^{1}

^{*}

^{1}

This work proposes an improved inertia weight update method and position update method in Particle Swarm Optimization (PSO) to enhance the convergence and mean square error of channel equalizer. The search abilities of PSO are managed by the key parameter Inertia Weight (IW). A higher value leads to global search whereas a smaller value shifts the search to local which makes convergence faster. Different approaches are reported in literature to improve PSO by modifying inertia weight. This work investigates the performance of the existing PSO variants related to time varying inertia weight methods and proposes new strategies to improve the convergence and mean square error of channel equalizer. Also the position update method in PSO is modified to achieve better convergence in channel equalization. The simulation presents the enhanced performance of the proposed techniques in transversal and decision feedback models. The simulation results also analyze the superiority in linear and nonlinear channel conditions.

Channel equalization [

PSO is proven as an efficient method to update the weights of equalizer in adaptive Equalization [

The positive constants ^{th} iteration.

At each iteration, the change in weights are calculated by Equation (1) and the weights are modified to new one using Equation (2). Initially, the weights are randomly selected from the search space for P number of particles. Using the randomly selected weights, the fitness function is calculated and based on it,

The PSO algorithm can be improved by modifying its inertia weight parameter and other parameters. Inertia weight parameter was initially introduced by Shi and Eberhart in [

PSO variants | Inertia weight equation | Initial parameters |
---|---|---|

Shi et al. [ | w_{i} = 0.3, w_{f} = 0.9 | |

Chatterjee et al. [ | w_{i} = 0.3, w_{f} = 0.9 & np = 0.7 | |

Feng et al. [ | w_{i} = 0.3, w_{f} = 0.9, z = 0.1 | |

Lei et al. [ | s = 0.8, β = n/m | |

Zheng et al. [ | n = current iteration | |

Jiao et al. [ | w_{initial} = 1.1, u = 1.0002 |

pulse. The channel output is added with the random additive white Gaussian noise (AWGN). The noise sequence has zero-mean and variance 0.001.

The raised cosine channel response is represented as

The factor W controls the amount of distortion. The effect of nonlinearities generated by the transmitter is modeled as three different nonlinear equations in (4), (5) and (6).

where

where

The error e(n) can be calculated as

where d(n) is the desired or training data. The adaptive algorithm updates the equalizer weights iteratively to minimize e^{2}(n). Since e^{2}(n) is always positive and gives the instantaneous power, it is selected as cost (fitness) function.

The system models [_{q} and summed to produce the output. The weights are trained to optimum value using adaptive algorithm. The output Z_{k} becomes

DFE is a nonlinear equalizer usually adopted for channels with severe amplitude distortion. The sum of the outputs of the forward and feedback part is the output of the equalizer. Decisions made on the forward part are sent back via the second transversal filter. The ISI is cancelled by deducting past symbol values from the equalizer output. The output of DFE is calculated as

The PSO based equalizer [

For LTE:

・ Error is estimated by comparing delayed version of each input sample with equalizer output

・ The mean square error function of each particle P is

・ Fitness value MSE(P) is minimized using PSO based optimization.

・ If the MSE of a particle is less than its previous value, term it as current local best value and its corresponding weight values as Pbest.

・ The minimum of MSE of all particles in every iteration is taken as global best value.

・ If the current global best value is better than the previous one, assign the corresponding tap weights to Gbest.

・ Calculated the change in position (Tap weights) of each particle using Equation (1).

・ Moved each particle (Tap weights) c_{q} in Equation (8) to new position by Equation (2).

・ Repeated the above steps for the number of iterations specified or stopped when the algorithm converges to an optimum value with least MSE value.

For DFE:

・ The coefficients are initialized randomly for forward and feedback filter.

・ In the first iteration, only forward filter is active and after calculating the error the output of the forward filter is feedback through feedback filter.

・ The output of equalizer is calculated by subtracting the output of forward and feedback filters.

・ The forward and feedback filter coefficients c_{q} and b_{i} in Equation (9) are updated based on Equations (1) and (2).

In most of the PSO variants the inertia value usually varies from high (1) to low (0). Initial search or global search requires high inertia value for particles to move freely in the search space. When inertia value gradually shifts to low, the search shifts from global to local to minimize MSE. The sudden shift of inertia weight from high to low after some initial steps minimizes the MSE better than gradual change of inertia value. The proposed algorithm uses a control function which suddenly shifts the inertia weight from high to low after a particular iteration as in Equations (10)-(12) and also shown in

The common factor used in all time varying inertia weight algorithms is^{th} iteration. This reduction produces optimum performance compared to existing inertia weight modified methods in terms of convergence speed and MSE.

In second modification, the position update

the convergence speed more than 20 iterations and is proved in simulation results.

where Pbest is the local best value of particle i till t^{th} iteration. Since the local best weight is the best of all weights till that iteration for the corresponding particle, it automatically speeds up the convergence. The global best in Equation (1) includes the global search in each iteration to avoid local minima.

