EnvironmentAdaptable Fast MultiResolution (EAFMR) optimization in largescale RFFPGA systems
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Abstract
Softwaredefined radio (SDR) can have high communication quality with a reconfigurable RF frontend. One of the main challenges of a reconfigurable RF frontend is finding an optimal configuration among all possible configurations. In order to efficiently find an optimal configuration, EnvironmentAdaptable Fast (EAF) optimization utilizes calculated signaltointerferenceandnoise ratio (SINR) and narrows down the searching space (Jun et al., Environmentadaptable efficient optimization for programming of reconfigurable Radio Frequency (RF) receivers, 2014). However, we found several limitations for applying the EAF optimization to a realistic largescale Radio FrequencyField Programmable Gate Array (RFFPGA) system. In this paper, we first investigated two estimation issues of RF impairments: a saturation bias of nonlinearity estimates and limited resources for RF impairment estimation. Using the estimated results, the SINR formula was calculated and used for the EnvironmentAdaptable Fast MultiResolution (EAFMR) optimization, which was designed by applying the EAF optimization to multiresolution optimization. Finally, our simulation setup demonstrated the efficiency improvement of the EAFMR optimization for a largescale RFFPGA.
Keywords
Softwaredefined radio (SDR) Reconfigurable RF frontend Largescale RFFPGA Estimation of RF impairments EnvironmentAdaptable Fast MultiResolution (EAFMR) optimizationAbbreviations
 DARPA
The U.S. Defense Advanced Research Projects Agency
 DoD
Department of Defense
 DoE
Design of experiments
 EAFMR optimization
EnvironmentAdaptable Fast MultiResolution Optimization
 EAF optimization
EnvironmentAdaptable Fast optimization
 FCC
Federal Communications Commission
 IF
Intermediate frequency
 IIP3
The thirdorder input intercept point
 IP3
The thirdorder intercept point
 LO
Local oscillator
 MR optimization
Multiresolution optimization
 MSB
Most significant bit
 RF
Radio frequency
 RFFPGA
Radio FrequencyField Programmable Gate Arrays
 SINR
Signaltointerferenceandnoise ratio
 SNR
Signaltonoiseratio
1 Introduction
Wireless communication is ubiquitous in our daily life using an excessive number of communication standards: 3G/4G/LTE cellular service, WiFi, bluetooth, and so on. The new paradigm of Internet of Things (IoT) has also accelerated the plethora of wireless communication [3]. The abundance of communication standards has recently increased interest in cognitive radio because its multistandard platform supports cost reduction of service suppliers and the efficient use of frequency spectrum.
First, cognitive radio allows for more affordable service to providers and subscribers. Previously, when a new communication standard comes out, service providers have spent a huge amount of expenditure for building a new infrastructure such as base stations and hardware systems corresponding to a new communication standard. Particularly in rural areas, it is a burdensome and timeconsuming process for service suppliers to include rural areas to the new service coverage due to a low user density. However, cognitive radio must have multistandard platforms in its hardware. Due to the dynamic property of cognitive radio, service providers can reuse the existing infrastructures when serving a new communication standard. Thus, service providers reduce the expensive cost and time for distributing the infrastructure for the new standard.
In addition, cognitive radio is an optimal solution for handling inefficient radio frequency (RF) spectrum utilization. The RF spectrum is a limited natural resource, and it is officially managed by Federal Communications Commission (FCC) [1]. Given the fixed standard assignment policy of government agencies, issues might arise for future wireless communication due to the excessive amount of communication standards. Since the RF spectrum occupation rate has a wide variation, in time and space, from 15 to 85% [1], cognitive radio can dynamically and intelligently change its communication channel and temporally utilize the portions of the unused RF spectrum (called a spectrum hole or white space) [7, 8, 17].
This multistandard platform of cognitive radio is achieved by implementing softwaredefined radio (SDR). In order to design the SDR architecture, there have been studies to design fixed wideband RF frontend. The fixed wideband RF frontend accommodates various signals on different RF spectrum bands by choosing a homodyne architecture. Also, the fixed wideband RF frontend generally requires to remove RF filters and to choose durable RF components with wideband signals [6]. However, the condition of wideband signals limits the choice of RF components and performance. In contrast, a reconfigurable RF frontend (also called reconfigurable Radio FrequencyField Programmable Gate Array (RFFPGA)) is more appropriate for the SDR architecture in terms of selectivity and sensitivity [9].
