International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637
Optimization of a System Using Steepest Descent Implementation Priya1, Yogesh Juneja2 Electronics and communication1, 2, PDM college of Engg1, 2 Email: priyavashist13@gmail.com1, yogeshjunejaer@gmail.com2 Abstract—Blind equalization using stochastic gradient descent in both batch and adaptive algorithms achieves parameter optimization through cost minimization. Stochastic descent algorithms require large number of iterations or long data samples to converge. This is a novel steepest descent batch algorithm that does not require data recycling. The proposed steepest descent batch implementation of both algorithms converge rapidly in a few iterations and deliver superior performance without the delay due to data recycling and refiltering. Index Terms- Blind equalization, frequency domain implementation, batch algorithm.
1. INTRODUCTION Adaptive filter is basically used for the signal processing techniques such as noise cancellation, array processing, line enhancement, PLL, system identification, adaptive equalization and spectral estimation etc[1]. Filtering is a signal processing operation whose objective is to process a signal in order to manipulate the information contained in the signal. A filter is a device that maps its input signal to another output signal facilitating the extraction of the desired information contained in the input signal. A digital filter is the one that processes discrete-time signals represented in digital format. For time invariant filters the internal parameters and the structure of the filter are fixed, and if the filter is linear the output signal is a linear function of the input signal. Once prescribed specifications are given, the design of time-invariant linear filters entails three basic steps, namely: the approximation of the specifications by a rational transfer function, the choice of an appropriate structure defining the algorithm, and the choice of the form of implementation for the algorithm. An adaptive filter is required when either the fixed specifications are unknown or the specifications cannot be satisfied by time-invariant filters. Strictly speaking an adaptive filter is a nonlinear filter since its characteristics are dependent on the input signal and consequently the homogeneity and additively conditions are not satisfied. Signals being processed by the filters . Adaptive Filter The block diagram of an adaptive filter is as shown in fig. 1 [2]. It is the adaptive algorithm that utilizes the coefficient updation according to the coefficient update equation of the form
…(1) Where ∆Wn is a correction that is applied toWn at time n to form a new Wn+1 at time (n+1). The key
component of adaptive algorithm is, how the correction ∆Wn to be formed [3].
Fig 1 Adaptive Filter
2. Adaptive Equalization Adaptive equalization is increasing popular in digital communications. However, with the ever-increasing demand for larger bandwidth and faster transmission speed, the increase in distortion in the channel is likely to be significant. Therefore, it is gainful to delve into adaptive equalization techniques to suppress distortion in a communication channel. Minimizing distortion will in turn, allow for longer haul communication before requiring a repeater, which will save infrastructure and equipment cost for a communication link. Adaptive filters are systems that can adjust themselves to different environments. It involves a process of filtering some input signal to match a desired response. It is very difficult for estimating both the channel order and the distribution of energy among the taps and even it is very difficult to predict the effect of the environment on these taps. so it necessary that the equalization process must be adaptive, means the equalizer need to be adapted very frequently with the changing environment. This includes two phases. Firstly the equalizer needs to be trained with some known samples in the presence of some desired response (Supervised Learning). After training the weights and various parameters associated with the equalizer structure is frozen to function as a detector. These two processes are
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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 frequently implemented to keep the equalizer adaptive. We call “the Equalizer is frozen” if we keep the adaptable parameters of the equalizer constant. The general operation of an adaptive equalizer in a communication channel is to track the variations of the channel response over time. This is achieved by sending a known training sequence through the channel and obtaining the difference when the output is subtracted from the known sequence. The difference is known as the prediction error. The computed error is then used to adjust the tap coefficients so that the channel response can be estimated. This is done recursively using an adaptive algorithm like the least mean square (LMS) algorithm. Once the equalizer has converged, the actual data can then be sent and received accurately as the channel response is known and compensated for by the equalizer. A typical digital communication system with adaptive equalizer is shown in Fig. 2
6. 7.
Linear Equalizer: processes the incoming signal with a linear filter MMSE equalizer: designs the filter to minimize E[|e|2], where e is the error signal, which is the filter output minus the transmitted signal.
