VIII. Coherence and Transfer Function Applications A. Coherence Function Estimates

VIII. Coherence and Transfer Function Applications A. Coherence Function Estimates Consider the application of these ideas to the specific problem of atmospheric turbulence measurements outlined in Figure VIII.. acceleration Output v(t) Vertical C.. Input u(t) Vertical gust velocity Figure VIII.. Airplane flying through atmospheric turbulence, where u(t) measured vertical gust velocity and v(t) measured vertical acceleration (acceleration in g's). Consider the spectral energy density and coherence functions (see Figures VIII. & VIII.3) for T 0 minute (600 sec) records of u(t) and v(t) in with a fundamental frequency band f 0.05 cps over the frequency range 0. to 4.0 cps. What are the degrees of freedom for these calculations? What is the zero coherence confidence level? Coherence Analysis High Coherence ( 0.8-0.9) in the range 0.3 < f <.0 cps Low Coherence @ f < 0.3 cps coherence low due to maneer loads induced by pilot, rather than atmospheric turbulence! i.e. contributions to y(t) from other inputs @ f >.0 cps low coherence due to reduced energy in response relative to measurement noise. Response energy decreases at higher frequencies due to decreasing plane response to turbulence and reduced turbulence energy. The ordinary coherence squared, defined as, (VIII.) where 0 is a statistical estimator related to the frequency response function as described next. 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown

Figure VIII. Power spectra of gust velocity and response acceleration. (Data from studies funded by the NASA Langley Research Center, Hampton, Virginia; Contract NAS -8538). Figure VIII.3 Coherence function between gust velocity and response acceleration. 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown

********************************************************************* EXAMPLES - The Extremes ********************************************************************* Extreme : Consider the ideal noise free, constant parameter system in Figure VIII.4, for which H and H. (VIII.) Figure VIII.4 In this model, a noise-free input produces a noise-free output. Thus the coherence squared is H (VIII.3) H Conclusion: The coherence between input and output in a noise-free system is ALWAYS equal to Extreme : Consider a "real world" situation in which the output v(t) is completely unrelated to the input u(t). Thus 0. Normally falls in the range 0 for one or more of the following reasons: (a) There is extraneous noise in the measurement of the input and output; (b) The system relating u(t) and v(t) is nonlinear; and/or (c) The output v(t) is due to u(t) as well as other unmeasured inputs. In these Anormal situations@ for a linear system (at each frequency f); can be thought of as the fractional part of the mean square output v(t ) due to u(t). Conversely the quantity - is the fractional part of the mean square output error due to sources other than u(t); and thus represents the level of noise in the system. To explore these ideas, consider the linear process in Figure VIII.5.in which () noise n(t) is added to the noise-free input u(t) at the input; and () noise m(t) is added to the noise-free output v(t) at the output. 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 3

Figure VIII.5 Schematic of linear system in which distinctly different noises are added to the noise-free input and output respectively. Under these circumstances, the respective measured input and output are u (t) v (t) u(t) + n(t) v(t) + m(t) And the respective auto- and cross-spectral density functions for the system are u + + + nn mm um + + + un vm nv + + + nu mv nm, (VIII.4) (VIII.5a) where H (VIII..5b) and H. (VIII..5c) Note that all possible correlations among signals and noise are allowed in this general case. So lets consider these special cases of interest. Case. No input noise [i.e. n(t) 0]; Output noise is uncorrelated (See Figure VIII.6) Figure VIII.6 Noise-free input and uncorrelated output noise. Thus Eqs. (VIII.5) are reduced to the following set of equations (VIII.6, 7, 8 and 9): ' u ( t ) u ( t ) (VIII.6a) ' v ( t ) v ( t ) + m ( t ) (VIII.6b) where ( f ) 0 vm u (VIII.7a) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 4

v + mm (VIII.7b) H H u (VIII.8a) H (VIII.8b) H (VIII.8) (VIII.9a) (VIII.9b) Thus the noise-free output spectrum can be computed using only the measured input and output ( i.e. u and! ). Therefore the spectrum of the noise at the output mm - mm can also be computed according to. This case is important because in reality () Inputs can be defined; and () Input noise can be minimized to simulate the conditions of the case. However we have no control over output noise because of () non-linearities and/or () unmeasured inputs! The ordinary coherence squared function for this case is (VIII.0) [ + mm ] u + mm, (VIII.) which compares to the previous noise-free case in which 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 5

