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23COH+ID

# 23COH+ID - Coherence Measurement Noise and System...

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Coherence, Measurement Noise and System Identification: SISO 1. Coherence Function(COH) 1 Assume and are both different from zero, meaning that both spectra are rich, and do not contain delta functions, meaning that both spectra have no deterministic parts with zero means, the coherence function between the input and the output of a system is defined by ) ( f S xx ) ( f S yy ) ( t x ) ( t y 1 ) ( ) ( ) ( ) ( 0 2 2 = γ f S f S f S f yy xx xy xy . For a constant parameter linear system, free of measurement noise, 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 = = = γ f S f H f S f S f H f S f S f S f xx xx xx yy xx xy xy . If and are completely unrelated, the coherence function will be zero. For , there arise such possible situations as: ) ( t x 2 γ < xy ) ( t y 1 < ) ( 0 f a) Extraneous noise is present in the measurements. b) The system relating and is not linear. ) ( t x ) ( t y c) is an output due to an input as well as to other inputs. ) ( t y ) ( t x d) Resolution bias errors are present in the spectral estimates. For linear systems, the coherence function γ can be interpreted as ) ( 2 f xy the fractional portion of the mean square value at the output ) ( t y , which is contributed by the input at frequency f . The value of the coherence function indicates how much of one record is linearly related to the other record. It does not necessarily indicate a cause-and- effect relationship between the two records. ) ( t x 2. Effects of Measurement Noise The standard single-input single-output system model with extraneous (meaning ‘not go through the system’) input and output measurement noises, m and , is shown below. ) ( t ) ( t n Now we can construct the simple relations: ) ( ) ( ) ( ), ( ) ( ) ( t n t v t y t m t u t x + = + = where are the measured input and output signals, and, u are the true input and output signals which are not measurable. Note that the measurement noises, are not measurable in the time domain together with x or , ) ( ), ( t y t x ) ( t ) ( ), ( t v t ) ( ), ( t y t ), ( n t m ) ( ), ( t v t u 1 Bendat & Piersol, Random Data , section 6.1, 2000

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MAE 591 RANDOM DATA ) ( t m ) ( t x ) ( t v ) ( t u ) ( f H + + ) ( t n ) ( t y either, but their autospectral densities can be estimated separately when the input u is turned off, leading to ) ( t 0 ) ( = t ( ) ( v . From now on, we assume for simplicity that extraneous noise terms are uncorrelated with each other and with the signals, i.e.
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23COH+ID - Coherence Measurement Noise and System...

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