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Fast tuning algorithm is required for antenna impedance networks especially with changing loads and operational conditions. In the case where many network combinations are possible it is important to have a tuning algorithm that can reduce the number of networks possible. This can be achieved by generating a look-up table of matching networks for various frequencies in the operating range. This approach will facilitate the control system to rapidly select a suitable tuning network as a function of frequency^{8}. The disadvantage of this approach is it’s not able to respond quickly to changes in the antenna’s impedance as a function of time without repeating the whole process again.

Genetic tuning algorithms, however, based on iterations converge on the best impedance networks^{23,25,26}. Such algorithms must initially perform many iterations to arrive at a satisfactory impedance solution with no need for explicit rules. But with continued operation such algorithms achieve matching with fewer iterations. The disadvantage of GA is it requires significant computational time.

It has been shown that AQGA is an efficient algorithm^{23,27,28,29,30} as it’s (i) less prone to being trapped in a localised optimum solution,(ii) requiring less iteration; and (iii) it converges more rapidly to a final solution. It is for these reasons the tuning mechanism chosen here is based on AQGA.

### Representation

AQGA is a quantum inspired genetic algorithm that is based on quantum computation and conventional GA. Its states are a superposition of qubits^{23,27}. A qubit can assume state |0 > or |1 > , or any superposition of the dual states defined by |(delta) > = ( i)|0 > + (j)|1 > , where parameters (i) and (j) represent complex numbers indicating the probability-amplitudes of the respective states. For a system of *n* qubits, the system can exhibit (2^{n}) states concurrently. Its representation is given by ^{4,30}

$$ left[ {left. {begin{array}{*{20}c} {i_{1} } \ {j_{1} } \ end{array} } right|left. {begin{array}{*{20}c} {i_{2} } \ {j_{2} } \ end{array} } right|left. {begin{array}{*{20}c} {i_{3} } \ {j_{3} } \ end{array} } right|left. {begin{array}{*{20}c} {i_{4} } \ {j_{4} } \ end{array} } right|left. {begin{array}{*{20}c} {i_{5} } \ {j_{5} } \ end{array} } right|left. {begin{array}{*{20}c} ldots \ ldots \ end{array} } right|begin{array}{*{20}c} {i_{n} } \ {j_{n} } \ end{array} } right] $$

(9)

where

$$ left| {i_{x} } right|^{2} + left| {j_{x} } right|^{2} = 1,quad x = 1, 2, 3, 4, 5, ldots ., n $$

(10)

One qubit chromosome in Eq. (9) can represent all possible states in the primary stages of evolution, whereas 2^{n} chromosomes are required in a classical system.

For the *T*-type impedance matching *LC*-network in Fig. 2, the components ((L_{1}), (L_{2}), and (C)) are coded in the form of Eq. (9). The chromosome (left[ {L_{1} ,L_{2} ,C} right]) comprises 30-bit qubits, where 10-bit qubits represent each component. The quantum algorithm’s challenge is to determine the minimum list of *M* items.

### Structure of AQGA

The structure of the AQGA for modifying the *T*-type impedance matching *LC*-network is thus^{4,30}:

(i) *Initialize the binary instants and the quantum population (do this for each generation)*

(ii) *Compute the fitness of each entity* (P_{Y}^{x})

(iii) *Categorize the entities related to the fitness amounts and store the best chromosome*

(iv) *Apply the quantum genetic operators on* (P_{Quantum}^{x})*and update qubit chromosome applying rotation matrix*

The initial component magnitudes of the *T*-type impedance matching *LC*-network are achieved from the previous section and are adjusted by random numbers where the magnitudes of the probability amplitudes are selected in a random fashion from the intervals of (L_{1} , L_{2} in left{ {10^{ – 12} ,10^{ – 6} } right}) and (C in left{ {10^{ – 14} ,10^{ – 7} } right}). Constituents of the quantum population are represented as

$$ P_{Quantum}^{x} = left[ {Q_{1}^{x} , Q_{2}^{x} , Q_{3}^{x} , Q_{4}^{x} , Q_{5}^{x} , ldots , Q_{m}^{x} } right] $$

(11)

where

$$ Q_{y}^{x} = left{ {i_{y,l}^{x} ,j_{y,l}^{x} } right}^{R} $$

(12)

(Q_{y}^{x}) is the (l{th}) qubit size in the (x{rm th}) generation of the (y{rm th}) constituent in the quantum population ((P_{Y}^{x})) and (m) is the population size. (P_{Y}^{x}) is generated from qubit chromosome (P_{Quantum}^{x}), and represented as

$$ P_{Y}^{x} = left[ {Y_{1}^{x} , Y_{2}^{x} , Y_{3}^{x} , Y_{4}^{x} , Y_{5}^{x} , ldots , Y_{m}^{x} } right] $$

(13)

where (Y_{y}^{x}) is the (x{rm th}) bit of the chromosome.

