Method and apparatus for implementing projections in singal processing applications

1. A method for generating a projection of a received signal y, said received signal comprising H, a signal of a source of interest; S, the signals of all other sources and multi-path versions of the source of interest and composed of vectors s 1 , s 2 , s 3 , . . . , s p ; and noise (n); the method comprising the steps of:

determining a basis matrix U composed of basis vectors u 1 , u 2 , . . . u p ;

storing elements of said basis matrix U;

generating a diagonal matrix from stored said elements of the basis matrix U;

generating one or more scalars from the diagonal matrix and from the basis vectors of the basis matrix U; and

applying the one or more scalars to the received signal to project the signal of the source of interest.

2. The method recited in claim 1 , further comprising the step of computing the basis vectors, wherein computing the basis vectors comprises the steps of:

A. assigning s 1 as a first vector of basis matrix U;

B. decomposing s 2 into a component which is in said basis matrix U and a component that is not (u 2 ); and

C. redefining the basis matrix U to incorporate basis vector u 2 .

3. The method recited in claim 2 , wherein said step of computing the basis vectors further comprises the steps of:

repeating steps B and C for each vector of S.

4. The method recited in claim 2 , wherein said step of computing the basis vectors further comprises the steps of:

comparing u i to a predetermined threshold and if u i is greater than said threshold, adding u i to the basis and repeating steps B and C for each vector of S, else ignoring the u i and continuing to repeat steps B and C.

5. The method recited in claim 2 , wherein said step of computing the basis vectors further comprises the steps of:

computing 1/σ i , where u i T u i =σ i ; and

storing u i and 1/σ i .

6. The method recited in claim 2 , wherein said step of computing the basis vectors further comprises the steps of:

computing

u

i

=

s

i

-

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

s

i

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

s

i

-

…

-

u

i

-

1

⁢

1

σ

i

-

1

⁢

u

i

-

1

T

⁢

s

i

;

storing u i and 1/σ i , wherein u i T u i =σ i ; and

repeating said computing and storing steps if u i is above a predetermine threshold, else ignoring this particular u i .

7. The method recited in claim 1 , wherein said step of applying comprises the step of calculating y perp with the following formula:

y

perp

=

y

-

U

⁡

[

1

σ

1

0

…

0

0

1

σ

2

…

0

⋮

⋮

…

⋮

0

0

…

1

σ

p

]

⁢

U

T

⁢

y

,

wherein y perp is a projected said received signal y, σ i =u i T u i , and u i is a basis vector of U.

8. The method recited in claim 7 , wherein said step of calculating y perp comprises the step of calculating y perp with the following formula:

y

perp

=

y

-

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

9. The method recited in claim 8 , further comprising the step of determining

y

s

=

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

,

wherein y s is a projected said signal of the source of interest.

10. A method for generating a projection of a received signal y, said received signal comprising H, a spread signal matrix of a source of interest; S, the spread signal matrix of all other sources of interest and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise n; the method comprising the steps of:

forming an orthogonal basis U of the matrix S, comprising:

A. assigning s i as a first basis vector u 1 ,

B. determining σ i , where u i T u i =σ i ,

C. storing u i ,

D. computing of inner products of the s i+1 and the u 1 through u i vectors,

E. multiplying said inner product with a respective scalar 1/σ i and thereby creating a first intermediate product,

F. scaling each respective basis vector u i by multiplying each respective first intermediate product with each respective basis vector u i ,

G. obtaining a vector sum from step F,

H. subtracting said vector sum from s i+1 to obtain the next basis vector u i+1 ,

I. comparing u i+1 to a predetermined value and if equal to or less than said value, discarding the u i+1 and going to step N,

J. storing u i+1 ,

K. determining an inner product of u T i+1 u i+1 ,

L. determining the reciprocal of step K which is 1/σ i+1 ,

M. storing 1/σ i+1 ,

N. incrementing i, and

O. conducting steps D through N until i=p, where p is the total number of said sources of interest;

generating a diagonal matrix from stored 1/σ i+1 values;

generating one or more scalars from the diagonal matrix and from the basis vectors of the orthogonal basis U; and

applying the one or more scalars to the received signal to project the source of interest.

