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A general matrix approach to residual spectra
Journal of Sound and Vibration (1980) 69(3), 487-490
LETTERS A GENERAL
MATRIX
1.
TO THE EDITOR APPROACH
TO RESIDUAL
SPECTRA
INTRODUCTION
In a recent note [l] the applicability of a matrix approach to the conditioning of the components of a multivariate random process was demonstrated, conditioning with respect to one or more components being separately treated, together with the conditioning of multivariate response components with respect to system normal co-ordinates. Although the approach adopted in reference [l] is the more appropriate for the purpose of bringing insight to a rather complex problem, the process of conditioning can also be embodied in a quite general analytical form. It is the object of the present note to demonstrate such a general approach. It is shown that suitable analysis of the matrices permits conditioning with respect to either particular physical co-ordinates, or particular normal co-ordinates, and that the notation of partitioned matrices removes the need to distinguish between conditioning with respect to one, or several co-ordinates. An important result covering the rank of conditioned spectral matrices is established. 2. CONDITIONING
WITH
TO ONE COMPONENT
RESPECT
The operation of conditioning has been fully described in reference [2]. Consider a multivariate random process described by the components xl(t), x*(t), . . . , x,,,(t). This may be represented by the m X 1 vector x = {x1, x2, . . . , x,,,}, and the corresponding set of spectral densities (direct and cross) may be represented by the matrix S which is square and of order m. It is now convenient to adopt a notation somewhat different from that of reference [l]. Suppose the vector i={x~, x3,. . . , x,} is conditioned with respect to the component x1. This consists of removing from each component of i the part which is fully coherent with xl. The conditioned vector is then denoted by i = (x2, 1, x3. 1, . . . , x,. 1}, but it is better here to treat instead, the complete uncoherent vector y={xt,x2.1,x3.1,...,xm.t1.
(1)
A simple relationship exists between the vectors y and x which can be obtained from the schematic block diagram given in reference [2], and this may be written in matrix form as 1 ioo 0x1 ._____: __.......___ _____ ____ __-.. A2ilO
y=
A3 . .
:.
Ill
0x2 io
1
: :
A,\0
0 .
o...;
“3
&y!J!?]_~,
(2)
.
x,
where hi = -Sii/Sii and 42) is an identity matrix of order (m - 1). This enables one to obtain the spectral density matrix associated with y by making use of the symbols 8 and S for the spectral densities of y and x respectively, and using the relationship 6 = H*SH’.
(3)
487 0022--460X/80/070487
+ 04 $02.00/O
0
1980 Academic Press Inc. (London) Limited
488
LETTERS
TO THE
EDITOR
Expansion of equation (3) and using the H indicated by equation (2) gives
s
...
0
11
0
s32--
s31s12
*.’
0
=
&-F
(4)
Sll
0
sm2__
s?llS12
. . .
S 11
SmlSI,
Smm--
S 11
which is the result obtaited in reference Cl] by a different approach. The form of the conditioned elements in S22 suggests that they could be obtained from the elements of the original matrix by a simple column ,transformation operation, which reduces to zero the cross-spectral density elements of S. Although use has not been made here of the sub-matrices M and I(2) defined in equation (2), their usefulness will become evident in section 3, which is concfrned with multiple conditioning. A typical element of Sz2 may be written in the form 1 Slj,
Sij, 1 = Sij -SilSiI
(5)
and if this matrix is next conditioned with respect to x2(t), a typical element may be written in the form skl.12
=
Sk,--sk2.
lsi~.ls2/.
(6)
1.
But, as already mentioned above, it is more convenient to carry out multiple conditioning in one operation, by making use of the submatrices formed by partitioning the original matrix S. 3.