The general parameters assigned for simulations are specified in

The PSO variants are analyzed for linear and nonlinear channel conditions.

Amplitude distortion W | Population size P | Window size ws | Acceleration coefficient ac_{1} | Acceleration coefficient ac_{2} | Tap size T |
---|---|---|---|---|---|

2.9 | 40 | 200 | 1 | 1 | 7 |

Channel used | LMS | Shi et al. | PSO-TVW2 | PSO-TVW3 | MPPSO-TVW3 | |||||
---|---|---|---|---|---|---|---|---|---|---|

MSE in dB | Convergence Rate | MSE in dB | Convergence rate | MSE in dB | Convergence rate | MSE in dB | Convergence rate | MSE in dB | Convergence rate | |

Linear channel LTE | −53 | 200 | −50 | 80 | −58 | 50 | −67 | 50 | −66 | 30 |

Linear channel DFE | −50 | 250 | −48 | 90 | −57 | 90 | −67 | 100 | −65 | 90 |

Nonlinear channel 1-LTE | −15 | 200 | −40 | 45 | −37 | 50 | −46 | 50 | −45 | 30 |

Nonlinear channel 1-DFE | −14 | 250 | −40 | 60 | −35 | 80 | −46 | 80 | −45 | 60 |

Nonlinear channel 2-LTE | −40 | 200 | −57 | 70 | −55 | 50 | −78 | 50 | −75 | 30 |

Nonlinear channel 2-DFE | −37 | 250 | −55 | 80 | −55 | 90 | −62 | 100 | −61 | 90 |

Nonlinear channel 3-LTE | −15 | 600 | −15 | 30 | −18 | 50 | −28 | 50 | −28 | 30 |

Nonlinear channel 3-DFE | −14 | 700 | −15 | 50 | −18 | 70 | −23 | 50 | −22 | 30 |

The minimum MSE achieved by the proposed techniques are also nearly achieved by the PSO variants suggested by shi et al. [

To clearly examine the superiority of the proposed MP-PSO over all PSO variants, MP-PSO based position modification as in Equation (13) is applied to all time varying PSO variants listed in

The proposed MP-PSO based PSO-TVW2 converges in 27^{th} iteration to its minimum MSE −59 dB. The MP-PSO based PSO-TVW1 converges in the 45^{th} iteration with minimum MSE of −60 dB as in

To find the optimum value intermediate iteration “N”, simulations are performed for different N values and are shown in

PSO variants | Linear channel | Nonlinear channel | ||
---|---|---|---|---|

MSE in dB | Convergence rate | MSE in dB | Convergence rate | |

Shi et al. [ | −57 | 80 | −59 | 90 |

Chatterjee et al. [ | −56 | 110 | −59 | 120 |

Feng et al. [ | −38 | 50 | −28 | 50 |

Lei et al. [ | −54 | 250 | −59 | 250 |

Zheng et al. [ | −25 | 25 | −25 | 25 |

Jia et al. [ | −20 | 10 | −18 | 10 |

PSO-TVW1 | −60 | 120 | −59 | 120 |

MP-PSO-TVW1 | −60 | 50 | −59 | 50 |

PSO-TVW2 | −59 | 45 | −59 | 50 |

PSO-TVW3 | −67 | 50 | −65 | 50 |

MP-PSO-TVW2 | −60 | 27 | −60 | 30 |

MP-PSO-TVW3 | −70 | 45 | −65 | 45 |

PSO variants | Without MP-PSO | With MP-PSO | ||
---|---|---|---|---|

MSE in dB | Convergence rate | MSE in dB | convergence rate | |

Shi et al. [ | −57 | 90 | −57 | 50 |

Chatterjee et al. [ | −56 | 110 | −56 | 50 |

Feng et al. [ | −38 | 50 | −40 | 30 |

Lei et al. [ | −54 | 250 | −58 | 50 |

Zheng et al. [ | −25 | 25 | −25 | 20 |

Jia et al. [ | −20 | 10 | −20 | 10 |

PSO-TVW1 | −60 | 120 | −60 | 50 |

PSO-TVW2 | −59 | 45 | −60 | 27 |

PSO-TVW3 | −67 | 50 | −70 | 45 |

and 50, it leads to optimal performance. If N is selected less than 40, MSE value is degraded and for greater values it delays the convergence.

