1. PLL Architecture

Fig. 1 shows an RF PLL architecture, composed of a phase frequency detector (PFD), CP, LF, voltage controlled oscillator (VCO), and multi-modulus divider (MMD). The PFD compares the reference clock $S_{\text{REF}}$ with the divided feedback clock $S_{\text{DIV}}$, generating UP and DN pulses proportional to their phase difference. These pulses drive the CP, converting them into a current $I_{\text{CP}}$ that is integrated by the LF to generate the control voltage $V_{\text{CTRL}}$ for the VCO. Subsequently, the frequency of the VCO output signal $S_{\text{VCO}}$ is divided by the MMD, and the resultant signal is fed back to the PFD, completing the feedback loop. A delta-sigma modulator (DSM) receives the frequency control word $D_{\text{FCW}}$ as the target value and generates a fractional division ratio $D_{\text{MMD}}$, which is supplied to the divider to perform fractional-$N$ operation ^{[7, 8]}.

Fig. 2 details the CP within the CP–PLL, comprising an up-current source (UCS) and a down-current source (DCS), both digitally programmable for post-fabrication trimming. UCS and DCS comprise $M_S$ identical unit current sources connected in parallel, with each gated by a dedicated on/off switch. A CP current bias (CPCB), programmed via the integer code $m_B \in \{1, \ldots, M_B\}$, sets the unit-current magnitude $I_{\text{unit}}(m_B)$, yielding $M_B$ discrete levels. With $m_{\text{UP}}$ and $m_{\text{DN}}$ units enabled in the UCS and DCS, respectively, the resulting currents are calculated as

where $m_{\text{UP}}$ and $m_{\text{DN}} \in \{1, \ldots, M_S\}$.

Recalling that the UP and DN pulses gate the CP as shown in Fig. 1, the instantaneous $I_{\text{CP}}$ is governed by the UP and DN pulse states and the programmed values of $I_{\text{UP}}$ and $I_{\text{DN}}$. During the UP pulse, CP sources current into LF; during the DN pulse, it sinks current from LF; otherwise, it remains ideally off. When averaged over a reference cycle, the net current is approximately proportional to the difference between the UP and DN pulse areas. Hence, the CP, combined with the PFD, delivers effective phase-detector gain. This gain, together with the LF and VCO gains, determines the PLL bandwidth. Increasing the unit current via the CPCB or enabling more units enhances this gain, widening bandwidth with the usual spur and stability tradeoffs.

In practice, $I_{\text{UP}}$ and $I_{\text{DN}}$ are trimmed independently, correcting residual mismatches or intentionally introducing small offsets between the UCS and DCS branches. A deliberate offset shifts the steady-state phase error away from the PFD dead zone, mitigating the phase-detector nonlinearity that leads to DSM noise folding ^[9]. To limit static phase error and reference spurs, the offset remains small. In addition, the discrete trimming codes as

$(m_{\text{UP}}, m_{\text{DN}}, m_B)$

determine the effective CP gain and thus the loop-bandwidth.

2. Definition of IPN

IPN quantifies the total phase noise power integrated within the specified frequency offset range and serves as a key figure of merit for PLLs. For a given single-sideband phase noise spectrum $L(f)$ (in dBc/Hz) over the offset range $f_1 \le f \le f_2$, the IPN is defined as

It correlates directly with the root mean squared timing jitter and overall PLL output spectral purity.

The IPN strongly depends on the PLL loop bandwidth $f_{\text{BW}}$, influencing the crossover between in-band and out-of-band noise sources. A narrow $f_{\text{BW}}$ reveals dominant in-band noise from the PFD and CP, whereas a wider $f_{\text{BW}}$ suppresses VCO noise but raises the in-band noise floor. Therefore, an optimal loop bandwidth that minimizes the IPN exists ^[10].

1. Sequential BO Method ($q = 1$)

Let $\mathcal{M}$ and $\mathbf{x} \in \mathbb{R}^p$ denote the discrete codebook of trimming configurations and a continuous relaxation of a code $\mathbf{m} \in \mathcal{M}$, respectively. We define a snapping operator $Q_{\mathcal{M}} : \mathbb{R}^p \to \mathcal{M}$ that maps a continuous query to the nearest feasible code (e.g., component-wise rounding). A physical measurement at a continuous query $\mathbf{x}$ returns

where $f : \mathcal{M} \to \mathbb{R}$ is the (unknown) code-level objective. Equivalently, the surrogate in BO models the composite function

which is piecewise-constant over the quantization cells induced by $Q_{\mathcal{M}}$.

