International Journal of Emerging Technologies and Engineering (IJETE) Volume 2 Issue 4, April 2015, ISSN 2348 – 8050 89 www.ijete.org Fixed Point LMS Adaptive Filter with Low Adaptation Delay INGUDAM CHITRASEN MEITEI Electronics and Communication Engineering Vel Tech Multitech Dr RR Dr SR Engg. College Chennai, India MR. P. BALAVENKATESHWARLU Electronics and Communication Engineering Vel Tech Multitech Dr RR Dr SR Engg. College Chennai, India Abstract In this paper, the behaviour of the Delayed LMS (DLMS) algorithm is studied. It is found that the step size in the coefficient update plays a key role in the convergence and stability of the algorithm. An upper bound for the step size is derived that ensures the stability of the DLMS. A novel partial product generator has been used for achieving lower adaptation delay and area-delay-power efficient implementation. The relationship between the step size and the convergence speed, and the effect of the delay on the convergence speed are also studied. The analytical results are supported by the computer simulations. The problem of the efficient realization of a DLMS transversal adaptive filter is investigated. Index Terms – Adaptive filters, delayed least mean square (DLMS) algorithm fixed-point arithmetic, least mean square (LMS) algorithm. INTRODUCTION Adaptive digital filters have been applied to a wide variety of important problems in recent years. Perhaps one of the most well-known adaptive algorithms is the least mean squares (LMS) algorithm, which updates the weights of a transversal filter using an approximate technique of steepest descent. Due to its simplicity, the LMS algorithm has received a great deal of attention, and has been successfully applied in a number of areas including channel equalization, noise and echo cancellation and many others. However, the feedback of the error signal needed to update filter weights in the LMS algorithm imposes a critical limitation on the throughput of possible implementations. In particular, the prediction error feedback requirement makes extensive pipelining of filter computations impossible. As a result of this, the design of high speed adaptive filters has been predominantly based on the adaptive lattice filter, which lends itself easily to pipelining. Recently it has been shown that it is possible to introduce some delay in the weight adaptation of the LMS algorithm. The resulting delayed least mean squares (DLMS) algorithm, which uses a delayed prediction error signal to update the filter weights, has been shown to guarantee stable convergence characteristics provided an appropriate adaptation step size is chosen. We present a modular pipelined filter architecture based on a time-shifted version of the DLMS algorithm. This pipelined architecture displays the most desirable features of both lattice and transversal form adaptive filters. As in an adaptive lattice filter, the computations are structured to be order recursive, resulting in a highly pipelined implementation. Also, the weights are updated locally within each stage. However, the equations being implemented actually correspond to a true transversal adaptive filter, and hence desirable properties of this structure are preserved. The modular pipeline consists of a linear array of identical processing elements (PES) which are linked together using both local and feedback connections. Each PE performs all the computations associated with a single coefficient of the filter. A significant advantage of the modular structure of the pipelined DLMS filter is that, unlike conventional transversal filters, the order of the filter can be increased by simply adding more PE modules to the end of the pipelined. The LMS algorithm has been commonly used in adaptive transversal filtering. In this algorithm, the estimated signal in each data interval is computed and subtracted from the desired signal. The error is then used to update the tap coefficients before the next sample arrives. In the figure, k = sample number, x = reference input, X = set of recent values of x, d = desired input, W = set of filter coefficients, ε = error output, f = filter impulse response, * = convolution, Σ = summation, upper box=linear filter, lower box=adaption algorithm There are two input signals to the adaptive filter: d k and x k which are sometimes called the primary and the reference input respectively. d k which includes the desired signal plus undesired interference and x k which includes the signals that are correlated to some of the undesired interference in d k . The filter is controlled by a set of L+1 coefficients or weights. W k =[w 0k ,w 1k ,…,w lk ] T represents the set or vector of weights, which control the filter at sample time k. where w lk refers to the l'th weight at k'th time.
