3. Specifying and Using Response Information¶
Theory of Instrument Response¶
Introduction¶
The Fourier Transform (\(t \rightarrow \omega\)) is defined by
\[X(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\]
while the Inverse Fourier Transform (\(\omega \rightarrow t\)) is given by
\[x(t)=\int_{-\infty}^{\infty}X(\omega)e^{+j\omega t}d\omega\]
Note that the choice of which transform (forward vs. reverse) has which sign of the complex exponent and which gets the \(\frac{1}{2\pi}\) scalefactor is optional.
For instance, some authors prefer to scale each transform by \(\frac{1}{\sqrt(2\pi)}\).
What is important is that the signs of the exponents in each transform must be opposite, and the product of their scalefactors must equal \(\frac{1}{2\pi}\).
Discrete Time Fourier Transform (DTFT)¶
In the Fourier transform pair above, both time (\(t\)) and frequency (\(\omega\)) are continuous parameters. In contrast, for signals sampled discretely in time, we may define the related Discrete Time Fourier Transform (DTFT) as
\[X(\omega)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\]
\[x[n]=\int_{0}^{2\pi}X(\omega)e^{+j\omega n}d\omega\]
where \(n\) is the discrete sample number, and \(\omega\) is still continuous.
Discrete Fourier Transform (DFT)¶
And finally, when both time and frequency are discrete, we define the Discrete Fourier Transform (DFT) pair by
\[X[k]=\frac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}\]
\[x[n]=\sum_{k=0}^{N-1}X[k]e^{+j2\pi kn/N}\]
Note that the popular Fast Fourier Transform (FFT) is a particular implementation of the DFT.
The Fourier Transform¶
Introduction¶
The Fourier Transform (\(t \rightarrow \omega\)) is defined by
\[X(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\]
while the Inverse Fourier Transform (\(\omega \rightarrow t\)) is given by
\[x(t)=\int_{-\infty}^{\infty}X(\omega)e^{+j\omega t}d\omega\]
Note that the choice of which transform (forward vs. reverse) has which sign of the complex exponent and which gets the \(\frac{1}{2\pi}\) scalefactor is optional.
For instance, some authors prefer to scale each transform by \(\frac{1}{\sqrt(2\pi)}\).
What is important is that the signs of the exponents in each transform must be opposite, and the product of their scalefactors must equal \(\frac{1}{2\pi}\).
Discrete Time Fourier Transform (DTFT)¶
In the Fourier transform pair above, both time (\(t\)) and frequency (\(\omega\)) are continuous parameters. In contrast, for signals sampled discretely in time, we may define the related Discrete Time Fourier Transform (DTFT) as
\[X(\omega)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\]
\[x[n]=\int_{0}^{2\pi}X(\omega)e^{+j\omega n}d\omega\]
where \(n\) is the discrete sample number, and \(\omega\) is still continuous.
Discrete Fourier Transform (DFT)¶
And finally, when both time and frequency are discrete, we define the Discrete Fourier Transform (DFT) pair by
\[X[k]=\frac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}\]
\[x[n]=\sum_{k=0}^{N-1}X[k]e^{+j2\pi kn/N}\]
Note that the popular Fast Fourier Transform (FFT) is a particular implementation of the DFT.
The Laplace Transform¶
Introduction¶
The Laplace Transform is defined by
\[X(\sigma,\omega)=\int_{-\infty}^{\infty}x(t)e^{-\sigma t}e^{-j\omega t}dt\]
If we make the substitution, \(s=\sigma + j\omega\), this becomes
\[X(s)=\int_{-\infty}^{\infty}x(t)e^{-s t}dt\]
Each point in the complex s-plane is associated with a frequency, \(\omega\) and an exponent \(\sigma\). Thus, each point in the s-plane describes a sinusoid of frequency \(\omega\) that is either exponentially growing (\(\sigma>0\)) or exponentially decaying (\(\sigma<0\)) with time.