This architecture of RFFPGA as shown in Fig. 1 maximizes the reconfigurability of SDR, and thus, RFFPGA is an attractive alternative for supporting a multistandard platform of cognitive radio in the future. Particularly, the enhanced reconfigurability in this RFFPGA architecture is directly related to increasing the reusability of this device. The RFFPGA reusability can save hardware redesign cost for updating wireless communication standards. In addition, RFFPGA can have a high performance solving the practicality issue of a cognitive radio because it can be dynamically tunable to a particular setting optimized to realtime communication environments in the field.
Note that RFFPGA operates on RF spectrums, and thus, it is distinguished from digital FPGA that operates on baseband spectrums. It is known that RFFPGA is beneficial with respect to reduced life cycle cost, reduced redesign cost, and service for multiple DoD platform.
RFFPGA can dynamically adapt to the communication environment onthefly retaining a high communication quality. RFFPGA is defined as a dynamically reconfigurable (programmable) RF transceiver. RFFPGA is composed of an array of reconfigurable RF component blocks that can have as many as 100 bits of control. As shown in Fig. 1, the RF component blocks are adjusted by switches that are deployed between the RF component blocks. The switches connect the RF component blocks that are chosen by the reconfiguration engine while bypassing the RF component blocks in the idle mode (i.e., the bypassed RF component blocks are not selected by the reconfiguration engine). This RFFPGA architecture is designed to maximize the flexibility of RFFPGA for adapting to varying RF environments. This flexibility of RFFPGA can allow to change the operating frequency band of the RF transceiver.
A reconfigurable RF frontend can flexibly select one of the RF components arranged in each component bank using switches and serially connect them in different architectures, including heterodyne and homodyne [5]. The ability to have many configurations in component banks supports a multistandard platform for currently existing standards. The component bank needs to include a number of configurations with different ranges to consider future standards. For future standards, the component banks contain many configurations that may not be accounted for current settings. When a reconfigurable RF frontend selects RF components in the component banks, it aims to satisfy the requirements of the given communication standard. Thus, a reconfigurable RF frontend supports a multistandard platform by dynamically changing its occupying RF spectrum of various standards for cognitive radio.
In order to operate a reconfigurable RF frontend, an important challenge is finding an optimal configuration among all possible configurations (a configuration is mapped to a combination of RF components in a reconfigurable RF frontends). In order to find the configuration, an optimization problem is defined as finding a configuration that minimizes its cost function (such as power consumption) while satisfying constraint requirements of a given communication standard (such as signaltonoise ratio (SNR)) [9]. Studies have aimed to solve this optimization problem in a reconfigurable RF frontend. Tao et al.’s study in [15] proposed twophase relaxation optimization that attempts to reduce the possibility of finding local optimal configurations. This process performs local relaxation optimization in two separate phases: In the first phase, the optimization finds a configuration of the highest SNR; in the second phase, starting from the found configuration, the optimization searches an optimal point using the relaxation optimization. This optimization has improved selectivity of optimal configuration, and thus, when the reconfigurable radio is initially programmed, several base configurations (i.e., default configurations) can be preprogrammed into it, based on the desired communication standards. However, it has a limited capability of reducing reconfiguration load for a dynamic communication environment, with blockers and large interferers appearing randomly at different frequencies. In this case, the reconfigurable RF frontend cannot be preprogrammed exhaustively, since there are too many possible scenarios to be considered. Rather, a fast algorithm to adapt the RF frontend to interferers in realtime is needed. Thus, a fast optimization algorithm—called EnvironmentAdaptable Fast (EAF) optimization—was proposed for speeding up the optimization of a reconfigurable RF frontend in a dynamic spectrum environment [9].
The EAF optimization improves its optimization efficiency by predicting configurations that are less likely to pass the communication quality requirement of a given standard. The prediction is performed based on signaltointerferenceandnoise ratio (SINR), which is calculated—as opposed to simulated —using RF impairment information. These are three steps of the EAF optimization: (1) RF impairments should be estimated when a reconfigurable RF frontend is manufactured. The estimators of RF impairments are designed based on the signal model of baseband signals. For RF impairment information, the estimation methods of RF impairments are designed as presented in [9, 10, 11]. (2) The SINR calculation formula for the EAF optimization is described in [11]. In the SINR calculation, impairing signal power due to each of the RF impairments—phase noise, IIP3, noise figure—was separately derived in terms of the power of the signal of interest and interference. The SINR calculation uses estimated RF impairments and a signal spectrum in a given communication environment. Thus, realtime blocker information is reflected in SINR calculations. (3) Finally, the EAF optimization uses the SINR calculation to identify the configurations that likely do not meet a SINR specification and to narrow down the search space of configurations. The EAF optimization method was shown to reduce the computational cost significantly, finding an optimal configuration after only five iterations instead of searching all possible configurations exhaustively. Therefore, the EAF optimization method can be successfully utilized as a heuristic to find a minimum power configuration of adequate SNR in an experimental case of a small reconfigurable RF system.
