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Description  |
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BACKGROUND OF THE INVENTION
The invention relates to low-power digital filtering with the use of
adaptive approximate filtering.
The increasing demand for battery-operated portable electronic devices has
made the minimization of power consumption a critical design criteria in
digital signal processing algorithm development. Applications for
low-power signal processing include cellular telephones, pagers, laptop
computers, portable Global Positioning System (GPS) receivers, portable
camcorders, and other wireless communications devices. The power
consumption of a digital system is proportional to the average number of
computations that are required per output sample to accomplish the desired
processing. Thus reducing the number of computations required to perform a
particular task directly results in reduced power consumption.
Power reduction in signal processing systems generally involves
optimization at all levels of the design abstraction including
consideration of process technology, logic and circuit design,
architecture design, and algorithm selection. Typically, optimization to
lower the power consumption is done statically at design time. However,
significant power gains can be achieved if the optimization is done
dynamically at run time by considering and adapting to time-varying signal
statistics.
A significant number of DSP functions involve frequency-selective digital
filtering in which the goal is to reject one or more frequency bands while
keeping the remaining portions of the input spectrum largely unaltered.
Examples include lowpass filtering for signal upsampling and downsampling,
bandpass filtering for subband coding, and lowpass filtering for
frequency-division multiplexing and demultiplexing. The exploration of
low-power solutions in these areas is therefore of significant interest.
To first order, the average power consumption, P, of a digital system may
be expressed as:
##EQU1##
where C.sub.i is the average capacitance switched per operation of type i
(corresponding to addition, multiplication, storage, or bus accesses),
N.sub.i is the number of operations of type i performed per sample,
V.sub.dd is the operating supply voltage, and f.sub.s is the sample
frequency.
Real-time digital filtering is an example of a class of applications in
which there is no advantage in exceeding a bounded computation rate. For
such applications, an architecture-driven voltage scaling process has
previously been developed in which parallel and pipelined architectures
can be used to compensate for increased delays at reduced voltages. This
strategy can result in supply voltages in the 1 to 1.5 V range by using
conventional CMOS technology. Power supply voltages can be further scaled
using reduced threshold devices. Circuits operating at power supply
voltages as low as 70 mV (at 300K) and 27 mV (at 77K) have been
demonstrated.
Once the power supply voltage is scaled to the lowest possible level, the
goal is to minimize the switched capacitance at all levels of the design
abstraction. At the logic level, for example, modules can be shut down at
a very low level based on signal values. Arithmetic structures (e.g.
ripple carry vs. carry select) can also be optimized to reduce transition
activity. Architectural techniques include optimizing the sequencing of
operations to minimize transition activity, avoiding time-multiplexed
architectures which destroy signal correlations, using balanced paths to
minimize glitching transitions, etc. At the algorithmic level, the
computational complexity or the data representation can be optimized for
low power.
Another method to reduce the switched capacitance is to lower the parameter
N.sub.i in equation (1). Efforts have been made to minimize N.sub.i by
intelligent choice of algorithm, given a particular signal processing
task. In the case of conventional filter design, the filter order is fixed
based on worst case signal statistics, which is inefficient if the worst
case seldom occurs. More flexibility may be incorporated by using adaptive
filtering algorithms, which are characterized by their ability to
dynamically adjust the processing to the data by employing feedback
mechanisms.
In accordance with the invention, it will be illustrated how adaptive
filtering concepts may be exploited to develop low-power implementations
for digital filtering.
SUMMARY OF THE INVENTION
Adaptive filtering algorithms have generally been used to dynamically
change the values of the filter coefficients, while maintaining a fixed
filter order. In contrast, the invention involves the dynamic adjustment
of the filter order. This process leads to filtering solutions in which
the filter output signal-to-noise ratio may be kept above a specified
minimum tolerable level, or solutions in which the stopband energy in the
filter output may be kept below a specified threshold, while using as
small a filter order as possible. Since power consumption is proportional
to filter order, the process of the invention achieves power reduction
with respect to a fixed-order filter whose output is similarly guaranteed
to have a signal-to-noise ratio above the specified level and a stopband
energy below the specified threshold. Power reduction is achieved by
dynamically minimizing the order of the digital filter.
