US5630208A — Adaptive multipath equalization

Claim 1

Independent Claim 266 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; selecting a grid of values {τ g } g (g=1, 2, . . . , G) of the time variable t lying in the interval -T c ≦t≦T c ; selecting a set of sampling times (τ' j ) j (j=1, . . . , J; J a selected integer≧2) for a correlation function R(τ), and determining the correlation function values R(τ' j )=x j , where the correlation function R(τ) is defined by ##EQU10## and T is a selected time satisfying T>>T c ; choosing complex numbers b g so that a sum ##EQU11## is minimized, where q is a selected positive number; arranging the magnitudes |b g | of the complex numbers b g in a sequence {|b gk |} gk in decreasing order of magnitude so that |b g1 |≧|b g2 |≧|b g3 |≧ . . . , and choosing H members of this sequence, where H is a selected positive integer; designating the time t=τ g1 as the time of arrival of the direct signal and the times t=τ gk (k=2, . . . , H) as the times of arrival of multipath signals; and interpreting a difference signal ##EQU12## as a signal from which at least one multipath signal has been removed.

Claim 2

Independent Claim 266 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; selecting a grid of values {τ g } g (g=1, 2, . . . , G) of the time variable t lying in the interval -T c ≦t≦T c ; selecting a set of sampling times (τ' j ) j (j=1, . . . , J; J a selected integer≧2) for a correlation function R(τ), and determining the correlation function values R(τ' j )=x j , where the correlation function R(τ) is defined by ##EQU10## and T is a selected time satisfying T>>T c ; choosing complex numbers b g so that a sum ##EQU11## is minimized, where q is a selected positive number; arranging the magnitudes |b g | of the complex numbers b g in a sequence {|b gk |} gk in decreasing order of magnitude so that |b g1 |≧|b g2 |≧|b g3 |≧ . . . , and choosing H members of this sequence, where H is a selected positive integer; designating the time t=τ g1 as the time of arrival of the direct signal and the times t=τ gk (k=2, . . . , H) as the times of arrival of multipath signals; and interpreting a difference signal ##EQU12## as a signal from which at least one multipath signal has been removed.

Claim 3

Dependent Claim 43 words

The method of claim 1, wherein said step of choosing said H members of said sequence {|b gk |} gk comprises the steps of: selecting a positive integer H; and choosing the largest H members of said sequence {|b gk |} gk .

Claim 4

Dependent Claim 48 words

The method of claim 1, wherein said step of choosing said H members of said sequence {|b gk |} gk comprises the step of choosing all members of said sequence {|b gk |} gk that satisfy the relation |b gk |≧δ, where δ is a selected positive number.

Claim 5

Dependent Claim 97 words

The method of claim 1, wherein said step of choosing said H members of said sequence {|b gk |} gk comprises the steps of: forming said sequence of values {Δb gk } gk , where Δb gk =|b gk |-|b g (k+1) |; determining a largest value μ(b) for which Δb gk ≧μ(b) for all numbers gk in the range 1≦gk≦gK for some positive integer K; and selecting said integer J=K and selecting said sequence {Δb gk } gk (k=1, . . . , K) as said J members of said sequence {Δb gk } gk .

Claim 6

Independent Claim 339 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU13## where T is a selected time satisfying T>>T c ; determining a first pair of distinct time values τ=τ 1 ,w and τ=τ 2 ,w which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ w ,sp =τ 2 ,w -τ 1 ,w in the range 0.4 T c ≦Δτ w ,sp ≦T c , and computing τ w ,avg =(τ 1 ,w +τ 2 ,w)/2; determining a second pair of distinct time values τ=τ 1 ,n and τ=τ 2 ,n which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ n ,sp =τ 2 ,n -τ 1 ,n in the range 0.01 T c ≦Δτ n ,sp ≦0.2 T c , and computing τ w ,avg =(τ 1 ,w +τ 2 ,w)/2, where Δτ n =τ n ,avg -τ peak and Δτ w =τ w ,avg -τ peak and τ peak is the value of τ, to be determined, for the peak of the correlation function magnitude; providing a collection {(Δτ w ,k,sim -Δτ n ,k,sim, Δτ n ,k,sim)} k of a selected number K of coordinate pairs of simulated values of the variables Δτ w -Δτ n and Δτ n , and determining real parameters a and b that provide an optimal approximation to this collection of coordinate pairs by the equation Δτ n ,sim =a(Δτ w ,sim -Δτ n ,sim)+b; determining the time value τ peak of the correlation function magnitude by the relation τ peak =τ n ,avg -a(Δτ w -Δτ n )-b; and interpreting the time value τ peak as the time a direct signal arrives at the receiver.