In contrast, a largescale RFFPGA poses a significant problem for the EAF optimization. A realistic RFFPGA is likely to be controlled by several hundred bits that select the component values. In such a largescale RFFPGA, it is not possible to directly measure and characterize the RF impairments of each of the possible configurations.
 (1)
RF impairment estimation in a largescale RFFPGA: we propose a method that estimates unknown RF impairments of the configurations that can not directly estimated using baseband signals in a largescale RFFPGA. The configurations with known estimates are used to measure other configurations with unknown RF impairments. This estimation method is useful because only a limited number of configurations in a largescale RFFPGA are estimated due to limited time and computation resources.
 (2)
environmentadaptable algorithm: when choosing an optimal configuration in a largescale RFFPGA, the EARMR algorithm reflects communication quality changes due to interference in the field. This interference effect is measured using the signaltointerferenceandnoise ratio (SINR) calculation in the algorithm.
 (3)
the improved timeefficiency of EAFMR optimization algorithm for a largescale RFFPGA: in the EAFMR algorithm, the EAF optimization synergizes with the multiresolution method in order to improve the time efficiency of the optimization process in a largescale RFFPGA.
In this paper, we investigate a modelbased approach to solve this problem in order to use the EAF optimization in largescale RFFPGA systems. First, we studied RF impairment estimation for a largescale RFFPGA system. We derived statistical signal models of RF impairments, and based on the statistical signal models, we designed estimators of RF impairments. We focused on two main problems while applying the estimators of RF impairments to a largescale RFFPGA system: two saturation areas of nonlinearity estimates and limited resources in RF impairment estimation procedures. The saturation areas of nonlinearity estimates are caused by a wide range of RF frontends. In order to solve the saturation, we designed a formula for changing the transmission signal power in an estimation procedure. Also, we considered limited resources in RF impairment estimation in a largescale RF frontend. Solving the limited resource problem, we used a design of experiments (DoE) approach that effectively selects sample configurations among the large number of all possible configurations in a largescale RFFPGA. We also used an interpolation method that obtains unknown component values from known component values of RF impairments calculated from sample configurations. Second, we derived a formula that calculates SINR in a given communication environment in terms of phase noise, nonlinearity, and noise figure. In order to improve the accuracy of the SINR calculation, the impairing signal power by each RF impairment was separately analyzed considering the frequency offset of interference and the power of the signal of interest and interference. We assumed that RF impairments are estimated, and a signal spectrum is obtained by a signal analyzer [9], a configuration of reconfigurable RF frontends that bypasses all filters and amplifiers. Third, using the SINR calculation, we designed the EAFMR optimization method that hastens an optimization process for finding an operable configuration in a largescale RFFPGA system. Simulation results showed that our EAFMR optimization method significantly improves its optimization speed compared with the multiresolution optimization method.

component: the element—such as amplifier, filter, or mixer—that composes a reconfigurable RF frontend. The components are placed and switched in RF component banks.

configuration: a cascade of the chosen components from RF component banks in a reconfigurable RF frontend. Configurations can be reconfigured by switching each component.

overall RF impairment: the metric that represents the overall RF impairment of a configuration.

component value: the metric that represents the RF impairment of a component.
2 RF impairment estimation in largescale RFFPGA systems
In this section, we investigate the estimation of RF impairments in a largescale RF frontend. RF impairments in a RF system are described in [16], and the basic insight of signal processing is explained in [1, 2, 8, 9]. The first problem associated with this large scale is that the large number of RF frontends span a wide range of RF impairment values. Due to the wide range of RF impairment values, the RF frontend is very likely to experience saturation in an estimation process when transmission power is fixed. In Section 2.1, we solve this problem by designing a method that adjusts the transmission power during the estimation procedure. The second problem is the large number of configurations in a largescale RF frontend. This problem is tackled by using a design of experiments (DoE) approach and an interpolation procedure as discussed in Section 2.2. These methods are important in a largescale RF frontend because we cannot simulate (or estimate) all configurations; we can only simulate a limited number of sample configurations. For example, a reconfigurable RF frontend with 63 bits of control means that the number of configurations is about 9.2 × 10^{18}, which results in an impractically timeconsuming estimation procedure. Our methods of RF impairment estimation in a largescale RF frontend are utilized to efficiently extract unknown properties of RF components from known properties that are calculated from simulated data. This method of RF impairment estimation is applied to a largescale RF frontend of 63 bits.