The idea of dynamically reducing cost (e.g. power consumption) while
maintaining a desired level of output quality (e.g. signal-to-noise ratio
or stopband energy in the filter output) emanates from the concept of
approximate processing in computer science. While approximate processing
concepts may be used to describe a variety of existing techniques in DSP,
communications, and other areas, there has recently been progress in
formally using these concepts to develop new DSP techniques. Since our
adaptive filtering technique falls into this category, the process used in
the invention is referred to as adaptive approximate filtering, or simply
approximate filtering.
The invention involves exploiting input signal statistics to significantly
reduce the power consumption in frequency-selective digital filters.
Rather than using a fixed filter order (as is the case in conventional
filtering), the filter order is forced to vary over time with the aim of
keeping the order as small as possible while ensuring that the ratio of
the passband power to the stopband power of the filter output is kept
above a specified threshold, or that the stopband power of the filter
output is kept below a specified level. Power consumption is reduced since
the number of operations required per output sample is dynamically
minimized rather than being constrained to use a fixed filter order
optimized for the worst case signal statistics. The invention has been
shown to be effective in reducing power consumption by an order of
magnitude or more in filtering applications involving speech signals.
Accordingly, the invention provides a method of digital filtering
comprising providing a filter response with a predetermined filter order;
receiving an input signal; producing a predetermined set of filtered
output samples defining an output signal from said input signal;
calculating the difference between the power associated with said input
signal and the power associated with said output signal; and varying the
filter order of said filter response based on the calculated difference of
power in order to change said filter response. The motivation for changing
the filter response is to reduce power consumption whenever possible.
In another embodiment of the invention there is provided a digital
filtering system comprising a filter with a means for providing a filter
response with a predetermined filter order, the filter response producing
a predetermined set of filtered output samples defining an output signal
from a received input signal. A means is provided for calculating the
difference between the power associated with the input signal and the
power associated with the output signal. A means is also provided for
varying the filter order of the filter response based on the calculated
difference of power in order to change the filter response.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a graph displaying the frequency response magnitudes for FIR
filters of orders N=20, 80 and 140;
FIG. 2 is a schematic diagram of an adaptive approximate FIR filter
implemented with a tapped delay line;
FIG. 3 is a schematic diagram of a cascade structure for an eighth-order
adaptive approximate IIR filter;
FIG. 4 is a functional block diagram of a filter and filtering technique in
accordance with the invention;
FIG. 5 is a graph showing a lot of the values of a FIR filter stopband
energy E.sub.SB [k] vs. filter order k;
FIG. 6 is a table which lists various measures obtained for the performance
of an approximate filter as applied to a FDM signal;
FIG. 7 is a graph showing plots of the evolution of filter order
corresponding to an FDM input signal;
FIG. 8 is a graph showing a plot of the average of relative power
consumption as a function of the silence duration relative to the duration
of the entire signal; and
FIG. 9A is a schematic diagram of a binary tree-structured filterbank of
highpass and lowpass filters implemented using the approximate filtering
technique of the invention; FIG. 9B is a graph showing a plot of the
filter order as a function of time; FIG. 9C is a graph showing a plot of
the stopband component of the input, x.sub.s [n], to demonstrate that the
filter order roughly tracks the stopband energy of the FDM input signal;
FIG. 10 is a schematic of an incremental refinement structure for an IIR
digital filter;
FIG. 11 is a graph having a plot of magnitude-squared frequency responses
for truncations of a 20th-order Butterworth filter with 3, 5, 7, 9, and 10
second-order sections;
FIG. 12 is a schematic diagram of a filter and an adaptation process for
updating the filter order after each new set of L output samples is
computed in accordance with the invention;
FIG. 13 is a graph having plots in which the solid curves represent ISNR
estimates as a function of L and N.sub.0 ;
FIG. 14 is a graph showing a performance profile for truncations of a
20th-order Butterworth filter with half-power frequency .pi./2; and
FIGS. 15A-15C show respective graphs of demultiplexing of FDM speech
signals using low-power frequency selective filtering for the passband
speech signal, stopband speech signal and the number of filter sections
utilized as a function of sample number.
DETAILED DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS
A frequency-selective digital filter may have either a finite impulse
response (FIR) or an infinite impulse response (IIR). It is well known
that IIR filters use fewer taps than FIR filters in order to provide the
same amount of attenuation in the stopband region. However, IIR filters
introduce nonlinear frequency dispersion in the output signals which is
unacceptable in some applications. For such cases it is desirable to use
symmetric FIR filters because of their linear phase characteristic.
