Generalized noise cancellation in a communication channel

A generalized noise canceler for canceling noise in a channel having noise composed of correlated noise components. The noise canceler is realized with a finite number of cascade networks, each network being composed of an inner product filter is series with another filter wherein both filters have characteristics determined from a eigenvector equations expressed in terms of the channel operator and noise operator.

FIELD OF THE INVENTION 
This invention relates generally to communication systems and, more 
specifically, to noise cancellation in a channel wherein the correlation 
of two different components of the noise are utilized to reduce the 
expected noise power of one of the components. 
BACKGROUND OF THE INVENTION 
In the book on information theory entitled "Information Theory and Reliable 
Communication", authored by R. G. Gallager and published by John Wiley, 
1968, Gallager shows how the capacity of a time-continuous channel with 
intersymbol interference and colored noise may be determined. The 
time-continuous channel of interest is shown in FIG. 1 wherein: channel 
100 has impulse response h(t); the input time signal 101 to channel 100 is 
s(t); one component of the output signal 103 of channel 100 is signal 102 
given by s.sub.o (t) (with s.sub.o (t) being the convolution of s(t) and 
h(t)); and the other component of output signal 103 is additive noise 104 
represented by n(t). Both components of output signal 103 are combined in 
summer 105. As shown in FIG. 2, which includes the frequency domain 
equivalent of FIG. 1, the first step disclosed by Gallager was that of 
filtering the channel output 103 with equalizer 201 to flatten the noise 
spectrum; equalizer 201 has a transfer function given by 
[N(.omega.)].sup.-1/2, where N(.omega.) is the spectrum of the noise. With 
reference to FIG. 3, the white noise model equivalent to FIG. 2 is shown 
wherein: the equivalent channel 301 is the original channel frequency 
transfer function H(.omega.) divided by the square-root of the noise 
spectrum ([N(.omega.)].sup.1/2), and the inputs to summer 305 are flat 
noise component 302 given by N.sub.o and channel output 303. Then Gallager 
determines the signal shapes that yield the least lost energy in 
transmission through the equivalent channel of FIG. 3. These optimum input 
signals form an orthogonal set that is complete in a restricted sense on 
the space of bounded energy signals at the channel input. Since the 
optimum input signals are the eigenfunctions of a singular value 
decomposition of the channel impulse response, the output signals are also 
orthogonal. Thus, the result of Gallager offered the tractable feature 
that the complex channel of FIG. 1 could be decomposed into an array of 
parallel scalar channels as illustrated in FIG. 4. 
In FIG. 4, a.sub.i (e.g., 401,402) is the coefficient in the series 
expansion in the input signals {.theta..sub.i (t)} that lose least energy 
on transmission through the equivalent channel 301: 
##EQU1## 
where {.theta..sub.i i(;)} are normalized to have unit energy. With this 
input, the channel output, s.sub.o (;), is given by 
##EQU2## 
where .lambda..sub.i.sup.1/2 .psi..sub.i (;) is the channel output when 
the channel input is .theta..sub.i (;). The functions {.psi..sub.i (;)} 
are normalized to have unit energy by using the normalization constant 
.lambda..sub.i, which is the channel gain (.lambda..sub.i =energy 
out/energy in) when .theta..sub.i (;) is transmitted. A receiver matched 
to recover s(;) would equalize s.sub.o (;) to eliminate the 
.lambda..sub.i.sup.1/2 factors. This would have the effect of producing 
noise in the coefficients of the final output (e.g. 407,408 from summers 
405 and 406, respectively) that are proportional to 
##EQU3## 
(e.g., 403,404) where N.sub.o is the flattened noise power spectral 
density. 
In a separate study as presented in "The optimum combination of block codes 
and receivers for arbitrary channels," authored by the present inventor J. 
W. Lechleider and published in the IEEE Trans. Commun., vol. 38, no. 5, 
May, 1990, pp. 615-621, Lechleider investigates transmission of short 
sequences of amplitude modulated pulses through a dispersive channel with 
colored, added noise. Lechleider found that the channel input sequences 
that led to the maximum ratio of mean output signal power to mean noise 
power are the solutions to a matrix eigenvalue problem similar to the 
integral equation eigenvalue problem considered by Gallager. The structure 
of this channel is much like a finite dimensional version of FIG. 4. 
