A. Acquiring Meaningful Gradient Estimates

The success of a gradient search strongly depends on the quality of its gradient estimations. Reliable estimates of the gradient for a given system performance metric can be obtained by evaluating the metric (dependent variable) at various values of load reflection coefficient or load impedance (independent variable). For valid gradient estimations, all system parameters (aside from impedance) must be held constant during the set of measurements used to compute the gradient; otherwise, the gradient will not reflect the impact of impedance on performance. This requirement conflicts with the adaptive nature of a cognitive radar since the radar may vary other parameters, such as the transmit frequency, bandwidth, and signal content during a single gradient estimate, all of which impact the evaluated performance metric and corrupt the observed relationship between metric and impedance. Note that this effect may not necessarily manifest as a change in the optimal impedance. Even in cases where the performance of various signals with respect to impedance differs only by some constant offset (sharing the same optimal impedance), this constant performance offset can be enough to skew the gradient in the direction that was evaluated while the better performing configuration was active.

The simplest approach to integrate spectral adaptation and circuit optimization is to throttle the rate of spectral adaptation such that the transmit configuration is held constant throughout each gradient evaluation, ensuring performance differences can be attributed solely to changes in impedance. However, such an approach would result in an unacceptable decline in the radar's ability to quickly respond to changes in the available spectrum, as multiple identical pulses would necessarily be transmitted, even if the chosen pulse is no longer ideal for the current spectral environment.

Alternatively, given a PRI that is substantially longer than the waveform length and a sufficiently fast impedance tuner, it is possible to optimize the transmit circuit for each pulse in loopback. By using a switch, the transmit antenna would be disconnected during the interval in which the receiver is gathering and assessing the radar returns, allowing a full optimization utilizing multiple performance evaluations of the next pulse to be performed prior to transmission. Unfortunately, impedance tuning technology capable of handling the high-power required of a radar transmitter that is also able to adjust multiple times over the course of a single PRI is not yet available. Additionally, optimizing in loopback during the “off” times of the radar negatively affects the power efficiency of the system, as the power used during optimization must be dissipated through nontransmissive means.

Given the state-of-the-art high-power impedance tuner used for the experiments in this article, a single impedance tuning operation requires approximately 30 ms. Given the PRI of 409.6 μs, assuming the radar can change transmit frequency on a pulse-to-pulse basis, over 70 changes in transmit frequency configuration can occur during a single impedance tuning operation, with many more changes in a full optimization, which requires multiple tuning operations.

Instead, each gradient estimation must be performed while accounting for the impacts of other changes to the system, isolating the relationship between impedance and performance. By evaluating the performance metric multiple times at each sampled impedance and tracking when each transmit configuration was used for transmission, the search can account for the performance impacts of each configuration and estimate the relationship between impedance and performance for the current set of transmit configurations (we refer to the number of performance evaluations per impedance as the measurement window).

An existing approach using this philosophy computes independent gradients for each available transmit configuration and combines the direction of these gradients in a weighted fashion based on the relative occurrences of the configurations, with configurations that are used more frequently having more influence on the result [24]. Unfortunately, this method ignores the relative magnitude and slope of each configuration's performance contours, producing an optimization result that minimizes the weighted distance from the optimum impedances of the various transmit configurations, rather than maximizing the overall average performance. Given this difference, it is recommended to optimize the average performance directly. We present our method for directly optimizing average performance in Section IV-A.

In order to evaluate average performance, it is necessary to establish what period of time should be considered during each iteration. This requires careful consideration of how the chosen measurement window for averaging will impact the performance of the search algorithm, especially as the cognitive radar's adaptation (and thus optimal averaging window) varies for different environments. If the measurement window is too small, it will not be possible to establish a consistent performance weighting across each impedance. If the measurement window is too large, the search will be slower and less responsive to changes in the cognitive radar's behavior. We discuss these impacts and an approach for selecting the optimal measurement window mid-optimization in Section IV-B.