2.1 Tackling a wide range of RF impairment values: bias in IIP_{3} estimate
where Δf_{(dB)}=10·log_{10}B_{(dB)}.
where K_{0}=P_{sig(dBm)} and \(\Delta K=\!\frac {1}{2} \!\left (\! P_{\!\text {sig} \text {(dBm)}} \,\, \left (NF_{0 \text {(dBm/Hz)}}\right.\right. \left.\left. + \Delta f_{\text {(dB)}} + NF_{\text {(dB)}} \right) \right)\).
Now, we specify the choice of transmission power P_{tx(dBm)}=P_{sig(dBm)}+ΔP_{(dB)} that should be used for IIP_{3} estimation to allow low bias.
where IIP_{3,k}, G_{ k }, are the IIP_{3} and gain of the kth component for k=1,2,⋯,N.
From this, we can assume that the nonlinearity of the last cascaded component is dominant, i.e., \(\frac {1}{{\text {IIP}}_{3}} \approx \frac {G_{1} \cdot G_{2} \cdot \cdots \cdot G_{N1}}{{\text {IIP}}_{3,N}}\). In that case, IIP_{3} is given as \({\text {IIP}}_{3 \text {(dBm)}} \approx \text {IIP}_{3,N \text {(dBm)}}  G_{1}^{N1} {~}_{\text {(dB)}}\) where \( G_{1}^{N1}\) is the overall gain of the first N−1 components: \( G_{1}^{N1} {~}_{\text {(dB)}} = G_{1 \text {(dB)}} + G_{2 \text {(dB)}} + \cdots + G_{N1\text { (dB)}} \).
2.2 Tackling a large number of configurations: DoE and interpolation methods
A significant challenge in applying the EAF optimization method to a largescale RF frontend is that RF impairment estimation is timeconsuming when the number of configurations is very large. In this section, we use modelbased design of experiments (DoE) and interpolation methods to drastically reduce the time needed for estimation.
2.2.1 Configuration subsets: fullfactorial design set, screening design set, and sample configuration set
2.2.2 Configuration subsets and RF impairment calculation
where G_{ k },IIP_{3,k}, and F_{ k } are the gain, IIP_{3}, and noise factor, respectively, of the kth component of a configuration (k=1,2,⋯,N), and G^{ A },IIP_{3} ^{ A }, and F^{ A } are the overall gain, IIP_{3}, and noise factor, respectively, of the configuration.
Thus, in order to obtain an unknown value of overall RF impairments, we need to obtain the component values of RF impairments. We will now explain how to obtain component values from sample configurations and how to select sample configurations based on a linear model of RF components in Section 2.2.3.
2.2.3 Linear model of RF impairments in a largescale RFFPGA
where X_{ k } represents the RF impairment of the kth component of a configuration, h_{ k } is a coefficient for the kth term, k=1,2,⋯,N, and Z represents the overall RF impairment of the configuration. The impairment can be either gain (X_{ k }=G_{ k } _{(dB)}), IIP _{3} (X_{ k }=1/IIP_{3}), or noise factor (X_{ k }=F_{ k }−1). For example, for (11), \(X_{k} = \frac {1}{{\text {IIP}}_{3,k}}\), h_{1}=1, h_{ k }=G_{1}·G_{2}·⋯·G_{k−1}, for k∈{2,⋯,N}.
where Y is the vector of measured RF impairment using simulation or actual measurement (each element corresponding to one sample configuration that is simulated) while the vector N represents measurement noise during the estimation process.
where l=1,2,3,⋯,C_{ k }.
Thus, we can define the matrix H in (15) composed of the row vector h^{′} in (16) corresponding to the measured sample configuration Y.