Because of the ubiquity of the form of FIG. 4, the idea of signaling so 
that the transmission model is a set of parallel, uncoupled subchannels 
with uncorrelated subchannel noises has come to be known as "Structured 
Channel Signaling," or SCS. Thus, SCS decomposes a complex vector channel 
into an ordered sequence of scalar sub-channels with uncorrelated 
sub-channel noise scalars. Because of this lack of correlation, no noise 
cancellation techniques can be used to further improve the signal-to-noise 
performance of the total channel. This places an upper bound on what can 
be achieved by noise cancellation techniques in SCS. 
As discussed by Widrow et al. in the paper "Adaptive noise cancelling: 
Principles and applications," Proc. IEEE, vol. 63, no. 12, pp 1692-1716, 
December, 1975, auxiliary measurements are made of noise that are 
correlated with the noise vector that is received with the signal in order 
to effect noise cancellation. The correlation is used to form a best 
estimate of the added noise that is subtracted from the received, noise 
corrupted signal. But, because of the formulation of SCS, SCS obviates any 
putative noise-cancellation improvement. 
As alluded to in the foregoing background, SCS is a modeling technique for 
dispersive channels that provides insight into channel performance. 
Moreover, SCS may also be used as a basis for the design of communication 
systems. By spreading signals over time and frequency, SCS offers some 
immunity to structured noise such as impulse noise and narrow-band noise. 
SCS also offers selective use of the best performing sub-channels for the 
most important subset of information to be transmitted. SCS subsumes 
generalized noise cancellation, which is a technique for exploiting the 
correlation of two different components of noise to reduce the expected 
noise power of one of the components. 
The art is devoid of teachings or suggestions, however, of a methodology 
and concomitant circuitry for generalized noise cancellation in a 
communication channel having correlated noise components. 
SUMMARY OF THE INVENTION 
The shortcomings of the prior art with respect to generalized noise 
cancellation is obviated, in accordance with the present invention, by a 
noise canceler realized with only a finite number of optimal noise 
functionals. The canceler and concomitant methodology of generalized noise 
cancellation utilize an analysis of the formulation and the solution of 
the SCS regime as the point of departure. 
Broadly, in one embodiment, the generalized noise canceler for processing a 
channel output signal composed of a signal plus noise to produce a 
filtered channel output signal, the noise being a linear combination of 
first and second noise components, includes a parallel arrangement of a 
plurality of cascade networks, each cascade network being responsive to 
the signal and the noise. Further, each cascade network is composed of an 
inner product filter circuit having a filter characteristic determined 
from a first eigenvector equation corresponding to minimization of noise 
in the first noise component, in series with a second filter having a 
characteristic determined from a second eigenvector equation related to 
the first eigenvector equation. All outputs of the second filters are 
combined to produce a filtered channel output signal. 
Broadly, in a second embodiment, a generalized noise canceler for 
processing a channel output signal composed of a signal plus noise to 
produce a filtered channel output signal, the noise being a linear 
combination of first and second noise components, includes a parallel 
arrangement of a plurality of cascade circuits, each cascade network being 
responsive to the signal and the noise. Further, each cascade network is 
composed of a first filter having a filter characteristic determined from 
a first eigenvector equation corresponding to minimization of the noise in 
the first noise component, a sampler in series with the first filter, and 
a second filter having a characteristic determined from a second 
eigenvector equation related to the first eigenvector equation. All 
outputs of the second filters are combined to produce a filtered channel 
output signal. 
The organization and operation of this invention will be understood from a 
consideration of the detailed description of the illustrative embodiments, 
which follow, when taken in conjunction with the accompanying drawing.

DETAILED DESCRIPTION 
The first part of this description presents a derivation of SCS for digital 
signaling of one-shot channels that is more general than those given 
previously in the prior art. The derivation is given in a Hilbert space 
setting that is applicable to a wide range of applications. Then a 
generalized form of optimum noise cancellation is presented for the same 
Hilbert space framework. Finally, the development is applied to obtain a 
methodology and circuitry for noise cancellation. 
