Method and apparatus for determining an inverse square root of a given positive-definite hermitian matrix
View Patent ↗Generally, a method and apparatus are provided for computing a matrix inverse square root of a given positive-definite Hermitian matrix, K. The disclosed technique for computing an inverse square root of a matrix may be implemented, for example, by the noise whitener of a MIMO receiver. Conventional noise whitening algorithms whiten a non-white vector, X, by applying a matrix, Q, to X, such that the resulting vector, Y, equal to Q·X, is a white vector. Thus, the noise whitening algorithms attempt to identify a matrix, Q, that when multiplied by the non-white vector, will convert the vector to a white vector. The disclosed iterative algorithm determines the matrix, Q, given the covariance matrix, K. The disclosed matrix inverse square root determination process initially establishes an initial matrix, Q 0 , by multiplying an identity matrix by a scalar value and then continues to iterate and compute another value of the matrix, Q n+1 , until a convergence threshold is satisfied. The disclosed iterative algorithm only requires multiplication and addition operations and allows incremental updates when the covariance matrix, K, changes.
1. A method for determining an inverse square root of a covariance matrix, K, to convert a non-white input vector to a white vector said method comprising the steps of:
establishing an initial matrix, Q 0 , in a memory by multiplying a scalar multiple, a, by an identity matrix, wherein said scalar multiple, a, is within a certain range; and
computing a next value of the matrix, Q n+1 , until a convergence threshold is satisfied where:
Q
n
+
1
=
3
2
Q
n
-
1
2
K
·
Q
n
3
,
wherein K is a covariance matrix, and Q n is a value of said matrix, Q, after n iterations; and
converting the non-white input vector to a white vector using said matrix, Q n , once said convergence threshold is satisfied.
2. The method of claim 1 , wherein said covariance matrix, K, corresponds to a space-time equalizer in a receiver.
3. The method of claim 1 ,wherein said range is expressed as:
0< a<tr ( K ) −1/2 .
4. The method of claim 1 , wherein said convergence threshold requires that the trace (I−KQ n 2 ) 2 <ε, where ε is a suitably chosen convergence value.
5. The method of claim 1 , wherein once said convergence threshold is satisfied, said matrix, Q n , when multiplied by a non-white input vector, will convert said input vector to a white vector.
6. A method for whitening an input vector, X, said method comprising the steps of:
establishing an initial matrix, Q 0 , by multiplying said scalar multiple, a, by an identity matrix, wherein said scalar multiple, a, is within a certain range; and
computing a next value of the matrix, Q n+1 , until a convergence threshold is satisfied where:
Q
n
+
1
=
3
2
Q
n
-
1
2
K
·
Q
n
3
,
wherein K is a covariance matrix, and Q n is a value of said matrix, Q, after n iterations; and
multiplying said input vector, X, by said matrix, Q n , once said convergence threshold is satisfied.
7. The method of claim 6 , wherein said covariance matrix, K, corresponds to a space-time equalizer in a receiver.
8. The method of claim 6 , wherein said range is expressed as:
0< a<tr ( K ) −1/2 .
9. The method of claim 6 , wherein said convergence threshold requires that the trace (I−KQ n 2 ) 2 <ε, where ε is a suitably chosen convergence value.
10. An apparatus for determining an inverse square root of a covariance matrix, K, to convert a non-white input vector to a white vector said apparatus comprising:
means for establishing an initial matrix, Q 0 , in a memory by multiplying a scalar multiple, a, by an identity matrix, wherein said scalar multiple, a, is within a certain range; and
means for computing a next value of the matrix, Q n+1 , until a convergence threshold is satisfied where:
Q
n
+
1
=
3
2
Q
n
-
1
2
K
·
Q
n
3
,
wherein K is a covariance matrix, and Q n is a value of said matrix, Q, after n iterations; and
means for converting the non-white input vector to a white vector using said matrix, Q n , once said convergence threshold is satisfied.
11. The apparatus of claim 10 , wherein said covariance matrix, K, corresponds to a space-time equalizer in a receiver.
12. The apparatus of claim 10 , wherein said range is expressed as:
0< a<tr ( K ) −1/2 .
13. The apparatus of claim 10 , wherein said convergence threshold requires that the trace (I−KQ n 2 ) 2 <ε, where Ε is a suitably chosen convergence value.
14. The apparatus of claim 10 , wherein once said convergence threshold is satisfied, said matrix, Q n , when multiplied by a non-white input vector, will convert said input vector to a white vector.
15. An apparatus for whitening an input vector, X, said apparatus comprising:
means for establishing an initial matrix, Q 0 , in a memory by multiplying said scalar multiple, a, by an identity matrix, wherein said scalar multiple, a, is within a certain range; and
means for computing a next value of the matrix, Q n+1 , until a convergence threshold is satisfied where:
Q
n
+
1
=
3
2
Q
n
-
1
2
K
·
Q
n
3
,
wherein K is a covariance matrix, K, and Q n is a value of said matrix, Q, after n iterations; and
means for multiplying said input vector, X, by said matrix, Q n , once said convergence threshold is satisfied.
16. The apparatus of claim 15 , wherein said covariance matrix, K, corresponds to a space-time equalizer in a receiver.
17. The apparatus of claim 15 , wherein said range is expressed as:
0< a<tr ( K ) −1/2 .
18. The apparatus of claim 15 , wherein said convergence threshold requires that the trace (I−KQ n 2 ) 2 <ε, where ε is a suitably chosen convergence value.