Given the dataset $\mathcal{D}_n = \{(\mathbf{x}_i, y_i)\}_{i=1}^n$ with $\mathbf{X}_n = [\mathbf{x}_1 \cdots \mathbf{x}_n]$ and $\mathbf{y}_n = [y_1 \cdots y_n]^\top$, we place a Gaussian process (GP) prior on $g(\cdot)$ over $\mathbb{R}^p$ and obtain the usual Gaussian posterior

with

where $\mathbf{k}(\mathbf{x}, \mathbf{X}_n) = [k(\mathbf{x}, \mathbf{x}_1) \cdots k(\mathbf{x}, \mathbf{x}_n)]$, $k(\cdot, \cdot)$ is a positive-definite covariance kernel on $\mathbb{R}^p$ (e.g., squared-exponential or Matérn) ^[11], $\mathbf{K}_n = [k(\mathbf{x}_i, \mathbf{x}_j)]_{i,j=1}^n$, $\mu_0 \in \mathbb{R}$ is a constant prior mean, and $\mathbf{1}_n$ is the $n$-dimensional all-ones vector. Accordingly, $\mu_n(\mathbf{x})$ is the predicted black-box function value, whereas $\sigma_n^2(\mathbf{x})$ quantifies its predictive uncertainty.

Based on (5), we use a standard acquisition, expected improvement (EI) as

where $f_{\max}^{(n)} = \max_{i \in \{1, \ldots, n\}} y_i$ and $(u)_+ = \max\{0, u\}$. Equivalently,

where $z_n(\mathbf{x}) = \left(\mu_n(\mathbf{x}) - f_{\max}^{(n)}\right) / \sigma_n(\mathbf{x})$ and $\phi(\cdot)$ and $\Phi(\cdot)$ denote the standard normal probability density and cumulative density functions, respectively. Then, we solve

where $\Omega \subset \mathbb{R}^p$ is the continuous search domain, snap the maximizer as

and perform a single physical measurement at $\mathbf{m}_{(n+1)}$,

Finally, we update the dataset by attaching the result to the original continuous query as

The selection loop (8)–(11) repeats until a stopping criterion is met.

In addition, we track the cumulative number of physical measurements, $B$. Under the snap-and-attach policy, if $\mathbf{x}_{(n+1)}^*$ snaps to a code that has been measured previously, we reuse its recorded measurement and do not increment $B$, although a new pair $\left(\mathbf{x}_{(n+1)}^*, y_{(n+1)}\right)$ is still added to the dataset. Accordingly, different continuous queries may snap to the same discrete code, i.e., $Q_{\mathcal{M}}(\mathbf{x}) = Q_{\mathcal{M}}(\mathbf{x}')$ for $\mathbf{x} \neq \mathbf{x}'$, while remaining distinct training inputs in the surrogate. Further discussion is provided in Section III-3.

2. Batch BO Method ($q > 1$)

In a batch setting, each BO iteration proposes $q$ continuous queries collected into a column-stacked matrix $\mathbf{X}^{(1:q)} = \left[ \mathbf{x}^{(1)} \cdots \mathbf{x}^{(q)} \right] \in \mathbb{R}^{p \times q}$. After $n$ iterations, the dataset size is $N = nq$, where $N$ denotes the current number of continuous query–response pairs and increases by $q$ after each batch update. Thus, we maintain $\mathcal{D}_N = \{(\mathbf{x}_i, y_i)\}_{i=1}^N$ with $\mathbf{X}_N = [\mathbf{x}_1 \cdots \mathbf{x}_N]$ and $\mathbf{y}_N = [y_1 \cdots y_N]^\top$. In contrast, we define $B$ as the cumulative number of physical measurements, which can be smaller than $N$ due to duplicate snapped codes.