An important family of symmetric FIR filters correspond to the symmetric
windowing of the impulse responses of corresponding ideal filters. For
example, a lowpass filter of this type has an impulse response given by:
##EQU2##
where w[n] is a symmetric N-point window. Such a filter has cutoff
frequency .omega..sub.c and can be implemented using a tapped delay line
with N taps. For the purposes of describing the invention, such a filter
is referred to as having filter order N.
FIG. 1 is a graph displaying the frequency response magnitudes of FIR
filters of orders N=20, 80 and 140, when w[n] is a rectangular window and
.omega..sub.c =.pi./2. It will be appreciated that the mean attenuation
beyond the cutoff frequency .omega..sub.c increases with filter order.
Furthermore, with respect to a tapped delay-line implementation 20 as shown
in FIG. 2, the taps 22 of the shorter filters are subsets of the taps of
the longer filters. This technique ensures that if the filter order is to
be decreased without changing the cutoff frequency, a segment 24 of the
tapped delay line for the higher order filter can simply be powered down.
The price paid for such powering down is that the average stopband
attenuation of the filter decreases.
Butterworth IIR filters are commonly used for performing
frequency-selective filtering in applications where nonlinear frequency
dispersion is tolerable. The frequency response magnitudes of such filters
do not suffer from the ripples which can be seen in the frequency response
magnitudes for FIR filters. These IIR filters are commonly implemented as
cascade interconnections of second-order sections 32, each of which
consists of 5 multiplies and 4 delays, as shown in FIG. 3. FIG. 3 is an
illustration of a cascade structure for an eighth-order IIR filter 30 as
the cascade of four second-order sections 32. For the purposes of
describing the invention, the order of a Butterworth IIR filter is
considered to be equal to twice the number of second-order sections in its
cascade implementation. An interesting property of IIR Butterworth filters
is that if the second-order sections are appropriately ordered, one may
sequentially power down the later second-order sections 34 without
changing the cutoff frequency of the filter. As in the case of the FIR
filter, the price paid for powering down of second-order sections is that
the average stopband attenuation of the filter decreases.
The details of the approximate processing process applied to low-power
frequency-selective filtering in accordance with the invention will now be
described. As described earlier, frequency-selective filters are used in
applications where the goal is to extract certain frequency components
from an input signal while rejecting others. Take for example a signal,
x[n], consisting of a passband component, x.sub.p [n], and a stopband
component, x.sub.s [n]. That is,
x[n]=x.sub.p [n]+x.sub.s [n]. (3)
If it were possible to cost-effectively measure the strength of the
stopband component, x.sub.s [n], from observation of x[n], it could be
determined how much stopband attenuation is needed at any particular time.
When the energy in x.sub.s [n] increases, it is desirable to increase the
stopband attenuation of the filter. This can be accomplished by using a
higher-order filter. Conversely, the filter order may be lowered when the
energy in x.sub.s [n] decreases.
The invention presents a practical technique, based upon adaptive filtering
principles, for dynamically estimating the energy fluctuation in the
stopband component, x.sub.s [n], and using it to adjust the order of a
frequency-selective FIR or IIR filter. As described in the previous
section, the decrease in filter order enables the powering down of various
segments of the filter structure. Powering down of the higher-order taps
has the effect of reducing the switched capacitance at the cost of
decreasing the attenuation in the stopband. Assuming that the FIR delay
line is implemented using SRAM, even the data shifting operation of the
higher-order taps can be eliminated through appropriate addressing
schemes.
FIG. 4 is a functional block diagram of a filter 40 and filtering technique
in accordance with the invention. The filter includes an input 41 which
receives an input signal x[n] and an output 42 which delivers an output
signal 42. A filter response module 43 having a variable filter order
transforms the input signal x[n] into the output signal y[n]. A filter
order update controller 44 initiates the change in filter order of the
filter response module 43. The filter order update varies the filter order
in response to the calculated difference between the energy (power)
associated with the input and output signals. Signal energy computational
devices 45 and 46 respectively calculate the energies associated with the
input and output signals. An adder 47 determines the energy difference
between the input and output signals.
