3.3 Pole Weighted Modal Vectors When comparing base vectors, at either the short or the long dimension, a pole weighted base vector can be constructed independent of the original UMPA(m,n,v) procedure used to estimate the poles and base vectors. For a given order v of the pole weighted vector, the base vector and the associated pole can be used to formulate the pole weighted vector as follows: { φ}r = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ λv r {ψ}r . . . λ2 r {ψ}r λ1 r {ψ}r λ0 r {ψ}r ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭r (3) The above formulation will be dominated by the high order terms if actual frequency units are utilized. Generalized frequency concepts (frequency normalization or z domain transform) are normally used to minimize this problem as follows: { φ}r = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ zv r {ψ}r . . . z2 r {ψ}r z1 r {ψ}r z0 r {ψ}r ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭r (4) zr = ej* π*( ωr/ ωmax) = ej*ωr*Δt (5) zmr = ej* π*m*( ωr/ ωmax) (6) In the above equations, Δt and ωmax can be chosen as needed to cause the positive and negative roots to wrap around the unit circle in the z domain without overlapping (aliasing). Normally, ωmax is taken to be five percent larger than the largest frequency identified in the roots of the matrix coefficient polynomial. 3.4 Pole Surface Consistency and Density A number of other pole presentation diagrams, related to the consistency diagram, such as pole surface consistency and pole surface density diagrams have proven useful for identifying modal parameters [43-44,56-57] and may be more powerful than the consistency diagram alone. Generally, pole surface density diagrams are more powerful than consistency or stability diagrams at locating similar pole vector estimates from all of the possible solutions represented in the consistency diagram. All of the poles from all solutions involved in the consistency diagram are ploted in the second quadrant of the s plane. Pole estimates that are located within a two dimensional theshold from each other are defined as participating in a pole cluster or a dense region of estimated poles in the s plane. The poles that are compared on these pole surface diagrams are generally limited to the poles identified on the consistency diagram (if some symbols are omitted from the consistency diagram, these poles will not be included on the pole surface diagram. The distribution of the poles that participate in a pole cluster can be used to find a single pole-vector estimate and the distribution can be used to estimate statistics related to the variance in the pole estimate. An example of a pole surface consistency diagram is given in Figure 3 and the companion pole surface density diagram in Figure 4. 368
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