Note that the laplace transform evaluated along the imaginary axis (where the attenuation parameter, \(s=0\)), reduces to the complex Fourier transform, \(X(\omega)\).
The laplace transform at point \(s\) is a measure of the similarlity between the input signal, \(x(t)\), and the corresponding exponentially growing/decaying sinusoid corresponding to that value of \(s\). A large value of \(X(s)\) corresponds to a strong similarity between the input signal and the sinusoid \(e^{-(\sigma + j\omega)t}\), indicating a strong presence of the sinusoid in the input signal.
In practice, we are often only interested in causal signals that begin at \(t=0\). Using the unit step function,
\[\begin{split}\begin{equation*}
u(t)=\begin{cases}
1 & t\ge0\\
0 & t<0
\end{cases}
\end{equation*}\end{split}\]
we may ensure causality by writing \(x(t)=u(t)x(t)\) , so that the Laplace Transform becomes
\[X(s)=\int_{0}^{\infty}x(t)e^{-s t}dt\]
Poles and Zeros¶
Suppose we have a data processing system (e.g., analog sensor + datalogger) that can be characterized by the linear differential equation,
\[a_{2}\ddot{y}(t)+a_{1}\dot{y}(t)+a_{0}y(t)=b_{2}\ddot{x}(t)+b_{1}\dot{x}(t)+b_{0}x(t)\]
where \(x(t)\) is the input signal (e.g., the ground motion), \(y(t)\) is the output signal (the signal recorded) and \(a_{k}\) and \(b_{k}\) are constant (time-invariant) coefficients. If we assume the system is causal, so that the signals + derivatives are all 0 for \(t<0\) , then the Laplace Transform of the equation gives
\[a_{2}s^{2}Y(s)+a_{1}sY(s)+a_{0}Y(s)=b_{2}s^{2}X(s)+b_{1}sX(s)+b_{0}X(s)\]
or
\[(a_{2}s^{2}+a_{1}s+a_{0})Y(s)=(b_{2}s^{2}+b_{1}s+b_{0})X(s)\]
from which we can write the transfer function of the system as
\[H(s) = \frac{Y(s)}{X(s)}=\frac{b_{2}s^{2}+b_{1}s+b_{0}}{a_{2}s^{2}+a_{1}s+a_{0}}\]
or more generally,
\[H(s) =\frac{\sum_{k=0}^{M}b_k s^n}{\sum_{k=0}^{N}a_n s^n}\]
This is the coefficient representation of the transfer function. It represents the transfer function as the ratio of two polynomials. The roots of the numerator polynomial are called ‘zeros’, while the roots of the denomenator polynomial are called ‘poles’.
Often, for analog stages, it is more convenient to factor the transfer function in terms of these poles and zeros:
\[H(s)=\frac{\Pi_{k=1}^{M} (s-z_{k})} {\Pi_{k=1}^{N} (s-p_{k})}\]
where \(z_{k}\) are the M zeros of the system, and \(p_{k}\) are the N poles.
Because the coefficients of the numerator and denominator polynomials are real, the corresponding roots (poles and zeros) must occur in complex conjugate pairs.
Thus, the poles and zeros are either real or form pairs that are symmetric with respect to the real axis in the complex \(s\)-plane. In addition, it can be shown that for systems that are stable and causal, the poles all have real parts \(\le 0\).
Recall that the Laplace transform variable is given by \(s=\sigma+j\omega\). Along the imaginary axis, \(\sigma=0\) and hence \(s=j\omega\). Thus, we may express the complex frequency response of the analog stage by calculating its polezero expansion
\[H(f)=A_0\frac{\Pi_{k=1}^{M} (s-z_{k})} {\Pi_{k=1}^{N} (s-p_{k})}\]
where \(s=j2\pi f\) [rad/s] or \(s=jf\) [Hz].
Thus, given the poles and zeros of an analog stage, in order to properly calculate the stage frequency response, we must know the units of \(s\) used to calculate the poles and zeros.