11. The method recited in claim 10 , wherein said computing step (D) is conducted in series.

12. The method recited in claim 10 , wherein said computing step (D) is conducted in parallel.

13. The method recited in claim 10 , wherein said multiplying step (E) is conducted in series.

14. The method recited in claim 10 , wherein said multiplying step (E) is conducted in parallel.

15. The method recited in claim 10 , wherein said scaling step (F) is conducted in series.

16. The method recited in claim 10 , wherein said scaling step (F) is conducted in parallel.

17. The method recited in claim 10 , wherein said storing step (C) also stores σ i .

18. The method recited in claim 10 , wherein said storing step (C) also stores 1/σ i .

19. The method recited in claim 10 , wherein said inner product step (K) is conducted in series.

20. The method recited in claim 10 , wherein said inner product step (K) is conducted in parallel.

21. The method recited in claim 10 , further comprising the step of determining

y

s

=

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

,

wherein y s is a projected said source of interest.

22. A method for generating a projection of a received signal y, said received signal comprising H, a spread signal matrix of a source of interest; S, the spread signal matrix of all other sources of interest and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the method comprising the steps of:

forming an orthogonal basis U of the matrix S, comprising:

A. assigning s 1 as a first basis vector u 1 ,

B. determining σ i , where u i T u i =σ i ,

C. storing u i ,

D. computing of inner products of the s i+1 and the u 1 through u i vectors,

E. multiplying said inner product with a respective scalar 1/σ i and thereby creating a first intermediate product,

F. scaling each respective basis vector u i by multiplying each respective first intermediate product with each respective basis vector u i ,

G. serially subtracting said intermediate product from s i+1 ,

H. utilizing the result from step G and subtracting the next incoming value of

u

i

⁢

1

σ

i

⁢

u

i

T

⁢

s

i

+

1

until all the values are processed,

I. obtaining the next basis vector u i+1 from step H,

J. comparing u i+1 to a predetermined value and if equal to or less than said value, discarding u i+1 and going to step O,

K. storing u i+1 ,

L. determining an inner product of u T i+1 u i+1 ,

M. determining the reciprocal of step K which is 1/σ i+1 ,

N. storing 1/σ i+1 ,

O. incrementing i, and

P. conducting steps D through O until i=p, where p is the total umber of said sources of interest;

generating a diagonal matrix from stored 1/σ i+1 values;

generating one or more scalars from the diagonal matrix and from the basis vectors of the orthogonal basis U; and

applying the one or more scalars to the received signal to project the source of interest.

23. The method recited in claim 22 , wherein said computing step (D) is conducted in series.

24. The method recited in claim 22 , wherein said computing step (D) is conducted in parallel.

25. The method recited in claim 22 , wherein said multiplying step (E) is conducted in series.

26. The method recited in claim 22 , wherein said multiplying step (E) is conducted in parallel.

27. The method recited in claim 22 , wherein said scaling step (F) is conducted in series.

28. The method recited in claim 22 , wherein said scaling step (F) is conducted in parallel.

29. The method recited in claim 22 , wherein said storing step (C) also stores σ i .

30. The method recited in claim 22 , wherein said storing step (C) also stores 1/σ i .

31. The method recited in claim 22 , wherein said inner product step (L) is conducted in series.

32. The method recited in claim 22 , wherein said inner product step (L) is conducted in parallel.

33. The method recited in claim 22 , further comprising the step of determining

y

s

=

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

,

wherein y s is a projected said source of interest.

34. An apparatus for generating a projection of received signal y, said received signal comprising H, a signal of a source of interest; S, the signals of all other sources and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the apparatus comprising:

means for determining a basis vector U;

means for storing elements of said basis vector U; and

means for generating a diagonal matrix from stored said elements of the basis vector U;

means for generating one or more scalars from the diagonal matrix and from the basis vector U; and

means for applying the one or more scalars to the received signal to project the signal of the source of interest.

35. An apparatus for generating a projection of a received signal y, said received signal comprising H, a spread signal matrix of a source of interest; S, the spread signal matrix of all other sources of interest and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the apparatus comprising:

means for forming an orthogonal basis U of the matrix S, comprising:

A. means for assigning s 1 as a first basis vector u i ,

B. means for determining σ i , where u i T u i =σ i ,

C. means for storing u i ,

D. means for computing of inner products of the s i+1 and the u 1 through u i vectors,

E. means for multiplying said inner product with a respective scalar 1/σ i and thereby creating a first intermediate product,

F. means for scaling each respective basis vector u i by multiplying each respective first intermediate product with each respective basis vector u i ,

G. means for obtaining a vector sum from step F,

H. means for subtracting said vector sum from s i+1 to obtain the next basis vector u i+1 ,

I. means for comparing u i+1 to a predetermined value and if equal to or less than said value, discarding this u i+1 and going to step N,

J. means for storing u i+1 ,

K. means for determining an inner product of u T i+1 u i+1 ,

L. means for determining the reciprocal of step K which is 1/σ i+1 ,

M. means for storing 1/σ i+1 ,

N. means for incrementing i,

O. means for conducting steps D through N until i=p, where p is the total number of said sources of interest;

means for generating a diagonal matrix from stored 1/σ i+1 values;

means for generating one or more scalars from the diagonal matrix and from the basis vectors of the orthogonal basis U; and

means for applying the one or more scalars to the received signal to project the source of interest.

