2.1. Methodology

In the proposed methodology, a sinusoidal sound source is used that generates plane waves in the tube. The sound pressure is measured at two locations in close proximity to the sample. The complex acoustic transfer function of the two microphones signal is determined and then used for calculating the normal - incidence complex reflection factor (r ), normal - incidence absorption coefficient (α ) and impedance ratio of the test material. It is found that the frequency range depends on the diameter of the tube and the distance between the microphone positions.

The normal - incidence complex reflection factor can be calculated using the formula

(1)

where x ₁ , distance between sample and the farther microphone location; Φ _r , phase angle of normal incidence reflection factor; H ₁₂ , transfer function from microphone one to two, defined by complex ratio p ₂ /p ₁ = S ₁₂ /S ₂₁ . H _R & H _I are the transfer function for the incident wave and transfer function for reflected wave.

The sound absorption coefficient can be calculated as:

(2)

2.2. Tube fabrication for transfer - function method

The impedance tube is made from a rigid, smooth, non-porous walls. To avoid vibration resonance in the working frequency range, the walls of the tube are made thick and heavy. The tube is made from brass and the holes for microphones are drilled to the required tolerance of ±0.2 mm. To seal the microphone hermetically, rubber seals are inserted at the holes where microphones are placed. A rigid plunger is fitted at the end of the tube to back the sample material. The rigid plunger is made of mild metal with a thickness of 23.8 mm. A membrane loud speaker is connected at the opposite end of the tube. The surface of the loudspeaker membrane covers two-thirds of the cross-sectional area of the impedance tube. Loudspeaker is contained in an insulating box in order to avoid airborne flanking transmission to the microphones as shown in Figure 2.

Fig. 2 Impedance tube schematic. 

2.3. Tube calculation for transfer function method

For sound absorption coefficient measurement, the generated sound signal shall be a plane wave. Therefore, the tube has to be long enough to ensure the development of plane sound waves between sound source and the sample. The standard ISO 10534-2 recommends a tube length of at least thrice the diameter, as non-plane waves will disappear at that distance. A length of at least 10-15 times the diameters is preferred (^{Seybert, 2010}). The upper working frequency is chosen in such a way as to avoid the occurrence of non-plane wave mode propagation and to assure accurate phase detection (^{RYU, n.d}.). The upper limiting frequency f _u can be calculated from the following condition.

(3)

The inner diameter of tube for this design is d = 104.8 mm which gives the upper liming frequency of f _u = 1.8 kHz. The limiting frequency depends on the distance between the microphones, s ₀ , as depicted in Figure 2, which should exceed 5% of the wavelength corresponding to the lower frequency interest. Therefore, s ₀ determines the lower limiting frequency f _l using the following condition.

(4)

Additionally, the following condition has to be satisfied.

(5)

For the selected distance s ₀ = 30 cm, the lower limiting frequency is f _l = 58 Hz. A larger spacing between the microphones has been chosen in order to enhance the accuracy of measurement (^{Suhanek, Jambrosic, & Domitrovic, 2008}). The spacing between the sound source and microphone x according to ISO 10534-2 should be

(6)

For this study, the spacing between the sound source and microphone is taken as x = 720 mm. The distance x ₂ between the test sample and the microphone nearest to it depends on the type of sample. To meet the highly asymmetrical nature of sample, the value is chosen as below.

(7)

Finally, the total tube length is given by

(8)

5.1. PSO based system

The solution to a complex equation with many numbers of unknowns can be solved by a number of methods. In this paper, a meta- heuristic method and neural network method are used to solve the equation. Particle Swarm Optimization is an intelligent algorithm which mimics the behaviour of birds flocking towards a source of food. The algorithm consists of entities called "particles" which are defined by their co-ordinates in the search space. In our problem, the search space is of five dimensions and each position provides a different value on substituting into the objective function. Meta-heuristic method considers error modelling through error minimization optimization, while neural networks approach trains the neurons to mimic the standard impedance tube. Particle Swarm Optimization is chosen because of its capabilities in solving complex non-linear systems and is used for correcting errors in different applications. In PSO, the particles move according to their velocities to different positions in the search space. Due to their movement, the position co-ordinates are altered and the fitness value of each particle changes. This happens in an iterative process. These velocities are updated (Fig. 3) in every iteration according to the mathematical Eq. (13).

(13)

where V _new is the velocity of the particle in the current iteration, V _old is the velocity of the particle in the previous iteration, c ₁ , c ₂ and w are tunable parameters, LB is local best solution of the particle and GB is the global best solution of the entire swarm. The localbest is the best solution obtained by an individual particle throughout the iterations. The globalbest is the best solution obtained yet by the algorithm. From the above equation it is observed that the particles move towards better solutions in a swarming nature. As the particles move in the search space, new solutions are obtained. For every new solution, the fitness value of the solution is found in order to determine if the new solution is better or worse compared to the previous solutions. The fitness function is derived from Eq. (12) as

(14)

where N is the total number of sample data and i is the sample number. Also the input variables are constrained in the penta-dimensional search space as given in Eqs. (15)-(19).

(15)

(16)

(17)

(18)

(19)

Fig. 3 PSO movement. 

At the end of all iterations, the global best solution is obtained which is used for compensating the error. The steps for the process are as follows

STEP 1: The following parameters are initialized for optimization

STEP 2: The particles are distributed randomly within the search space

STEP 3: The fitness value for all the particles are found using the fitness function

STEP 4: The local best and global best values are found from the initial solution

STEP 5: In each iteration,

(vi) STEP 6: After completion of all the iterations, the global best solution is obtained and is considered for error compensation model.