In stationxml, these units are specified by the PzTransferFunctionType element within the PolesZerosType response stage:
<Stage number="1"> <PolesZeros> ... </OutputUnits> <PzTransferFunctionType>LAPLACE (RADIANS/SECOND)</PzTransferFunctionType> <NormalizationFactor>1.0</NormalizationFactor> <NormalizationFrequency unit="HERTZ">1.0</NormalizationFrequency>
where the possibile values for PzTransferfunctionType are:
“LAPLACE (RADIANS/SECOND)”
“LAPLACE (HERTZ)”
“DIGITAL (Z-TRANSFORM)” (Discussed in next section)
Note also the NormalizationFactor
The z-Transform¶
The z-Transform is defined by
\[X(z)=\sum_{n=0}^{\infty}x[n]z^{-n}\]
where
\[ \begin{align}\begin{aligned}z=re^{j\omega}\\z^{-n}=r^{-n}e^{-j\omega n}\end{aligned}\end{align} \]
Notice that on the unit circle, where \(|z|\equiv |r|=1\) and \(z=e^{j\omega}\) , the z-transform reduces to the discrete Fourier transform (DTFT):
\[X(e^{j\omega})=\sum_{n=0}^{\infty}x[n]e^{-j\omega n}\]
z-transforms of linear time-invariant (LTI) systems described by difference equations play an important role in signal processing
The General form of a difference equation is:.
\[\sum_{k=0}^{N}a_{k}y[n-k]=\sum_{k=0}^{M}b_{k}x[n-k],\]
where \(a_{0}\ne0\) (e.g., the coefficient of y[n] can’t be zero)
Taking the z-transform of both sides,
\[\sum_{k=0}^{N}a_{k}z^{-k}Y(z)]=\sum_{k=0}^{M}b_{k}z^{-k}X(z)\]
or
\[Y(z)=\frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}}X(z)\]
From this we can write the (system) transfer function
\[H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}}\]
The transfer function is the z-transform of the system impulse response, h[n], or
\[H(z)=\sum_{n=0}^{\infty}h[n]z^{-n}\]
The transfer function can also be factored in terms of poles and zeros (for \(b_{0}\ne0\))
\[H(z)=\frac{b_{0}}{a_{0}}\frac{\Pi_{k=1}^{M}(1-c_{k}z^{-1})}{\Pi_{k=1}^{N}(1-d_{k}z^{-1})}\]
where \(c_{k}\) are the M zeros of the system, and \(d_{k}\) are the N poles.
For a system to be both stable and causal, its poles must lie inside the unit circle, or \(|d_{k}|<1 k=1,N\)
StationXML Examples¶
Broadband seismic recorder
This example stationxml was created using the steps shown in StationXML Tools.
Streckeisen STS-2 + REF TEK RT130
<?xml version='1.0' encoding='UTF-8'?> Response type: PolesZerosResponseStage, Stage Sequence Number: 1 <?xml version='1.0' encoding='UTF-8'?> <FDSNStationXML xmlns="http://www.fdsn.org/xml/station/1" schemaVersion="1.0"> <Source>demo</Source> <Module>ObsPy 1.1.0</Module> <ModuleURI>https://www.obspy.org</ModuleURI> <Created>2020-02-07T22:26:54.123236</Created> <Network code="US"> <Station code="ANMO"> <Latitude unit="DEGREES">34.945911</Latitude> <Longitude unit="DEGREES">-106.457199</Longitude> <Elevation unit="METERS">1820.0</Elevation> <Site> <Name>Albuquerque, New Mexico, USA</Name> </Site> <CreationDate>1970-01-01T00:00:00</CreationDate> <Channel code="BHZ" locationCode="10"> <Latitude unit="DEGREES">34.945911</Latitude> <Longitude unit="DEGREES">-106.457199</Longitude> <Elevation unit="METERS">1820.0</Elevation> <Depth unit="METERS">0.0</Depth> <Response> <InstrumentSensitivity> <Value>941864732.6931986</Value> <Frequency>1.0</Frequency> <InputUnits> <Name>M/S</Name> <Description>Velocity in Meters per Second</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> </InstrumentSensitivity> <Stage number="1"> <PolesZeros> <InputUnits> <Name>M/S</Name> <Description>Velocity in Meters per Second</Description> </InputUnits> <OutputUnits> <Name>V</Name> <Description>Volts</Description> </OutputUnits> <PzTransferFunctionType>LAPLACE (RADIANS/SECOND)</PzTransferFunctionType> <NormalizationFactor>3.4684e+17</NormalizationFactor> <NormalizationFrequency unit="HERTZ">1.0</NormalizationFrequency> <Zero number="0"> <Real minusError="0.0" plusError="0.0">0.0</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Zero> <Zero number="1"> <Real minusError="0.0" plusError="0.0">0.0</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Zero> <Zero number="2"> <Real minusError="-15.15" plusError="-15.15">-15.15</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Zero> <Zero number="3"> <Real minusError="-176.6" plusError="-176.6">-176.6</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Zero> <Zero number="4"> <Real minusError="-463.1" plusError="-463.1">-463.1</Real> <Imaginary minusError="-430.5" plusError="-430.5">-430.5</Imaginary> </Zero> <Zero number="5"> <Real minusError="-463.1" plusError="-463.1">-463.1</Real> <Imaginary minusError="430.5" plusError="430.5">430.5</Imaginary> </Zero> <Pole number="0"> <Real minusError="-0.037" plusError="-0.037">-0.037</Real> <Imaginary minusError="-0.037" plusError="-0.037">-0.037</Imaginary> </Pole> <Pole number="1"> <Real minusError="-0.037" plusError="-0.037">-0.037</Real> <Imaginary minusError="0.037" plusError="0.037">0.037</Imaginary> </Pole> <Pole number="2"> <Real minusError="-15.64" plusError="-15.64">-15.64</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Pole> <Pole number="3"> <Real minusError="-97.34" plusError="-97.34">-97.34</Real> <Imaginary minusError="-400.7" plusError="-400.7">-400.7</Imaginary> </Pole> <Pole number="4"> <Real minusError="-97.34" plusError="-97.34">-97.34</Real> <Imaginary minusError="400.7" plusError="400.7">400.7</Imaginary> </Pole> <Pole number="5"> <Real minusError="-374.8" plusError="-374.8">-374.8</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Pole> <Pole number="6"> <Real minusError="-520.3" plusError="-520.3">-520.3</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Pole> <Pole number="7"> <Real minusError="-10530.0" plusError="-10530.0">-10530.0</Real> <Imaginary minusError="-10050.0" plusError="-10050.0">-10050.0</Imaginary> </Pole> <Pole number="8"> <Real minusError="-10530.0" plusError="-10530.0">-10530.0</Real> <Imaginary minusError="10050.0" plusError="10050.0">10050.0</Imaginary> </Pole> <Pole number="9"> <Real minusError="-13300.0" plusError="-13300.0">-13300.0</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Pole> <Pole number="10"> <Real minusError="-255.097" plusError="-255.097">-255.097</Real> <Imaginary minusError="0.0" plusError="0.0">0.0</Imaginary> </Pole> </PolesZeros> <StageGain> <Value>1500.0</Value> <Frequency>1.0</Frequency> </StageGain> </Stage> <Stage number="2"> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="3"> <Coefficients> <InputUnits> <Name>V</Name> <Description>Volts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>1.0</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">102400.0</InputSampleRate> <Factor>1</Factor> <Offset>0</Offset> <Delay>0.0</Delay> <Correction>0.0</Correction> </Decimation> <StageGain> <Value>629129.