104 6.3 Performance of MSE MIMO-DFE based on SM-AP algorithm for different projection order. 104 6.4 BER Performance of MIMO-DFE based on SM-AP algorithm for different projection order. 105 6.5 MSE performance of MIMO-DFE based on SM-AP algorithm with upper boundγ. 57 4.3 SS-MSE and the number of DFE updates based on the SM-AP algorithm for different values of γ.
Introduction
Literature Survey
The adaptive MIMO channel equalization using RLS algorithm was presented in [32], whose performance can be further improved by minimizing the interference of non-diagonal channel elements. A low complexity soft-input soft-output DFE for the MIMO system was reported in [38].
Motivation
Problem Formulation
Thesis Contributions
Thesis Organization
The performance of the proposed schemes has been investigated with different values of µ, upper bound γ on the estimation error and also projection order P in case of AP algorithm. The performance of the equalizer is investigated for a MIMO receiver in a multi-path fading environment as experienced in the indoor and pedestrian environment.
Major Impairments in Wireless Channels
This frequency shift is called the Doppler shift, which is proportional to the speed of the mobile unit. Doppler shift in a multipath propagation environment expands the bandwidth of multipath waves, which is a consequence of the frequency dispersion of the channel.
Adaptive Channel Equalization
Decision Feedback Equalizer
DFE avoids amplification noise, where the channel has null spectrum, due to the use of combination of FF and FB filters. The error is used to determine the direction in which the filter weights should be changed in order to approach an optimal set of values.
Adaptive Algorithms
- LMS Algorithm
- RLS Algorithm
In an adaptive process, the crane weights are automatically adjusted to the error produced. The RLS algorithm is a special case of the Kalman filter, using the least squares approximation to optimize the equalizer coefficients [22].
MIMO System
Channel Capacity
Channel capacity is a measure of how much information can be transmitted and received with a negligible probability of error. The average performance for a MIMO system is given by C=E. 2.17), where M elements of the channel noise vector are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and common variance σw2, N symbols that make up the transmitted signal vector are i.i.d.
Channel Models
Summary
In this chapter, we investigate the performance of an adaptive equalizer for decision feedback using algorithms for reusing data across time-dispersed wireless channels.
Introduction
Data-reusing Algorithms
The Affine Projection Algorithm
The Binormalized Least Mean Square Algorithm
Adaptive DFE Model
A block diagram of a generic adaptive DFE consisting of a linear feedback filter (FFF), a linear feedback filter (FBF) and a nonlinear decision maker is shown in Fig.
The Affine Projection Algorithm based DFE
- Computational Complexity
- Simulation Results and Discussion
- Performance of DFE with AP and NLMS as Adaption Algorithms
- Performance of AP Algorithm based DFE with Different Values of Step
- Performance of AP Algorithm based DFE with Different Filter Lengths 35
- Computational Complexity Issues
- Simulation Results and Discussion
- Performance of DFE using BNLMS, NLMS and AP (P = 2) Algorithm. 40
The comparison of BER performance of DFEs based on NLMS algorithm with DFEs based on AP algorithm of projection order 2, 3 and 4 is shown in Fig. The simulation results show improved performance of DFEs based on the AP algorithm over the scheme based on the NLMS algorithm. The BER performance comparison of BNLMS, NLMS and AP (P = 2) based on the DFE algorithm is shown in Fig.
Summary
In wireless communication scenarios, adaptive filtering algorithms are used to update the equalizer coefficients and track the channel variations. Introducing set-membership (SM) filtering into an adaptive algorithm reduces the computational complexity with an overall improvement in system performance due to data-selective updates [ 45 ]. As a standard procedure, the filter coefficients are updated such that the output estimation error is upper bounded by a prescribed threshold.
Data-selective Algorithms
- Set-Membership Filtering
- Set-membership Normalized LMS Algorithm
- Set-membership Affine Projection Algorithm
- Set-membership Binormalized LMS Algorithm
The exact membership setψk is the intersection of the constraint sets over all the available time instants i= 0,1, .., k. The membership setψk is expressed as:. where ψk−Pk is the intersection of the first k−P + 1 constraint sets. So, mathematically, the objective function for SM-AP algorithm can be evaluated as described below.