2.2.4 DoE for choosing optimal sample configurations
We select a set of optimal sample configurations among the configurations in the screening design set using the design of experiments (DoE) approach. The DoE method selects an optimal set of m sample configurations (equivalently, the matrix H in (18) corresponding to the m selected sample configurations) using a Doptimal design approach. In the matrix H, each row corresponds to one of sample configurations. The Doptimal design approach iteratively updates a set of sample configurations in order to increase the determinant H^{ T }·H, i.e., minimize the logdeterminant of noise variance matrix in (15) to find optimal sample configurations. The matrix H is updated until its conditional number—the ratio of the maximal eigenvalue of H over the minimal eigenvalue of H—is small. The condition of a small conditional number prevents an error in Y from causing a large error in θ in 15 The implementation details of the DoE method for choosing sample configurations are described in Algorithm 1 (Appendix).
2.2.5 Simulation results
The estimated gain and IIP_{3}, \(\widehat {Y}\) in (18), of sample configurations are plotted on the yaxis against true gain and IIP_{3}Y of the sample configurations on the xaxis in Fig. 7a and b respectively. The simulation results show that the estimates of \(\widehat {Y}\) proportionally increase with true values of \(\widehat {Y}\). (At high gain values, around 100 dB, the estimates begin to saturate in Fig. 7a. Also, configurations where at least one amplifier is bypassed have IIP_{3} estimates which are about 3 dB lower in Fig. 7b.)
The estimated gain and IIP_{3}, \(\widehat {\mathbf {\theta }}\) of the three amplifiers, are plotted on the yaxis against true gain and IIP_{3}θ of the three amplifiers on the xaxis in Fig. 7c and d respectively. The estimates \(\widehat {\mathbf {\theta }}\) proportionally increase with true values of θ. (The estimated IIP_{3} of the first and second amplifiers are about 3 dB lower than that of the third amplifier in Fig. 7d.)
Finally, the estimated gain and IIP_{3}, \(\widehat {\mathbf {Z}}\), of the remaining configurations in the screening design set are plotted on the yaxis against true gain and IIP_{3}Z on the xaxis in Fig. 7e and f respectively. The estimates \(\widehat {\mathbf {Z}}\) proportionally increase with true values of Z.
2.2.6 Interpolation method
In order to extend the RF impairments estimated for configurations in the screening design set to the fullfactorial design set, the interpolation method is applied. Recollect that the configurations in the screening design set only use a few of the most significant bit (MSB) bits of each component, with the other bits frozen to zero. The assumption made here is that the bits of each component define a real number (or set of real numbers) and that the true RF impairment is a continuous function of this (set of) real numbers.
The gain and IIP_{3} characteristics of amplifiers in the fullfactorial design set on the yaxis are obtained by applying the interpolation method to the corresponding estimators of the screening design set. Figure 9a, b, which shows gain and IIP_{3} respectively on the yaxis, shows close agreement with the true values on the xaxis.
The obtained RF impairment estimates using the interpolation method are applied to calculate the overall RF impairments of the configurations in the fullfactorial set using the formula (10), (11), and (12). The calculated overall gain and IIP_{3} of 150 configurations in a full factorial set on the yaxis are plotted against the true values on the xaxis in Fig. 9c, d respectively while estimates of gain and IIP_{3} obtained by simulation of the same configurations representing a fullfactorial set are plotted in Fig. 9e, f. The results demonstrate that unknown RF impairments of the configurations in a full factorial design set are successfully obtained by applying the interpolation method.
2.2.7 Efficiency of the design of RF impairment estimation for a largescale RFFPGA
Comparison of time consumption for RF impairment estimation
RF impairments  Gain  IIP_{3} 

Simulation running time  41 s  42 s 
for a configuration  
The number of simulations  10^{19} runs  10^{19} runs 
for fullfactorial designs  
The number of simulations  2.6×10^{5} runs  2.6×10^{5} runs 
for screening designs (2 MSBs of  
components are reconfigured)  
The number of simulations for  384 runs  384 runs 
sample configurations  
Total simulation time for  1.3×10^{13} years  1.33×10^{13} years 
full factorial designs  
Total simulation time for  3×10^{3} h  3.1×10^{3} h 
screening designs  
Total simulation time for  4.37 h  4.45 h 
sample configurations 
3 SINR calculation in a largescale RFFPGA
where P_{ S } is the signal power, P_{ phn }, P_{ip3}, and P_{ n } are the impairing signal powers caused by phase noise impairment, nonlinearity impairment, and thermal noise impairment, respectively.