B. Time-Varying Optimal Circuit Configuration

While very rapid, frequently recurring transmit adaptations are handled by the proposed averaging approach, more infrequent variations that result in significant shifts in the optimal circuit configuration can impact the search algorithm over longer time periods. Gradient searches typically operate to convergence; that is, the algorithm has some step size that is decremented over time until the search attempts to decrease step size below the specified lower limit. When the step size reaches this limit, a final optimum value is selected, as implemented in our existing circuit optimization algorithms [13], [19]–[21], [24], [31]. These algorithms lack a method for performing additional optimization in response to changes to the optimal circuit configuration.

Alternate approaches for gradient-based optimization addressing time-varying performance contours exist under the umbrella of online optimization, such as the works of Mokhtari [22] and Dixon [23]. In these approaches, the gradient algorithm attempts to track a time-varying optimal solution with the least amount of error possible. Like other gradient algorithms, these assume each individual gradient evaluation is performed on a fixed set of performance contours and, if adapted to use our averaging technique, could be applied to a cognitive radar.

However, these methods have some disadvantages for our specific application. Generally, these methods are most effective if changes in the optimal solution are relatively continuous or smooth. For a cognitive radar, we expect the optimal solution following a change in operating band will often be uncorrelated to the previous optimal solution, as these changes will be driven by the external spectral environment. Additionally, in situations where the optimal configuration temporarily becomes static, such as in the absence of interfering devices, gradient calculations would continue to be performed unnecessarily. In our physical system, this requires adjusting the system away from the optimal impedance, resulting in an undesirable loss of performance during the gradient measurement period.

Instead, when possible, we prefer to converge to a fixed solution and wait for any changes to the cognitive radar's behavior that would suggest a need to resume the optimization process. We have previously demonstrated the Earth Mover Distance (EMD) as a metric to quantify the radar's behavior over time, and we have correlated this metric to potential performance improvements that can be used to establish a threshold to trigger additional optimization [24].

However, this previous work [24] does not address the difficulties that arise if the optimal configuration changes while its convergent gradient-based algorithm is active. Consider a scenario where the algorithm has nearly converged to a solution, but the optimal solution suddenly moves to the other side of the search space. In this situation, the algorithm's current small step size will cause the algorithm to progress very slowly to the new solution, resulting in poor responsiveness to the change in radar transmission.

To mitigate the impact of this situation, we allow the algorithm to reconsider its decision to decrease its step size, and instead increase the step size if it determines it is no longer near the current optimal solution. We discuss our specific approach to implementing this capability in Section IV-C.

A. Average Performance Gradient Search

A gradient search is performed to maximize the output power of the amplifier. It has been shown that the gradient algorithm tends to perform well for real-time circuit optimizations compared with some other typical algorithms [32]. The search adjusts the positions, labeled ${n_1}$ and ${n_2}$, of the discs atop two resonant cavities in the second-generation evanescent-mode cavity tuner of Semnani [18] depicted in Fig. 4. The values of ${n_1}$ and ${n_2}$ are specified in 0.5 μm increments from the highest position.

Fig. 4.

Second-generation evanescent-mode cavity tuner of [18] (figure reprinted from [21]). Impedance adjustments are performed by raising and lowering the discs connected to the linear actuators. The extension lengths of the actuators are described by the parameters ${n_1}$ and ${n_2}$.

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The gradient search is similar to the search approach presented by Baylis [31], with two differences: 1) the search is applied in the $({{n_1},{n_2}})$ plane, adjusting the fundamental control elements of the tuner, rather than a characterized optimization of load reflection coefficient ${{\rm{\Gamma }}_L},$ and 2) the performance metric is the average output power over recent transmit configurations weighted based on the relative frequency of occurrence of each configuration.