Set-membership Affine Projection Algorithm based DFE
Simulation Results and Discussion
- Performance of SM-AP Algorithm based DFE with Different Values of
In this study, the performance of SM-AP algorithm based DFE is compared for different values of convergence factor µ. SS-MSE and the number of updates associated with SM-AP algorithm based DFE for different. The comparison of BER performance of SM-AP algorithm based DFE with different values of convergence factor µ is shown in Fig.
Set-Membership Binormalized LMS Algorithm based DFE
Convergence Analysis
Let the desired signal be , . where zk is concatenated input signal vector and nk is measurement noise. a) measurement noise is Gaussian with zero mean and variance σn2, (b) input signal vector and measurement noise are independent, and (c) variance of input signal vector is σz2. Our goal is to analyze the convergence behavior of the coefficient vector w as a function of convergence factor µ. E[PE[Q]] where P and Q are terms in the numerator and denominator, respectively, showing independence of P and Q, as well as a 1st order approximation of Eh.
Computational Complexity
It should be noted that the main complexity terms of SM-AP algorithm based DFE lies in calculating the matrix inversion, which is of order O(P3), where P is the projection order. The proposed technique offers less computational power compared to SM-AP algorithm based equalization schemes.
Simulation Results and Discussion
- Performance Comparison with other Data-selective Algorithms based
- Performance of SM-BNLMS Algorithm based DFE with Different Val-
- Performance of SM-BNLMS Algorithm based DFE with Different Val-
The performance of the SM-BNLMS Algorithm-based DFE is studied for different values of upper bounds on estimation error. In this study, the performance of SM-BNLMS Algorithm-based DFE is compared for different convergence factors. The convergence performance of the proposed SM-BNLMS algorithm-based DFE as a function of convergence factor is shown in Fig.
Summary
Furthermore, the SM-BNLMS algorithm-based DFE presented in this chapter converges faster, provides much better BER performance, and requires almost 10% fewer number of updates than the SM-NLMS algorithm-based methods. Moreover, its overall performance is more or less similar to the SM-AP algorithm (with projection order 2) based technique, but with reduced complexity due to the non-existence of matrix inversion component. However, higher projection orders SM-AP algorithm based DFE performs better but with more complexity.
Multiple-input Multiple-output System
Adaptive MIMO DFE Model
Affine Projection Algorithm based Adaptive MIMO-DFE
Simulation Results and Discussion
- Performance of MIMO-DFE using AP and NLMS as Adaption Algorithm 87
In addition, the MIMO-DFE with AP algorithm of projection order 2 and 3 is observed to exhibit similar BER performance. The MSE and BER performance of this MIMO channel equalizer using the AP algorithm is studied for different values of step sizeµ. b) Vehicle Test Environment-B Figure 5.7: MSE performance of MIMO-DFE with AP algorithms with different step size values. Furthermore, the BER performance of this MIMO-DFE is observed to degrade significantly for a step size greater than 0.1.
Binormalized LMS Algorithm based Adaptive MIMO-DFE
Computational Complexity
It can be seen that the calculation involved for updating in the proposed scheme is higher than the scheme based on the NLMS algorithm.
Simulation Results and Discussion
- Performance Comparison of MIMO-DFE with BNLMS, NLMS and AP
It is observed that for a given test environment, the MIMO channel equalizer using the BNLMS algorithm converges faster than the equalizer based on the NLMS algorithm and has a convergence rate almost similar to that of the equalizer using the SM-AP algorithm of projection order 2. BNLMS BER performance comparison , MIMO-DFE based on NLMS and AP algorithm with step size µas 0.1 is shown in Fig. Furthermore, it is found that although the BER performance of MIMO-DFE based on the BNLMS algorithm is similar to the equalizer based on the NLMS algorithm, for SNR values below 12 dB in Ped-A and 4 dB in Veh-B;.
Summary
This chapter examines the channel equalization scheme using adaptive MIMO-DFE based on the SM-AP algorithm and the SM-BNLMS algorithm for frequency-selective MIMO channels. It has been found that the proposed MIMO channel equalization scheme based on SM-AP and SM-BNLMS algorithm shows almost comparable performance to data-reusing AP and BNLMS algorithm-based adaptive MIMO-DFE, respectively, but with a significant reduction in computational power due to selectively update the data. The algorithm-based adaptive MIMO-DFE schemes examined in the previous chapter have been found to offer better convergence and BER performance than NLMS-based equalizers, with a marginal increase in computational complexity.