Claim 7

Independent Claim 339 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU13## where T is a selected time satisfying T>>T c ; determining a first pair of distinct time values τ=τ 1 ,w and τ=τ 2 ,w which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ w ,sp =τ 2 ,w -τ 1 ,w in the range 0.4 T c ≦Δτ w ,sp ≦T c , and computing τ w ,avg =(τ 1 ,w +τ 2 ,w)/2; determining a second pair of distinct time values τ=τ 1 ,n and τ=τ 2 ,n which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ n ,sp =τ 2 ,n -τ 1 ,n in the range 0.01 T c ≦Δτ n ,sp ≦0.2 T c , and computing τ w ,avg =(τ 1 ,w +τ 2 ,w)/2, where Δτ n =τ n ,avg -τ peak and Δτ w =τ w ,avg -τ peak and τ peak is the value of τ, to be determined, for the peak of the correlation function magnitude; providing a collection {(Δτ w ,k,sim -Δτ n ,k,sim, Δτ n ,k,sim)} k of a selected number K of coordinate pairs of simulated values of the variables Δτ w -Δτ n and Δτ n , and determining real parameters a and b that provide an optimal approximation to this collection of coordinate pairs by the equation Δτ n ,sim =a(Δτ w ,sim -Δτ n ,sim)+b; determining the time value τ peak of the correlation function magnitude by the relation τ peak =τ n ,avg -a(Δτ w -Δτ n )-b; and interpreting the time value τ peak as the time a direct signal arrives at the receiver.

Claim 8

Dependent Claim 41 words

The method of claim 5, wherein said step of determining said optimal approximation to said collection of coordinate pairs comprises the step of determining said parameters a and b that minimize a sum ##EQU14## where q is a selected positive number.

Claim 9

Dependent Claim 200 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU15## where T is a selected time satisfying T>>T c ; (2) determines a sequence of N time values (N a selected integer≧2) {τ m } m , each of which is estimated to correspond to a time of arrival of a direct signal or of a multipath signal at the receiver; (3) selects a sequence of sampling times {τ' j } j (j=1, . . . , J; J a selected integer≧2) for the correlation function R(τ) and determines the correlation function values R(τ' j )=x j ; (4) determines complex numbers b m so that a sum ##EQU16## is minimized, where q is a selected positive number; and (5) interprets a difference signal ##EQU17## as a signal from which at least one multipath signal has been removed.

Claim 10

Dependent Claim 200 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU15## where T is a selected time satisfying T>>T c ; (2) determines a sequence of N time values (N a selected integer≧2) {τ m } m , each of which is estimated to correspond to a time of arrival of a direct signal or of a multipath signal at the receiver; (3) selects a sequence of sampling times {τ' j } j (j=1, . . . , J; J a selected integer≧2) for the correlation function R(τ) and determines the correlation function values R(τ' j )=x j ; (4) determines complex numbers b m so that a sum ##EQU16## is minimized, where q is a selected positive number; and (5) interprets a difference signal ##EQU17## as a signal from which at least one multipath signal has been removed.

Claim 11

Dependent Claim 68 words

The apparatus of claim 7, wherein said signal processor determines said sequence of time values {τ m } m by: forming a logarithm of a Fourier transform, log{Φ(s(t))} of said received signal; forming an inverse Fourier transform X(t)=Φ -1 {log {Φ(s(t))}}; and identifying a time value t=τ 1 for which the magnitude |X(t)| has a peak, as a time at which a multipath signal arrives at said receiver.

Claim 12

Dependent Claim 56 words

The apparatus of claim 7, wherein said signal processor determines said sequence of time values {τ m } m by (i) identifying at least two slope transition points for said correlation function R(τ) and (ii) for each slope transition point identified, identifying at least one of said time values τ m with this slope transition point.

Claim 13

Dependent Claim 107 words

The apparatus of claim 9, wherein said signal processor identifies said slope transition point by: forming an error function e(τ;τ 0 ;τ 1 ;q) defined by e(τ-τ.sub.0 ;τ.sub.d ;r)=|R(τ-τ.sub.0 -τ.sub.d)|.sup.r -|R(τ-τ.sub.0 +τ.sub.d)|.sup.r, where τ d is a time value that ranges over a selected sequence of distinct time values {t' u } u in the range -T c ≦t' u ≦T c , and r is a selected positive number; determining at least one value τ=τ' for which the error function e(τ-τ 0 ;τ d ;r) has a slope transition point; and interpreting the time value τ' as a slope transition point for the correlation function R(τ).