First, the tuner is set to its initial candidate $({{n_1},{n_2}})$ value, and the output power is evaluated $N$ times, dictated by the current measurement window. This set of measurements typically includes multiple transmit configurations. Next, the tuner is moved to the nearest neighbor to the right, located at $({{n_1} + {D_n},{n_2}})$, where ${D_n}$ is the neighboring-point distance. The output power is evaluated again using the same process as at the candidate impedance. This is repeated for the nearest above neighbor at $({{n_1},{n_2} + {D_n}})$, as shown in Fig. 5(a). Any measurements from transmit configurations that were not encountered at each of the three points are not usable and are discarded; they cannot be used with the averaging method described below.

Fig. 5.

(a) Evaluation of neighboring $({{n_1},{n_2}})$ points, reprinted from [19], (b) location of the next candidate $({{n_1},{n_2}})$ point, reprinted from [24].

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Using the power values measured at the candidate and the two neighbors, the average power of the $K$ usable configurations can be evaluated at each point as

View Source \begin{equation*} {\bar{P}\ \left(i \right) = \mathop \sum \limits_{k = 1}^K {w_k}\frac{1}{{{M_{k,i}}}}\mathop \sum \limits_{m = 1}^{{M_{k,i}}} P\left({k,m,i} \right)\ } \tag{1} \end{equation*}

$$\begin{equation*} {\ {w_k} = \frac{{\mathop \sum \nolimits_{i = 1}^3 {M_{k,i}}}}{{\mathop \sum \nolimits_{\kappa = 1,i = 1}^{K,3} {M_{\kappa,i}}}}.} \tag{2} \end{equation*}$$ View Source \begin{equation*} {\ {w_k} = \frac{{\mathop \sum \nolimits_{i = 1}^3 {M_{k,i}}}}{{\mathop \sum \nolimits_{\kappa = 1,i = 1}^{K,3} {M_{\kappa,i}}}}.} \tag{2} \end{equation*}

This process ensures that the average power evaluation for each point can be compared coherently across the three gradient estimation points, as the weighting assigned to each configuration is the same for all three points. This approach is required, as the simpler method of naively averaging the performance obtained at each impedance incorrectly assumes that each configuration is used the same number of times at each impedance. Failure to account for this variance results in a gradient that does not accurately reflect the impact of impedance tuning alone. For instance, if the highest performing configuration were encountered unusually often at the upper neighboring point, the resulting gradient would be skewed upward. Additionally, it is clear from (1) that any configurations that are not observed at each impedance must be discarded, as the value of ${M_{k,i}}$ would be zero for at least one value of $i$, resulting in an undefined average power. Using our technique, the gradient of the average power can then be estimated as follows:

View Source \begin{equation*} {\nabla \bar{P} \approx {{\hat{n}}_1}\frac{{\bar{P}\left(2 \right) - \bar{P}\left(1 \right)}}{{{D_n}}} + {{\hat{n}}_2}\frac{{\bar{P}\left(3 \right) - \bar{P}\left(1 \right)}}{{{D_n}}}.} \tag{3} \end{equation*}

The unit vector in the direction of the average power gradient, $\nabla \widehat {\bar{P}}$, gives the direction of steepest ascent for average power:

View Source \begin{equation*} {\ \widehat {\bar{P}} = \frac{{\nabla \bar{P}}}{{\nabla {{\bar{P}}_2}}}.\ } \tag{4} \end{equation*}

Following the estimation of this gradient, the search proceeds one step distance ${D_s}$ in the direction of $\widehat {\bar{P}}$, and the average power value is measured at this new candidate during the complete measurement window of $N$ measurements. This step is shown in Fig. 5(b). If the average power for the new candidate is higher than for the previous candidate, the process is repeated beginning at the new candidate. If the average power for the new candidate is lower than for the previous candidate, ${D_s}$ is divided by two, and a candidate in the direction of $\widehat {\bar{P}}$ at the new distance ${D_s}$ is evaluated. The process continues until ${D_s} < {D_r}$, the resolution distance. If the optimum location changes, this step size reduction may need to be reversed; we discuss a method to accomplish this in Section IV-C.