Set-Membership Affine Projection Algorithm based MIMO-DFE
Comparison of Computational Complexity
It can be seen that the calculation involved per update in the proposed scheme is higher than SM-NLMS algorithm based scheme. However, SM-AP algorithm based MIMO-DFE requires less number of updates than the scheme with SM-NLMS algorithm, which makes the total computational load. Moreover, in the implementation of SM-AP algorithm-based MIMO-DFE, it is observed that the main computational complexity terms are in computing the matrix inversion, which is of order O(P3), where P is the projection order.
Simulation Results and Discussion
- Performance Comparison of MIMO-DFE using SM-NLMS, AP and
- Performance of SM-AP Algorithm based MIMO-DFE for Different
- Performance of SM-AP Algorithm Based MIMO-DFE for Different
- Performance of SM-AP Algorithm based MIMO-DFE with Different
The convergence as well as BER performance of SM-AP algorithm based MIMO-DFE is studied for different projection order (P) values in both Ped-A and Veh-B environments. Here, the performance of SM-AP Algorithm based MIMO-DFE is investigated for different values of convergence factor. The number of updates in SM-AP algorithm-based MIMO-DFE for different values of convergence factor is shown in Table 6.6.
SM-BNLMS Algorithm based Equalizer for MIMO Channel
Computational Complexity Issues
Here we compare the computation involved in implementing the proposed MIMO-DFE using the SM-BNLMS algorithm with existing SM-NLMS and SM-AP algorithm-based techniques. However, smoothing with the SM-BNLMS algorithm converges significantly faster than the scheme with the SM-NLMS algorithm and the number of required updates is relatively less in the case of SM-BNLMS, so the total computational burden is slightly higher for SM-NLMS, while better convergence is obtained in the case of SM-BNLMS. In addition, it is noted that the complexity per update in the proposed MIMO channel equalizer is significantly less compared to SM-AP algorithm based MIMO-DFE.
Results and Discussion
- Performance of Different Data-selective Algorithms based MIMO-DFE 112
- Performance of SM-BNLMS Algorithm based MIMO-DFE with Dif-
It is further noted that MIMO-DFE based on the BNLMS algorithm has slightly better performance for SNR below 12 dB due to the absence of SM filtering, but for higher SNR the performance of the two schemes is almost comparable. Simulations are performed for γ-values asγ1=√. 5σn, with convergence factor µset to shows the convergence performance of SM-BNLMS algorithm based MIMO-DFE for different γ values. Threshold for percentage reduction in updates Estimation error Pedestrian-A Vehicle-B. b) Vehicle Test Environment-B Figure 6.12: BER performance of the proposed MIMO-DFE with different γ values.
Summary
The work in this thesis focused primarily on reduced complexity compensation for frequency-selective channels. Summary of the works reported in this thesis is presented in the remainder of this section. The channel equalization technique is introduced, the latest literature in this area is discussed, and the problem is formulated in chapter 1.
Contributions
Directions for Future Work
Feuer, “Convergence and performance analysis of the normalized LMS algorithm with uncorrelated Gaussian data,” IEEE Trans. Diniz, "Convergence analysis of the binormalized data-reusing LMS algorithm," IEEE Transactions on Signal Processing, vol. Werner, "Set-membership binormalized data-reusing LMS algorithms," IEEE Transactions on Signal Processing, vol.
Multipath Propagation
Block Diagram of a Generic DFE
A Generic MIMO System Model
Geometrical Interpretation of Weight Updating using BNLMS Algorithm
Block diagram of a generic adaptive DFE model
Convergence Performances of DFE using AP and NLMS as Adaption Algorithms
BER Performance of DFE using AP and NLMS as Adaption Algorithms
MSE Performance of AP Algorithm based DFE with Different Values of Step Size
BER Performance of AP Algorithm based DFE with Different Values of Step Size
MSE Performance of Affine Projection Algorithm based DFE with Different Filter
BER Performance of Affine Projection Algorithm based DFE with Different Filter
MSE Performance of Binormalized LMS Algorithm based DFE with Different Values
BER Performance of Binormalized LMS Algorithm based DFE with Different Values
Graphical Visualization of the Updating Procedure of SM-NLMS Algorithm