I.1 STRUCTURED CHANNEL SIGNALING 
The model of a communications system that is used in this discussion is 
illustrated in FIG. 5. This model is a generalization of the time 
continuous model studied by Gallager. The data 501 to be transmitted is a 
single number, a, that modulates a signal S (502), which is taken to be an 
element of a Hilbert space. For example, the signal space may consist of 
all functions of time that are square-integrable on a given segment of the 
time line. The channel, C (503), is a map from one Hilbert space to 
another. The space of received signals may, in fact, have a different form 
for the inner product of two elements than the transmitted signal space 
does. For example, the transmitted and received signal spaces may have 
different dimensionalities. It is assumed that there are no input signals 
that have no output. This is no loss of generality; such signals would not 
be used in practice. The noise, n (504), is an element of the output space 
that is selected at random. Only the second order statistics of this noise 
are used. There may be a subset of the receiver input space that cannot be 
reached by any transmitted signal. This subspace is referred to as the 
null output space. The part of n that lies in the null output space may be 
correlated with the part that lies in its orthogonal complement. This 
correlation is exploited by the receiver in both SCS and noise 
cancellation. The receiver 506 is linear, providing an estimate, a (507) 
of the transmitted signal. This estimate may be written in the form 
EQU a.dbd.&lt;R, aS.sub.o +n&gt; (3) 
where the parentheses &lt;&gt; indicate Hilbert space inner product, R is an 
element of the output signal space, as are S.sub.o and n. The channel 
operator may be explicitly included in equation (3) to yield 
EQU a.dbd.(R,aCS+n).dbd.a(R,CS)+(R,n) (4) 
The mean-square error in this estimate is 
EQU e.dbd.&lt;(a-a).sup.2 &gt;.dbd.&lt;a.sup.2 &gt;[(R,CS)-1].sup.2 +&lt;(R,n).sup.2 &gt;](5) 
The received signal power is 
EQU &lt;a.sup.2 &gt;.dbd.&lt;a.sup.2 &gt;(R, CS).sup.2 (6) 
In SCS the signal, S, and the receiver vector, R, are jointly selected to 
maximize the signal-to-noise ratio (SNR) at the receiver output, i.e., 
##EQU4## 
However, this ratio can be made arbitrarily large by using distortionless 
transmission ((R,CS)-1) and arbitrarily large transmitted power. Thus, it 
is necessary to constrain the transmitted power to, say, &lt;a.sup.2 &gt;P while 
determining the best combination of S and R. Thus, a modified SNR is 
considered, namely, .lambda..sub..mu. given by 
##EQU5## 
where .mu. is a parameter similar to a Lagrange parameter; it will be 
chosen to make the transmitted power equal to &lt;a.sup.2 &gt;P so that the 
second term in the numerator on the right in equation (7) vanishes. In 
equation (7), parentheses have been used to indicate inner products in the 
input space. 
Now, it is noted that &lt;(R ,n).sup.2 &gt; is a symmetric quadratic form in R 
and, consequently, may be written in the form (R,NR), where N is a 
symmetric operator. This converts equation (7) to the form 
##EQU6## 
Now, consider the combination of S and R that maximizes .lambda..sub..mu. 
when .mu. is chosen so that the the input power is &lt;a.sup.2 &gt;P. First, 
keep R fixed and vary S in the sense of the calculus of variations and set 
the result equal to zero to get a stationary condition. This yields 
EQU [.alpha.-.lambda..sub..mu. (.alpha.-1)]CS-.lambda..sub..mu. NR.dbd.0 (9) 
where 
EQU .alpha..dbd.(R,CS) (10) 
is a measure of the distortion incurred in transmission. Next, hold S fixed 
in equation (8) and vary R. Then, stationarity requires that 
EQU &lt;a.sup.2 &gt;[.alpha.-.lambda..sub..mu. (.alpha.-1)]CS-.lambda..sub..mu. 