For the SINR calculation in a largescale RF frontend, we combined the previous estimation methods of RF impairments for obtaining P_{ phn } and P_{ip3} and our new approach to extend the estimation methods to a largescale RF frontend. The P_{ phn } and P_{ip3} in the SINR calculation can be calculated from the estimation methods of RF impairments introduced in [9, 10, 11]. Using the estimation methods, the parameters of RF impairments were directly obtainable for all available configurations of a smallscale reconfigurable RF frontend. However, the designed methods cannot be utilized in a largescale reconfigurable RF frontend due to the limitation of time resource as discussed in Section 4.3. Thus, for the SINR calculation in a largescale reconfigurable RF frontend, we use the estimated RF impairments obtained by applying the DoE approach and the interpolation method. The details of the designed estimation method of RF impairments for a largescale reconfigurable RF frontend were introduced in Section 2.2.
4 EAFMR optimization in a largescale RFFPGA
4.1 Optimization problem in a largescale RFFPGA
where the vectorvalued x indicates one of the available configurations for a reconfigurable RF frontend, and each element of x represents the parameter value of an individual component. F(x) is a cost function such as power consumption of x, which needs to be minimized, while g_{ k }(x)≥G_{ k }(1≤k≤K) describe constraints such as SINR.
However, this optimization model in (20) is not suitable to describe the optimization problem in a largescale reconfigurable RF frontend due to the explosively increased size of search space. With the increased search space in a largescale reconfigurable RF frontend, the limited time resource causes a bottleneck for a practical use of the previously suggested optimization methods. Practically, the amount of time consumed to obtain SINR using simulation (or measurement) in an optimization process dramatically increases in a largescale reconfigurable RF frontend. Thus, one of the important factors to be considered in a largescale reconfigurable RF frontend is reducing the number of simulations (or measurements) for obtaining SINR in an optimization process.
where H(n) is a function of the suggested total number n of simulations (or measurements) in a given optimization algorithm. In our example of a largescale RF frontend with 63 bits of control, the optimization problem in (21) is given by defining that F(x) is the power consumption of the configuration x, g_{1}(x) is the SINR of x, G_{1} is the SNR threshold of the given communication standard, K=1, and H(n) is given as log_{2}(n).
We note that according to the choice of optimization algorithms, an optimal configuration \(\widehat {\mathbf {x}}\) that satisfies the optimization problem in (21) can vary because there is a trade of the costs F(x) and H(n). If a designed algorithm runs more iterations of algorithms to find an optimal configuration, the cost of F(x) decreases but the cost H(n) increases, and vice versa. Thus, the optimal configurations of different optimization algorithms may not be the same according to how to design the algorithms.
Solving the defined optimization problem, we introduce the EnvironmentAdaptable Fast MultiResolution (EAFMR) optimization designed for a largescale RFFPGA to adapt to the dynamic conditions in communication.
4.2 EAFMR optimization
We designed the EAFMR optimization method [9] by applying the EAF optimization that utilizes the SINR calculation to the multiresolution optimization.
The multiresolution optimization primarily has a similar structure as the twophase relaxation optimization [15]. In the first phase, the multiresolution optimization iteratively finds the configuration of maximal SINR. In the second phase, starting from the found configuration, the multiresolution optimization finds an optimal configuration that has the lowest power consumption satisfying the SNR specification for a given communication standard.
The multiresolution optimization improve the twophase relaxation optimization for a largescale reconfigurable RF frontend. The multiresolution optimization narrows down the search space into screening design sets of available configurations in a largescale RF frontend. The size of screening design sets is determined by the number of active reconfiguration knob bits in all RF components. The fewer active bits of knobs, the more sparsely the characteristic values of the configurations in the screening design sets are distributed. The multiresolution optimization tries to search for two types of configurations in the two phases, from small to large active bits. If there is no improvement for one cycle of iterations for all configurations, the iterative reconfiguration is stopped. The pseudocode of the multiresolution algorithm is given in Algorithm 3 (Appendix).
In our EAFMR optimization method, we applied the primary algorithmic structure of the multiresolution optimization, and we utilized the calculated SINR in order to reduce the number of configurations to be simulated. When the SINR calculation of the configurations does not meet the predefined SINR threshold, the configurations are trimmed out in the list of configurations to be simulated. The pseudocode of the multiresolution algorithm is given in Algorithm 4 (Appendix).