CONDITIONING
WITH
RESPECT
TO SEVERAL
VARIABLES
For simultaneous conditioning with respect to the set %= {XI, x2,. . . , x,} of components, the remaining set {x,+~, . . . , x,} may as above be denoted by i. The conditioning may again be interpreted as obtaining by linear filtering a set of components (7)
Y={Xl,X2,...,Xp,Yp+l,~..,Ym}
through the transformation x=Hx,
(8)
where Icl, is a pth order identity matrix, I C2jis a (m -p)th order identity matrix and M is an as yet undetermined (m-p) X p submatrix of filter coefficients. To impose the required conditioning, the components of M must be determined such that the cross-spectral matrices connecting f and {yP+r, . . . , y,} vanish: i.e., so that
where 8 is as before the spectral matrix of y, Srr is the spectral matrix of components i, and S22 is the spectral matrix of {yP+l, . . . , y,,,}. The matrix S in equation (9) is the spectral matrix of S after conditioning with respect to x1, . . . , xp and may be denoted by SC.,...,,,.
LETTERS
489
TO THE EDITOR
From the relation
it follows that ~22=M*S11M’+S21MI+M*S1Z+S22, ,. $1 = M”S11 +!&I.
(11) (12)
The conditioning requirement that S2r should vanish thus yields M* = -S21SYI’. Substitution of this solution for M* into equation (11) then finally yields S( .X1,_..,X,)= 922 = S22- s*Kl%,
(13)
which is a very neat generalization of formula (5). Individual elements of the residual spectrum can be obtained by evaluating the matrix products in equation (13). For instance, when m = 4 and p = 2, the off-diagonal element in Sz2 is s31s22&4 s34
-
s32s21s14
-
s31s12s24
-
S12~21
+
s32slls24 9
SllS22
(14)
which is usually denoted S34. r2 because it is the residual cross-spectrum between components y3 and y4 after conditioning with respect to x1 and x2. This formula holds irrespective of whether the components Xi are physical co-ordinates or normal coordinates. 4.
RANK
OF RESIDUAL
SPECTRAL
MATRICES
There are many problems in engineering in which a response spectral density matrix of order m may have a rank r such that r < m. This may arise either from the nature of the excitation or from the dynamic characteristics of the system. Under these circumstances there exist interdependences between the elements of the spectral density matrix, and these can be represented by means of equations of constraint which have been discussed in reference [3]. It is important to be aware of the effect of conditioning on the rank of the response spectral matrix in such a case, for the circumstances imply essential limitations on multiple conditioning. The rank r of any response spectral matrix S” will be less than the number m of co-ordinates in cases where the response arises from a multivariate excitation having fewer than m components, and also when x includes certain combinations of physical and normal co-ordinates. However, it is clear from equation (13) that the set of conditioning components %must always have a non-singular spectral matrix Sll, and hence the rank of Sll must be equal to p. The defined form of H in the case of conditioning as implied by equation (10) ensures that det H = 1, and therefore it follows that rank Sy = rank S” = r. From the special form of equation (9) it is evident that ,. I rank Sy = rank Sll + rank SZ2;
(15)
(16)
as the rank of S,, is p it follows from equations (15) and (16) that the rank of the residual spectral matrix S2, is r - p.
490
LETTERS TO THE EDITOR
Thus an m-variate random process, having a spectral matrix of rank r, cannot meaningfully be conditioned with respect to more than r-l of its (independent) components. This conclusion applies whether the latter are physical quantities or normal co-ordinates. Department of Mechanical Engineering, Universityof Glasgow, Glasgow G12 8QQ, Scotland
D. B.
MACVEAN
S. MAHALINGAM
J. D.
(Received 29 November 1979)
ROBSON
REFERENCES 1. S. MAHALINGAM, D. B. MACVEAN and J. D. ROBSON 1980 68,3 13-3 16. Residual spectral densities: a matrix approach. 2. C. J. DODDS and J. D. ROBSON 1975 Journal of Sound and
Journal of Sound and Vibration
Vibration, 42, 243-249. Partial coherence in multivariate random processes. 3. S. MAHALINGAM, D. B. MACVEAN and J. D. ROBSON 1980 Journal of Sound and Vibration 69,461-476. Interdependence of the spectral densities of multiple responses.