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International Journal of Emerging Technologies and Engineering (IJETE)
Volume 2 Issue 4, April 2015, ISSN 2348 – 8050
89
www.ijete.org
Fixed Point LMS Adaptive Filter with Low Adaptation Delay
INGUDAM CHITRASEN MEITEI Electronics and Communication Engineering
Vel Tech Multitech Dr RR Dr SR Engg. College
Chennai, India
MR. P. BALAVENKATESHWARLU Electronics and Communication Engineering
Vel Tech Multitech Dr RR Dr SR Engg. College
Chennai, India
Abstract In this paper, the behaviour of the Delayed LMS
(DLMS) algorithm is studied. It is found that the step
size in the coefficient update plays a key role in the
convergence and stability of the algorithm. An upper
bound for the step size is derived that ensures the
stability of the DLMS. A novel partial product generator
has been used for achieving lower adaptation delay and
area-delay-power efficient implementation. The
relationship between the step size and the convergence
speed, and the effect of the delay on the convergence
speed are also studied. The analytical results are
supported by the computer simulations. The problem of
the efficient realization of a DLMS transversal adaptive
filter is investigated.
Index Terms – Adaptive filters, delayed least mean
square (DLMS) algorithm fixed-point arithmetic, least
mean square (LMS) algorithm.
INTRODUCTION Adaptive digital filters have been applied to a wide
variety of important problems in recent years. Perhaps
one of the most well-known adaptive algorithms is the
least mean squares (LMS) algorithm, which updates the
weights of a transversal filter using an approximate
technique of steepest descent. Due to its simplicity, the
LMS algorithm has received a great deal of attention,
and has been successfully applied in a number of areas
including channel equalization, noise and echo
cancellation and many others.
However, the feedback of the error signal needed to
update filter weights in the LMS algorithm imposes a
critical limitation on the throughput of possible
implementations. In particular, the prediction error
feedback requirement makes extensive pipelining of
filter computations impossible. As a result of this, the
design of high speed adaptive filters has been
predominantly based on the adaptive lattice filter, which
lends itself easily to pipelining. Recently it has been
shown that it is possible to introduce some delay in the
weight adaptation of the LMS algorithm. The resulting
delayed least mean squares (DLMS) algorithm, which
uses a delayed prediction error signal to update the filter
weights, has been shown to guarantee stable
convergence characteristics provided an appropriate
adaptation step size is chosen.
We present a modular pipelined filter architecture
based on a time-shifted version of the DLMS algorithm.
This pipelined architecture displays the most desirable
features of both lattice and transversal form adaptive
filters. As in an adaptive lattice filter, the computations
are structured to be order recursive, resulting in a highly
pipelined implementation. Also, the weights are updated
locally within each stage. However, the equations being
implemented actually correspond to a true transversal
adaptive filter, and hence desirable properties of this
structure are preserved. The modular pipeline consists of
a linear array of identical processing elements (PES)
which are linked together using both local and feedback
connections. Each PE performs all the computations
associated with a single coefficient of the filter. A
significant advantage of the modular structure of the
pipelined DLMS filter is that, unlike conventional
transversal filters, the order of the filter can be increased
by simply adding more PE modules to the end of the
pipelined.
The LMS algorithm has been commonly used in
adaptive transversal filtering. In this algorithm, the
estimated signal in each data interval is computed and
subtracted from the desired signal. The error is then used
to update the tap coefficients before the next sample
arrives.
In the figure, k = sample number, x = reference input,
X = set of recent values of x, d = desired input, W = set
of filter coefficients, ε = error output, f = filter impulse
response, * = convolution, Σ = summation, upper
box=linear filter, lower box=adaption algorithm
There are two input signals to the adaptive
filter: dk and xk which are sometimes called
the primary and the reference input respectively.
dk which includes the desired signal plus undesired
interference and xk which includes the signals that are
correlated to some of the undesired interference in dk.
The filter is controlled by a set of L+1 coefficients or
weights.
Wk=[w0k,w1k,…,wlk]T
represents the set or vector of weights, which control
the filter at sample time k. where wlk refers to the l'th