4.3 Search space of optimization methods
Search space of optimization methods
Optimization  Search space  No. simulations 

method  per iteration when  
K=3, C_{ k }=3, N(k,l)=7  
Exhaustive search  \(O\left (2^{\sum _{l} \sum _{k} N(k, l)}\right)\)  2^{3×3×7}=2^{63}=9.22×10^{18} 
Simulated annealing  \(O\left (2^{\sum _{l} \sum _{k} N(k, l)}\right)\)  ≤ 9.22×10^{18} 
search  
Local relaxation search  \( O\left (\sum _{k} 2^{\sum _{l} N(k, l)}\right)\)  3×2^{3×7}=6.29×10^{6} 
EAF optimization  \( O\left (\sum _{k} 2^{\sum _{l} N(k, l)}\right)\)  ≤ 3×2^{3×7}=6.29×10^{6} 
Multiresolution search  \(O\left (\sum _{l} \sum _{k} 2^{N(k, l)}\right)\)  1.15×10^{3} 
EAFMR optimization  \(O\left (\sum _{l} \sum _{k} 2^{N(k, l)}\right)\)  ≤ 1.15×10^{3} 
The EAF optimization and the EAFMR optimization have the same search space as the local relaxation and the multiresolution search, respectively. However, these optimization methods reduce the search spaces by eliminating candidate configurations that have SINR calculation under a threshold. The size of reduced search space depends on the communication environment in a field.
Reducing search space is a critical factor in the optimization of a largescale RF frontend due to practical time consumption. For this reason, it is impractical to apply classical methods to a largescale RF frontend. However, the search space is dramatically reduced from exponential increase in exhaustive search to linear increase in multiresolution search in terms of the number of configurations and the number of components as shown in Table 2. Therefore, multiresolution search and the EAFMR optimization method are practically applicable in a largescale RF frontend.
H(n) in (21) of optimization methods
Optimization method  H(n) after M iteration assuming C_{ k }=L and N(k,l)=N_{0}  H(n) after one iteration (M=1), when K = 3, C_{ k } = 3 and N(k,l)=7 

Exhaustive search  L·K·N_{0}+log_{2}M  63 
Simulated annealing  ≤L·K·N_{0}+log_{2}M  ≤ 63 
search  
Local relaxation  L·N_{0}+log_{2}K+log_{2}M  33.28 
search  
EAF optimization  ≤L·N_{0}+log_{2}K+log_{2}M  ≤33.28 
Multiresolution  N_{0} + log_{2}L+log_{2}K+log_{2}M  10.17 
search  
EAFMR optimization  ≤N_{0}+log_{2}L+log_{2}K+log_{2}M  ≤ 10.17 
In the next section, we are going to show simulation results of multiresolution search and the EAFMR optimization.
4.4 Simulation
We finally demonstrate the performance of our EAFMR optimization using a Matlab simulation.
4.4.1 Simulation setting
We verified the performance of the designed EAFMR optimization in a largescale RFFPGA (Fig. 2) using Matlab Simulink. In the simulation, we used the communication standard IEEE 802.11g, and the required SNR specification is 11.5 dB. The transmitted signals have the bandwidth 20 MHz and the power − 67 dBm. The RFFPGA has 63 bits, which implies approximately 10^{19} configurations. There are two fixed RF filters, one fixed IF filter, and three 21 bit amplifiers. The power consumption range of all possible configurations is from 11.85 to 107.85 mW. We tested two optimization methods, multiresolution and EAFMR, on the scenario, which has one interferer of − 30 dBm at 20 MHz frequency offset from the signal of interest.
4.4.2 Simulation results
Simulation results of no. simulations
Optimization  Phase I  Phase II  Total number 

method of  
simulations  
Multiresolution optimization  816  1356  2172 
MREAF optimization  1  1  2 
Simulation results of optimization
Optimization  Power (mW)  SINR (dB)  Total number 

method  of simulations  
Multiresolution optimization  12.52  11.63  2172 
MREAF optimization  20.98  15.55  2 
Both optimization methods found optimal configurations that satisfy the SNR specification (11.5 dB) as shown in Table 5. In terms of power consumption, we found that there is a tradeoff between higher power consumption and the number of simulations. The power consumption of the EAFMR’s optimal configuration is around 21 mW compared to around 12 mW (multiresolution’s optimal configuration). The power consumption of the EAFMR’s optimal configuration is almost twice of that of multiresolution search. However, the maximal power consumption of all possible configurations is larger than 100 mW. Compared to the maximal power consumption, the EAFMR optimization found an optimal configuration that has a power consumption in a reasonable range of energy efficiency.
