The Application of Frequency-Weighting to Improve
Filtering and Smoothing Performance
Garry A. Einicke
CSIRO
Technology Court, Pullenvale, QLD 4069, Australia
garry.einicke @ csiro.au
Abstract: Pioneering research on the perception of sounds at
different frequencies was conducted by Fletcher and Munson in
the 1930s. Their work led to a standard way of weighting
measured sound levels within investigations of industrial noise
and hearing loss. Frequency weightings have since been used
within filter and controller designs to manage performance
within bands of interest. This paper introduces iterative
frequency weighted filtering and smoothing procedures. It is
assumed that filter and smoother estimation errors are generated
by a first-order moving-average (AR1) system. This AR1 system
is identified and used to apply a frequency weighting function
within the design of minimum-variance filters and smoothers. It
is shown under prescribed conditions that the described solutions
result in nonincreasing error variances. An example is described
which demonstrates improved mean-square-error performance.
Keywords—Frequency weighting; filtering; smoothing
I. INTRODUCTION
Frequency weightings are used within filter and controller
designs to manage performance within bands of interest. For
example, Grimble and El Sayed [1] employ a stable,
minimum-phase frequency shaping function to weight the
average power of the error spectral density in the design of
least squares and H
polynomial filters. Zang, Bitmead and
Gevers defined frequency weighting functions as ratios of
prediction error and exogenous signal spectra in an iterative
controller design [2]. Pots, Petersen and Kelkar chose a
frequency weighting function to attenuate noise in a specified
frequency band within a robust controller [3]. Perceptual
weighting functions based on ratios of linear prediction
coefficients can be used within speech codecs [4].
This paper investigates the application of frequency
weighting to improve on the performance of filters and
smoothers which rely on inexact assumptions. Parameter
uncertainty is not modeled explicitly as that approach is
described within papers on H
estimation - see [5], [6] and the
references therein.
This paper investigates the application of frequency
weighting to improve on the performance of filters and
smoothers which rely on inexact assumptions. Parameter
uncertainty is not modeled explicitly as that approach is
described within papers on H
estimation - see [5], [6] and the
references therein.
The paper is organized as follows. Section II defines the
problem of interest and derives the minimum-variance
frequency-weighted filter and smoother solutions. Procedures
for designing a frequency weighting system are described in
Section III. It is shown that if the estimation error is assumed
to be generated by a high-pass moving average order-1 (MA1)
system then the sequence of frequency-weighted error
variances is nonincreasing. Finally, a scalar example is
presented which demonstrates that repeated iteration of
frequency weighting can provide performance benefits. The
conclusion follows in Section IV.
Fig. 1. The frequency-weighted estimation problem. The objective is to
design a filter H that produces estimates ˆ y which minimize the energy of the
frequency-weighted estimation error y .
II. FREQUENCY WEIGHTED SMOOTHER AND FILTER
A. Problem Definition
Let w = [w
1
, …, w
N
]
T
, w
k
R, represent a stochastic input
sequence over an interval of length N with a symmetric
distribution function and { } 0
k
Ew ,
2 2
{ }
k w
Ew V , where (.)
T
denotes the transpose and E{.} denotes the expectation
operator. Assume that there exists a second independent
measurement noise sequence v = [v
1
, …, v
N
]
T
, v
k
R, with
{ } 0
k
Ev ,
2 2
{ }
k v
Ev V and { } 0
k k
Ewv . Consider a system
G: R ĺ R having the realization
1 0
, 0
k k k
x Ax Bw x
, (1)
k k
y Cx , (2)
where
k
x R
n
is an internal state and A R
nɯn
, B R
nɯ1
, C
R
1ɯn
. Suppose that observations
k k k
z y v (3)
\ are available. Filter and smoother solutions H: R ĺ R
are desired that produce estimates ˆ
k
y of
k
y from the
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