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="4"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.000976562</Numerator> <Numerator>0.00244141</Numerator> <Numerator>0.00488281</Numerator> <Numerator>0.00854492</Numerator> <Numerator>0.0136719</Numerator> <Numerator>0.0205078</Numerator> <Numerator>0.0292969</Numerator> <Numerator>0.0393066</Numerator> <Numerator>0.0498047</Numerator> <Numerator>0.0600586</Numerator> <Numerator>0.0693359</Numerator> <Numerator>0.0769043</Numerator> <Numerator>0.0820312</Numerator> <Numerator>0.0839844</Numerator> <Numerator>0.0820312</Numerator> <Numerator>0.0769043</Numerator> <Numerator>0.0693359</Numerator> <Numerator>0.0600586</Numerator> <Numerator>0.0498047</Numerator> <Numerator>0.0393066</Numerator> <Numerator>0.0292969</Numerator> <Numerator>0.0205078</Numerator> <Numerator>0.0136719</Numerator> <Numerator>0.00854492</Numerator> <Numerator>0.00488281</Numerator> <Numerator>0.00244141</Numerator> <Numerator>0.000976562</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">102400.0</InputSampleRate> <Factor>8</Factor> <Offset>0</Offset> <Delay>0.00013672</Delay> <Correction>0.00013672</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="5"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.225586</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">12800.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.00046875</Delay> <Correction>0.00046875</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="6"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.225586</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">6400.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.0009375</Delay> <Correction>0.0009375</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="7"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.225586</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">3200.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.001875</Delay> <Correction>0.001875</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="8"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.225586</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">1600.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.00375</Delay> <Correction>0.00375</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="9"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>0.000244141</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.225586</Numerator> <Numerator>0.193359</Numerator> <Numerator>0.12085</Numerator> <Numerator>0.0537109</Numerator> <Numerator>0.0161133</Numerator> <Numerator>0.00292969</Numerator> <Numerator>0.000244141</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">800.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.0075</Delay> <Correction>0.0075</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="10"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>-7.15032e-07</Numerator> <Numerator>-5.60109e-06</Numerator> <Numerator>-2.62179e-06</Numerator> <Numerator>-4.31403e-05</Numerator> <Numerator>-4.64771e-06</Numerator> <Numerator>1.43006e-06</Numerator> <Numerator>2.34769e-05</Numerator> <Numerator>1.43006e-06</Numerator> <Numerator>-5.27932e-05</Numerator> <Numerator>-0.000366692</Numerator> <Numerator>0.000376107</Numerator> <Numerator>0.000854226</Numerator> <Numerator>3.05081e-05</Numerator> <Numerator>-0.00127621</Numerator> <Numerator>-0.000910951</Numerator> <Numerator>0.00127669</Numerator> <Numerator>0.00215165</Numerator> <Numerator>-0.000461554</Numerator> <Numerator>-0.00333765</Numerator> <Numerator>-0.00140933</Numerator> <Numerator>0.00377072</Numerator> <Numerator>0.00419414</Numerator> <Numerator>-0.00264288</Numerator> <Numerator>-0.00720121</Numerator> <Numerator>-0.000644006</Numerator> <Numerator>0.009184</Numerator> <Numerator>0.00608445</Numerator> <Numerator>-0.00857824</Numerator> <Numerator>-0.0127401</Numerator> <Numerator>0.00398