The gradient search parameter values used in this article in terms of distance in the $({{n_1},{n_2}})$ plane are included in Table I. Additional parameters affected by the supplementary algorithms are discussed in the next sections.

TABLE I Gradient Search Parameters

B. Dynamic Measurement Window

1) Impact of Measurement Window on Search Performance

As mentioned in Section III-A, the average performance gradient search relies on a measurement window parameter $N$ that dictates the number of measurements that are performed at each tested impedance. As we will show, this parameter impacts the speed at which the algorithm can converge, as well as how consistently the algorithm can converge to the same result. Additionally, the optimal measurement window is influenced by the current RFI environment.

To investigate the effects of $N$ on the search's behavior, we performed multiple searches under various RFI scenarios at 3.3 GHz for multiple values of $N$. For each scenario, we run 100 searches with starting locations uniformly distributed in a grid throughout the $({{n_1},{n_2}})$ plane.

We first consider the SDRadar operating in the presence of a tone sweeping through a 60 MHz range centered at 3.3 GHz. A spectrogram of this RFI pattern and resulting SDRadar transmit waveforms are shown in Fig. 6. A load-pull of the average performance using a large measurement window ($N\ = \ 300$) is shown as the contours in Fig. 7. The low values of output power near the top right of the plot are due to the high reflectivity of the impedance tuner in this $({{n_1},{n_2}})$ region.

Fig. 6.

Waterfall plot displaying the encountered RFI (swept tone) and selected SDRadar transmit waveforms (horizontal chirp pulses) over time (vertical axis) and frequency (horizontal axis).

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Fig. 7.

Final positions selected by the search algorithm with different measurement windows, shown with average power contours based on the transmit configurations used in Fig. 6 and $N\ = \ 300$ for comparison. The maximum load-pulled average performance (without interpolation) is 19.38 dBm at $({{n_1},{n_2}}) = \ ({3850,\ 4180})$. The classic search is unable to consistently navigate the power contours, resulting in unreliable convergence across the search space.

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Results for this scenario using measurement windows of 15 and 40, as well as a traditional gradient search that does not account for the varying SDRadar center frequency and bandwidth (Classic Search ($N\ = \ 1$)) are shown in Figs. 7–9. Fig. 8 is a histogram describing how many searches obtain final average power values within the bins along the horizontal axis. This plot shows that the standard gradient search, which is not built to account for the change in transmit frequencies, fails to achieve large power values on multiple occasions or even to a consistent, poorly performing impedance, whereas the average search consistently converges near the optimal impedance.

Fig. 8.

Final achieved average RF output power for the classic gradient search, average performance search with $N\ = \ 15$, and average performance search with $N\ = \ 40$ for the scenario of Fig. 6. The classic approach is unable to reliably find the optimum performance, with the search convergence criteria quickly being triggered by inconsistencies during candidate point comparisons.

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Fig. 9.

Search duration for the classic gradient search, average performance search with $N\ = \ 15$, and average performance search with $N\ = \ 40$ for the scenario of Fig. 6. Using a measurement window $N$ that is too small to capture the full selection of transmit configurations results in extremely large measurement times, as the search must repeatedly attempt to gather enough information for each gradient estimate.

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This variation in convergence consistency is also evident in Fig. 7, which shows the final impedance obtained by each search. Clearly, consistency of the final location is improved by our averaging technique, with the classic search's final impedances widely distributed throughout the search space. The inability of the classic search to navigate toward the optimum is attributed to the gradient estimation errors introduced by the radar's varying transmit configuration, as previously discussed.

Fig. 9 compares the search durations of the classic search and the $N\ = \ 15$ and $N\ = \ 40$ average performance searches. In addition to gradient estimation errors, the classic search also demonstrates early convergence due to excessive step size decrements. This early convergence arises from aggressive step size reductions that occur any time a new candidate impedance is evaluated at a transmit configuration that provides worse performance than the configuration of the preceding candidate.