Simulation results and cost function in the optimization problem (21)
Optimization  F(x)  H(n)  F(x)+H(n) 

method  
Multiresolution optimization  12.52  11.08  33.60 
MREAF optimization  20.98  1  21.98 
5 Conclusion
In this paper, we investigated the application of the EAF optimization in a largescale RFFPGA. To estimate RF impairments in a largescale RF frontend, we solved two main problems. First, we solved the saturation of nonlinear estimates due to a wide range of RF frontends. To avoid the saturation problem, we obtained a formula for adjusting transmission power for an estimation procedure. Second, we solved a limited estimation resources problem. Because of a large number of configurations, it is not possible to directly measure and characterize the RF impairments of all possible configurations. We extended the estimation procedure to a largescale RFFPGA using the design of experiments (DoE) approach and the interpolation method. Finally, we designed an EAF multiresolution (EAFMR) optimization method in which the EAF optimization method was applied to a multiresolution (MR) optimization. Simulation results showed that our EAFMR optimization requires only two iterations while multiresolution optimization takes more than 2000 iterations. There is a tradeoff between efficiency and local optimum. However, as an optimal configuration exists, finding a global optimum is not necessary. The main focus of our research is attaining low computational cost (fast convergence) in optimization for a realtime application. Therefore, using this algorithm, reconfigurable RF frontends can move forward to a reliable multistandard platform for the needs of future communication systems.
In the future, we want to investigate spectral strategies that help to prune the space of RF configurations to be explored (“search space”) and rules to automatically rank preferred RF configurations. In order to narrow down the search space, we plan to study the heuristic rules that are applicable using the reconfigurability of filters and local oscillators (LOs). While reconfiguring filters and LOs, we assume that the characteristics of RF components (e.g., filters’ center frequency and bandwidth) and also the signal spectrum are accessible. When a blocker appears, in order to improve communication quality, in addition to the choice of filters, etc., we can also change either radio frequency (RF) or intermediate frequency (IF) of the RF system based on the blocker information in the signal spectrum. For example, the blocker signal can be effectively eliminated if it is placed on the stopband of a notch filter in the frequency domain. In order to satisfy this condition, the IF of a reconfigurable RF frontend can be changed by reconfiguring LOs considering the characteristics of the notch filter. The heuristic strategies for reconfiguring filters and LOs will help a reconfigurable RF frontend improve the speed of an optimization process in dynamic communication environments.
6 \thelikesection Appendix
In Algorithm 3, x^{(current)},SNR^{(current)},Power^{(current)} represent a current configuration, the simulated SNR, and power of the current configuration, respectively. nBit_{1} and nBit_{2} are the maximal bits of control for reconfiguring components in phase I and phase II, respectively. nBit_{1}=nBit_{2}=4 for our simulation setting. In line 2 to 17, a configuration with maximal simulated SNR (phase I) is found. In line 6, a current component c is updated to the next available component. In line 7, a list of configurations to be simulated is obtained by reconfiguring component values at the current component c of a current configuration x^{(current)}. The resolution for the component reconfiguration is given as n. In line 9, SNR(x) is obtained by simulating configurations of x∈X. In line 18 to 35, a configuration that has the lowest power while meeting a condition SNR(x)≥SNR_{Spec} (Phase II) for x∈X is found.
In Algorithm 4, x^{(current)},SNR^{(current)},Power^{(current)} represent a current configuration, the simulated SNR and power of the current configuration, respectively. In line 1, nBit is the maximal bits of control for reconfiguring components in phase II, and nBit=4 for our simulation setting. In line 2 to 10, in order to find a configuration with maximal simulated SNR (phase I), calculated SINR is utilized. In line 11 to 36, a configuration that has the lowest power while meeting a condition SNR≥SNR_{Spec} (phase II) is found. In line 14, a current component is moved to the next available component and updated to c. In line 15, a list of configurations to be simulated is obtained by reconfiguring component values at the current component c of a current configuration x^{(current)}. The resolution for the component reconfiguration is given as n.
Notes
Acknowledgments
This study is sponsored by the DARPA RFFPGA (Radio FrequencyField Programmable Gate Arrays) program under Grant HR00111210005. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.
Funding
This study was funded by the DARPA under Grant HR00111210005.
Availability of data and materials
Not applicable.
Authors’ contributions
MJ and RN designed the estimation and optimization of the EAFMR method, demonstrated the method using simulation, and wrote the manuscript. SY contributed to foundational work at the beginning stages of the project. FW, MA, and MS contributed to the implementation of the framework of the circuit specifications and optimization methods for a largescale RFFPGA. TM and XL provided advice and insight on the reconfigurable RFFPGA architecture and optimization. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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