An IFS has been proposed for image compression and signal modeling, and is capable of producing family functions with different fractal dimensions (FDs). It is a finite set for contraction mappings, and has been used to create images, various waveforms and patterns for medical image classification and biomedical signal analysis [10,11]. IFS is implemented with similarity maps, and the resulting data are self-similar. The significant removal of redundancy is related to the self-similarity of natural patterns. In modeling the pattern of a function or data sequence x[t], t = 1, 2, 3, , N, the pth interpolation map W_p, p = 1, 2, 3, , P, can be presented as

For each map, W_p maps the data sequence x_p[t] onto the subsequences with N_p sampling data in the interval [N_p1, N_p2], and the maps can be constructed side by side. The remaining map parameters c_p, d_p, and f_p can be solved by minimizing the sum of squared errors between the transformed data and the original data in the range of the pth map, and can be justified by the Collage Theorem [12,13]:

where, N_p = N_p2 – N_p1 + 1and N = N₁ + N₂ + N₃ + ⋯ + N_p + ⋯ + N_P. To improve the constraint, non-linear interpolation is used to adjoin the data among the interpolation points, which makes the model more flexible for processing non-linear and irregular signals. Non-linear terms as sinusoidal functions can be added to W_p. The non-linear interpolation function can be represented as

A fractal pattern of ECG signal can be adjoined with several segments such as P-R interval, P-R segment, Q-R-S complex, S-T segment, or Q-T interval. The nonlinear function with fractal dimension (FD) will change the ECG signals into fractal patterns at different scale parameters. FD must be a parameter between 1 and 2 for processing one-dimensional signals. The non-linear interpolation function, FDT with FD, can be modified as

where D is a FD parameter (1 < D < 2). The remaining map parameters c_p, d_p, f_p, and g_p can be solved by

Apply sequence data x_p [t] are the sampling data from the ECG signals. The fractal patterns can be reconstructed as [14]

Equations (5) and (6) are used to extract the features from the ECG signals, and Equation (9) is utilized to construct the fractal patterns of cardiac arrhythmias.

The PNN consists of four layers, including the input layer, hidden layer, summation layer, and output layer, that has the parallelism distributed process, learning, and pattern recognition ability. In this study, we have considered the sequence data x_p[t], t=1, 2, 3, , N_p, with P segments from the ECG signal, and a fractal pattern F can be represented as

The vector F can be also combined as F = [j₁₁, j₁₂, j₁₃, , j_1N1|j₂₁, j₂₂, j₂₃, , j_2N2|…|j_P1, j_P2, j_P3, , j_PNP] = [f₁, f₂, f₃, , f_i, , f_N], i = 1, 2, 3, , N, N = N₁ + N₂ + N₃ + + N_P, and each feature f_i is connected to the input nodes of input layer. The number of input nodes is equal to the number N. The number of hidden nodes H_k (k = 1, 2, 3, , K) is equal to the number of training data, while the number of summation nodes S_j and output nodes O_j (j = 1, 2, 3, , m) equals to the types of cardiac arrhythmias. Therefore, its architecture can be easily to determined without any trial-and-error procedure. The weights (connecting the kth hidden node and the ith input node) and (connecting the jth summation node and the kth hidden node) are determined by K input-output training pairs. In the hidden node, a normalized Gaussian function H_k is applied to the Euclidean distance E_d between the unknown pattern F and K training patterns. The function Hk is inversely proportional to the distance E_d. If E_d approaches zero, the unknown pattern F is similar to any training data. This concept can be used for analyzing pattern relations. The output O_j can be computed by

where the weights are created by training data F(k) = [f₁(k), f₂(k), f₃(k), , f_N(k)], k=1, 2, 3, , k, , K;

the weights are the desired outputs associated with each stored pattern. The value of will be equal to “1” or “0”. The value will be set to be “1” when the kth training data belonged to the jth class; s_k is the smoothing parameter. Finally, the maximum output O_max = [O₁, O₂, O₃, , O_j, , O_m] indicates the type of cardiac arrhythmias.