Claim 14

Dependent Claim 105 words

The apparatus of claim 9, wherein said signal processor identifies said slope transition point by: determining the one-sided derivatives R'(τ) + =(dR/dτ).sub.τ+0 and R'(τ) - =(dR/dτ).sub.τ-0 of said correlation function R(τ) at a selected sequence of points τ=τ k (k=1, 2, . . . ) in an interval -T c ≦τ k ≦T c ; and when two one-sided derivatives R'(τ) + and R'(τ) - satisfy the relation |R'(τ k ) + -R'(τ k ) - |≧δ for at least two of the points τ k in this sequence, where δ is a selected positive constant, designating these points δ k as slope transition points.

Claim 15

Dependent Claim 272 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU18## where T is a selected time satisfying T>>T c ; (2) selects a sequence of values {τ g } g of the time variable τ lying in the interval -T c ≦τ≦T c , (3) selects a sequence of sampling times {τ' j } j (j=1, . . . , J; J a selected integer≧2) for the correlation function R(τ) and determines the correlation function values R(τ' j )=x j ; (4) determines complex numbers b g so that a sum ##EQU19## is minimized, where q is a selected positive number; (5) arranges the magnitudes |b g | of the complex numbers b g in a sequence {|b gk |} in decreasing order of magnitude so that |b g1 |≧|b g2 |≧|b g3 |≧ . . . , and chooses H members of this sequence, where H is a selected positive integer; (6) designates the time t=τ g1 as the time of arrival of the direct signal and at least one of the times t=τ gk (k=2, . . . , H) as the times of arrival of multipath signals; and (7) interprets a difference signal ##EQU20## as a signal from which at least one multipath signal has been removed.

Claim 16

Dependent Claim 272 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU18## where T is a selected time satisfying T>>T c ; (2) selects a sequence of values {τ g } g of the time variable τ lying in the interval -T c ≦τ≦T c , (3) selects a sequence of sampling times {τ' j } j (j=1, . . . , J; J a selected integer≧2) for the correlation function R(τ) and determines the correlation function values R(τ' j )=x j ; (4) determines complex numbers b g so that a sum ##EQU19## is minimized, where q is a selected positive number; (5) arranges the magnitudes |b g | of the complex numbers b g in a sequence {|b gk |} in decreasing order of magnitude so that |b g1 |≧|b g2 |≧|b g3 |≧ . . . , and chooses H members of this sequence, where H is a selected positive integer; (6) designates the time t=τ g1 as the time of arrival of the direct signal and at least one of the times t=τ gk (k=2, . . . , H) as the times of arrival of multipath signals; and (7) interprets a difference signal ##EQU20## as a signal from which at least one multipath signal has been removed.

Claim 17

Dependent Claim 42 words

The apparatus of claim 12, wherein said signal processor chooses said H members of said sequence {|b gk |} gk by: (1) selecting a positive integer H; and (2) choosing the largest H members of said sequence {|b gk |} gk .

Claim 18

Dependent Claim 45 words

The apparatus of claim 12, wherein said signal processor chooses said H members of said sequence {|b gk |} gk by choosing all members of said sequence {|b gk |} gk that satisfy the relation |b gk |≧δ, where δ is a selected positive number.

Claim 19

Dependent Claim 94 words

The apparatus of claim 12, wherein said signal processor chooses said H members of said sequence {|b gk |} gk by: forming a sequence of values {Δb gk } gk , where Δb gk =|b gk |-|b g (k+1) |; determining a largest value μ(b) for which Δb gk ≧μ(b) for all numbers gk in the range 1≦gk≦gK for some positive integer K; and selecting said integer H=K and selecting the sequence {Δb gk } gk (k=1, . . . , K) as said H members of said sequence {Δb gk } gk .

Claim 20

Dependent Claim 349 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU21## where T is a selected time satisfying T>>T c ; (2) determines a first pair of distinct time values τ=τ 1 ,w and τ=τ 2 ,w which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ w ,sp =τ 2 ,w -τ 1 ,w in the range 0.4 T c ≦Δτ w ,sp ≦T c , and computes τ w ,avg =(τ 1 ,w +τ 2 ,w)/2; (3) determines a second pair of distinct time values τ=τ 1 ,n and τ=τ 2 ,n which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ n ,sp =τ 2 ,n -τ 1 ,n in the range 0.01 T c ≦Δτ n ,sp ≦0.2 T c , and computes τ w ,avg =(τ 1 ,w +τ 2 ,w)/2, where Δτ n =τ n ,avg -τ peak and Δτ w =τ w ,avg -τ peak and τ peak is the value of τ, to be determined, for the peak of the correlation function magnitude; (4) provides a collection {(Δτ w ,k,sim -Δτ n ,k,sim, Δτ n ,k,sim)} k of a selected number K of coordinate pairs of simulated values of the variables Δτ w -Δτ n and Δτ n , and determines real parameters a and b that provide an optimal approximation to this collection of coordinate pairs by the equation Δτ n ,sim =a(Δτ w ,sim -Δτ n ,sim)+b; (5) determines the time value τ peak of the correlation function magnitude by the relation τ peak =τ n ,avg -a(Δτ w -Δτ n )-b; and (6) interprets the time value τ peak as the time a direct signal arrives at the receiver.