Thus for > 0 mm. However because of (VIII.9) u < (VIII.) So we can use measurements to compute the Acoherent output spectrum@ or Further from above (t) mm and we can compute the spectrum of the output noise (VIII.3) - (- mm v ) (VIII.4) ************************************************************** Case. No output noise [m(t) 0]; Input noise is uncorrelated (See Figure VIII.7) Figure VIII.7. Uncorrelated noise is added only to the input. (t) u(t) + n(t), (VIII.5a) with un 0. (t) v(t) (VIII.5b) u + nn v H (VIII.6a) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 6

H * [ ] H (VIII.6b) u Alternatively it can be shown from the definitions of From above > (VIII.8) H v H * H S u that. (VIII.7) * H and can be computed from u (t), (t). Also since and > v H > nn u - (VIII.9) (VIII.0) The spectra of both the noise-free input and the noise at the input can be computed. However, do not apply this model if output noise is expected. The ordinary coherence squared in this case is v [ + nn] (VIII.a) + nn (VIII.b) Thus, like case, the ordinary coherence squared function can be used to determine the spectrum of the noise-free input according to (VIII.) AND the input noise spectrum according to (- ) nn (VIII.3) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 7

The Effects of eneral Extraneous Noise Under ideal conditions, the measured spectral density function u + nn (VIII.4a) v + mm (VIII.4b) u H H (VIII.5) u H v - nn (VIII.6) Note H u v but need to know the noise. H f( nn ), - mm - nn not f( mm ), H f( mm, mn ) The coherence squared, based on measurements, is always less than the true coherence as shown below û ˆ (VIII.7) [ + nn ][ + mm] + nn + mm nn mm + < Set output noise mm 0. B. Frequency Response Function Estimates Consider a single-input output linear system u(t) v(t) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 8

v(t) 0 h( τ ) u(t -τ ) dτ h( τ ) 0 for τ < 0 It follows that i.e. physically realizable v(t) v(t + τ ) Taking expected values of both sides (zero Similarly R R 0 h( α ) h( β ) u(t - β ) u(t + τ -α ) dα d β u, v ) input/output auto-correlation function (t) 0 ( τ ) h( α) h( β ) R 0 h( α) R Fourier Transforms of the above in terms of one-sided-spectra ( τ -α) dα ( τ + β -α ) dα d β H _ H (VIII.8a) H > H (VIII.8b) Consider the ratio of the two different ways of computing the squared gain factor: H H (VIII.9) Different estimates of the squared gain factor: () Ĥ ) (VIII.30a) which is a biased estimate except at 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 9

() Ĥ (VIII.30b) which is a biased estimate if there is input noise n(t) However, if m(t) is the only output noise, then the ratio of () and () is an unbiased estimate. Ĥ Ĥ (VIII.30c) The sample frequency response function is Ĥ Ĥ e +i ˆ θ (VIII.3) the bias error sources in this case are: () Inherent in the estimation process; () Nonlinear and/or time varying system parameters (i.e. non-stationarity); (3) Bias in the power- and cross-spectral density estimates; (4) Extraneous noise measured at input (incoherent noise does not cause bias); (5) Contributions to the measured output from other unmeasured inputs correlated with the measured input; (6) Bias due to propagation time delays. Remarks: () In general there is bias in the estimation process. That is } E E{ E{ } H } E{ Ĥ (VIII.3) E{ Ĥ} H, thus Ĥ is a biased estimator as shown below. 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 0

In practice this source of error can be made relatively small with a good choice of degrees of freedom. () The linearity assumption is often violated if the operating range is sufficiently wide. Nevertheless it should be noted that the application of equation (a) above to nonlinear systems represents the "best" linear approximation (in the least squares sense) of the frequency response function of the system under the specified input and output conditions. (3) Note: The bias of spectral density estimates can be large in and around spectral peaks. However, these errors can be suppressed by choosing f s small enough to define spectral peaks. (4) In the case, where extraneous noise at input is uncorrelated with input or output signal (i.e. Case ), (see Figure VIII.8). Figure VIII.8 Extraneous noise at the input is uncorrelated with input or output signal. Consider input signal u(t) + n(t) (t) for the case in which n(t) passes through system, but u(t) E{ u } (VIII.33) E{ } E{H } H (VIII.34) E{ } + nn + nn does. If all other bias errors are ignored, then with the normalized bias error E{Ĥ } nn ε b - + nn -- (VIII.35) H mm + nn downward bias Thus the estimated frequency response function, Ĥ, is biased downwards by an amount eb - 0.09 equal to the noise to signal ratio at input e.g. N/S 0.0 or (5) When noise m(t) at output is correlated with measured input u(t) (Figure VIII.9) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown

Figure VIII.9 Noise at the output m(t) is correlated with the measured input u(t); u(t) is unmeasured input. u(t) - measured input; u'(t) - an unmeasured input; v(t) - output; and v'(t) - measured output, and from the definition of second edition of Bendat and Piersol) that Ĥ above, it can be shown (Eq. 7.3 of section 7... of the u u (VIII.36) If all other bias errors are ignored, then E{H} H - u v - u u v H u [- ] [- u u u ] v (VIII.37) Note, further that if u and are uncorrelated, then u u u 0 (VIII.38) E{Ĥ} H which demonstrates that unmeasured uncorrelated noise at the input causes no bias. 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown

Random errors (see Figure VIII.0) If H is unknown Figure VIII.0 A system with noise z(t) at measured output that is uncorrelated with input or output. Let u(t) - input v(t) - output z(t) - noise uncorrelated with input or output (i.e. residual extraneous noise at the output related only to the measurement only) v'(t) measured output z(t) v (t) - v(t) (VIII.39) The Finite Fourier Transform for z(t) is z(t) v (t) - 0 h( τ ) u(t -τ ) dτ Z(f, T) V (f,t) - H U(f, T) + [- Ĥ U(f, T) + Ĥ U(f,T)] where Ĥ v - a biased estimator or Z(f, T) Ẑ(f,T) + U(f,T) [Ĥ(f,T) - H], (VIII.40) where Ẑ (f,t) V (f,t) - Ĥ(f,T) U(f, T) (VIII.4) with Ẑ best estimate of noise It follows that + Ĥ - H zz ẑẑ (VIII.4) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 3

If (VIII.4) is smoothed so that all the terms, including the ẑ ẑ estimate, have n f s T degrees of freedom (DOF), then it will be distributed according to n zz χ n (VIII.43) zz The two terms on the RHS of (VIII.4) are statistically independent, thus their DOF will be additive according to χ n χ n n + n-n χ n n where DOF for term (a) of (VIII.4) n is reduced by number of constraints on From (VIII.40) and (VIII.4) Ẑ (f,t) F(Ĥ), ẑ ẑ. where Ẑ(f, T) is defined in terms of Ĥ (not H), which implies independent constraints on Ẑ ; one each for the real and imaginary part. Hence ẑẑ will have less DOF than zz, thus n zz χ n zz (VIII.44a) n Ĥ - H zz y n χ (VIII.44b) ẑẑ v v u v but we know [- ] so (VIII.44a) becomes n [- ] v v u v χ n- zz y n- (VIII.45) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 4

F,n n y / n y / n n n n- F,n- rˆ Ĥ - H F n- rˆ (n- ) Ĥ - H (VIII.46) [- ˆu v ] v v [- ˆ ],n-; v n ft; n- α (VIII.47) _ ( ft-) ft For a ratio of estimators with Χ n distributions we have a new distribution From (VIII.46), if H is unknown, then the confidence intervals for H with confidence coefficient -α, can be determined by a quantity rˆ such that the variance Var[ Ĥ] rˆ H is a ratio of χ distributed estimates so the appropriate distribution function is v F,n-;α input spectrum 00 α percentage point of an F-distribution with n and n n- degrees of freedom, f n f measured output spectrum s T T f Ts s ˆ v sample estimate of the ordinary coherence between u(t) and v (t) Approximate (-α) confidence intervals [ Ĥ - rˆ H Ĥ + rˆ ] [ ˆ φ - ˆ φ φ ˆ φ + ˆ] φ (VIII.48) (See B&P for Tables) ˆ - φ sin rˆ Ĥ 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 5