In Equation (11), the smoothing parameter s₁ = s₂ = s₃ = = s_K = s (s ¹ 0) would refine the classification accuracy. The non-linear optimization method, such as gradient descent method, steepest descent method or Newton-Raphson method [15-17], is employed to adjust the parameter s and minimize the error with iteration procedures. It is intended to minimize the predicted squared error function, and is updated by using pattern learning as

where T_j is the desired output for training pattern, h is the learning rate, and q is the iteration number. The PNN based classifier has a dynamic and fast adaptation ability with continuity add-in or delete-off training data by automatically tuning the desired outputs and parameters s of hidden nodes.

The field-programmable gate array (FPGA) is a semiconductor device that can be configured by the customer or the designer after manufacturing. It has the array architecture of logical elements, which can be used to design any logical functions to implement the given applications, such as an application-specific integrated circuit (ASIC). FPGA is also a programmability device, like programmable ROMs, and its programming techniques include antifuse-based device (Programmed Once) and static-memory-based device (Reprogrammed an Unlimited Number of Times). A typical FPGA consists of configurable logic blocks (CLB), input/output blocks (IOB), and programmable interconnects [4-9]. Each CLB can design the logical functions and latching data with combinational logic and sequential logic (AND, OR, Flipflops, and Registers). IOBs provide the interface between external pins and internal logics. Programmable intercomnects link the CLBs, I/O pins, and other resources on-chip memory through the routing paths, where interconnection among these blocks can be programmed by using hardware description language (HDL). Then, the designs can transferred the descriptions to gate-level netlists and dataflowflow on a chip. The prototype device can be implemented, tested, debugged, and modified as needed in a short design cycle.

Custom-design platforms have been provided to the FPGA implementation, such as Verilog, ModelSim, and Quartus II, and its development environment can simulate and verify the logic designs before initiating the compilation process [18-20]. However, HDL is a low level textbased programming language used for FPGA hardware design. Currently, LabVIEW extends the graphical programming tools to FPGA-based hardware. It has parallel execution, data acquisition, and floating-point arithmetic (Data Word Length: 8, 16, and 32 bit) functions to process signals on FPGA, and embeds specific functions for math, signal analysis/generation, comparison logic, linear/ non-linear control, analog and digital I/O, and timing. In this workspace, designers can quickly develop specific and reliable embedded systems containing FPGA-chips, real-time processor, and human machine interface (HMI). Graphical user interface for windows application has shorter design cycle, reprogrammability, and flexibility, which is much better than text-based languages. Under this development environment, we use saturation arithmetic functions to configure the combinational logics without prior complex digital design or electronic design automation (EDA) tools, and then to embed intelligent algorithms on the compact chip.

An ECG signal is measured with the modified limb lead II (ML II), and its typical waveform consists of the Pwave, QRS-complex, and T-wave. The QRS-complex provides distinct information in monitoring heartbeats, which can be used to discriminate the arrhythmic types. Centered on the R-wave peak, the QRS-complex can be divided into the Q-R segment and R-S segment (P = 2). For 50 sampling points (25 points before R-peak and 25 points after R-peak, N = N₁ + N₂ = 50), the remaining map parameters can be solved by Equations (7) and (8) in the Matlab workspace. By using FD between 1 and 2, the fractal patterns are constructed with Equations (5), (6), and (9). The FDTs with fractal dimension D = 1.6 is chosen in this study. For a QRS-complex of normal beat (Patient Number: MIT-103), related parameters c_p, d_p, f_p, and g_p, p = 1, 2, are computed as shown in Table 1. The coefficients of four terms are assigned to construct FDT functions with 4 remaining map parameters for Q-R segment and R-S segment, respectively. Following the multiplication and addition of the FDT, the fractal fractures can be computed. However, the logical blocks only support the four fundamental operations of arithmetic. In Equation (5), special function must be expanded a function into finite terms of Maclaurin series as

where and, w and are the constant coefficients; R, r = 0, 1, 2, , R, is the number of series term (R = 10 in this study); z_r, r = 0, 1, 2, , R, represent the series coefficients and are also constant values. These coefficients can be computed and completed to verify the expanded function accuracy in the Matlab workspace, as shown in Table 1.