To notify the computational complexity, all time varying inertia weight modification methods in

PSO variants | Inertia weight equation | Initial parameters | Complexity |
---|---|---|---|

Shi et al. [ | w_{i} = 0.3, w_{f} = 0.9 | mxMUL + 4xmxADD + mxDIV | |

Chatterjee et al. [ | w_{i} = 0.3, w_{f} = 0.9 & np = 0.7 | mxMUL + 4xmxADD + mxDIV + mxPOW(np) | |

Feng et al. [ | w_{i} = 0.3, w_{f} = 0.9, z = 0.1 | 2xmxMUL + 4xmxADD + mxDIV | |

Lei et al. [ | s = 0.8, β = n/m | mxMUL + 2xmxADD + 2xmxDIV | |

Zheng et al. [ | n = current iteration | mxDIV + POW(0.3) | |

Jia et al. [ | w_{initial} = 1.1, u = 1.0002 | (n(n + 1)m)/2xMUL | |

PSO-TVW1 | mxMUL + mxADD | ||

PSO-TVW2 | N = 40 | mxADD + mxDIV + (m − n)xDIV | |

PSO-TVW3 | N = 40 | mxADD + mxDIV + (m − n)xDIV + mxPOW(0.5) |

suggested by Zheng et al. has complexity nearer to PSO-TVW2 but its performance is poor. The proposed modifications have less complexity compared to all existing variants.

The parameter values and choices of the PSO algorithm have high impact on the efficiency of the method, and few others have less or no effect. The analysis is done with respect to six key parameters namely, the intermediate iteration value N, the data window size ws, the acceleration constants ac_{1} and ac_{2}, the population size P, number of tap weights T and distortion factor W. The effect of the basic PSO parameters swarm size or number of particles, window size, number of tap weights and acceleration coefficients are analyzed in [

On average, an increase in the number of particles will always provide a better search and faster convergence. In contrast, the computational complexity of the algorithm increases linearly with population size, which is more time consuming. In _{1} and ac_{2} control the rate at which the respective local and global optima are reached. Setting the acceleration coefficients to a minimum value slows down the convergence speed. The local search and global search are best when the summation of acceleration coefficients become ac_{1} + ac_{2} < 4 in adaptive equalization. The acceleration coefficients greater than 1 also seem to give the best performance. For equal value of acceleration constants, the algorithm converges fastest to its lowest MSE value. The MSE calculated on iterations is the average of the MSE over the window; a large window size increases the complexity per iteration and time consumption. From

The tap weights are problem dependent. As given in

Population P | Tap weights T | Window size WS | Acceleration coefficients c_{1} and c_{2} | |||||
---|---|---|---|---|---|---|---|---|

P | MSE in dB | T | MSE in dB | WS | MSE in dB | c_{1} | c_{2} | MSE in dB |

10 | −52 | 5 | −53 | 32 | −65 | 1 | 1 | −68 |

20 | −64 | 7 | −65 | 256 | −68 | 1 | 2 | −58 |

40 | −68 | 9 | −70 | 512 | −69 | 2 | 1 | −56 |

60 | −72 | 13 | −74 | 1024 | −70 | 2 | 2 | −67 |

Intermediate iteration N | Convergence rate (iterations) | MSE in dB for PSO-TVW2 | MSE in dB for PSO-TVW3 |
---|---|---|---|

10 | 30 | −24 | −31 |

20 | 40 | −45 | −49 |

30 | 45 | −52 | −58 |

40 | 50 | −58 | −68 |

50 | 60 | −52 | −59 |

60 | 70 | −56 | −57 |

Eb/No. in dB | LTE | DFE | ||||
---|---|---|---|---|---|---|

For W = 3.7 MSE in dB | For W = 3.1 MSE in dB | For W = 2.9 MSE in dB | For W = 3.7 MSE in dB | For W = 3.1 MSE in dB | For W = 2.9 MSE in dB | |

5 | −17 | −18 | −20 | −17 | −18 | −25 |

10 | −24 | −26 | −27 | −24 | −27 | −35 |

15 | −25 | −27 | −33 | −28 | −30 | −47 |

20 | −27 | −35 | −40 | −35 | −37 | −60 |

25 | −33 | −44 | −53 | −38 | −45 | −75 |

30 | −37 | −54 | −58 | −46 | −62 | −88 |

The MSE is computed with different amplitude distortion that leads to different eigen value spread. An increase in amplitude distortion degrades the MSE performance. The performance degradation is not severe in proposed PSO based algorithms compared to existing algorithms. The MSE performance of DFE is better than the LTE structure. But the number of iterations required for convergence is less in LTE compared to DFE except for PSO-TVW2.

In this work, an enhanced PSO based channel equalization is proposed to improve convergence and mean square error of equalizer for adaptive equalization. The proposed time varying PSO algorithms, PSO-TVW2, PSO-TVW3 and MP-PSO improve the convergence speed much better than other existing variants in linear and nonlinear channels. All the existing PSO variants have improved convergence speed when enhanced with position based modification MP-PSO. MP-PSO based PSO-TVW1 is less in complexity and MP-PSO based PSO-TVW2 is fast in convergence. The proposed modifications reduce the computational complexity and also increase the convergence speed without compromising the MSE. Also the convergence is guaranteed within 50 iterations for all independent runs.

Diana, D.C. and Rani, S.P.J.V. (2016) Enhancement in Channel Equalization Using Particle Swarm Optimization Techniques. Circuits and Systems, 7, 4071-4084. http://dx.doi.org/10.4236/cs.2016.712336