Claim 21

Dependent Claim 349 words

Apparatus for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the apparatus comprising: a receiver that receives a signal s(t) that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; a signal processor, connected to the receiver, that: (1) forms a correlation function R(τ) defined by ##EQU21## where T is a selected time satisfying T>>T c ; (2) determines a first pair of distinct time values τ=τ 1 ,w and τ=τ 2 ,w which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ w ,sp =τ 2 ,w -τ 1 ,w in the range 0.4 T c ≦Δτ w ,sp ≦T c , and computes τ w ,avg =(τ 1 ,w +τ 2 ,w)/2; (3) determines a second pair of distinct time values τ=τ 1 ,n and τ=τ 2 ,n which produce equal values of the correlation function magnitude |R(t)|, using a difference Δτ n ,sp =τ 2 ,n -τ 1 ,n in the range 0.01 T c ≦Δτ n ,sp ≦0.2 T c , and computes τ w ,avg =(τ 1 ,w +τ 2 ,w)/2, where Δτ n =τ n ,avg -τ peak and Δτ w =τ w ,avg -τ peak and τ peak is the value of τ, to be determined, for the peak of the correlation function magnitude; (4) provides a collection {(Δτ w ,k,sim -Δτ n ,k,sim, Δτ n ,k,sim)} k of a selected number K of coordinate pairs of simulated values of the variables Δτ w -Δτ n and Δτ n , and determines real parameters a and b that provide an optimal approximation to this collection of coordinate pairs by the equation Δτ n ,sim =a(Δτ w ,sim -Δτ n ,sim)+b; (5) determines the time value τ peak of the correlation function magnitude by the relation τ peak =τ n ,avg -a(Δτ w -Δτ n )-b; and (6) interprets the time value τ peak as the time a direct signal arrives at the receiver.

Claim 22

Dependent Claim 38 words

The method of claim 16, wherein said signal processor determines said optimal approximation to said collection of coordinate pairs by determining said parameters a and b that minimize a sum ##EQU22## where q is a selected positive number.

Claim 23

Independent Claim 254 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU23## where T is a selected time satisfying T>>T c ; determining a sequence of N time values (N a selected integer≧2) {τ m } m , each of which corresponds to a time of arrival, lying in the interval -T c ≦t≦T c , of a direct signal or of a multipath signal at the receiver, wherein the time values τ m are determined by (i) forming the logarithm of a Fourier transform log{Φ(s(t))} of the received signal s(t), (ii) forming an inverse Fourier transform X(t)=Φ -1 {log {Φ(s(t))}}, and (iii) identifying at least one value t=τ m (m=1, 2, . . . , N) for which the magnitude |X(t)| has a peak; selecting a set of sampling times {τ' k } k (k=1, . . . , K; K a selected integer≧2) for the correlation function R(τ), and determining the correlation function values R(τ' k )=x k ; choosing complex numbers b m so that a sum ##EQU24## is minimized, where q is a selected positive number; and interpreting a difference signal ##EQU25## as a signal from which at least one multipath signal has been removed.

Claim 24

Independent Claim 254 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU23## where T is a selected time satisfying T>>T c ; determining a sequence of N time values (N a selected integer≧2) {τ m } m , each of which corresponds to a time of arrival, lying in the interval -T c ≦t≦T c , of a direct signal or of a multipath signal at the receiver, wherein the time values τ m are determined by (i) forming the logarithm of a Fourier transform log{Φ(s(t))} of the received signal s(t), (ii) forming an inverse Fourier transform X(t)=Φ -1 {log {Φ(s(t))}}, and (iii) identifying at least one value t=τ m (m=1, 2, . . . , N) for which the magnitude |X(t)| has a peak; selecting a set of sampling times {τ' k } k (k=1, . . . , K; K a selected integer≧2) for the correlation function R(τ), and determining the correlation function values R(τ' k )=x k ; choosing complex numbers b m so that a sum ##EQU24## is minimized, where q is a selected positive number; and interpreting a difference signal ##EQU25## as a signal from which at least one multipath signal has been removed.