The quantum matrix is transformed into a binary matrix in the measurement operation. As done in other quantum systems a single solution is extracted from the quantum matrix while preserving all other configurations. The magnitude of the qubit is determined according to its probability pairs (left| {i_{y,l}^{x} } right|^{2}) and (left| {j_{y,l}^{x} } right|^{2}). Population of binary entities is built from the quantum population (P_{Quantum}^{x}). Each qubit is observed for any destruction in the qubit chromosome. Diversity in population is achieved by generating a random number. |1 > state is recorded whenever the magnitude of the random number is bigger than the corresponding chromosome probability amplitude. If the magnitude of the random number is smaller compared to the corresponding chromosome probability amplitude |0 > state will be observed.

In the evaluation phase the fitness of the current and the best chromosome is given by (fleft( {CH_{beta }^{alpha } } right)) and (fleft( {CH_{best} } right)), respectively, where (CH_{best}) is the (alpha{rm th}) bit of the best chromosome. (fleft( {CH_{best} } right)) is obtained from

$$ fleft( {CH_{best} } right) = maximumleft{ {fleft( {CH_{1}^{alpha } } right), fleft( {CH_{2}^{alpha } } right), fleft( {CH_{3}^{alpha } } right), fleft( {CH_{4}^{alpha } } right), fleft( {CH_{5}^{alpha } } right), ldots , fleft( {CH_{m}^{alpha } } right)} right} $$

(14)

For the *T*-type impedance matching *LC*-network, the population is evaluated by

$$ fleft( {L_{1} ,L_{2} ,C} right) = left( {1 + left| {Z_{source} – Z_{input} } right|^{2} } right)/left| {Gamma_{source} } right| $$

(15)

where (Z_{input}) is the population’s actual input impedance, and (Gamma_{source}) is the source’s actual reflection-coefficient. The aim of the tuning algorithm is to determine the values of (L_{1}), (L_{2}), and (C) to meet the matching conditions. This is followed by updating the *x**th* population of qubit chromosomes (P_{Quantum}^{x}) by applying quantum rotation.

Qubit chromosome of (P_{Quantum}^{x}) with the best fitness is chosen for each binary chromosome (P_{Y}^{x}). Then qubit chromosomes are sorted according to the values of the fitness in every iteration executed. To avoid convergence to a local maximum, the selection strategy used here was to extract the optimum and part of “*not so good*” entities. In this way global optimisation is achieved. A uniform quantum crossover operation is applied thereafter to the chosen entities. This is followed by mutating the probability amplitude of each qubit chromosome to generate new entities. Finally, the quantum chromosome is updated by using quantum rotation. The rotation matrix is represented by

$$ Rleft( varphi right) = left[ {begin{array}{*{20}c} {{sin}left( varphi right)} & {cosleft( varphi right)} \ {cosleft( varphi right)} & { – sinleft( varphi right)} \ end{array} } right] $$

(16)

where (varphi) represents the angle of the rotation. The (l{rm th}) qubit in the (x{rm th}) generation is

$$ Q_{y}^{x} = left{ {i_{y,l}^{x} ,j_{y,l}^{x} } right}^{R} $$

(17)

which is updated as

$$ left[ {begin{array}{*{20}c} {i_{y,l}^{x} } \ {j_{y,l}^{x} } \ end{array} } right]_{new} = Tleft( {varphi_{y,l}^{x} } right)left[ {begin{array}{*{20}c} {i_{y,l}^{x} } \ {j_{y,l}^{x} } \ end{array} } right]_{old} $$

(18)

where (varphi_{y,l}^{x}) is the corresponding rotation angle of (Q_{y}^{x}).

$$ Q_{y,new}^{x} = left{ {i_{y,l}^{x} ,j_{y,l}^{x} } right}_{new}^{R} $$

(19)

$$ Q_{y,old}^{x} = left{ {i_{y,l}^{x} ,j_{y,l}^{x} } right}_{old}^{R} $$

(20)

$$ varphi_{y,l}^{x} = d_{y,l}^{x} left| {varphi_{y,l}^{x} } right| $$

(21)

where (left| {varphi_{y,l}^{x} } right|) is its amplitude and (d_{y,l}^{x}) is its sign. (d_{y,l}^{x}) is dependent on Ψ = (i_{y,l}^{x}). (j_{y,l}^{x}). Table 1 shows the relation between the fitness condition (d_{y,l}^{x}) and “Ψ”, where the mark sensing Ψ > 0 stands *Re*(Ψ).*Im*(Ψ) > 0, and Ψ < 0 denotes for *Re*(Ψ).*Im*(Ψ) < 0. In case of Ψ = 0, i.e. (i_{y,l}^{x}) = 0 or (j_{y,l}^{x}) = 0, there are different (d_{y,l}^{x}) amounts, whose details are presented in Table 1. The symbolism ‘*’ illustrates arbitrary state, symbol “1” in the columns demonstrates for the clockwise rotation of (d_{y,l}^{x}), and “0” in the columns corresponded to (d_{y,l}^{x}) indicates that rotation is not accomplished. The rotation operation helps AQGA to converge rapidly to find a solution.

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