C. Estimating Tidal Transfer Function Error Assume that the observed tide v'(t) is the outcome of a linear process, characterized by the true transfer function H and corresponding weighting function h(t), plus some uncorrelated random noise at the output m(t) as shown below. u(t) h(t) /H v(t) + v'(t) (VIII.49a) m(t) The input u(t) in this case is the noise-free astronomical forcing, which is specified in terms of the equilibrium tide. The measured response of this system v'(t) is the sum of the noise-free tidal response v(t) and the noise at the output m(t) according to (t) v(t) + m(t) (VIII.49b) The tidal response v(t) is described here in terms of the harmonic constants, (amplitude H n and phase K n respectively) for each tidal constituent of the equilibrium tide (with it s unique frequency f n ). Here we seek an estimate of the uncertainty in the estimates of H n, K n in the presence of noise. In the frequency domain (VIII.49b) becomes v + where the 's are one-sided spectral density functions. We have shown that since mm v(t) 0 h( β ) u(t- β ) d β, H (VIII.50) A corresponding relation is u ' where Ĥ is a transfer function estimate. Ĥ v, (VIII.5) 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 6

Here is the complex one-sided cross-spectral density function, C + i Q, (VIII.5) where C and Q are the co- and quadrature-spectrum respectively. In polar notation, Eq. (VIII.50) becomes where and Thus and -iθ -i Φ e (C tan H e + Q [Q ) / C / ] - θ. H θ Φ., (VIII.53) Since is noise-free, the uncertainty in computing estimates of, θ uy and H, Φ is equivalent. There are several references on estimating uncertainties for H and Φ. From the Appendix B of Munk and Cartwright (966), transfer function estimates true transfer function T and a random noise according to Here / Ĥ e -iν mm u v ' / Ĥ are written in terms of the T + ε H e -i ε is a complex random variable, whose (a) real and + ε. imaginary parts have approximate normal probability distributions with zero mean and (b) whose variance in the fundamental frequency band f is given by / [ mm f ]/ [ f ] 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 7 p σ, where p f e T; in which f e is the effective bandwidth used in the spectral computation and T is the length of series. (Note that [ mm f]/p is the mean square error in estimating a mean square value in the presence of band limited white noise).

We normalizeσ / by the magnitude of the true transfer function H / and find ' '' σ σ H ( f ) [ mm f ]/ [ f ] They compute the probability density function and the 95% confidence limits for the normalized variables Ĥ /H and β Φˆ - Φ in terms of σ. (See Figure VIII.). u v p, Figure VIII.The upper diagrams show the probability distributions of (left) ρ sample admittance/true admittance, and (right) θ sample phase - true phase, for stated values of the noise parameter σ. Lower diagrams show the 95% confidence limits of the same quantities and also the mean r.m.s. values of ρ, all plotted against σ. (From Munk and Cartwright, 966). 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 8

D. Summary of Error Relations Error Relations (a) Sample Mean --- is unbiased Random Error: In general Var [u] T T -T when τ» T; T τ - R T 0 T ( τ ) dτ ; Var [u] - R ( τ ) dτ Normalized Mean Square Error: Band-limited white noise (BT > 5) ε u σ u BT µ where B and τ o is the integral time scale τ o (b) Sample Mean Square --- is unbiased Var [u ] T T T -T - τ - (R T (R ( τ ) + µ R ( τ ) + µ R u u ( τ )) dτ ( τ )) dτ ; whenτ Normalized Mean Square Error: Band limited white noise (BT 5) max» T Var u [ ^] R (O) BT τ o ε BT T 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 9

(c) Sample Cross-Covariance --- is unbiased Var[ Rˆ ( τ )] T T T α - [R -T T [R - ( α) R ( α) R ( α) + R ( α) + R ( α + τ ) R ( α + τ ) R vu vu ( α -τ )] dα ( α -τ )] dα ; for τ» T Normalized Mean Square Error: Band limited white noise BT > 5 ε BT + R (0) R (0), R ( τ ) (d) where Sample Spectral Density Var[ ] b[ ft " -8π - τ R ( τ ) e f ] 4 -iπ f τ dτ " Normalized mean square error E{ - } ε (-α) confidence interval ft 4 f + 576 " n χ n; α / n < χ Similar for cross-spectral density. n;-α / n f s T (e) Sample Coherence ˆ / + ˆ - w ln tan h - 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown 0

(-α) confidence interval [tanh w - (n- ) - - σ w α / z ) < - tanh (w - (n- ) + σ w zα / ] Frequency Response: Ĥ (a biased ratio of Χ distributed variables) rˆ F n- (-α) confidence interval [- ˆ,n-; α ] [( Ĥ - rˆ) H ( Ĥ + rˆ)] [ ˆ θ - ˆ θ θ ˆ θ + ˆ], θ where ˆ - θ sin rˆ Ĥ 7 February 008 Chapter VIII Coherence & Transfer Apps 8Wendell S. Brown