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Description  |
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FIELD OF THE INVENTION
This invention relates to the field of digital filtering and more
particularly to the field of adaptive least means squares digital
filtering.
BACKGROUND OF THE INVENTION
Digital filtering involves manipulating one or more digital input signals
in order to provide one or more digital output signals. Digital signals
are a series of digital values wherein each valve represents the magnitude
of a particular signal, such as an electrical signal, at a particular
time. In the case of electrical signals, a digital input signal can be
produced by sampling the electrical signal periodically and converting
each sampled value of the signal to a digital value. Similarly, a digital
output signal can be converted to an analog electrical output signal by
converting the series of digital values to analog values.
A digital filter can correspond to a known analog filter. For example, it
is possible to construct a digital low-pass filter which operates in a
manner equivalent to an analog low-pass filter having a resistor and a
capacitor. The output signal of such a digital low-pass filter will be the
input signal having high frequency components attenuated therefrom. An
example of a digital low-pass filter can be represented by the following
equation:
Y[n]=(Y[n-1]+X[n])/2
Y[n] represents a digital value indicative of the magnitude of the present
value of the digital filter output. Y[n-1] represents a digital value
indicative of the magnitude of the most recent digital filter output (i.e.
the output of the digital filter for the previous iteration). X[n]
represents a digital value indicative of the magnitude of the present
input to the digital filter. For the low-pass digital filter represented
by the above equation, the magnitude of the output equals one half of the
sum of the magnitudes of the next most recent output and the input.
The flexibility of digital filters makes it possible in some instances to
adjust the parameters of a filter during operation. For example, if a
digital filter is implemented using computer software, then the equation
that governs operation of the filter can be easily modified or replaced,
thus changing characteristics of the filter. A digital filter which can be
modified during operation is called an "adaptive digital filter". Adaptive
digital filters are very useful in applications where the characteristics
of the signal being filtered change over time.
A difficulty with adaptive digital filters is establishing criteria for
modifying the filters. It is desirable to modify the filters in response
to actual changes in the signal and to not modify the filters in response
to spurious noise in the system. The modification mechanism needs to be
relatively quick in order to respond adequately to actual signal changes.
At the same time, the modification mechanism needs to be relatively slow
in order to be resistant to spurious noise.
Techniques for theoretically determining the optimum rate at which an
adaptive digital filter should be allowed to change are set forth in W. A.
Gardner, "Nonstationary Learning Characteristics of the LMS Algorithm,"
IEEE Trans, on Circuits and Systems, Vol. CAS-34, No. 10, pp. 1199-1207,
October 1987 and in B. Widrow, J. M. McCool, M. G. Larimore, C. R.
Johnson, "Stationary and Nonstationary Learning Characteristics of the LMS
Adaptive Filter," Proc. IEEE, Vol. 64, No. 8, pp. 1151-1162, August 1976.
However, the Gardner and Widrow results are only theoretical because they
require values of parameters which cannot be measured in real-world
applications of adaptive digital filter.
There are also a variety of ad-hoc techniques for determining the optimum
rate of change for an adaptive digital filter, such as that shown in U.S.
Pat. No. 4,349,889 to van den Elzen et. al titled "NON-RECURSIVE FILTER
HAVING ADJUSTABLE STEP-SIZE FOR EACH ITERATION". The majority of these ad
hoc techniques adjust the rate according to a measure of the filter error
(i.e. the difference between the actual output and the idealized output of
the filter). However, this measurement does not distinguish between
spurious noise, which should be ignored and hence should cause the rate of
change of the adaptive filter to decrease, and an actual change in the
signal being tracked, which should cause the rate of change of the
adaptive filter to increase.
SUMMARY OF THE INVENTION
According to the present invention, a weight adjustment unit for adjusting
the weights of an adaptive digital filter according to one or more input
signals to the digital filter and according to an error signal indicative
of the difference between the actual and desired outputs of the digital
filter uses a first low-pass filter for low-pass filtering a signal
indicative of the product of the error signal and the one or more input
signals, a squarer, for squaring the output of the first low-pass filter,
a second low-pass filter for low-pass filtering the output of the squarer
to extract the D.C. component thereof, a third low-pass filter for
low-pass filtering a signal indicative of the output of the error signal
squared to extract the D.C. component thereof, a dividing unit for
dividing the output of the second low-pass filter by the output of the
third low-pass filter to provide a loop bandwidth signal, and a weight
calculation unit for providing values for one or more weights of the
adaptive digital filter according to the value of the loop bandwidth
signal and according to the previous values of the weights.