36. An apparatus for generating a projection from a received signal y, said received signal comprising H, a spread signal matrix of a source of interest; S, the spread signal matrix of all other sources of interest and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the apparatus comprising:

means for forming an orthogonal basis U of the matrix S, comprising:

A. means for assigning s 1 as a first basis vector u i ,

B. means for determining σ i , where u i T u i =σ i ,

C. means for storing u i ,

D. means for computing of inner products of the s i+1 and the u 1 through u i vectors,

E. means for multiplying said inner product with a respective scalar 1/σ i and thereby creating a first intermediate product,

F. means for scaling each respective basis vector u i by multiplying each respective first intermediate product with each respective basis vector u i ,

G. means for serially subtracting said intermediate product from s i+1 ,

H. means for utilizing the result from step G and subtracting the next incoming value of

u

i

⁢

1

σ

i

⁢

u

i

T

⁢

s

i

+

1

until all the values are processed,

I. means for obtaining the next basis vector u i+1 from step H,

J. means for comparing u i+1 to a predetermined value and if equal to or less than said value, going to step O,

K. means for storing u i+1 ,

L. means for determining an inner product of u T i+1 u i+1 ,

M. means for determining the reciprocal of step K which is 1/σ i+1 ,

N. means for storing 1/σ i+1 ,

O. means for incrementing i,

P. means for conducting steps D through O until i=p, where p is the total number of said sources of interest;

means for generating a diagonal matrix from stored 1/σ i+1 values;

means for generating one or more scalars from the diagonal matrix and from the basis vectors of the orthogonal basis U; and

means for applying the one or more scalars to the received signal to project the source of interest.

37. A method for generating a projection of a received signal y, said received signal comprising H, a signal of a source of interest; S, the signals of all other sources and multi-path versions of the source of interest and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the method comprising the steps of:

determining a basis matrix U composed of basis vectors u 1 , u 2 . . . , u p ;

storing elements of said basis matrix U;

generating a diagonal matrix from stored said elements of the basis matrix U;

generating one or more scalars from the diagonal matrix and from the basis vectors of the basis matrix U;

applying the one or more scalars to the received signal to project the signal of the source of interest: and

determining

y

s

=

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

,

wherein y s is a projected said signal of the source of interest.

38. An apparatus for generating a projection from a received signal y, said received signal comprising H, a signal of a source of interest; S, the signals of all other sources and composed of vectors s 1 , s 2 , s 3 . . . , s p ; and noise (n); the apparatus comprising:

means for determining a basis vector U;

means for storing elements of said basis vector U;

means for generating a diagonal matrix from stored said elements of the basis vector U;

means for generating one or more scalars from the diagonal matrix and from the basis vector U;

means for applying the one or more scalars to the received signal to project the signal of the source of interest; and

means for determining

y

s

=

u

1

⁢

1

σ

1

⁢

u

1

T

⁢

y

-

u

2

⁢

1

σ

2

⁢

u

2

T

⁢

y

-

…

⁢

⁢

u

p

-

1

⁢

1

σ

p

-

1

⁢

u

p

-

1

T

⁢

y

-

u

p

⁢

1

σ

p

⁢

u

p

T

⁢

y

,

wherein y s is a projected said signal of the source of interest.

39. A system, comprising:

means for generating a first matrix from a received signal, wherein the received signal comprises a plurality of signals;

means for generating a second matrix from the first matrix, wherein the second matrix is a substantially orthogonal basis of the first matrix;

means for storing values used in generating the second matrix;

means for generating a diagonal matrix from stored said values;

means for generating one or more scalars from the diagonal matrix and from the second matrix; and

means for multiplying the one or more scalars to the received signal to project the received signal substantially orthogonal to said plurality of signals.

40. A method, comprising:

generating a first matrix from a received signal, wherein the received signal comprises a plurality of signals;

generating a second matrix from the first matrix, wherein the second matrix is a substantially orthogonal basis of the first matrix;

storing values used in generating the second matrix;

generating a diagonal matrix from stored said values;

generating one or more scalars from the diagonal matrix and from the second matrix; and

multiplying the one or more scalars to the received signal to project the received signal substantially orthogonal to said plurality of signals.