Figure 1 shows the structure of FDT logical block, in which 13 adders, 26 multipliers, and 17 constant coefficients are required to implement each FDT. Its CLB can design the function of feature extraction with combinational logical elements, and constant coefficients are stored in the memory elements. When each sampling data is applied to the FDT, each fractal feature is computed with multiplication and addition. This parallelism process completes the fractal pattern, and then each feature is applied to the PNN logical block.

In this section, we focus on classifier design and collection of the annotated ECG beats for PNN training data. The ECG signals are obtained from the MIT-BIH arrhythmia database, including patient numbers: 100, 103, 107, 109, 111, 118, 119, 124, 200, 202, 207, 209, 212, 213, 214, 217, 221, 231, 232, and 233 [21]. ECG signals have various morphological information and waveforms, which can be classified into seven categories, including normal beat (·), premature ventricular contraction (V), atrial premature beat (A), right bundle branch block beat (R), left bundle branch block beat (L), paced beat (P), and fusion of paced and normal beat (F). Centered on the R-wave peak, the QRS-complex is divided into the Q-R segment and R-S segment. The FDTs with FD are utilized to construct various fractal patterns as shown in Figure 2. With data self-similarity, the preprocess resulting in the fractal patterns are similar for the same category, which can reduce the requirement of training data. The total number of fractal patterns are selected to be 1-, 6-, 2-, 3-, 4-, 4-, and 2-set data (K = 22) for the seven categories, respectively. These associated patterns could be expressed as weights, k = 1, 2, 3, , 22, i = 1, 2, 3, , 50, between the input and hidden layer. The weights, m = 1, 2, 3, , 7, between the hidden and summation layer are encoded as binary values with signal “1” denoting the seven categories while the rest of the weights are zero. The smoothing parameter s was adjusted by using the gradient descent method with 22-set training data. The optimal parameter s = 0.10942 can be computed in the Matlab workspace. For the convergent condition (Squared Error £ 10^–4), PNN converges to the nearest local minimum for less than five learning cycles. The related data of the PNN-based classifier are shown in Table 2.

In Figure 5, the overall structure of the proposed classifier divides into three stages: 1) signal preprocessing; 2) fractal feature extraction; and 3) heartbeat recognition with PNNbased classifier. In the preprocessing stage, ECG signals are acquired by using the amplifier and filter. The bandpass filter is used to remove unwanted frequency components, which comprise supply line frequency interference (50 Hz/60 Hz), baseline wander, and muscle noises. Thus, these noises do not affect the performance of the proposed classifier. ECG records are composed by a modified limb lead II (ML II) sampled at 360 Hz. Then R-peak waves are detected by the Peak detection algorithm. It begins by scanning for local maxima in the absolute value of ECG data. For certain window, the search continues to look for next larger value. If this search finishes without finding a larger maximum, the current maximum is assigned as the R-peak wave. The averaging window was chosen to be roughly the width of a typical QRS-complex. This window is at least 150 ms wide to allow for the wide QRS-complexes produced by V-heartbeats [22-24]. Centered on the detected R-peak, 25 sampling data are acquired including both Q-R segment (140 ms) and R-S segment (140 ms), respectively. In the second stage, these sampling data are converted to digital form (Signed 8-bit, 16-bit, or 32-bit Length Data) and then sent directly to the FDTs. The FDT CLBs with parallelism process are used to extract features and construct the fractal pattern. Finally, the classifier module integrates hidden-node, summation-node, and outputnode CLBs for cardiac arrhythmias recognition.

