 as the max-UL gain decreases from to to and the max-DL gain decreases from to to as the number of antennas at the BS increases. This behavior is attributed to the relative contribution of passive beamforming at the IRS compared to active beamforming at the BS. As the ratio of the number of IRS elements to the number of antennas at the BS increases, the performance of the system becomes more dependent on the passive beamforming at the IRS and sensitive to proper optimization of the IRS phase shifts. Hence, the performance improvement due to joint IRS design becomes more critical.

While the WMMSE algorithm used in Section III-B converged within a few iterations, zero forcing (ZF) beamforming can also be used as heuristic in order to obtain the beamforming vectors with less complexity [27]. Fig. 11 shows the uplink-downlink trade-off regions when ZF beamforming is used for FDD and TDD with . For FDD, compared to the results of Fig. 8, the rate regions shrink when ZF is used as the maximum achievable downlink rate is reduced from 2.8 bit/sec/Hz to 2.52 bit/sec/Hz and thus WMMSE yields an improvement of . Similarly, WMMSE yields an improvement of in terms of the maximum achievable uplink rate. The max-DL gain falls slightly from under WMMSE to for ZF beamforming. Similarly, the max-UL gain falls slightly from under WMMSE to for ZF beamforming. Hence, for ZF, the improvement in performance due to joint design compared with the uplink/downlink fixed design is similar to WMMSE. Similarly for TDD, the rate regions shrink when ZF is used as the maximum achievable downlink rate is reduced from 3.1 bit/sec/Hz to 2.8 bit/sec/Hz and thus WMMSE yields an improvement of . The maximum achievable uplink rate shows an improvement of when WMMSE is adopted compared with ZF. Furthermore, under ZF the max-UL and max-DL gains are similar to those when WMMSE is adopted for beamforming. Consequently, while ZF has lower complexity and performance compared to WMMSE, however, the relative performance gains of joint-design are about the same as the WMMSE.

V Conclusion

This paper has investigated joint uplink-downlink IRS design for both TDD and FDD MU-systems. A weighted-sum problem (WSP) to maximize the uplink and downlink rates accounting for the flexibility needed when assigning unequal resources to uplink/downlink transmissions was formulated and solved. For TDD, a joint design reduces the overhead required for updating the IRS as well as relaxes the constraints on time synchronization between the IRS and BS. A joint design is also essential to the operation of FDD systems as an individual design is not physically realizable in FDD. Extensive numerical results were used to quantify the improvement in performance due to the joint design compared to fixed uplink/downlink heuristic designs and the ideal slicing benchmark. Moreover, the performance loss due to the joint design with respect to allowing individual IRS design for uplink/downlink transmissions was quantified. Subsequently, we investigated the effect of different parameters on the performance improvement due to the joint design. This improvement was enhanced by increasing the ratio of the number of IRS elements to the number of BS antennas. In FDD, the gains achieved by the joint design compared with the alternative schemes were significant in all considered scenarios and thus the joint design has significant benefits for FDD in all scenarios. For TDD, the joint design yields substantial benefits compared to fixed uplink/downlink when the ratio of the number of IRS elements to the number of BS elements is large and/or when the user weights are independent. Otherwise, in TDD, the fixed UL design is almost optimal and is less complex than the fixed DL design, joint design, or the individual design.

Appendix A

In this appendix the non-convex WSP in (11) is optimized. A BCD algorithm is adopted and optimization variables are decomposed into four blocks: , , , and .

A-a Updating the beamforming vectors and power control for fixed IRS configuration

For fixed , the WSP in (11) decouples into a separate downlink sub-problem in (9) and uplink sub-problem in (10).

A-A1 Updating transmit beamforming vectors

when is fixed, optimizing the transmit beamforming vectors in (9) corresponds to a DL-WSR maximization problem. As in [22], by introducing the auxiliary variables and and with , , the DL-WSR problem in (9) is equivalent to the WMMSE problem written as

(15)
subject to

where the MSE associated with user is where

and the estimated DL symbol at user

is given by .

The equivalent WMMSE problem lends itself to a tractable iterative algorithm as the problem in (15) is convex in the optimization variables when the other variables are fixed to the values of their previous iteration. Proceeding similar to [22], the problem in (15) is optimized by iterating over the following update rules for each user :

(16a)
(16b)
(16c)

and is the optimal Lagrange multiplier chosen such that complementary slackness associated with the sum power constraint is achieved [22]. By iterating over (16a), (16b) and (16c), Theorem 3 of [22] guarantees convergence to a stationary point of the (15).

A-A2 Updating receive beamforming vectors

when and are fixed, the objective in the uplink sub-problem of (10), is maximized by the MMSE receiver [27]. For each user , the receive beamforming vector is given as

(17)

A-A3 Updating uplink power control

when and are fixed, fractional programming (see [33], [34]) is used to optimize in (10). Similar to [33], the Lagrangian dual reformulation is applied by introducing the auxiliary variable . The power control problem of (10) is equivalent to:

(18)
subject to

For fixed , setting the derivative of the objective in (18) with respect to and solving for , yields for each which is given by

(19)

By applying the quadratic transform and proceeding similar to [33], the power control update for each user is obtained as

(20)

where is obtained using (19) and the auxiliary variables are updated as:

(21)

Iterating over the updates in (21), (19) and (20) yields a fixed point of the power control problem in (18) as established in [34].

A-B Updating the IRS configuration for fixed beamforming vectors and uplink power control

For fixed , and , ignoring terms that don’t depend on , the WSP in (11) simplifies to:

(22)
subject to

Here for , is given as

(23)

where

(24a)
(24b)

The constraint of the optimization problem in (22) defines a search space characterized by a product of complex circles, thus representing a Riemannian submanifold of [25]. Moreover, the function is differentiable. Consequently, a stationary point of the optimization problem in (22) can be obtained using the Riemannian conjugate gradient (RCG) algorithm [8, 25].

For any point on a manifold , the tangent space denoted as comprises all the tangent vectors, each of which defines a search direction that can be used to optimize an objective function. The Riemannian gradient denoted as represent the direction along which the objective function experiences the steepest increase and in the case of the complex circle manifold (CCM) of (22), is given by