1. Conversion of Test Patterns Using FFT

In order to apply ML to circuit-level adaptive testing, this paper resolves the limitation that test patterns have different lengths depending on the circuit. For instance, the number of input test patterns in ISCAS '85 benchmark circuits ranges from 32 to 233. Previous studies addressed this issue by training a model individually for Another limitation is that using raw data has limited each circuit. However, training multiple models increases both computational time and memory consumption expressiveness. As emphasized in ^[15], the performance of ML algorithms is significantly influenced by the quality of data representation. Test patterns, which are represented as simple binary sequences, lack features that reflect structural or statistical patterns relevant to fault detection. Therefore, ML can be struggled to identify meaningful features, limiting its effectiveness in learning fault-relevant representations. In order to facilitate effective training, it is helpful to convert input data into a feature space where test patterns related to fault prediction become more distinguishable. Without such conversion, the training process may require more data and longer time to achieve comparable performance. Therefore, we propose a method that converts test patterns into fixed-size spectral images using a two-dimensional (2-D) FFT. As shown in Fig. 1, the conversion process consists of five steps as follows:

• Transform all 0s in the test pattern into -1 to suppress the DC component.

• Add zero-padding to the test pattern to align with the target image resolution.

• Reshape the test pattern into a 2-D matrix to enable spatial representation.

• Apply the 2-D FFT to convert the test pattern into a frequency-domain spectral image.

• Normalize the FFT results to the range [0, 1] to prevent numerical instability.

The spectral image obtained from the above process is used as input to the CNN. Since all test patterns are converted to a uniform size, a single CNN can be applied across different circuits without modification. Therefore, the proposed method eliminates the need for circuit- specific retraining. In addition, the convolutional layers in effectively extract spatial features from spectral images. Fig. 2 shows the images obtained by converting the test patterns of the ISCAS'85 benchmark circuits using the proposed method. As can be seen, although the length of the test patterns ranges from about 40 to over 300 binary signals, the proposed method enables them to be transformed into images of the same size.

Compared using test patterns directly, the conversion into spectral images enables the CNN to extract more informative features. Experimental results demonstrate that our proposed method is suitable for efficient adaptive testing across different circuits. To enhance the spectral image, we propose a method that converts test patterns into a multi-channel spectral image. Fig. 3 shows a visual comparison between the single-channel and multi-channel spectral image conversion processes. After applying the 2-D FFT to the test pattern, the real, imaginary, and magnitude parts can be obtained. The single-channel spectral image includes only the magnitude part, whereas the multi-channel includes the real, imaginary, and magnitude parts. In other words, after each part is normalized, they are then arranged as separate channels to form a unified spectral image. Since each part reflects different characteristics in the frequency domain, the multi-channel spectral image provides CNNs with a broader set of spectral information. Specifically, the magnitude part captures frequency strength, while the real and imaginary parts contain phase-related characteristics. The following section presents extensive experimental evaluations. These include a performance comparison of the proposed method with MLP and experimental evaluations. These include a performance comparison of the proposed method with MLP and conventional CNNs and an analysis of the impact of single-channel and multi-channel spectral images.

Fig. 1. Conversion process of a test pattern into a 2-D spectral image.

Fig. 2. 2-D spectral images on ISCAS '85 benchmark circuits.

Fig. 3. Process comparison between single-channel spectral image and multi-channel spectral image generation.

1. Implementation Details

For evaluation, we used the ISCAS'85 benchmark circuits in conjunction with the Atalanta automatic test pattern generator (ATPG) ^[16] and the FSIM fault simulator ^[17]. The dataset consists of test patterns as inputs and corresponding labels with the number of stuck-at faults detected by each pattern. Moreover, regression models were trained using the Adam optimizer ^[18] with a learning rate of 1e-4 for 200 epochs and a batch size of 32. All spectral images were resized to a fixed resolution of $28\times 28$ to ensure uniform input dimensions for the CNNs. The objective was to predict the number of detected faults for each test pattern based on the spectral images. We propose a lightweight CNN, which consists of two convolutional layers with $3\times 3$ kernels and ReLU activation functions, each followed by $2\times 2$ max pooling. The convolutional layers are followed by a fully connected layer with 128 hidden dimensions and a single output dimension. The term Proposed in the experimental results refers to the CNN described above. The accuracy metric used in this study quantifies the agreement between the CNN regres-sion outputs and the ground-truth fault expectations. Eq. (1) defines the accuracy as follows:

where $N$ is the number of test patterns, $V^y_i$ denotes the ground-truth expected fault count for the $i$-th test pattern, and $V^{pred}_i$ is the predicted value generated by the CNN-based regression model.