Meanwhile, the average search with $N\ = \ 15$ was found to require long convergence times in many cases. This is because the small value of $N$ does not permit successful gradient calculations at many transmit configurations. This in turn results in repeated measurements of gradient points to obtain enough information for an average gradient calculation. Additionally, even when a gradient estimate is completed, the underlying average power measurements may not be representative of the true average performance, as transmit configurations that were not observed at all the required impedances cannot be included in the averaging process, as discussed in Section IV-A with (1) and (2). This can cause the search to step away from the true optimum or traverse the search space in a scattered, inconsistent manner.

These observations suggest the existence of a measurement window “sweet spot,” below which the search time increases dramatically while the convergence consistency declines, and above which the search time increases gradually with diminishing returns on convergence consistency. Once the measurement window is large enough to consistently encapsulate the typical cognitive radar behavior over subsequent measurement intervals, there is little to gain from increasing the measurement window.

This sweet spot can be located by correlating the search time with the number of discarded measurements (i.e., measurements associated with configurations that satisfy ${M_{k,i}} = \ 0$ for some $i$ for each gradient estimate. Fig. 10 shows that the search convergence time is minimized for the RFI scenario of Fig. 6 when the percentage of measurements utilized by the gradient calculations reaches its peak, as defined the equation:

$$\begin{equation*} {U\ = \frac{1}{{3N}}\ \mathop \sum \limits_{\kappa \ = \ 1,i\ = \ 1}^{K,3} {M_{\kappa,i}}} \tag{5} \end{equation*}$$ View Source \begin{equation*} {U\ = \frac{1}{{3N}}\ \mathop \sum \limits_{\kappa \ = \ 1,i\ = \ 1}^{K,3} {M_{\kappa,i}}} \tag{5} \end{equation*} where $3N$ is the total number of measurements, $K$ is the number of usable configurations, and the summation provides the number of usable measurements. In this case, the search is most efficient when the minimum measurement utilization ratio over 100 trials reaches ∼95% at a measurement window near $N\ = \ 20.$ This trend indicates that the percentage of measurements utilized by the gradient calculations can be used as a metric to adapt the measurement window in real time during the search process.

Fig. 10.

Correlation of measurement utilization ratio with search convergence time for the RFI scenario of Fig. 6. The optimal measurement window for this RFI scenario is around 20, as the minimum utilization ratio begins to saturate around 95%.

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Furthermore, we find that the optimal measurement window varies with RFI. To demonstrate, consider the RFI scenario of Fig. 11. In this situation, we find that the optimal measurement window providing a minimum utilization ratio of ∼95% is near $N\ = \ 40,$ with a corresponding minimization in search convergence time, as shown in Fig. 12. It appears from these two examples that a utilization ratio of ∼95% is useful to provide convergence as efficiently and consistently as possible.

Fig. 11.

Waterfall plot displaying a second RFI scenario and selected SDRadar transmit waveforms over time (vertical axis) and frequency (horizontal axis). This RFI scenario is the scenario of Fig. 6 with an additional tone hopping between two channels.

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Fig. 12.

Correlation of measurement utilization ratio with search convergence time for the RFI scenario of Fig. 11. The optimal measurement window for this RFI scenario is around 40, as the minimum utilization ratio begins to saturate around 95%.

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2) Iterative Optimization of Dynamic Measurement Window

Given the variation in measurement utilization ratio for a fixed measurement window and RFI scenario, we propose a dynamic measurement window algorithm that seeks to maintain a utilization ratio, $U,$ between 93 and 97%. We desire that the algorithm errs on the side of overshooting the optimal measurement window, as the increase in search time attributed to undershooting the optimal window is significantly higher than overshooting based on the search times shown in Figs. 10 and 12. As such, the window selection algorithm is based on the additive increase/multiplicative decrease approach of the transmission communication protocol's method of avoiding network congestion by adjusting its congestion window [33], which also incurs asymmetric costs on either side of the optimal window value [34].