Claim 25

Dependent Claim 34 words

The method of claim 18, further comprising the step of identifying at least one of said time values τ m with a time of arrivel of a direct signal or of a multipath signal.

Claim 26

Independent Claim 241 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU26## where T is a selected time satisfying T>>T c ; determining a sequence of N time values (N a selected integer≧2) {τ m } m , each of which corresponds to a time of arrival, lying in the interval -T c ≦t≦T c , of a direct signal or of a multipath signal at the receiver, wherein the time values τ m are determined by (i) identifying at least two slope transition points for the correlation function magnitude |R(τ)|, and (ii) for each slope transition point identified, identifying at least one time value τ m with each slope transition point; selecting a set of sampling times {τ' k } k (k=1, . . . , K; K a selected integer≧2) for the correlation function R(τ), and determining the correlation function values R(τ' k )=x k ; choosing complex numbers b m so that a sum ##EQU27## is minimized, where q is a selected positive number; and interpreting a difference signal ##EQU28## as a signal from which at least one multipath signal has been removed.

Claim 27

Independent Claim 241 words

A method for compensating for the presence of one or more multipath signals in an electromagnetic signal received at a receiver, the method comprising the steps of: receiving a signal s(t) at a receiver that is the result of a transmitted message signal with frequency ω and having a shape signal p(t) with frequency f c =1/T c ; forming a correlation function R(τ) defined by ##EQU26## where T is a selected time satisfying T>>T c ; determining a sequence of N time values (N a selected integer≧2) {τ m } m , each of which corresponds to a time of arrival, lying in the interval -T c ≦t≦T c , of a direct signal or of a multipath signal at the receiver, wherein the time values τ m are determined by (i) identifying at least two slope transition points for the correlation function magnitude |R(τ)|, and (ii) for each slope transition point identified, identifying at least one time value τ m with each slope transition point; selecting a set of sampling times {τ' k } k (k=1, . . . , K; K a selected integer≧2) for the correlation function R(τ), and determining the correlation function values R(τ' k )=x k ; choosing complex numbers b m so that a sum ##EQU27## is minimized, where q is a selected positive number; and interpreting a difference signal ##EQU28## as a signal from which at least one multipath signal has been removed.

Claim 28

Dependent Claim 34 words

The method of claim 20, further comprising the step of identifying at least one of said time values τ m with a time of arrivel of a direct signal or of a multipath signal.

Claim 29

Dependent Claim 108 words

The method of claim 20, wherein said step of identifying said slope transition point comprises the steps of: forming an error function e(τ-τ 0 ;τ d ;r) defined by e(τ-τ.sub.0 ;τ.sub.d ;r)=|R(τ-τ.sub.0 -τ.sub.d)|.sup.r -|R(τ-τ.sub.0 +τ.sub.d)|.sup.r, where τ d is a time value that ranges over a selected sequence of distinct time values {t' u } u in the range 0≦t' u ≦T c , and r is a selected positive number; determining at least one value τ=τ' for which the error function e(τ-τ 0 ;τ d ;r) has a slope transition point; and interpreting the time value τ' as a slope transition point for said correlation function R(τ).

Claim 30

Dependent Claim 112 words

The method of claim 20, wherein said step of identifying said slope transition point comprises the steps of: determining the one-sided derivatives R'(τ) + =(dR/dτ).sub.τ+0 and R'(τ) - =(dR/dτ).sub.τ-0 of said correlation function R(τ) at a selected sequence of points τ=τ k (k=1, 2, . . . ) in an interval -T c ≦τ k ≦T c ; and when two one-sided derivatives R'(τ k ) + and R'(τ k ) - satisfy the relation |R'(τ k ) + -R'(τ k ) - |≧δ for at least two of the points τ k in this sequence, where δ is a selected positive constant, designating these points τ k as slope transition points.

Annuity Stage	Window Opens	Payment Deadline	Surcharge Period Until	Estimated Fee	Status
3.5 Years (Year 4)	2000-05-12	2000-11-11	2001-05-13	$2000.00	Window Closed / Lapsed
7.5 Years (Year 8)	2004-05-12	2004-11-11	2005-05-13	$3760.00	Window Closed / Lapsed
11.5 Years (Year 12)	2008-05-12	2008-11-11	2009-05-13	$7700.00	Window Closed / Lapsed

Adaptive multipath equalization

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