In an exemplary embodiment of the invention, the low-pass filters are
single-pole IIR filters. In still another embodiment, the squarer provides
to the second low-pass filter the sum of the squares of the output of the
first low-pass filter and a single loop bandwidth is provided. In still
other embodiments, processing for each weight is performed separately so
that different loop bandwidth values are provided for each weight. In
other embodiments of the invention, the first low-pass filter is a Weiner
filter and a value indicative of spurious noise is subtracted from the
input to the second low-pass filter.
A feature of the present invention is the ability to change the weights of
the adaptive digital filter relatively rapidly in response to changes in
the signal being filtered while at the same time changing the weights of
the adaptive digital filter relatively slowly in response to spurious
noise. The use of IIR digital low-pass filters for some embodiments
simplifies the design. Determining a single loop bandwidth simplifies
calculations in cases where the possible values for the weight adjustment
are symmetrical about a mean squared error axis. Similarly, determining a
different loop bandwidth for each weight is advantageous in cases where
the possible values for the weight adjustment are not symmetrical about a
mean squared error axis. Using the Weiner filter and subtracting a value
indicative of spurious noise from the input to the second filter can
improve performance in some instances.
Other advantages and novel features of the present invention will become
apparent from the following detailed description of the invention when
considered in conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1A is a schematic block diagram illustrative of an adaptive digital
filter configured according to an exemplary embodiment of the present
invention.
FIG. 1B is a graph illustrating a relationship between an analog filter
impulse response and a response of the adaptive digital filter shown in
FIG. 1A.
FIG. 2 is a graph illustrating the relationship between coefficients of a
digital filter and the error 10 output of the digital filter.
FIG. 3 is a plurality of graphs illustrating the relationship between
coefficients of a digital filter and the error output of the digital
filter.
FIG. 4 is a schematic block diagram illustrating a high frequency
communication channel using an adaptive digital filter according to an
exemplary embodiment of the invention.
FIG. 5A is a schematic block diagram of apparatus according to an exemplary
embodiment of the invention for adjusting the weights of an adaptive
digital filter.
FIG. 5B diagrammatically illustrates a plurality of prefilters 122-1 . . .
122-1N for the lowpass prefilter 122 of FIG. 5A.
FIG. 6A is a schematic block diagram of apparatus according to another
exemplary embodiment of the invention for adjusting the weights of an
adaptive digital filter.
FIG. 6B diagrammatically illustrates a plurality of prefilters 142-1 . . .
142-N for the lowpass Weiner prefilter 142 of FIG. 6A.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Referring to FIG. 1A, an adaptive digital filter 20 processes a digital
version of an input signal X and provides a digital version of a signal Y
as an output. The precise means by which the adaptive filter 20 is
implemented (i.e. software, digital hardware, or a combination thereof)
depends upon a variety of functional factors known to one skilled in the
art which will become more apparent in light of the discussion which
follows.
The magnitude of the present (i.e. most recently sampled) value of the
signal X is a single digital quantity indicated as X[n]. The digital
quantity X[n] is provided to a first time delay unit 22 which has a single
digital output value indicated as X[n-1]. The quantity X[n-1] is the
magnitude of a sampled value of the signal X sampled just prior to the
X[n] sample. In other words, X[n] represents a single digital value
indicative of the most recently sampled value of the signal X and X[n-1]
represents a single digital value indicative of the next most recently
sampled value of the signal X. The first time delay unit 22 therefore
represents providing a time delay of one sample period.
The signal X[n-1] is provided to a second time delay unit 23 which provides
an output X[n-2] which represents a single digital value indicative of a
sampled value of the signal X sampled just prior to the X[n-1] sample.
Similarly, the signal X[n-2] is provided to a third time delay unit 24
which has an output indicated as X[n-3] which represents a single digital
value indicative of a sampled value of the signal X sampled just prior to
the X[n-2] sample. The time delay units 22-24 each delay propagation of
the digital sample values of the signal X by one sample period. Therefore,
the time delay units 22-24 act as a shift register shifting therethrough
digital values indicative of successive sampled values of the signal X.