2. Experimental Results and Analysis

To assess the effectiveness of the proposed method for adaptive testing, extensive experiments were conducted across the circuits. The experiments aim to investigate three key aspects: (1) the impact of FFT dimensionality on model performance, (2) the effectiveness of multi-channel spectral images, and (3) the competitive-ness of the proposed lightweight CNN in comparison with MLP and conventional CNNs.

First, we compared the impact of 1-D FFT and 2-D FFT in the conversion of single-channel spectral images derived from the magnitude part. 1-D FFT refers to the FFT using the 1-D test pattern during the conversion of spectral images, excluding the reshaping process. Both images were used to train CNNs with the same hyperparameters. As shown in Table 1, using the 2-D FFT for spectral image conversion resulted in improved model performance across all circuits. The average prediction accuracy increased from 94.92% to 95.20%, showing a gain of 0.28%. These results suggest that spectral images using 2-D FFT offer richer spatial and frequency-domain features, which allow the CNN to extract more informative representations for fault prediction. Therefore, the 2-D FFT was used in all subsequent experiments.

Second, we compared the impact of single-channel and multi-channel spectral images. The single-channel spectral images include only the magnitude, while the multi-channel images incorporate magnitude, real, and imaginary parts from the FFT result. Table 2 shows comparative results across circuits when using single-channel 2-D spectral images and multi-channel 2-D spectral images. Contrary to the hypothesis in Section II, the single-channel spectral images achieved higher prediction accuracy overall. Specifically, the single-channel spectral images achieved 94.93% prediction accuracy, while the multi-channel spectral images achieved 94.42% accuracy, representing a performance drop of 0.51%. One explanation for these results is that the magnitude, real, and imaginary parts are highly correlated. Therefore, the correlation between these parts may introduce redundant information into the input, limiting the ability to benefit from additional channels. As shown in Fig. 4, the model using multichannel spectral images initially achieved higher prediction accuracy during early epochs. However, the performance converged and eventually declined compared to the model using single-channel spectral images, suggesting that excessive representation similarity across channels may interfere with robust learning. Therefore, the 2-D FFT and single-channel configuration was used in all subsequent experiments.

Lastly, we compared the proposed CNN with MLP and conventional CNNs in terms of model complexity and computational efficiency. The evaluation includes VGG16, ResNet20, and MobileNetV3, which are widely adopted as baselines in image-based deep learning tasks, as well as MLP. In addition, the number of trainable parameters and floating-point operations per second (FLOPs) were used as metrics for comparison. As shown in Table 1, the proposed model consists of 0.42M parameters and requires 2.28 MFLOPs. In contrast, MobileNetV3, which is optimized for lightweight applications, requires 2.97M parameters and 18.64 MFLOPs. VGG16 and ResNet20 require 0.61M and 0.27M parameters, respectively, but their computational costs remained higher than that of the proposed model. Furthermore, MLP requires 40$\times $ more parameters and nearly 30$\times $ more MFLOPs than the proposed model. These results demonstrate that the proposed CNN achieves an optimal trade-off between the number of parameters and computational costs. In addition, Table 4 shows a comparison of the prediction accuracy of each model when trained with single-channel spectral images using 2-D FFT. The proposed model achieved the highest average accuracy of 95.20% across circuits. In contrast, MobileNetV3 achieved 94.06%, while VGG16 and ResNet20 showed significantly lower average accuracies of 84.46% and 81.33%, respectively. These results suggest that the performance is not directly proportional to architectural complexity or the number of parameters. MLP showed an average accuracy of 92.71% but required significantly more parameters and computational cost than both conventional CNNs and the proposed model, as shown in Table 1. As a result, converting test patterns into spectral images using the FFT proved effective for adaptive testing. Moreover, the use of a lightweight CNN enabled high predictive performance while requiring fewer parameters and computational resources compared to more complex models. These results demonstrate that the combination of FFT-based spectral image conversion and lightweight CNNs offers a scalable and efficient approach for adaptive testing.

Fig. 4. Training graphs of models using multi-channel and single channel based spectral images.