NR.dbd.0 (11) 
Equations (9) and (11) must be satisfied if R and S are optimum 
receiver-signal pair. Before proceeding with the solution of these 
equations, .alpha. and .mu. are now determined. To do this, first take an 
inner product of equation (11) with R to get, after using equation (10), 
##EQU7## 
Now, use equation (12) in equation (7) and assume that the transmitted 
power constraint is satisfied. The result is 
EQU .lambda..sub..mu. (1-.alpha.).dbd.0 (13) 
The only non-trivial solution of this equation is 
EQU .alpha..dbd.1 (14) 
so that the optimal solution uses distortionless transmission. The value of 
.mu. is now easily determined by using equation (14) in equation (9) and 
taking the inner product of the result with S to get 
##EQU8## 
Equations (14) and (15) permit writing equations (9) and (11) in the form 
EQU PC.sup.T R-S.dbd.0 (16) 
and 
EQU &lt;a.sup.2 &gt;CS-.lambda.NR.dbd.0. (17) 
Using equations (16) in (17) then yields the following eigenvalue problem: 
EQU P&lt;a.sup.2 &gt;CC.sup.T .theta..sub.i .dbd..lambda..sub.i N.theta..sub.i (18) 
where .theta..sub.i and .lambda..sub.i are the i.sup.th 
eigenvector-eigenvalue pair. If it is assumed that P&lt;a.sup.2 &gt;, which is 
the total transmitted power, is unity, .lambda..sub.i becomes the SNR for 
unit transmitted power and equation (18) becomes 
EQU CC.sup.T .theta..sub.i .dbd..lambda..sub.i N.theta..sub.i. (19) 
At this point it can be assumed that N is positive definite. However, 
recall that there might be a subspace of the receiver input space that is 
inaccessible by transmission through the channel. This implies that 
C.sup.T may have a null subspace so that there might be eigenvectors of 
equation (19) that correspond to a zero eigenvalue. The null sub-space of 
C.sup.T is, of course, spanned by these eigenvectors. The vectors {C.sup.T 
.theta..sub.i } span the range of C.sup.T, or, equivalently, the domain of 
C. Hence, write 
EQU .psi..sub.i .dbd.C.sup.T .theta..sub.i (20) 
so that the .psi..sub.i span the input signal space. Using equation (20)in 
equation (19) yields 
EQU N.sup.-1 C.psi..sub.i .dbd..lambda..sub.i .theta..sub.i (21) 
or, operating on both sides with C.sup.T, 
EQU C.sup.T N.sup.-1 C.psi..sub.i .dbd..lambda..sub.i .psi..sub.i. (22) 
Thus, the {.psi..sub.i } also satisfy an eigenvalue problem. The form of 
equation (22) is the same as that of the integral equation used by 
Gallager. Thus, Gallager's eigenfunctions maximize the SNR at the receiver 
input for constrained transmitted power. 
Because of the symmetry of the operators in equation (22), the {.psi..sub.i 
} form a complete orthogonal family on the space of input signals. 
Equation (22) leads to: 
EQU (.psi..sub.i, .psi..sub.j).dbd.P.delta..sub.ij (23) 
by standard arguments, where .delta..sub.ij a Kronecker delta. Using the 
definition of the {.psi..sub.i } that was given by equation (20) in 
equation (23) gives an orthogonality principle for the {.theta..sub.i }, 
but with a weighting operator: 
EQU (.theta..sub.i, CC.sup.T .theta..sub.j).dbd.P.delta..sub.ij. (24) 
Using equation (19) in (24) yields another form for this principle: 
##EQU9## 
when .lambda..sub.i .noteq.0. Recalling the definition of N, equation (25) 
implies that 
##EQU10## 
Thus, the noise scalar at the output of receiver .theta..sub.i is 
uncorrelated with the noise at the output of receiver .theta..sub.j. It is 
important to note that equation (23) says nothing about the component of 
the noise vector that lies in the orthogonal complement of the space 
spanned by the {.theta..sub.i } with non-zero .lambda..sub.i. 
An orthogonality principle for the channel outputs corresponding the 
optimal channel inputs can also be derived. To do this, take the inner 
product of both sides equation (22) with .psi..sub.j to get 
EQU (C.psi..sub.j, N.sup.-1 C.psi..sub.i).dbd..lambda..sub.i P.delta..sub.ij ( 
27) 
where equation (23) has been used. 
The most important orthogonality principle resulting from the eigenvalue 
problem is that the optimum channel output signal set {C.psi..sub.i }, is 
biorthogonal in the optimum receiver vector set {.theta..sub.i }. This 
principle implies that the separate sub-channels formed by the 
{.theta..sub.i, .psi..sub.i } pairs are not coupled to each other. To 
arrive at this bi-orthogonality, recognize that equation (24) may be 
written in the form 
EQU (.theta..sub.i, C.psi..sub.j).dbd.P.delta..sub.ij (28) 
by employing equation (20). Equation (28) says that separate information 
may be sent on each of the sub-channels without any interference between 
them. When this is combined with uncorrelated sub-channel noise, a complex 
channel with colored noise can be used as separate, unrelated subchannels. 