The quantities X[n], X[n-1], X[n-2], and X[n-3] represent the four most
recently sampled values of the X signal.
The value X[n] is tapped and provided to a first weight multiplier 26.
Similarly, the values X[n-1], X[n-2], and X[n-3] are tapped and provided
to second, third, and fourth weight multipliers 27, 28, 29, respectively.
The first weight multiplier 26 provides a digital output signal indicative
of the magnitude of the X[n] quantity multiplied by a first weight W1. The
second weight multiplier 27 provides an output signal indicative of the
product of the quantity X[n-1] and a second weight, W2. The third weight
multipliers 28 provides an output signal indicative of the product of
X[n-2] and a third weight, W3. The fourth weight multiplier 29 provides an
output signal indicative of the product of X[n-3] and a fourth weight, W4.
The digital multiplications are performed by means known to one skilled in
the art. The outputs of the weight multipliers 26-29 are provided to a
summer 30, which adds the outputs of the weight multipliers 26-29 and
provides the sum as a single digital output value, Y[n]. The value Y[n] is
a present value of the signal Y that is output by the adaptive filter.
Operation of the adaptive filter 20 is as follows: For each sample period,
a digital signal equal to the present sampled value of the signal X is
provided to the first time delay unit 22. The time delay units 22-24 cause
the X[n] signal from the previous iteration to become X[n-1], X[n-1] from
the previous iteration to become X[n-2], and X[n-2] from the previous
iteration to become X[n-3]. The quantities X[n], X[n-1], X[n-2], and
X[n-3] are then multiplied by the weight multipliers 26-29, respectively.
The outputs of the weight multipliers 26-29 are input to the summer 30
which provides the output signal Y[n]. At the next sample period, the
process begins again and a new Y[n] is determined and then output. The
quantity Y[n] can be expressed mathematically using the following
equation:
Y[n]=W1*X[n]+W2*X[n-1]+W3*X[n-2]+W4*X[n-3] (1)
It is worth noting that although the adaptive filter 20 is illustrated
having the input X[n] shifted successively through the time delay units
22-24 once for each sample period, the invention described hereinafter can
be practiced with an adaptive digital filter that is provided each
iteration with a number of separate input signals rather than a single
input signal delayed a number of times. In other words, it would be
possible to provide the weight multipliers 26-29 with the present value of
four separate input signals from four different sources, i.e. X1[n],
X2[n], X3[n], and X4[n], instead of X[n], X[n-1], X[n-2], and X[n-3].
Therefore, the discussion which follows is applicable to either a single
input delayed a number of times (e.g. X[n], X[n-1], X[n-2], etc.), the
present value of multiple inputs (e.g. X1[n], X2[n], X3[n], etc.), or some
combination thereof.
Referring to FIG. 1B, a graph 40 illustrates the impulse response of the
adaptive filter 20. The graph 40 has an analog plot 42 thereon indicative
of a response of an analog filter to an impulse input (not shown). The
plot 42 corresponds approximately to the Y signal. Four bars 46-49 are
superimposed on the plot 42. The length of the bar 46 corresponds to the
magnitude of the weight W1. Similarly, the lengths of the bars 47-49
correspond to the magnitudes of the weights W2, W3, and W4, respectively.
The bars 46-49 illustrate that, by adjusting the weights W1, W2, W3, and
W4, the response of the adaptive digital filter 20 to a digital impulse
input can be made to approximate the impulse response of the analog filter
shown by the plot 42. That is, the weights W1, W2, W3, and W4 can be set
such that an impulse digital input to the adaptive digital filter 20 will
produce an output, Y[n], that approximates the plot 42.
The adaptive digital filter 20 is so-named because the values of the
weights W1, W2, W3, and W4 can change as a function of time, thereby
adapting the filter 20 to a variety of conditions during operation. An
adaptive digital filter weight adjustment unit 32, shown in FIG. 1A,
adjusts the weights of the adaptive digital filter 20. For instance, the
weights can be changed so that the adaptive filter 20 approximates the
impulse response of a different analog filter than that shown by the plot
42 of FIG. 1B. Changing the weights as a function of time is useful for a
variety of applications where the nature of the signal and/or the nature
of the noise superimposed on the signal can vary with time, such as in the
high frequency communication channel application described hereinafter and
used to illustrate the present invention.