Fig. 3 Adaptive equalization tree
3. SIMULATION SET UP AND RESULTS
Fig. 2 Digital transmission system with equalizer
Adaptive equalizers are either supervised or unsupervised. The equalizers with unsupervised training are called blind equalizers. The classification of the equalizers is shown in the fig. 3 Several equalizer types are listed below: 1. 2.
3.
4.
5.
Zero Forcing Equalizer: approximates the inverse of the channel with a linear filter. Decision Feedback Equalizer: augments a linear equalizer by adding a filtered version of previous symbol estimates to the original filter output [5,6]. Blind Equalizer: estimates the transmitted signal without knowledge of the channel statistics, using only knowledge of the transmitted signal's statistics [4]. Adaptive Equalizer: is typically a linear equalizer or a DFE. It updates the equalizer parameters (such as the filter coefficients) as it is processes the data. Typically, it uses the MSE cost function; it assumes that it makes the correct symbol decisions, and uses its estimate of the symbols to compute , which is defined above. Viterbi Equalizer: Finds the maximum likelihood (ML) optimal solution to the equalization problem. Its goal is to minimize the probability of making an error over the entire sequence.
Channel Equalization using steepest gradient method: The method of steepest descent is a celebrated optimization procedure for minimizing the value of a cost function J(n) with respect to a set of adjustable parameters W(n) The steepest descent algorithm uses the approximation H = 2I in newton's update equation. u> 0 is used to control the rate of convergence. The general update equation is.
…(2)
…(3) In other words, the ith parameter of the system is altered according to the derivative of the cost function with respect to the ith parameter. Collecting these equations in vector form, we have
…(4) The current equalizer taps vector is W(n) and the next sample equalizer taps vector weight is W(n+1), We could estimate the W(n+1) vector by this approximation: 2. The gradient is a vector pointing in the direction of the change in filter coefficients that will cause the greatest increase in the error signal. Because the goal is to minimize the error, however, the filter
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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 coefficients updated in the direction opposite the gradient; that is why the gradient term is negated. 3. The constant is a step-size. After repeatedly adjusting each coefficient in the direction opposite to the gradient of the error, the adaptive filter should converge. Fig. 4 shows the Frequency response of the channel. Fig. 5 demonstrates the Frequency response of the equalizer with steepest descent algorithm. Fig. 6 depicts the eye diagram at the input of the equalizer with steepest descent algorithm. Fig. 7 depicts the eye diagram at the output of the equalizer with steepest descent algorithm. Fig. 8 shows the convergence rate of mean square error at zero noise value.
Fig.6 Eye diagram at the input of the equalizer
Fig. 7 Eye diagram at the output of the equalizer
Fig. 4 Frequency response of the channel
Fig.8 Convergence rate of mean square error at zero noise value
4. CONCLUSION
Fig.5 Frequency response of the equalizer
Equalization techniques compensate for the time dispersion introduced by communication channels and combat the resulting inter-symbol interference (ISI) effect. In other words, the overall our system model, which is a cascade connection of the channel and equalizer, provides nearly an ideal transmission medium that the information source signals can be sent through. Given a channel of unknown impulse response, the purpose of an adaptive equalizer is to operate on the channel output such that the cascade connection of the channel and the equalizer provides an approximation to an ideal transmission medium.
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International Journal of Research in Advent Technology, Vol.2, No.6, June 2014 E-ISSN: 2321-9637 REFERENCE [1]. S. Qureshi, Adaptive equalization", IEEE Communications Magazine, vol. 73, no. 9, pp. 1349-1387, Sept 1985. [2]. S. Haykin, Adaptive Filter Theory", PrenticeHall, 3rd Ed., 1996. [3]. Thomas Drumright, Adaptive Filtering", Spring 1998. [4]. Weerackody, V. ; Kassam, S.A. ; Laker, K.R., “Convergence analysis of an algorithm for blind equalization”, IEEE Transactions on Communications, Vol. 39 , Issue: 6, Pag: 856 – 865, 1991. [5]. Mathew, G. ; Farhang-Boroujeny, B. ; Wood, R.W., "Design of multilevel decision feedback equalizers”, IEEE Transactions on Magnetics, Vol. 33, Issue: 6, Pag: 4528 – 4542, 1997. [6]. Bakhtiar Qutub Ali,” A New Blind Equalization Scheme based on Principle of Minimal Disturbance”, king fahd university of petroleum and minerals, May 2004.
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