I.1.1 CHANNEL CAITY 
This section uses the SCS model to determine the capacity of the channel 
under the assumption of Gaussian noise. Then the individual scalar 
sub-channels of the SCS model have Gaussian noise. Since the capacity of 
any scalar channel is bounded below by the channel that has Gaussian noise 
with the same mean-square value as noise that is actually present (e.g. 
see the text entitled "Transmission of Information" by R. M. Fano, John 
Wiley, 1973), a lower bound on the capacity of the channel is obtained by 
making the Gaussian assumption. 
It is shown that the channel capacity is achieved when only a finite number 
of the optimal transmitted {.phi..sub.n } are employed in modulating data, 
so that the optimal transmitted signal space is finite-dimensional. Thus, 
in a very broad class of communications channels are, effectively, 
finite-dimensional channels. This simplifies filtering and signal 
processing generally. This detailed description shows how to implement 
this optimal and simplest form of signal processing. 
With the assumption of Gaussian noise for the sub-channel, the sub-channel 
capacity for the m.sup.th channel may be written in the form 
##EQU11## 
where, from equation (I.1.26), the noise on the m.sup.th sub-channel is 
EQU .sigma..sup.2 .sub.m .dbd.&lt;(.theta..sub.m, n).sup.2 &gt; (2) 
and where the information power has been written in the form A.sup.2.sub.m 
for convenience. 
Since the noise components on different sub-channels are uncorrelated, the 
capacity of the overall channel, C, can be written as the sum of the 
capacities of the individual sub-channels: 
##EQU12## 
To maximize the capacity of the channel, the power on each of the 
subchannels, A.sup.2.sub.m, must be chosen appropriately. Of course, the 
sum of the sub-channel powers must be constrained during the optimization. 
To optimize, first assume that there are only a finite number of 
sub-channels that are used, say M. Then write (3) in the form 
##EQU13## 
The values of the A.sup.2.sub.m 's cannot effect the second sum on the 
right in equation (4), so the A.sup.2.sub.m 's should be chosen to 
maximize the first summation in order to maximize the channel capacity. By 
the Schwartz inequality, the first summation is maximized if all of the 
logarithms are equal. This can only be achieved if 
EQU A.sup.2.sub.m +.sigma..sup.2.sub.m .dbd.k, (5) 
where k is some constant. To determine k, sum both sides of equation (5) 
over m to get 
##EQU14## 
If the transmitted power is constrained to be A.sup.2, i.e., 
##EQU15## 
in equation (6), the following obtains: 
##EQU16## 
With this, obviously positive, value for k, equation (5) yields 
##EQU17## 
In particular, this equation must be true for m.dbd.M, so that 
##EQU18## 
From equation (I.1.26) it is known that the mean square noise on the 
subchannels is inversely proportional to the eigenvalues of a strictly 
positive definite compact operator. These eigenvalues can be arranged in 
descending order in a sequence that has zero as a limit point even though 
zero is not an eigenvalue, because of the properties of compact operators. 
Thus, the sequence {.sigma..sup.2.sub.m } is a non-decreasing, unbounded 
sequence. Consequently, the sequence 
##EQU19## 
is a monotonically non-increasing unbounded sequence. As a result, for all 
M exceeding some minimal value, 
##EQU20## 
so that equation (10) dictates that A.sup.2.sub.M should be negative for 
all M exceeding the minimum value, which cannot follow. Consequently, the 
sequence {A.sup.2.sub.m } of optimal transmitted signal powers on the SCS 
sub-channels must terminate at the largest M for which 
##EQU21## 
and the capacity of the channel is then given by equation (4). 
I.2 NOISE CANCELLATION 
This section generalizes the noise cancellation theory advanced by Widrow, 
et al., in the article entitled "Adaptive noise canceling: Principles and 
application," as published in Proc. IEEE, vol. 63, no. 12 pp. 1692-1716, 
December, 1975, to a Hilbert space setting. The essential idea in noise 
cancellation is that the space that a noise vector lies in may be divided 
into two orthogonal subspaces and the correlation of the components of the 
noise in these subspaces can be exploited to reduce the magnitude of the 
noise in one of the subspaces. This is tantamount to decorrelating the 
components of the noise in the two subspaces. 
