5. Multisine excitations for MIMO measurements
5.1 MIMO multisine design
Can the results obtained for TITO systems (4.4.22) be generalized to the full MIMO case?
The problem is that the orthogonalization of the inputs and using only single random amplitude for all of the inputs and experiments (albeit with different weighting) can introduce constraints, which influence the bias. The answer is that it is indeed the case.
The traditional input matrix (4.2.2) built from random excitations is not a good choice. Even in the absence of disturbing output noise the FRF measurements will vary from one realization to the other. The reason is the fluctuation of the inverse matrix in (4.2.5), due to the randomness of the excitations. Note, that the matrix (UUH)-1 fluctuates even when random phase multisines are used, contrary to the measurements on SISO systems [162].
Example 5.1.1. Fluctuation of the input matrix for the random multisine measurements. We illustrate the problem with the simplest 2-dim case:
=
) ( ) (
) ( ) ) (
( (2)
2 )
1 ( 2
) 2 ( 1 )
1 ( 1
2 U l U l
l U l l U
U (5.1.1)
Omitting for brevity the frequency index we have:
=
2 1 2
2 S D
S
H D U
U (5.1.2)
where Dk = |Uk(1)
|2 + |Uk(2)
|2 is deterministic, and off-diagonal S = U1(1)Ū2(1) + U1(2)Ū2(2) is a zero mean random term. Consequently:
−
−
= −
−
1 2 2 2 1 1 2
2 | |
) 1
( S D
S D S D D UH
U (5.1.3)
and the random off-diagonal terms introduce additional fluctuation in the FRF measurements. Note also the
expression of the determinant in the denominator. Its small values can further amplify the variance.
noise noise multi e
Var G >Var G >Var G , because =
∑
eJ=(2.2.24) (4.3.1) randomly fluctuates for the noises, but is deterministic for the (random phase) multisines.
In the MIMO case this wisdom won’t be valid, because as mentioned earlier
1
1 ( ) ( )
ˆUU− l = UNUHN −
S in (4.2.5) will contain random components for all three considered inputs.
For static non-linearities there is no leakage and variances will be comparable. For dynamic non-linear systems the leakage of the Gaussian noise will increase the variance comparing to the periodic input signals. In the comparison of the periodic noise and the random phase multisine periodic noise turns out a bit better, because its input matrix is better conditioned.
Example 5.2.1. Condition number of the periodic noise and the random phase multisines. Consider for illustration and simplicity the case N = 2 and let us assume that signals are independent over the channels, and that the amplitudes for the periodic noise are similarly distributed in different input channels, with E{Akm} = A = 1. the random phase multisines and the periodic noise respectively, where all the phases are uniformly distributed on the unit circle and the amplitudes Akm are exponentially distributed with unit expected value. All the considered random variables are independent.
The condition number equals κ(U) = ||U|| ||U-1|| [84], and let choose for the investigation the Frobenius norm, i.e.:
||U||2=Σkm |ukm|2. In this simple case the respective inverse matrices are:
with the determinants:
21 The condition numbers become then:
2 independent, exponentially distributed variables. Clearly for (5.1.6) to be singular not only the phazors must be colineated as for (5.1.5), but also the random amplitudes should match, which is an event of lower probability than the singularity of (5.1.5). On the other hand (5.1.6) can be excessively large without the colineation of the phases, simply when the amplitudes are small. Simulations show that the average condition number for the periodic noise is a bit better (simulations indicate a rough factor of 2, not really a difference).
If an input matrix built from random multisines does not work well, then what? Hadamard matrix makes the experiments the simplest; no computation is required to find the amplitudes for the new experiments. Using Hadamard matrix for the input matrix (4.2.3) makes it possible to Crest Factor optimize the input signals once for all, because the subsequent change of sign in an input channel does not influence the phases of the already optimized signal.
Another issue to consider is that although only Hadamard matrix is proposed in the literature to minimize the noise influence [88], it can be used with full impact only for the system with N = 2K inputs. Approximate design fares already not so well for linear systems, and is dubious when the Best Linear Approximation comes into question. One could use in (4.3.7) another orthogonal (or unitary) matrix, e.g. the Fourier matrix (DFT matrix), defined for any dimension (i.e. number of inputs). This choice however introduces already additional computation to the amplitudes.
First we will present a negative result telling that (4.4.20-4.4.22) cannot be generalized fully.
This will lead to the introduction of a new input matrix design for which the equivalence and the optimality will be proved in the general case.
Theorem 5.1.1: Input matrix design (4.3.7) does not generalize to the general MIMO Volterra system of arbitrary dimension and order. When a Volterra MIMO system (Def.
4.1.1) of arbitrary order, with a number of inputs N > 2, is measured with random orthogonal inputs (2.2.10), with Hadamard or Fourier matrix, acc. to (4.3.7), the bias on the Best Linear Approximation (in the kth signal channel), besides terms mentioned in Theorem 4.5.1 will also contain some (not many) other terms as well.
Proof: In Appendix A.4
Example 5.1.1: Some of the kernels adding to the bias, beside the “normal” bias terms (4.5.4) from Th. 4.5.1.:
N = 4, α =3, k =1, kernel (j1j2j3) = (234) for both Fourier and Hadamard matrix, k =2, kernel (j1j2j3) = (123) for Fourier matrix,
k =2, kernel (j1j2j3) = (134) for both Fourier and Hadamard matrix, k =3, kernel (j1j2j3) = (124) for both Fourier and Hadamard matrix, k =4, kernel (j1j2j3) = (233) for Fourier matrix, etc.
Generally it can be seen that Fourier matrix introduces more bias terms, than the Hadamard matrix. On the other hand Fourier matrix can be applied for an arbitrary (e.g. odd) number of inputs, but the Hadamard matrix cannot.
Note: Using orthogonal inputs amounts to a modified increased bias. Beside this all the evaluation leading to the additive non-linear noise source model remains valid; consequently in this case we also have the additive noise model, albeit with different Best Linear Approximation and non-linear variance.
Note: Although direct analytic comparison between the influence of the Fourier and the Hadamard cases (with respect to the level of the non-linear variance) does not seem realistic (different algebraic operations involved), low order calculations show that they eliminate roughly similar number of (similar) kernels, consequently, if the number of inputs permits, Hadamard matrix could be proposed as a simpler one for the calculations.