We use the same GP surrogate $g(\cdot)$ as in the sequential case, defined in (4), and consider the joint GP posterior over the batch $\mathbf{X}^{(1:q)}$ is given by ^[11]

where

and

Here, $\mathbf{K}(\mathbf{A}, \mathbf{B})$ is the Gram matrix with $[\mathbf{K}(\mathbf{A}, \mathbf{B})]_{ij} = k(\mathbf{a}_i, \mathbf{b}_j)$. Given (12), the batch extension of EI, namely $q\text{EI}$, is defined as

where $f_{\max}^{(N)} = \max_{i \in \{1, \ldots, N\}} y_i$. For $q > 1$, $q\text{EI}$ is approximated numerically (e.g., via Monte Carlo) ^[4]. Then, we solve

where $\Omega^q = \Omega \times \cdots \times \Omega$ for the $q$-fold product set, and snap each column to the discrete configurations as

Within a batch, multiple continuous queries may snap to the same discrete code, and some codes may have been measured before. Let $\mathcal{U}_{n+1}$ be the set of unique codes in $\mathbf{M}_{(n+1)}$ that have not been measured up to iteration $n$. We perform one physical measurement for each $\mathbf{u} \in \mathcal{U}_{n+1}$ to obtain $y = f(\mathbf{u})$. For every $j \in \{1, \ldots, q\}$, we assign to the dataset the response associated with its snapped code as

which may reuse a previously recorded value and therefore does not necessarily increment $B$. Then, we update the dataset by pairing each response value with the original continuous query as

Therefore, the dataset grows by $q$ entries per iteration (total size of $(N + q)$), whereas the number of physical measurements in that iteration equals $|\mathcal{U}_{n+1}| \le q$. Accordingly, over $n$ iterations, we have $B = \sum_{i=1}^n |\mathcal{U}_i| \le N$. The GP surrogate is derived on $\mathcal{D}_N$, so it leverages $N = nq$ continuous query–response pairs, even when only $B$ unique physical measurements are performed. With this update rule, the batch-selection loop (14)–(17) repeats until a stopping criterion is met.

As mentioned in Section I, joint batch selection spreads early evaluations across multiple promising regions while still prioritizing high-value candidates, which reduces redundancy and can approach the measurement efficiency of sequential BO ^[3].

3. Addressing Quantization-Induced Duplicates

Quantization via $Q_{\mathcal{M}}(\cdot)$ can map different continuous queries to the same discrete code, which may cause previously evaluated codes to be selected again. A simple alternative is to build the surrogate directly on the discrete codebook, using the snapped code as the input and the corresponding measurement as the output. This discrete-input formulation is often natural in trimming problems because the DUT can only be programmed with discrete codes and physical measurements are obtained only at those code settings. Accordingly, the dataset update takes the form

Under this formulation, if quantization yields a code that has already been evaluated, the candidate must be dropped (or replaced) because it does not provide a new input–output pair for updating the surrogate.

In contrast, our proposed snap-and-attach policy optimizes in the continuous domain and pairs each continuous query with the measurement at its snapped code. Therefore, even when multiple continuous queries snap to the same previously measured code, we can reuse the recorded measurement without performing additional physical measurements and still add new training pairs to the surrogate, as in (17). In additional comparative experiments not reported here, the simple dropping strategy degraded performance relative to snap-and-attach. This result suggests that retaining these additional continuous-labeled pairs improves sample efficiency and surrogate calibration without increasing the physical measurement budget.

1. IPN Measurement Setup

Fig. 3 illustrates the setup to automatically measure the IPN of the RF PLL. A DC power supply delivers a stable voltage to the motherboard, which powers the DUT board and serial peripheral interface (SPI) control module. The control personal computer (PC) runs the batch BO procedure that proposes trimming-code candidates. Then, the selected trimming-code vector is written to the DUT over SPI, thereby setting $I_{\text{CP}}$ and, in turn, the PLL loop bandwidth.

After applying the trimming codes, the corresponding phase noise spectrum is measured using a phase-noise analyzer (PNA). A Python-based automation framework manages SPI configuration, measurement control, and IPN evaluation. The PNA trace is transferred to the PC over general purpose interface bus, stored, and post-processed to compute IPN according to (2), allowing fully automated and repeatable IPN measurements across various boards and trimming code configurations.

To establish a per-board reference for the best attainable IPN through a full sweep of the trimming codes, we exhaustively evaluated three DUT boards (A, B, and C). The sweep range per code was limited to 20 representative values, yielding $20^3 = 8,000$ measurements per DUT board. Figs. 4(a), (b), and (c) show the measured IPN with the trimming codes for the three DUT boards. These figures show that while the overall trimming behavior is consistent across boards, the specific code combination that minimizes IPN differ for each board due to process-induced variations. Specifically, the optimal codes for boards A, B, and C are $(19, 2, 14)$, $(20, 4, 20)$, $(18, 3, 19)$, respectively, and their corresponding minimum IPNs are $-37.94$, $-38.60$, and $-38.01\text{ dBc}$, respectively. This confirms that post-fabrication trimming is an effective means to optimize PLL noise performance while compensating for silicon variation.