The dynamic measurement window selection utilizes several parameters: the initial window size, the maximum iterative window increase, the target utilization ratio range, and thresholds placed on the number of allowed consecutive iterations that the utilization ratio is allowed to be outside of the target range before adjustments are made. If $U$ is above the target range for more than the allowed period, $N$ is decremented by one until the target utilization ratio is met. This gradual decrement helps to avoid severe performance costs associated with undershooting the target utilization ratio range. If $U$ is below the target range for more than the allowed period, $N$ is incremented at a faster rate, controlled by how far $U$ is below 1 (100%), up to the maximum allowed increase. In the case, the new measurement window is described by the following equation:

$$\begin{equation*} {N\ \left({n + 1} \right) = \ N\left(n \right) + {I_{{\rm{max}}}}\left({1 - U\left(n \right)} \right)} \tag{6} \end{equation*}$$ View Source \begin{equation*} {N\ \left({n + 1} \right) = \ N\left(n \right) + {I_{{\rm{max}}}}\left({1 - U\left(n \right)} \right)} \tag{6} \end{equation*} where $N(n)$ is the measurement window at iteration $n$, ${I_{{\rm{max}}}}$ is the maximum allowed window increase, and $U(n)$ is the utilization ratio at iteration $n$, where $0 \leq U(n) \leq 1,$ with 0 indicating all measurements were discarded and 1 indicating that no measurements were discarded. This fast increase allows quick attainment of a range where the optimization can make meaningful decisions about the system's performance.

It is required that the utilization ratio fall outside of the range for multiple consecutive iterations to filter out unnecessary adjustments that would be triggered by anomalies in the utilization ratio, such as those caused by the sudden introduction of a new interferer mid-window, whose effects on the cognitive radar's behavior would still be well described by the current window size had the interferer been present for the entire window.

A flowchart of the dynamic measurement window algorithm is included in Fig. 13, and Table II describes the algorithm parameters.

Fig. 13.

Flowchart of the dynamic measurement window selection process.

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TABLE II Dynamic Measurement Window Parameters

C. Step Size Convergence

In typical gradient algorithms, as described in [13], [19]–[21], [24], [31] and Section III-A, instances where a performance reduction is observed after stepping to the next candidate point cause the step size parameter ${D_s}$ to be decreased. The assumption underlying this action is that a point with improved performance has been stepped over, and that the optimum lies somewhere between the previous and current candidate points. The remainder of the algorithm functions similar to a binary search, narrowing in on the actual optimal point until the step size falls below a preset convergence threshold: the resolution-distance parameter ${D_r}$. This assumption results in desired behavior if the optimal impedance is stationary over time.

However, this stationary assumption may be violated while optimizing impedance on a cognitive radar transmitter that is quickly changing its transmission frequency content. For instance, the radar's transmission frequency content may change after the search algorithm has begun to decrement its step size, resulting in a different optimal impedance not necessarily near the currently used impedance. In this situation, the algorithm must increase its step size to quickly reach the new optimum. While momentum-based methods such as classical momentum [35] and Nesterov acceleration [36] are often used in gradient applications for similar effect [37], their benefit for time-varying environments is less certain [38].

Instead, instances where the optimal impedance may have moved can be detected by monitoring how many consecutive steps with improved performance are made after a step size reduction. This trend provides a sense of performance momentum, rather than the trajectory momentum of other approaches. Assuming halved step sizes, if more than two consecutive steps observe improved performance, then it is expected that the optimal point no longer lies in a region that has been overstepped. In this case, the step size can be doubled. For quick recovery, this doubling action can be repeated until a decrease in performance is observed (indicating overstep and a need to halve the step size) or the maximum allowed step size is reached. To determine search convergence and end the search, the existing minimum step-size threshold technique of [13], [19]–[21], [24], [31] is used. Parameters related to step size convergence are included in Table III.