The proposed classifier is tested with the software simulation, in order to examine the implementation of FPGA modules. After the designated modules, the designated circuit can be programmed into the FPGA chip through the download cable. In feature extraction, fractal features are extracted by using 50 FDTs. Then 50 features are reconstructed into one fractal pattern. Since the FDT module is a simplified module, it is worth noting that each FDT function must be first examined before programming them into the FPGA chip. The average fractal patterns from the same category are obtained from the selected patients. For training data, fractal patterns are computed by FDT module and high-level programming language (HLPL) in the Matlab workspace, respectively. Figure 6 shows the comparison fractal patterns in FDT module and Matlab workspace. The asterisk-line stands for the computed data with HLPL, and plus-sign-line is the computed data with FDT module Through cursory observation, nice inosculations can be seen between them, such as the fractal patterns of normal heartbeat, V-heartbeat, and F-heartbeat (Patient Numbers: 100, 107, 119, and 200). The differences are less than 0.2 as shown by the circle-line in Figure 6. This confirms that the proposed FDT module has high confidence of computation performance for reconstructing fractal patterns.

The learning performance of the proposed classifier was tested with 22-set training data. In the learning stage, the optimal parameter s can be computed in the Matlab workspace. The proposed method has a fast learning process with slight iteration for adjusting parameter s. It takes 0.195 seconds to classify the 22-set training data into 7 categories. Then this optimal value was used to determine the parameter s of 22 hidden-node modules and was obtain to minimize the misclassification errors. Under the optimal parameter, decision boundaries can become increasingly non-linear, the network approaches a nearest neighboring classifier. In the recalling stage, all outputs

approach the desired targets for 7 categories. The output values of the proposed classifier are shown in Figure 7. This confirms that the proposed classifier has nice learning performance.

Note: 1) No. 1: ·; 2) No. 2-7: V; 3) No. 8-9: A; 4) No. 10-12: L; 5) No. 13-16: R; 6) No. 17-20: P; 7) No. 21-22: F.

Clinical diagnostic subjects have multiple cardiac arrhythmias such as supraventricular ectopic beat, ventricular ectopic beat, bundle branch ectopic beat, fusion, and paced beats. For example, patient number 200 has normal heartbeats and V-heartbeats with annotation labels in 1.5 minute segment. Test results reveal that the accuracy is 94% as shown in Table 3. The processes recognized 38 V-heartbeats with 6 failures, and the expected sensitivity as the fraction of category V correctly classified is 84.2%, and the specificity for normal heartbeats is 100%. The results confirm that the major category is the premature ventricular contraction. Patient number 217 has V-heartbeats, P-heartbeats, and fusion heartbeats (F). As shown in Table 3, test results confirm that the major category is P. The processes recognized 94 P-heartbeats with 5 failures, the sensitivities for ectopic beats is 94.7%, and the accuracy is 95%. Through the experimental tests, the proposed classifier can also recognize multiple cardiac arrhythmias with good accuracy in the FPGA module.

In this study, the parameters of FDT functions have been directly computed by Collage theorem. Each FDT CLB has the same structure; it only assigns the segmented remaining map parameters and series coefficients. Most of them are repeatable uses. Owing to the enhancement in features by the FDT functions, the number of training data, data storage, and processing needs can be reduced. For classification applications, artificial neural network (ANN) has been presented for this study. However, it has some limitations including very slow learning process, need iteration for updating weights, and need to determine the network architecture such as the number of hidden layers and hidden nodes. The weights, input-layer to hidden-layer and hidden-layer to output-layer, are always nonzero values. This will increase the requirement of the addition and multiplication operations and memory storage. In addition, sigmoid activation functions of hidden and output nodes [25] are difficult to implement the CLB mod-

ules, including the arithmetic process and memory storage. The proposed classifier provides a promising way for implementing the portable bio-monitor and telemedicine.