The full sweep also suggests that the IPN-loss landscape is moderately smooth yet nonconvex, and it exhibits anisotropic sensitivity across the trimming dimensions with local variations induced by discretization and measurement variability. This qualitative structure motivates using a kernel that is flexible enough to capture local irregularities while remaining sufficiently smooth for stable acquisition optimization.

We employed a GP surrogate with an automatic relevance determination (ARD) Matérn-5/2 covariance kernel, and kernel hyperparameters were re-estimated during the BO loop. We selected Matérn-5/2 because it provides a practical balance between smoothness and flexibility for measurement-driven optimization. The Matérn family controls smoothness via $\nu$ and recovers the radial basis function (RBF) kernel as $\nu \to \infty$ ^[12]. In our quantized trimming setting, Matérn-5/2 avoids overly strong smoothness assumptions of RBF while remaining sufficiently smooth for stable acquisition optimization. Prior BO studies have also noted that the GP prior, including the kernel choice, can materially affect optimization performance ^[13]. In additional comparative experiments not reported here, we further verified this choice empirically by comparing Matérn-5/2 against Matérn-3/2 and RBF under the same BO settings, observing consistently improved mean and tail performance for Matérn-5/2.

2. Comparison Methodology

Because batch BO proposes $q$ continuous queries per iteration (which are snapped to discrete trimming codes), iteration counts are not directly comparable across different $q$ values. Therefore, we report IPN as a function of the cumulative number of physical measurements, denoted $B$, rather than the number of iterations. Under the snap-and-attach method, each batch contributes at most $q$ physical measurements, exactly $|\mathcal{U}_n|$, the number of unique, previously unmeasured codes after snapping, while still adding $q$ entries to the dataset by attaching each observed value to its originating continuous query. If a query snaps to a code that was measured earlier, we reuse its recorded measurement and do not count an additional measurement toward $B$.

Before applying batch BO, we conducted two initial measurements to initialize the surrogate model. These initial measurements were included in the total count. Across batch-BO settings with different batch sizes $q$, we used a fixed total physical-measurement budget of $B = 70$, including the two initial measurements.

3. Performance of Batch BO

Figs. 5(a), (b), and (c) show the IPN performance loss of batch BO for varying batch sizes $q$ on boards A, B, and C, respectively, measured relative to the per-board optimum IPN from the full trimming-code sweep combinations. The solid and the dotted lines represent the mean and the 95th percentile losses, respectively. Performance loss reflects the increase in IPN for a given method compared to the per-board optimum from the full sweep, i.e.,

In addition, a non-monotonic effect of $q$ on performance is observed. For small $B$, a moderate $q$ accelerated IPN reduction by testing multiple, diverse candidates early. As $B$ increases, smaller $q$ sizes became more effective due to frequent model updates that enhance surrogate adaptability. By contrast, a large $q$ (e.g., $q = 16$) performs poorly at large $N$ because many within-batch decisions are made without intermediate measurement feedback, yielding redundant or suboptimal measurements.

Selecting the batch size $q$ is practically important in queued single-instrument tuning because it governs the balance between diversification and update cadence under a fixed physical-measurement budget $B$. However, prescribing a universal optimal $q^*$ is nontrivial since the preferred tradeoff can be DUT-board dependent, reflecting board-to-board variations in the trimming landscape as well as measurement repeatability. Accordingly, rather than claiming a single $q^*$, we report results across a range of $q$ values under the same budget $B$ to identify a practical operating range. As a simple heuristic suggested by the trend in Fig. 5, we recommend avoiding overly large batches, in practice, a moderate $q$ is preferable in the early exploration regime, while smaller $q$ is more suitable in the refinement regime to preserve frequent surrogate updates.