One technique for updating the weights of an adaptive digital filter is the
"least-means squares" technique, wherein the weights of the filter are
updated according the magnitude of the error of the filter output. The
error of the filter output is the difference between the actual filter
output and the desired filter output. The following equation illustrates
the least-mean squares technique for calculating new values for the
weights W1, W2, W3, and W4:
W[n+1]=W[n]+2ue[n]X[n] (2)
The quantities W and X represent vectors where the vector W has components
W1, W2, W3, etc. and the vector X has components X1, X2, X3, etc. Writing
the above equation in vector notation is equivalent to writing the
following separate equations:
W1[n+1]=W1[n]+2ue[n]X1 (3)
W2[n+1]=W2[n]+2ue[n]X2 (4)
W3[n+1]=W3[n]+2ue[n]X3 (5)
W4[n+1]=W4[n]+2ue[n]X4 (6)
If the values for the X vector are from the adaptive digital filter 20 of
FIG. 1A, then X1=X[n], X2=X[n-1], X3=X[n-2], and X4=X[n-3]. However, as
discussed above, in general it is possible for X1, X2, X3, etc. to
represent the present value of separate input values.
The quantity e[n] in the above equations represents the present value of
the error of the filter and is given by the following equation:
e[n]=d[n]-Y[n] (7)
where d[n] equals the present value of the desired output and Y[n] equals
the present value of the actual output.
The quantity u in the above equations is the rate that the weights of the
adaptive filter 20 are allowed to change. This quantity is deemed the
"loop bandwidth" of the filter 20. The magnitude of u controls the
operation of the weight adjustment unit 32 and hence controls the
performance of the adaptive digital filter 20. As u becomes larger, the
loop bandwidth increases, thus making the adaptive digital filter 20 more
responsive to the magnitude of e[n]. As u decreases, the loop bandwidth
decreases and the adaptive digital filter 20 is less responsive to the
magnitude of e[n].
There is a trade-off between having a wide loop bandwidth (i.e. the weights
are allowed to change at a relatively rapid rate) and a narrow loop
bandwidth (i.e. the weights are allowed to change at a relatively slow
rate). Having a wide loop bandwidth that allows the weights to change
rapidly is useful when the nature of the signal changes rapidly and the
filter 20 must adapt quickly to the new signal. However, the wide loop
bandwidth is disadvantageous in instances where changes in the error
signal are due to noise. The weights will be erroneously changed due to
the noise. A narrow loop bandwidth, on the other hand, prevents the
weights from being changed incorrectly due to spurious noise, but may
unacceptably slow down desirable weight changes when signal changes occur.
Referring to FIG. 2, a graph 50 illustrates adjustment of two weights W1
and W2 in response to the value of the mean-square error (MSE) of the
output signal Y[n]. A first axis 52 of the graph 50 represents the
magnitude of W1. A second axis 54 represents the magnitude of W2. A third
axis 56 represents the magnitude of the MSE. A three dimensional surface
58 describes possible value combinations for W1, W2, and MSE. The optimal
values for W1 and W2, indicated on the graph 50 as Wopt, corresponds to
the lowest part of the three dimensional surface 58, i.e. the part of the
surface 58 corresponding to the smallest value for the MSE.
A line 60 drawn on the surface 58 shows the transition of the weights W1,
W2 from Wstart, which indicates the starting weights, to Wopt, which
indicates the optimal weights. For each sample iteration, W1 and W2 are
adjusted according to Eq. (2), above. The rate at which W1 and W2 converge
on Wopt is determined by the loop bandwidth, the magnitude of u. That is,
the number of iterations required for the weights to change from Wstart to
Wopt is determined by the loop bandwidth of the adaptive digital filter
represented by the surface 58. The direction and steepness of a vector
drawn tangential to the surface 58 at a point is deemed the gradient of
the surface 58 at that point. The steeper the gradient at a particular
point, the more quickly W1 and W2 will converge to Wopt.
There are two phenomena which would cause W1 and W2 to change after the
Wopt operating point is reached. The first is spurious noise in the
system. The second is a change in the surface 58, i.e. a change in the
signal being filtered. If u is kept relatively small (i.e. a narrow loop
bandwidth), then the effect of the noise will be kept to a minimum. For
each iteration, the values of W1 and W2 will change very slightly and any
noise in the system will not greatly effect W1 and W2. Furthermore, on
average, the noise signals will tend to cancel each other. Since a narrow
loop bandwidth makes W1 and W2 change slowly, the effects of the noise on
W1 and W2 will be minimal.