Suppose that a (random) noise vector, n, lies in a Hilbert space, H, that 
is comprised of two orthogonal subspaces, H.sub.1 and H.sub.2. The noise 
is thus a linear combination of components in each subspace: 
EQU n.dbd.n.sub.1 +n.sub.2 .dbd.P.sub.1 n+P.sub.2 n, (1), 
where P.sub.k is the orthogonal projection of H onto H.sub.k, i.e., 
EQU P.sub.k H.dbd.H.sub.k, (2) 
and 
EQU P.sub.j P.sub.k .dbd.P.sub.k .delta..sub.ij. (3) 
The correlation of the components of the noise in the two subspaces is 
expressed by covariance operators that are defined by 
EQU &lt;(P.sub.k n.sub.i P.sub.j n)&gt;.dbd.TR[&lt;P.sub.j n(P.sub.k n).sup.T 
&gt;].dbd.TR[P.sub.j &lt;nn.sup.T &gt;P.sub.k ].dbd.TRN.sub.jk (4) 
where TR indicates the trace and 
EQU N.sub.jk .dbd.P.sub.j NP.sub.k (5) 
is the covariance operator for the noise components in the subspaces. Now, 
suppose that it is desired to minimize the noise in H by exploiting the 
correlation between n.sub.1 and n.sub.2. To do that, form a linear 
combination of n.sub.1 and a vector (called Ln.sub.2) in H.sub.1 that is 
linearly related to n.sub.2. Thus, form a new vector n.sub.0 in H.sub.1 of 
the form 
EQU n.sub.0 .dbd.n.sub.1 +Ln.sub.2 (6) 
where L is a linear operator with domain H.sub.2 and range H.sub.1, i. e., 
EQU L.dbd.P.sub.1 LP.sub.2. (7) 
Of course, L should be chosen to minimize the mean-square value of n.sub.0 
: 
EQU &lt;(n.sub.0,n.sub.0)&gt;.dbd.TR[n.sub.0 n.sub.0.sup.T ].dbd.TR[(n.sub.1 
+Ln.sub.2)(n.sub.1 +Ln.sub.2) .sup.T ]. (8) 
By a standard variational procedure, the optimum operator, call it L.sub.0, 
is given by 
EQU L.sub.0 .dbd.-N.sub.12 N.sub.22.sup.-1 .dbd.-P.sub.1 N.sub.12 
N.sub.22.sup.-1 P.sub.2. (9) 
A filter, F, that operates on n and produces the minimal noise (n.sub.0) in 
H.sub.1 is given by 
EQU F.dbd.P.sub.1 +L.sub.0 .dbd.P.sub.1 (I-N.sub.12 N.sub.22.sup.-1 P.sub.2). 
(10) 
To see that n.sub.0 and n.sub.2 are uncorrelated, use n.sub.0 .dbd.Fn and 
then, 
EQU &lt;(n.sub.0, n.sub.2)&gt;.dbd.&lt;TR[n.sub.0.sup.T n.sub.2 ].dbd.TR[&lt;n.sub.2 
n.sub.0.sup.T &gt;]. (11) 
But, 
EQU TR[&lt;n.sub.2 n.sub.0.sup.T &gt;].dbd.TR[&lt;n.sub.2 n.sup.T (P.sub.1 
-N.sub.22.sup.-1 N.sub.21)&gt;].dbd.TR[N.sub.21 -N.sub.21 ].dbd.0. (12) 
I.3 NOISE CANCELLATION IN STRUCTURED CHANNEL SIGNALING 
This section demonstrates that SCS incorporates optimum noise cancellation 
as described in the preceding section. In fact, noise cancellation is a 
limiting case of SCS. The starting point is the observation that the space 
of channel outputs is a (possibly) proper subspace of the receiver input 
space. First, a formula is developed for the operator that projects the 
receiver input space onto the channel output space. To do this, first 
write the channel outputs in the form 
EQU S.sub.0 .dbd.CS (1) 
where S.sub.0 is a generic channel output that is caused by input signal S. 