225</Numerator> <Numerator>0.0186261</Numerator> <Numerator>0.0052052</Numerator> <Numerator>-0.0209407</Numerator> <Numerator>-0.0181629</Numerator> <Numerator>0.0166669</Numerator> <Numerator>0.0322447</Numerator> <Numerator>-0.00346588</Numerator> <Numerator>-0.0429528</Numerator> <Numerator>-0.0193265</Numerator> <Numerator>0.044309</Numerator> <Numerator>0.0497909</Numerator> <Numerator>-0.0294164</Numerator> <Numerator>-0.0826078</Numerator> <Numerator>-0.00934166</Numerator> <Numerator>0.107552</Numerator> <Numerator>0.0816604</Numerator> <Numerator>-0.10311</Numerator> <Numerator>-0.204208</Numerator> <Numerator>-3.12231e-05</Numerator> <Numerator>0.390432</Numerator> <Numerator>0.589958</Numerator> <Numerator>0.390432</Numerator> <Numerator>-3.12231e-05</Numerator> <Numerator>-0.204208</Numerator> <Numerator>-0.10311</Numerator> <Numerator>0.0816604</Numerator> <Numerator>0.107552</Numerator> <Numerator>-0.00934166</Numerator> <Numerator>-0.0826078</Numerator> <Numerator>-0.0294164</Numerator> <Numerator>0.0497909</Numerator> <Numerator>0.044309</Numerator> <Numerator>-0.0193265</Numerator> <Numerator>-0.0429528</Numerator> <Numerator>-0.00346588</Numerator> <Numerator>0.0322447</Numerator> <Numerator>0.0166669</Numerator> <Numerator>-0.0181629</Numerator> <Numerator>-0.0209407</Numerator> <Numerator>0.0052052</Numerator> <Numerator>0.0186261</Numerator> <Numerator>0.00398225</Numerator> <Numerator>-0.0127401</Numerator> <Numerator>-0.00857824</Numerator> <Numerator>0.00608445</Numerator> <Numerator>0.009184</Numerator> <Numerator>-0.000644006</Numerator> <Numerator>-0.00720121</Numerator> <Numerator>-0.00264288</Numerator> <Numerator>0.00419414</Numerator> <Numerator>0.00377072</Numerator> <Numerator>-0.00140933</Numerator> <Numerator>-0.00333765</Numerator> <Numerator>-0.000461554</Numerator> <Numerator>0.00215165</Numerator> <Numerator>0.00127669</Numerator> <Numerator>-0.000910951</Numerator> <Numerator>-0.00127621</Numerator> <Numerator>3.05081e-05</Numerator> <Numerator>0.000854226</Numerator> <Numerator>0.000376107</Numerator> <Numerator>-0.000366692</Numerator> <Numerator>-0.00041031</Numerator> <Numerator>2.52645e-05</Numerator> <Numerator>0.000261821</Numerator> <Numerator>0.000120602</Numerator> <Numerator>-9.99854e-05</Numerator> <Numerator>-0.000162312</Numerator> <Numerator>-9.79595e-05</Numerator> <Numerator>-2.94355e-05</Numerator> <Numerator>-3.09847e-06</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">400.0</InputSampleRate> <Factor>2</Factor> <Offset>0</Offset> <Delay>0.125</Delay> <Correction>0.125</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> <Stage number="11"> <Coefficients> <InputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </InputUnits> <OutputUnits> <Name>COUNTS</Name> <Description>Digital Counts</Description> </OutputUnits> <CfTransferFunctionType>DIGITAL</CfTransferFunctionType> <Numerator>-1.09889e-05</Numerator> <Numerator>-1.99798e-05</Numerator> <Numerator>-3.29668e-05</Numerator> <Numerator>-4.39561e-05</Numerator> <Numerator>-4.79522e-05</Numerator> <Numerator>-4.09589e-05</Numerator> <Numerator>-1.8981e-05</Numerator> <Numerator>1.8981e-05</Numerator> <Numerator>6.7932e-05</Numerator> <Numerator>0.000118881</Numerator> <Numerator>0.000158842</Numerator> <Numerator>0.000174826</Numerator> <Numerator>0.000157843</Numerator> <Numerator>0.000104895</Numerator> <Numerator>2.49751e-05</Numerator> <Numerator>-6.49352e-05</Numerator> <Numerator>-0.00014086</Numerator> <Numerator>-0.000178822</