The quantity d[n], which represents the energy (power) differential between
the input signal x[n] at input 41 and the output signal y[n] at output 42,
is obtained as:
d[n]=E.sub.x [n]-E.sub.y [n] (4)
where
##EQU3##
The filter order for sample period n, Order[n], is updated at each sample
period. One method for the update process is to choose order[n] to be the
smallest positive integer which guarantees that the stopband energy, Q[n],
of the output signal will be maintained below a specified threshold
.gamma.. Assuming that the stopband portion of the input spectrum is
essentially flat (this flatness constraint may be relaxed considerably
without detrimental effects), the stopband energy in the output can be
estimated as:
Q[n]=.alpha.d[n]E.sub.SB [Order[n]] (7)
where .alpha. is a proportionality constant, and E.sub.SB [k] represents
the stopband energy in the frequency response, H.sub.k (.omega.), of the
kth order filter. That is,
##EQU4##
where SB denotes the stopband region. Since for every sample period this
method requires an expensive search over the stored values of E.sub.SB
[k], a more efficient strategy which incrementally updates the most recent
filter order has been designed. In this case, the stopband energy in the
output is estimated as:
Q[n]=qd[n]E.sub.SB [Order[n-1]]. (9)
The decision rule for choosing Order[n] is then given by:
##EQU5##
where .alpha., .gamma., .delta., and N.sub.0 are application-specific
parameters. It should be noted that in this case the filter order is
changed at most by N.sub.0 during each sample period.
The parameters .delta. and N.sub.0 in equation (10) control the sensitivity
of the time evolution of the filter order. The choice of the parameter L
in equations (5) and (6) involves a tradeoff between suppression of
sensitivity to local fluctuations and preservation of the possible
time-varying nature of the signal energy. For the case of FIR filters, it
is also observed that when the value of L is less than the maximum filter
order, there is no extra storage required to compute E.sub.x [n] beyond
that required for the filter implementation. On the other hand, excess
storage is always required to update E.sub.y [n].
The arithmetic cost of the update process can be easily shown to involve
five multiplications, five additions, one table lookup from a small memory
module, and simple control. This cost is roughly equivalent to that of
increasing the FIR filter order by five or the IIR filter order by two.
This, for example, means that net power savings can be expected in the FIR
case if for significant periods of time the dynamic FIR filter order
decreases by more than five with respect to the maximum filter order. The
overhead of multiplication is reduced to one multiplication instead of
five per update if absolute value operations are used to compute E.sub.x
[n] instead of magnitude-squared operations.
In the context of FIR filters, simulations of the approximate filtering
technique in accordance with the invention have been used to show that
reduction in power consumption by an order of magnitude is achieved over
fixed-order filter implementations when the stopband energy of the output
signal is stipulated to remain below a given threshold .gamma.. The
context for most of these simulations is frequency-division demultiplexing
of pairs of speech waveforms.
With respect to the application of speech signals, each of the speech
signals used in simulations was sampled at 8 KHz and normalized to have
maximum amplitude of unity. Each signal corresponds to a complete sentence
with negligible silence at its beginning and end.
Each digitized speech waveform was pre-filtered to have a maximum frequency
of 1.5 KHz. A guard band of 1 KHz was used in multiplexing a reference
speech signal (corresponding to the sentence, "that shirt seems much too
long,") with each of the other speech signals. The reference signal always
occupied the 0 to 1.5 KHz band, while the other signals always occupied
the 2.5 KHz to 4 KHz band.
Demultiplexing involves lowpass filtering (cutoff frequency 2 KHz) to
isolate the reference speech signal. The approximate filtering technique
was used to perform this lowpass filtering for each of the 10 FDM signals.
The parameter values in equation (10) were chosen to be:
##EQU6##
The family of FIR filters used in these simulations corresponds to
equation (2) with w[n] rectangular. The values of the stopband energy
E.sub.SB [k] vs. filter order for this case are plotted in the graph of
FIG. 5.
In the table of FIG. 6 there are listed various measures obtained for the
performance of the approximate filter as it was applied to each FDM
signal. The first column contains the sentence number for the stopband
component of the input signal. The second and third columns respectively
list the minimum and maximum filter orders used by the approximate filter
in each case. The final column shows the relative power consumption of the
approximate filter with respect to a fixed-order filter which is
guaranteed to keep the stopband energy in the output below .gamma. for all
times. It is observed that the adaptive technique of the invention reduces
the power consumption on average by a factor of 5.9.