Figs. 6(a), (b), (c) compare the IPN loss trajectories of fixed-$q$ batch BO and the $q$-switching batch-size schedule as a function of the number of distinct candidates on boards A, B, and C. The solid and dotted lines denote the mean and the 95th percentile losses, respectively. The $q$-switching schedule, denoted as batch BO ($q$ changes at $\tau$th physical measurement), starts with $q = 8$ and switches to $q = 2$ when the cumulative number of physical measurements reaches $\tau$. We evaluated at $\tau = 15, 20, 25$, and $30$. Table 1 summarizes the mean and 95th percentile losses at 70 distinct candidates. From this table, the $q$-switching schedule generally improved or maintained both the mean and the 95th percentile performances relative to fixed-$q$ baselines. In particular, switching from a larger $q$ early to a smaller $q$ later mitigated the 95th percentile degradation observed with a large fixed $q$, while achieving performance comparable to or better than a small fixed $q$.

Given the observed board-to-board variability, adaptive $q$ schedules may offer additional benefits beyond fixed-$q$ batch BO. While the simple $q$-switching schedule evaluated in this work provided consistent improvements or maintained performance across the tested boards, the optimal switching rule and schedule design are likely to depend on practical factors such as the measurement budget, noise level, and the local structure of the trimming response surface. A more systematic design of adaptive $q$ schedules and their empirical characterization across different boards and measurement conditions remains an important direction for future work.

Fig. 7 shows the elapsed time per proposed candidate (sec.) as a function of the number of generated candidates. For each board (A, B, and C), we repeated the experiment 1,000 times by generating 100 candidates per execution, regardless of whether quantization produced duplicate codes, and the results were aggregated across all three boards. The solid lines denote the mean per-candidate elapsed time and the shaded regions indicate the 5th–95th percentile range across executions. Batch BO selects $q$ candidates simultaneously in each iteration. As $q$ increases, acquisition-function optimization becomes more complex due to higher dimensional search space. In addition, the per-iteration computational overhead can increase because GP refitting and hyperparameter optimization are performed after every evaluation.

However, because batch BO generates $q$ candidates at once per iteration, the computation time per candidate decreases for the same total number of candidates. This trend is quantitatively confirmed by the execution-time measurements. Compared with sequential BO, the per-candidate computation time is reduced by a factor of approximately 1.33, 2.06, 3.46, and 6.64 for $q = 2, 4, 8$, and 16, respectively. These results indicate improved candidate-generation efficiency as $q$ increases. Moreover, the percentile range decreases with larger $q$, indicating that the computation time becomes more stable as $q$ increases.

4. Performance Comparison with Other Methods

In this section, we compared batch BO with sequential BO and genetic algorithm (GA). Both batch and sequential BO began with two initial configurations to initialize the surrogate model, followed by specific acquisition routines. For the GA, reflecting its population-based operation, we initialized it with 10 distinct configurations and, at each generation, produced 5 new candidate configurations ^[14], filtering out any duplicates from GA and BO.

Figs. 8(a), (b), and (c) depict the IPN performance loss for different batch sizes $q$ on boards A, B, and C, respectively. In addition, the solid and the dotted lines represent the mean and the 95th percentile losses, respectively. The batch BO consistently outperforms the GA across all three boards. For a small $B$, batch BO with $q = 8$ achieves a noticeably lower loss than sequential BO and GA. This indicates that batch BO can be effective when the measurement budget is limited.

For larger $B$, the mean IPN loss shows minimal differences between batch and sequential BO. However, on boards A and C, batch BO achieves a considerably lower 95th percentile losses. Table 2 summarizes the IPN performance losses of batch BO, sequential BO, and GA at $B = 40$. For example, on board A, the 95th percentile loss is less than 2.0 dB for batch BO, approximately 4.0 dB for sequential BO, and approximately 7.5 dB for GA. This gap may reflect the ability of batch BO to select candidates jointly under the GP posterior, accounting for correlations and diversifying early evaluations across distinct promising regions. In turn, this reduces the susceptibility to unfortunate initializations and local optima. Our empirical evaluation used only three DUT boards, so the findings may not fully generalize to other PLL designs or broader process, voltage, and temperature variations. Validation on a larger and more diverse set of boards is left for future work.

5. Limitations and Future Work

Our measurements were conducted under nominal operating conditions, and a comprehensive sweep over process, supply-voltage, and temperature (PVT) variations was not performed due to the substantial additional effort required for hardware characterization. Future work will validate the observed batch-size behavior under PVT variations and across different PLL architectures, and will develop more systematic guidelines for selecting $q$ under such conditions.

In addition, given the observed board-to-board variability, systematically designing and validating adaptive $q$ schedules across different boards and measurement conditions remains an important direction for future work. Finally, we will investigate alternative acquisition functions and multi-objective formulations that consider figures of merit beyond IPN in RF-PLL design.