However, a small u will inhibit the system's ability to react quickly to a
change in the nature of the signal. If the signal is such that the Wopt
point changes rapidly, it is desirable to have a relatively wide loop
bandwidth, i.e. a large u, in order to allow the weight adjustment unit 32
of the adaptive digital filter 20 to adapt sufficiently to the new signal.
Systems wherein Wopt changes as a function of time are deemed
"non-stationary" systems.
Referring to FIG. 3, a plurality of graphs 70, 80, 90, 100 illustrate
different ways that the signal being filtered can change for
non-stationary systems. In the graph 70, the surface that describes W1,
W2, and MSE is shown translating in a vertical direction from one position
to another. In this case the value of Wopt does not change, but the MSE
associated therewith does. In the graph 80, the surface that describes W1,
W2, and MSE changes from one position to another, thereby causing Wopt to
change from Wopt1 to Wopt2. The number of iterations that are required to
adjust W1 and W2 to the new value of Wopt is a function of the loop
bandwidth of the filter. As the loop bandwidth increases, the number of
iterations decreases.
The graphs 90, 100 show a top view of surfaces describing W1, W2, and MSE.
For the graphs 90, 100, the MSE is the vertical axis coming out of the
page. The darker lines of the graphs 90, 100 indicate the contour of a
first surface and the lighter lines of the graphs 90, 100 indicate the
contour of a second surface. The graphs 90, 100 show that the surfaces
describing W1, W2, and MSE can change shape and be non-symmetrical about
an axis corresponding to the value of the MSE. The graph 90 shows that the
surface, from the top view, can change shape from a circle to an oval.
Similarly, the graph 100 illustrates that the surface can change from an
oval having an axis with a first orientation to an oval having an axis
with a second orientation. Note that the graphs 70, 80, 90, 100 only show
a small subset of the number of possible signal changes for a
non-stationary system.
Referring to FIG. 4, a schematic block diagram 110 illustrates a high
frequency communication channel 112 that is modeled using an adaptive
digital filter 114. At one end of the channel 112 a transmitter (not
shown) transmits a signal. At the other end of the channel 112, a receiver
(not shown) receives the signal. The difference between the transmitted
signal and the received signal is distortion introduced by the channel
112. The adaptive digital filter 114 is provided to counteract the
distortion introduced by the channel 112 by filtering the received signal.
The difference between the received signal and the filtered signal is
determined at a summing junction 116, which outputs an error signal that
is fed back to the adaptive filter 114. The adaptive filter 114 attempts
to drive the error signal to zero.
The high frequency communication channel 112 represents high frequency
radio communication passing through the Earth's upper atmosphere. The
channel 112 is labeled "Raleigh Fading" to denote the specific
characteristics of this type of communication. Raleigh Fading refers to a
relatively slow time-variant change in the transmission characteristics of
the channel 112. Generally, the change of the characteristics of a Raleigh
fading channel is similar to transition in signal characteristics
illustrated by the graph 80 of FIG. 3.
The adaptive digital filter 114 is provided at the receiving end of the
channel 112 to model the characteristics of the Raleigh Fading channel
112. Any received communication signals can thereby be filtered at the
receiver to remove distortion introduced by the channel 112. The adaptive
digital filter 114 adapts at periodic intervals when a PN sequence, having
known signal characteristics, is transmitted over the channel 112. The
error is calculated by taking the difference between the received PN
sequence and an idealized PN sequence filtered through the adaptive filter
114.
The adaptive digital filter 114 is similar to the adaptive digital filter
20 shown in FIG. 1A. The output of the adaptive filter 114 equals the
input multiplied by a first weight plus the input delayed one sample
period multiplied by a second weight plus the input delayed two sample
periods multiplied by a third weight and so on. During the adaptation
phase (when the PN sequence is transmitted), the weights are adjusted by
an adaptive digital filter weight adjuster using equation (2), above. The
error is determined by comparing an idealized version of the PN sequence
with the received PN sequence filtered through the filter 114. The value
of u, which also varies with time, determines the rate at which the
weights are allowed to change in response to the magnitude of the error
signal.