LETTERS A GENERAL
MATRIX
1.
TO THE EDITOR APPROACH
TO RESIDUAL
SPECTRA
INTRODUCTION
In a recent note [l] the applicability of a matrix approach to the conditioning of the components of a multivariate random process was demonstrated, conditioning with respect to one or more components being separately treated, together with the conditioning of multivariate response components with respect to system normal co-ordinates. Although the approach adopted in reference [l] is the more appropriate for the purpose of bringing insight to a rather complex problem, the process of conditioning can also be embodied in a quite general analytical form. It is the object of the present note to demonstrate such a general approach. It is shown that suitable analysis of the matrices permits conditioning with respect to either particular physical co-ordinates, or particular normal co-ordinates, and that the notation of partitioned matrices removes the need to distinguish between conditioning with respect to one, or several co-ordinates. An important result covering the rank of conditioned spectral matrices is established. 2. CONDITIONING
WITH
TO ONE COMPONENT
RESPECT
The operation of conditioning has been fully described in reference [2]. Consider a multivariate random process described by the components xl(t), x*(t), . . . , x,,,(t). This may be represented by the m X 1 vector x = {x1, x2, . . . , x,,,}, and the corresponding set of spectral densities (direct and cross) may be represented by the matrix S which is square and of order m. It is now convenient to adopt a notation somewhat different from that of reference [l]. Suppose the vector i={x~, x3,. . . , x,} is conditioned with respect to the component x1. This consists of removing from each component of i the part which is fully coherent with xl. The conditioned vector is then denoted by i = (x2, 1, x3. 1, . . . , x,. 1}, but it is better here to treat instead, the complete uncoherent vector y={xt,x2.1,x3.1,...,xm.t1.
(1)
A simple relationship exists between the vectors y and x which can be obtained from the schematic block diagram given in reference [2], and this may be written in matrix form as 1 ioo 0x1 ._____: __.......___ _____ ____ __-.. A2ilO
y=
A3 . .
:.
Ill
0x2 io
1
: :
A,\0
0 .
o...;
“3
&y!J!?]_~,
(2)
.
x,
where hi = -Sii/Sii and 42) is an identity matrix of order (m - 1). This enables one to obtain the spectral density matrix associated with y by making use of the symbols 8 and S for the spectral densities of y and x respectively, and using the relationship 6 = H*SH’.
(3)
487 0022--460X/80/070487
+ 04 $02.00/O
0
1980 Academic Press Inc. (London) Limited
488
LETTERS
TO THE
EDITOR
Expansion of equation (3) and using the H indicated by equation (2) gives
s
...
0
11
0
s32--
s31s12
*.’
0
=
&-F
(4)
Sll
0
sm2__
s?llS12
. . .
S 11
SmlSI,
Smm--
S 11
which is the result obtaited in reference Cl] by a different approach. The form of the conditioned elements in S22 suggests that they could be obtained from the elements of the original matrix by a simple column ,transformation operation, which reduces to zero the cross-spectral density elements of S. Although use has not been made here of the sub-matrices M and I(2) defined in equation (2), their usefulness will become evident in section 3, which is concfrned with multiple conditioning. A typical element of Sz2 may be written in the form 1 Slj,
Sij, 1 = Sij -SilSiI
(5)
and if this matrix is next conditioned with respect to x2(t), a typical element may be written in the form skl.12
=
Sk,--sk2.
lsi~.ls2/.
(6)
1.
But, as already mentioned above, it is more convenient to carry out multiple conditioning in one operation, by making use of the submatrices formed by partitioning the original matrix S. 3.