To develop the projection, first operate on both sides of (1) with C.sup.T 
to obtain 
EQU C.sup.T S.sub.0 .dbd.C.sup.T CS. (2) 
Since C does not annihilate any channel inputs, C.sup.T C is 
positive-definite and consequently may be inverted in equation (2) to 
obtain 
EQU (C.sup.T C).sup.-1 C.sup.T S.sub.0 .dbd.S. (3) 
Now, operate on both sides of equation (3) with C to obtain 
EQU C(C.sup.T C).sup.-1 C.sup.T S.sub.0 .dbd.S.sub.0. (4) 
Since equation (4) is true for all channel outputs, the operator on the 
left must be an identity operator on the space of channel outputs. Since 
the domain of C.sup.T is the receiver input space, it follows that the 
operator 
EQU P.sub.1 .dbd.C(C.sup.T C).sup.-1 C.sup.T (5) 
projects the receiver input space onto the channel output space, which is 
now designated by H.sub.1. That P.sub.1 is an orthogonal projection 
follows directly from the fact that it is obviously symmetric and 
idempotent. 
Equation (5) tells how to construct the operator that projects any receiver 
input onto the space of channel outputs, which is called H.sub.1, from a 
knowledge of the channel characteristics. To see what the implications of 
this construction are, consider equation (I.1.19) for the optimum receiver 
vectors. The receiver eigenvectors, including those for .lambda..sub.i 
.dbd.0, span the receiver input space, of which H.sub.1 is a proper 
subspace. The eigenvectors corresponding to .lambda..sub.i .dbd.0 lie in 
the orthogonal complement to H.sub.1, which will be called H.sub.2. To see 
this, generically index with a zero all eigenvectors and eigenvalues 
corresponding to a zero eigenvalue. Then, equation (I.1.19) becomes, for 
eigenvectors corresponding to zero eigenvalue, 
EQU CC.sup.T .theta..sub.0 .dbd.0. (6) 
Operate on both sides of equation (6) with C(C.sup.T C).sup.-2 C.sup.T to 
obtain 
EQU P.sub.1 .theta..sub.0 .dbd.0. (7) 
Thus, those .theta..sub.i that correspond to zero eigenvalues lie 
completely in H.sub.2. For the non-zero .lambda..sub.i, operate on both 
sides of equation (I.1.19) with P.sub.1 to obtain 
EQU CC.sup.T .theta..sub.i .dbd..lambda..sub.i P.sub.1 N.theta..sub.i. (8) 
Comparison of equations (8) and (I.1.19) now yields 
EQU P.sub.1 N.theta..sub.i .dbd.N.theta..sub.i (9) 
which equation implies, on comparison with (I.1.19), 
EQU P.sub.2 N.theta..sub.i .dbd.0, (10) 
where another orthogonal projection has been defined by 
EQU P.sub.2 .dbd.I-P.sub.1, (11) 
where I is the identity operator on the receiver input space and P.sub.2 is 
projection onto H.sub.2. Use equation (11) to express .theta..sub.1 in 
equation (10), with the results 
EQU P.sub.2 N(P.sub.1 .theta..sub.i)+P.sub.2 N(P.sub.2 .theta..sub.i).dbd.0 
(12) 
or, 
EQU N.sub.21 P.sub.1 .theta..sub.i .dbd.-N.sub.22 P.sub.2 .theta..sub.i, (13) 
so that 
EQU P.sub.2 .theta..sub.i .dbd.-N.sub.22.sup.-1 N.sub.21 P.sub.1 .theta..sub.i. 
(14) 
Thus, if a noise vector, n, is passed through a matched filter, 
.theta..sub.i, the sampled output of the filter is 
EQU (.theta..sub.i, n).dbd.(P.sub.1 .theta..sub.i, n)+(P.sub.2 .theta..sub.i, 
n). (15) 
Employ equation (14) in the second term on the right in equation (15) with 
the results 
EQU (.theta..sub.i,n)+(P.sub.1 .theta..sub.i -N.sub.22.sup.-1 N.sub.21 P.sub.1 
.theta..sub.i, n).dbd.(F.sup.T P.sub.1 .theta..sub.i,n).dbd.(P.sub.1 
.theta..sub.i,Fn).dbd.(.theta..sub.i, FN) (16) 
where F is given by equation (I.2.10) and the definition of the adjoint of 
an operator has been used. This equation is similar to the optimum noise 
filter of section (I.2), but only one of the eigenvectors is involved. 