Referring to FIG. 5A an exemplary embodiment of an adaptive digital filter
weight adjuster unit 120 for the adaptive digital filter 114 is shown. The
weight adjuster unit 120 is provided with an input signal indicative of
the present value of the error signal, e[k], and the present value of the
received signal, x[k]. These signals are processed each iteration to
provide a new value of u, the loop bandwidth. The value of u and the value
of w[k] are then used to determine the value of a new weight, w[k+1].
As shown in FIG. 5A the quantity 2*e[k]*x[k] is provided to a prefilter
122. For this embodiment, the prefilter 122 is shown as a single pole IIR
digital low-pass filter, although it will be understood by one of ordinary
skill in the art that other types of digital low-pass filters could be
used instead. The filter constant, p, determines the bandwidth of the
low-pass filter 122. The filter 122 is such that the bandwidth is narrow
enough to filter out spurious noise and wide enough to detect
non-stationary behavior. The exact value for p depends upon the nature of
the signal being monitored, the noise which distorts the signal, and a
variety of functional factors which can be determined by one of ordinary
skill in the art in light of discussion contained herein.
The quantity x[k], which is part of the input to the filter 122, is a
vector quantity that indicates separate input values corresponding to
x[k], x[k-1], x[k-2], etc. Therefore, as diagrammatically illustrated in
FIG. 5B, the filter 122 represents a plurality of separate filters, 122-1
. . . 122N wherein one filter filters x[k], another filter filters x[k-1],
etc. Of course, as discussed above, the input could be in the form of
X1[k], X2[k], X3[k], wherein each represents a present value of an input
signal from separate sources. However, for the adaptive filter illustrated
herein, there is only a single input source and the vector x[k] represents
a single digital signal.
The multiple outputs of the filter 122 are combined at a squaring unit 124,
which provides the sum of the squares of each of the separate outputs of
the filter 122. The D.C. component of the output of the squaring unit 124
is the power of the signal gradient. The output of the squaring unit 124
is provided to a low-pass filter 126, which can be a single-pole IIR
filter similar to the low-pass filter 122. Of course, it will be
understood by one of ordinary skill in the art that other types of filters
could be used instead. The filter 126 attempts to extract the D.C.
component of the signal and hence has a fairly narrow bandwidth. However,
the bandwidth is wide enough so as to make the delay of the filter 126 not
be excessive. The output of the filter 126 is a signal indicative of the
signal gradient power.
The present value of the error signal, e[k], is squared and provided to a
low-pass filter 128, which is similar to the filter 126. The magnitude of
the D.C. component of the squared error signal is indicative of the signal
error power. The bandwidth of the filter 128 is made relatively narrow in
order to provide the D.C. component of the input signal as an output.
However, just as with the low-pass filter 126, the bandwidth of the filter
128 is not so narrow as to create an unacceptable delay.
The output of the filter 128, the error power, is provided to a division
unit 130, which is also provided with the output of the filter 126, the
gradient power. The division unit 130 divides the gradient power by the
error power and outputs u, the loop bandwidth. The loop bandwidth of the
system is proportional to the ratio of the gradient signal power to the
error power. The value of u is provided to a weight calculation unit 132,
which calculates a new value for the weight for the adaptive digital
filter 114 using u and using the previous value of the weight.
Referring to FIG. 6A, an alternative embodiment of a weight adjuster unit
140 is shown. A low-pass filter 142 is similar to the low-pass filter 122
of FIG. 5A. Namely, as diagrammatically illustrated in FIG. 6B, low-pass
Weiner filter 142 may comprise a plurality N of prefilters 142-1 . . .
142-N. The filter 142, however, is a Weiner filter, which is a known type
of digital low-pass filter that provides an optimum signal to noise ratio
at the output. The filter constant, p, varies with time and equals
1-2u.sigma..sub.x.sup.2, as shown in FIG. 6A at an input to the filter
142. The symbol .sigma..sub.x.sup.2 represents the magnitude of the power
of the input signal, x, squared and then averaged. This is determined by
squaring the magnitude of x[k] and then providing the result to a
conventional digital low-pass filter (not shown), such as a single-pole
IIR filter. The low-pass filter averages the values of x[k] squared.
Th | | |