CONDITIONING
WITH
RESPECT
TO SEVERAL
VARIABLES
For simultaneous conditioning with respect to the set %= {XI, x2,. . . , x,} of components, the remaining set {x,+~, . . . , x,} may as above be denoted by i. The conditioning may again be interpreted as obtaining by linear filtering a set of components (7)
Y={Xl,X2,...,Xp,Yp+l,~..,Ym}
through the transformation x=Hx,
(8)
where Icl, is a pth order identity matrix, I C2jis a (m -p)th order identity matrix and M is an as yet undetermined (m-p) X p submatrix of filter coefficients. To impose the required conditioning, the components of M must be determined such that the cross-spectral matrices connecting f and {yP+r, . . . , y,} vanish: i.e., so that
where 8 is as before the spectral matrix of y, Srr is the spectral matrix of components i, and S22 is the spectral matrix of {yP+l, . . . , y,,,}. The matrix S in equation (9) is the spectral matrix of S after conditioning with respect to x1, . . . , xp and may be denoted by SC.,...,,,.
LETTERS
489
TO THE EDITOR
From the relation
it follows that ~22=M*S11M’+S21MI+M*S1Z+S22, ,. $1 = M”S11 +!&I.
(11) (12)
The conditioning requirement that S2r should vanish thus yields M* = -S21SYI’. Substitution of this solution for M* into equation (11) then finally yields S( .X1,_..,X,)= 922 = S22- s*Kl%,
(13)
which is a very neat generalization of formula (5). Individual elements of the residual spectrum can be obtained by evaluating the matrix products in equation (13). For instance, when m = 4 and p = 2, the off-diagonal element in Sz2 is s31s22&4 s34
-
s32s21s14
-
s31s12s24
-
S12~21
+
s32slls24 9
SllS22
(14)
which is usually denoted S34. r2 because it is the residual cross-spectrum between components y3 and y4 after conditioning with respect to x1 and x2. This formula holds irrespective of whether the components Xi are physical co-ordinates or normal coordinates. 4.
RANK
OF RESIDUAL
SPECTRAL
MATRICES
There are many problems in engineering in which a response spectral density matrix of order m may have a rank r such that r < m. This may arise either from the nature of the excitation or from the dynamic characteristics of the system. Under these circumstances there exist interdependences between the elements of the spectral density matrix, and these can be represented by means of equations of constraint which have been discussed in reference [3]. It is important to be aware of the effect of conditioning on the rank of the response spectral matrix in such a case, for the circumstances imply essential limitations on multiple conditioning. The rank r of any response spectral matrix S” will be less than the number m of co-ordinates in cases where the response arises from a multivariate excitation having fewer than m components, and also when x includes certain combinations of physical and normal co-ordinates. However, it is clear from equation (13) that the set of conditioning components %must always have a non-singular spectral matrix Sll, and hence the rank of Sll must be equal to p. The defined form of H in the case of conditioning as implied by equation (10) ensures that det H = 1, and therefore it follows that rank Sy = rank S” = r. From the special form of equation (9) it is evident that ,. I rank Sy = rank Sll + rank SZ2;
(15)
(16)
as the rank of S,, is p it follows from equations (15) and (16) that the rank of the residual spectral matrix S2, is r - p.
490
LETTERS TO THE EDITOR
Thus an m-variate random process, having a spectral matrix of rank r, cannot meaningfully be conditioned with respect to more than r-l of its (independent) components. This conclusion applies whether the latter are physical quantities or normal co-ordinates. Department of Mechanical Engineering, Universityof Glasgow, Glasgow G12 8QQ, Scotland
D. B.
MACVEAN
S. MAHALINGAM
J. D.
(Received 29 November 1979)
ROBSON
REFERENCES 1. S. MAHALINGAM, D. B. MACVEAN and J. D. ROBSON 1980 68,3 13-3 16. Residual spectral densities: a matrix approach. 2. C. J. DODDS and J. D. ROBSON 1975 Journal of Sound and
Journal of Sound and Vibration
Vibration, 42, 243-249. Partial coherence in multivariate random processes. 3. S. MAHALINGAM, D. B. MACVEAN and J. D. ROBSON 1980 Journal of Sound and Vibration 69,461-476. Interdependence of the spectral densities of multiple responses.