Now, recall from sections (I.1) and (I.1.1) that the optimum linear 
receiver that achieves the maximum capacity for a channel is of the 
general form 
##EQU22## 
That is, for any signal x in the receiver's input space, the receiver's 
output, s.sub.x, which lies in the channel's input space, is of the 
general form 
##EQU23## 
Using equation (16) in equation (18) yields 
##EQU24## 
so that the optimum SCS receiver is comprised of the optimum noise 
cancellation filter, F, followed by a bank of matched filters with sampled 
outputs that drive filters that have responses equal to the optimal basis 
vectors of the input space. 
It should now be noted that as the channel operator approaches the 
characteristics of a projection operator, i.e., as C.fwdarw.P.sub.1, as it 
might through equalization, that the channel outputs approach the channel 
inputs from equation (I.1.20), i.e., P.sub.1 .theta..sub.i 
.fwdarw..psi..sub.i. Consequently, as C.fwdarw.P.sub.1, equation (19) 
becomes 
##EQU25## 
so that the optimum SCS receiver becomes an optimum noise canceler. Thus, 
an optimum noise canceler has been determined in the form of a finite bank 
of matched filters with sampled outputs that drive filters with responses 
that are the optimal SCS input basis signals. 
A circuit implementation of a generalized noise canceler 600 in accordance 
with equation (18) is shown in FIG. 6 wherein the signal plus noise (s and 
n--reference numeral (601)) serve as the input to a parallel arrangement 
of M cascade networks 610, . . ., 620. Each cascade network (e.g., cascade 
610) is similarly arranged and includes an inner product filter circuit 
(e.g. element 611) determined in correspondence to a given .theta..sub.k 
(e.g., .theta..sub.1) in series with a filter (e.g. element 612) having a 
filter characteristic determined from a corresponding .psi..sub.K (e.g., 
.psi..sub.1). The outputs of the cascade networks 610, . . . , 620 are 
summed in summer 631 to produce output signal s.sub.x plus n.sub.0 
(reference numeral 602). The .theta..sub.k 's and .psi..sub.k 's are 
determined from the solution of the eigenvalue relation set forth in 
equations I1.21 and I.1.22, respectively, given the channel operator C and 
the noise operator N. An inner product is defined for the given Hilbert 
space; generally for realizable circuits the output of each inner product 
circuit (e.g., element 611) is a scalar value generated periodically. Each 
scalar value serves as the input signal to the corresponding filter having 
an impulse response given by the .psi..sub.k 's. 
To illustrate one embodiment of an inner product circuit depicted FIG. 6, 
reference is now made to FIG. 7 wherein, for example, inner product 
circuit 611 of FIG. 6 is composed of filter 711, having an impulse 
response given by .theta..sub.1, followed by a sampler 713 sampling every 
.tau. seconds. In the specific embodiment of FIG. 7, noise canceler 700 is 
composed of a plurality of cascade networks 710, . . . , 720 for which 
cascade network 710 is exemplary; network 710 includes: matched filter 711 
having an impulse response given by .theta..sub.1 ; sampler 713 sampling 
every .tau. seconds to produce a sampled output value; and matched filter 
712 having an impulse response given by .psi..sub.1 for processing the 
sampled output value. 
It should be noted that the number of cascade networks in either FIG. 6 or 
FIG. 7 is finite even if the channel is capable of transmitting signals 
from an infinite dimensional space. When the channel is a projector this 
is an important result. It says that the optimum noise canceler will 
select a finite dimensional subspace of the projector range that is the 
best subspace for the representation of information in the given noise 
environment. The subspace is best in the sense that it permits the 
representation of the maximal amount of information in a signal in the 
sense of information theory. The optimal noise canceler thus not only 
cancels noise, it also chooses the optimum subspace, including the 
dimensionality of the subspace, for the representation of information. 
Although the foregoing detailed description has focused on time-invariant 
channels, it is also within the contemplation of one of ordinary skill in 
the art that the foregoing development may be readily modified to 
encompass time-varying channels. In addition, rather than just treating 
time-domain signal spaces, one of ordinary skill in the art may readily 
contemplate adapting the theoretical development to further encompass 
spatial signal spaces. 
It is to be understood that the above-described embodiments are simply 
illustrative of the principles in accordance with the present invention. 
Other embodiments may be readily devised by those skilled in the art which 
may embody the principles in spirit and scope. Thus, it is to be further 
understood that the circuit arrangements described herein are not limited 
to the specific forms shown by way of illustration, but may assume other 
embodiments limited only by the scope of the appended claims.