PSO has several simulation parameters which will define the movement of the particles through the search space. Hence the ideal parameters for optimization should be found by repeated optimization with different simulation parameters.

5.2. Neural network based error corrector

Artificial Neural networks (ANN) consist of groups of entities called neurons which mimic the operations of a biological neuron. These artificial neurons are interconnected and they possess the ability to learn and recognize patterns. Their activity is similar to the human brain. A trained ANN can recognize patterns that are too complex for humans and other computer techniques. The trained ANN can provide results to "what if" scenarios and predict results when sufficient information is provided for analysis. In the brain, knowledge is acquired by forming new interconnections between the neurons. Every new memory triggers a new connection in the brain. The stronger the connection, the better the memory. Usually the connection strength is improved by repeated training. In the same way, the artificial neurons are interconnected in a complex network and the network is altered with new experiences. ANN consists of number of layers of individual neurons. The neurons in a single layer communicate with the neurons in the neighbouring layers. The number of layers also determines the extent of learning of the network. More layers will facilitate more robust network but it slows down the learning process. New learning's are stored as weighting factor in the connections. These weighting factors are corrected through a technique called back propagation which is a feedback mechanism. The pattern recognition ability if the ANN differs with the type of neural network and feedback mechanism. Several classes of neural networks are considered for our problem.

5.2.1. Feed forward network (FFN)

This is the simplest type of ANN where the information flows in only one direction, and there are not any loops in the network, as shown in Figure 4. The first layer is called as the input layer because it receives the input values and the last layer is called the output layer because it provides the results of the neural network. The layers in the middle are called hidden layers, because they represent the internal states of the network and are concealed from the inputs and outputs. Each neuron connection is initialized by a random weight which affect the data flowing across the neurons. The weights are adjusted in a feedback mechanism in an iterative process called back propagation. The iterative process continues until sufficient accuracy is reached.

Fig. 4 Structure of feed forward network. 

5.2.3. Probabilistic Neural Network (PNN)

A Probabilistic Neural Network is a feed forward neural network which has a statistical algorithm called kernel discriminant analysis and has an additional layer called summation layer in addition to the previous three layers (Fig. 5). PNN shows faster learning when compared with standard FFN and has inherently parallel structure.

Fig. 5 PNN structure. 

The measurement from the designed impedance tube and frequency are fed as the inputs to the network. The performance of the neural network is dependent on the total number of input datasets. The datasets are collected by repeated experiment on the impedance tube with different materials.

5.3. Testing and validation

The simulation parameters chosen for PSO are given in Table 2.

Since, the method is a stochastic method, there is bound to be minor deviation in the results and runtime. Hence the optimization is repeated for many times with different parameter set and the best value out of these optimizations is considered. The error values after optimization with different values of c ₁ , c ₂ for input dataset is given below in Table 3.

The above table shows that the parameter changes only have a very minor change in the performance of the algorithm. This shows the robustness of the algorithm in solving this problem. The optimized values of the error compensating parameters are given in Table 4.

Figure 6 shows the convergence characteristics of PSO attaining an early convergence and its effectiveness in solving the problem. The standard deviation from the best solution is almost none which shows that PSO reaches global minima in almost all the trials.

Fig. 6 Convergence characteristics of PSO. 

The solution set given in Table 4 is used to correct the error in the system to a certain extent. In this way the overall accuracy of the impedance tube is improved. The results with a test data is given in Table 5. Here C is the PSO aided error compensated measurement. C is calculated using Eqs. (9) and (12). From Table 5 it is clear that the average error has dropped by nearly 37%. This shows the effectiveness of error compensation technique with a meta-heuristic approach.

To test using neural method, the input data set is expanded by making large number of measurements at intermediate frequencies to obtain sufficient training of the neural network. Figure 7 shows the effectiveness of neural network method in error compensation for different number of input dataset.

Fig. 7 Performance of neural network with varying number of datasets. 

Neural performance improves with increased input dataset. The neural network is trained with the expanded dataset and the training parameters are chosen as in Table 6.

With the above training parameters, the performance of training, validation and testing were observed as 0.168972, 0.11176 and 0.006083 respectively. Figure 8 shows the error histogram and Figure 9 shows the performance of the neural network training while Figure 10 displays the validation of neural network.

Fig. 8 Error histogram of trained neural network. 

Fig. 9 Performance of neural network training. 

Fig. 10 Neural network validation. 

With the expanded input dataset, the neural networks modelled error compensation produced an average error of 0.0013 improving the accuracy of the impedance tube by 69.7%. When the trained neural network was used to compensate the errors of the impedance tube with a test dataset, the result was worse compared to the system without error compensation. The test data is tested with the three types of neural network aided error compensator and the results are shown in Table 7. From Table 7 it can be inferred that the system with RBFN neural network has the least error rate when compared to other neural networks. However, neural network has not provided better error correction for the sample under study. It is clear that the average error has increased nearly twice. Hence neural networks require lot of input datasets for effective operation, which is effort consuming.

A comparison of PSO and different neural network modelling reveals that PSO provides good error compensation at all frequencies, while neural network performs only at the mid frequency range. At other frequencies, the error is higher than the original data as evident from Figure 11. Similar experiments were conducted on other materials and the accuracy of the output increased by a significant number, at all frequencies (Table 8). In the table, HL indicates cotton/linen blend aka Halbleinen. With neural network model, the error is reduced for soft felt, whereas increased for other materials, and hence is not a reliable system. PSO modelled system provides better accuracy for all samples and hence is robust and reliable.

Fig. 11 Comparative analysis. 

Similar tests were conducted with 20 samples and the average mean square error (MSE) obtained for each method is given in Table 9.