TABLE III Step Size Convergence Parameters

Note that comparing the average performance between candidate points has the same transmit configuration observation requirements as gradient estimations. That is, at least one transmit configuration must have been observed at both candidate points in order to make a valid comparison. In instances where no valid comparison can be made, we maintain both the record of consecutive performance improvements and the current step size.

D. Cognitive Radar Behavior Transition Detection

Once the search algorithm converges, it is necessary to continue observing the cognitive radar's behavior for any changes (such as a change in transmission frequency content) that may appreciably reduce performance, warranting reoptimization of the load impedance. As discussed in [39], the EMD is used to quantify the amount of change in the cognitive radar's chosen transmit frequencies over time by producing transmit frequency probability distributions from the transmitted waveforms and evaluating the distance between these distributions.

Once the search converges, the transmit frequency distribution that was observed during the final measurement window of the search is used to represent the current “optimized” configuration. Afterwards, additional transmit frequency distributions are continually evaluated using the same measurement window, and the normalized EMD between the current distribution and the optimized configuration is determined. If the current and optimized configurations have a normalized EMD greater than 0.1, then the search algorithm is reactivated to handle the new behavior. Unlike when starting the initial search process, the reactivated search begins with ${D_s}$ that is one-fourth of the allowed maximum. However, the search can immediately increase the step size if needed. This approach allows the search to more quickly converge if the needed adjustment is small, or to quickly scale up the step size if it is evident that a larger adjustment is needed.

One advantage of this algorithm is its flexibility. Because it does not solely rely on a look-up table, the algorithm actually can achieve the best performance available from the system over a variety of environmental conditions or system changes. However, we have previously shown that look-up tables can speed the optimum identification in circuit gradient searches [20], and such improvements may also be possible in this scenario. However, a look-up table will need to be more diverse in this situation, due to the large number of possible transmit center frequency and bandwidth combinations in this problem.

A. Test Configuration

To demonstrate the circuit optimization of the previous section in conjunction with the SDRadar setup of Section II, a randomly varying RFI environment was generated and presented to the cognitive radar. The possible RFI patterns were selected to produce a wide variety of distinct SDRadar transmissions (narrow/wideband, with varying offsets from the band center frequency) and optimal measurement windows within a 100 MHz bandwidth. The chosen RFI pattern was switched at random time intervals according to the distribution of Fig. 14. Additionally, the SDRadar operating band hopped across five different operating bands in the United States radar S-band allocation, triggered at randomly varying intervals uniformly distributed from 2 to 25 s.

Fig. 14.

Probability distribution of the time interval between RFI pattern transitions, excluding transitions that result in an operating band change. This distribution arises from evaluating if the RFI pattern should be changed every 0.5 s, with the likelihood of a change occurring increasing each time the RFI pattern is not changed. The probability of an RFI change occurring begins at 10% and increases by 10 percentage points each time the RFI pattern is not adjusted, returning to 10% once a change occurs.

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Prior to the test, the optimal impedance was predetermined for each of the possible SDRadar operation bands with no RFI present, and the radar utilizing the entire 100 MHz band. This was used to define a baseline performance metric:

View Source \begin{equation*} {{P_{{\rm{base}}}}\ \left(t \right) = \ P\left({{Z_{{\rm{opt,\ full}}}}\left({{t_{{\rm{prev}}}}} \right),\ t} \right)} \tag{7} \end{equation*}

$$\begin{equation*} {R\ \left(t \right) = \sqrt[4]{{\frac{{P\left(t \right)}}{{{P_{{\rm{base}}}}\left(t \right)}}}}.} \tag{8} \end{equation*}$$ View Source \begin{equation*} {R\ \left(t \right) = \sqrt[4]{{\frac{{P\left(t \right)}}{{{P_{{\rm{base}}}}\left(t \right)}}}}.} \tag{8} \end{equation*}

The optimal impedance was also predetermined for each of the allowed RFI patterns at each operation band. This information was used for assessment of algorithm performance by postcomparison with algorithm results.