To gain further insight into the source for this power reduction, in FIG. 7
the nature of the adaptation performed by the technique of the invention
in the case of one of the FDM signals is illustrated. One of the curves 70
shows the evolution of the filter order as a function of time while the
other curve 72 shows the normalized energy profile of the stopband energy
of the input signal as a function of time. Clearly, the variations in
filter order roughly follow the energy variations of the stopband signal.
In particular, the most power savings is achieved during the silence
regions of the stopband component signal.
Longer periods of speech communication generally include significantly
larger fractions of silence periods than an individual sentence. To factor
this into the analysis, the simulations were repeated while inserting
additional silence at the end of each speech signal. The average (over all
10 cases) of the relative power consumption is displayed in the graph of
FIG. 8 as a function of the silence duration relative to the duration of
the entire signal. As expected, the power reduction improves as the
relative amount of silence is increased.
Data compression techniques for voice signals often use a binary
tree-structured filterbank of highpass and lowpass filters, as depicted in
FIG. 9A. Each of these filters may be implemented using the approximate
filtering technique of the invention. To illustrate the potential for
power savings in the first stage of the subband decomposition, an
approximate FIR lowpass filter was applied to a speech signal, x[n],
corresponding to the sentence, "that shirt seems much too long." The
time-varying FIR filter order used by the technique of the invention is
shown in the plot of FIG. 9B. The plot in FIG. 9C shows the stopband
component of the input, x.sub.s [n], to demonstrate that the filter order
roughly tracks the stopband energy of the input signal.
The invention as described heretofore relates to an approximate processing
method for reducing the power consumption of frequency-selective filters.
The method utilizes adaptive processing principles for the run-time
control of the number of stages utilized from an IIR or FIR filter
structure that has a type of incremental refinement property. In
particular, the filter structure must consist of a series of stages (such
as individual second-order sections) ordered in such a way that the
effective transfer function after the inclusion of each successive stage
results in increased attenuation in the stopband region(s) while
maintaining close to unity gain in the passband region(s). The adaptation
mechanism used with such a structure is designed to minimize the number of
stages that must be applied at any given time while ensuring that the
filter output satisfies a specified criterion. Minimization of the number
of stages utilized at any given time is desirable because of the resulting
savings in power consumption by the underlying hardware. Accordingly, the
invention has been described with respect to applying the process to an
FIR structure.
In accordance with another aspect of the invention, the process of the
invention will now be described as applied to an IIR filter structure.
This process uses a slightly different adaptation criterion than the one
described for the FIR implementation. Rather than seeking to keep the
stopband power in the filter output below a specified level, the technique
as applied to IIR filters is designed to keep the output SNR (ratio of the
passband power to the stopband power in the filter output) below a
specified level. It will be appreciated that both adaptation criteria are
equally applicable to IIR as well as FIR cases. Furthermore, results are
presented on the convergence of the order of the adapted filter to the
correct filter order when the input signal satisfies the underlying
theoretical assumptions of the process and the input SNR is known
a-priori.
Consider the case of a Butterworth filter of order 2 M.sub.0. A cascade
structure for this filter consists of a serial connection of M.sub.0
second-order Direct-Form II sections, as shown in FIG. 10. FIG. 10 is a
schematic of an incremental refinement structure 100 for an IIR digital
filter. Each section corresponds to a pair of conjugate poles of the
Butterworth filter and two zeros (both located at z=-1). Denoting the
frequency response of the order-2 M.sub.0 Butterworth filter by H.sub.M0
(.omega.), it follows that:
H.sub.M.sbsb.0 =G.sub.1 (.omega.)G.sub.2 (.omega.)G.sub.3 (.omega.) . . .
G.sub.M.sbsb.0 (.omega.) (12)
where G.sub.i (.omega.) denotes the frequency response of the ith
second-order section in the cascade structure of FIG. 10. It can be
furthermore assured that G.sub.i (0)=1. If only the first N sections
(N.ltoreq.M.sub.0) of the cascade structure in FIG. 10 are utilized, the
resulting order-2N truncated Butterworth filter has the frequency response
H.sub.N (.omega.), given by:
##EQU7##
The Butterworth pole pairs are assigned to each of the second-order
sections so that as the number of second-order sections is increased, the
attenuation in the stopband of the filter also increases, while keeping
the passband gain close to unity. An empirical strategy for making such a
pole-pair assignment is as follows: the pole pair for the M.sub.0 th
section is selected first as the one which results in .vertline.H.sub.M0-1
(.omega.)).vertline. having the smallest maximum deviation (from unity) in
the passband. From the remaining pole pairs, the pair for the (M.sub.0
-1)st section is selected as the one which results in .vertline.H.sub.MO-2
(.omega.).vertline. having the smallest maximum deviation (from unity) in
the passband.
The process is continued backwards in an analogous manner until each
section has been assigned its corresponding pole pair. To illustrate,
consider the application of this strategy to a 20th order Butterworth
filter with half-power frequency of .pi./2. The functions
.vertline.H.sub.N (.omega.).vertline..sup.2 obtained in this case are
shown in the graph of FIG. 11. FIG. 11 shows a plot of magnitude-squared
frequency responses for truncations of a 20th-order Butterworth filter
with 3, 5, 7, 9, and 10 second-order sections. It should be observed that
as the number of sections (N) is increased, the average attenuation in
most of the stopband also increases. On the other hand, the filter gain
remains close to unity in most of the passband.
Suppose that a stationary input x[n] with power spectrum S.sub.x (.omega.)
is filtered using the N-section truncated Butterworth filter of FIG. 10,
where 1.ltoreq.N.ltoreq.M.sub.0, to obtain an output y[n]. The
signal-to-noise ratio, SNR, is defined as the ratio of the power in the
passband to the power in the stopband. Specifically, the input SNR may be
defined as
##EQU8##
Correspondingly, the output SNR is defined as
##EQU9##
Ideally, one would like to select N to be the smallest value for which
OSNR[N].gtoreq.OSNR.sub.tol (20)
where OSNR.sub.tol is the minimum tolerable output SNR for the application.
Furthermore, if the input is non-stationary, OSNR[N] will be time varying
and consequently the filter order would have to adapt over time to reduce
power consumption. Of course, this requires an adaptation framework whose
overhead is low relative to the expected savings in power consumption.
FIG. 12 is a schematic diagram of a filter 120 and an adaptation process
for updating filter order after each new set of L output samples is
computed in accordance with the invention. The number of sections utilized
in the filter's incremental refinement structure is updated for every new
set of L output samples.
The update procedure involves the calculation of input and output
signal-power estimates followed by the application of a decision module
122 shown in FIG. 12. The module 122 uses the signal-power estimates to
form an estimate of the temporally local ISNR. This ISNR estimate is then
used as the basis for selecting the filter order to be applied in
computing the next set of L output samples. The precise formulation of the
decision rule is based upon the following set of assumptions: (1) S.sub.x
(.omega.)) in the passband is arbitrary, (2) S.sub.x (.omega.) in the
stopband is white but with unknown power, (3) S.sub.x (.omega.) in the
transition band is negligible, and (4) .vertline.H.sub.N
(.omega.).vertline..sup.2 in the passband is effectively equal to 1.
Consider a situation where a N.sub.0 -section (N.sub.0 .ltoreq.M.sub.0)
truncated Butterworth filter is applied to a stationary input x[n] to
obtain the output y[n]. ISNR and OSNR [N.sub.0 ] are expressed in terms of
the input signal power P.sub.x and the output signal power P.sub.y
[N.sub.0 ]. Accordingly,
##EQU10##
Next, assume that S.sub.x (.omega.) is an unknown constant
.sigma..sup.2.sub.SB in the stopband, it is negligible in the transition
band, and it is arbitrary in the passband. It then follows that
##EQU11##
Furthermore, we note that
##EQU12##
Assuming that .vertline.H.sub.N0 (.omega.).vertline..sup.2 is effectively
equal to 1 in the passband,
##EQU13##
which may be rearranged to yield
##EQU14##
Substituting this expression for .sigma..sup.2.sub.SB into equation (23)
results as
P.sub.y.sup.SB [N.sub.0 ]=(Px-Py[N.sub.0 ])P.sub.h.sup.SB [N.sub.0 | | |