B. Measurement Results

Fig. 15 shows the SDRadar's chosen transmissions over the course of an experimental period lasting 6 min. These transmissions are represented as a frequency utilization percentage for each measurement window processed during the experiment; that is, frequencies that were used in every transmitted chirp within the algorithm's current measurement window are marked as 100% and frequencies that were never used in any chirp within the window are marked as 0%. This provides an indication of the frequencies that were being evaluated at each search operation (performance or EMD measurement).

Fig. 15.

SDRadar transmit frequency utilization in response to time-varying RFI. Large jumps correspond to SDRadar operating band changes triggered by insufficient available transmit bandwidth, while smaller variations correspond to minor adaptations to RFI changes within the current operating band.

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Fig. 16 shows the improvement in maximum detectable radar range obtained by the algorithm in comparison to the baseline metric of (7), as calculated in (8), the maximum improvement that could be obtained, and time periods when the optimization algorithm was active or idle, using the method of Section IV-D. The EMD values found during the experiment, used to determine when the optimization should become active, are shown in Fig. 17.

Fig. 16.

Percent increase in maximum detectable radar range obtained by the algorithm (blue) compared to the ideal result (orange). Regions shaded in green indicate periods when the average performance gradient search algorithm was active, while unshaded regions indicate periods when the search converged to a final impedance tuner setting and the optimization is waiting for a significant shift in the utilized frequencies before resuming the gradient search algorithm using the method of Section IV-D.

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Fig. 17.

Normalized EMD between transmit frequency utilization at previous search convergence and current utilization. Instances where the distance crosses the threshold of 0.1 cause the average performance gradient search algorithm to resume from the current impedance.

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These measurements show that the algorithm is consistently able to find the optimal performance, with some time delay as the algorithm responds to RFI changes, resulting in an average realized performance improvement of 3.29% over the baseline, compared to the optimal improvement of 3.77% on average. The rate at which our method is able to adapt to changes in the environment (as indicated by how quickly the achieved improvement approaches the best possible improvement following an environmental change) is impacted by the amount and rate of environmental change, preventing more general convergence analysis. For instance, the largest and often slowest possible improvements are associated with transitions from an operating band of 3.5 to 3.1 GHz, where an amplifier optimized for 3.5 GHz would perform quite poorly at 3.1 GHz. Additional improvements to the rate of adaptation are expected by using a look-up table, as shown in our previous work [19], [20], [40]. Furthermore, we expect the average improvement obtained by this method to become more pronounced when applied to wider band dynamic systems that are capable of frequent, larger jumps in operating frequency.

In some instances, the algorithm appears to outperform the expected optimal performance, such as at 187 s. However, in these cases, the output power observed by the algorithm differs from the premeasured optimum performance by less than the margin of error for what we observe when returning to a certain configuration (<0.1 dB variation). These variations are due to changes in temperature and minor inconsistencies in SDR performance when adapting to various frequent bands.

In other instances, the algorithm obtains performance below the expected baseline performance. The lesser deficits are also attributed to small power differences below our margin of error. Larger deficits are due to the impedance being optimized for the specific circumstances prior to the band hop, while our baseline metric assumes no RFI. In these cases, it is possible for the more specific optimized impedance to perform worse at the new operating band than the baseline impedance.

Finally, the measurement window used by the algorithm throughout the experiment is shown in Fig. 18. These results demonstrate that the dynamic window algorithm correctly chooses sudden, significant window increases when necessary, along with gradual decreases when it is clear that the window can be reduced without degrading the search algorithm's performance.

Fig. 18.

Dynamic measurement window versus time. The window adjusts throughout the experiment period in response to changes in RFI and the resulting behavior of the cognitive radar. Instances where the radar behavior transmits more consecutive unique pulses without repeat require larger measurement windows to accurately sample the radar's performance for optimization.

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