Numerator> <Numerator>-0.00016084</Numerator> <Numerator>-8.59142e-05</Numerator> <Numerator>3.29668e-05</Numerator> <Numerator>0.000163837</Numerator> <Numerator>0.000268733</Numerator> <Numerator>0.000310691</Numerator> <Numerator>0.000263737</Numerator> <Numerator>0.00013087</Numerator> <Numerator>-6.09391e-05</Numerator> <Numerator>-0.00026074</Numerator> <Numerator>-0.000408593</Numerator> <Numerator>-0.000448554</Numerator> <Numerator>-0.000353648</Numerator> <Numerator>-0.000135864</Numerator> <Numerator>0.000155845</Numerator> <Numerator>0.000438563</Numerator> <Numerator>0.000623379</Numerator> <Numerator>0.000638365</Numerator> <Numerator>0.000456546</Numerator> <Numerator>0.000108891</Numerator> <Numerator>-0.000315686</Numerator> <Numerator>-0.000694309</Numerator> <Numerator>-0.000903101</Numerator> <Numerator>-0.00085415</Numerator> <Numerator>-0.000533469</Numerator> <Numerator>-7.99164e-06</Numerator> <Numerator>0.000581421</Numerator> <Numerator>0.00105695</Numerator> <Numerator>0.00125675</Numerator> <Numerator>0.00108792</Numerator> <Numerator>0.000559443</Numerator> <Numerator>-0.000201799</Numerator> <Numerator>-0.000983021</Numerator> <Numerator>-0.00154047</Numerator> <Numerator>-0.00167733</Numerator> <Numerator>-0.0013037</Numerator> <Numerator>-0.000484518</Numerator> <Numerator>0.000571431</Numerator> <Numerator>0.00155645</Numerator> <Numerator>0.00215685</Numerator> <Numerator>0.00214287</Numerator> <Numerator>0.00145855</Numerator> <Numerator>0.00025075</Numerator> <Numerator>-0.00115385</Numerator> <Numerator>-0.00233568</Numerator> <Numerator>-0.00290311</Numerator> <Numerator>-0.0026174</Numerator> <Numerator>-0.00148752</Numerator> <Numerator>0.000215785</Numerator> <Numerator>0.002014</Numerator> <Numerator>0.00335166</Numerator> <Numerator>0.00376825</Numerator> <Numerator>0.00304597</Numerator> <Numerator>0.0013037</Numerator> <Numerator>-0.001009</Numerator> <Numerator>-0.0032208</Numerator> <Numerator>-0.00463139</Numerator> <Numerator>-0.0047233</Numerator> <Numerator>-0.00334667</Numerator> <Numerator>-0.000793211</Numerator> <Numerator>0.00224477</Numerator> <Numerator>0.00486516</Numerator> <Numerator>0.00620583</Numerator> <Numerator>0.0057273</Numerator> <Numerator>0.00340861</Numerator> <Numerator>-0.000199801</Numerator> <Numerator>-0.00409193</Numerator> <Numerator>-0.00707596</Numerator> <Numerator>-0.00812791</Numerator> <Numerator>-0.00672831</Numerator> <Numerator>-0.00307194</Numerator> <Numerator>0.00192309</Numerator> <Numerator>0.00682721</Numerator> <Numerator>0.010091</Numerator> <Numerator>0.0105175</Numerator> <Numerator>0.00766437</Numerator> <Numerator>0.00206594</Numerator> <Numerator>-0.00483219</Numerator> <Numerator>-0.01101</Numerator> <Numerator>-0.0144376</Numerator> <Numerator>-0.0136934</Numerator> <Numerator>-0.00847457</Numerator> <Numerator>0.000173827</Numerator> <Numerator>0.010004</Numerator> <Numerator>0.018085</Numerator> <Numerator>0.0215935</Numerator> <Numerator>0.0186664</Numerator> <Numerator>0.00910094</Numerator> <Numerator>-0.0053287</Numerator> <Numerator>-0.0210541</Numerator> <Numerator>-0.0333958</Numerator> <Numerator>-0.0376226</Numerator> <Numerator>-0.030137</Numerator> <Numerator>-0.00949755</Numerator> <Numerator>0.0229931</Numerator> <Numerator>0.063304</Numerator> <Numerator>0.10534</Numerator> <Numerator>0.142124</Numerator> <Numerator>0.167226</Numerator> <Numerator>0.176134</Numerator> <Numerator>0.167226</Numerator> <Numerator>0.142124</Numerator> <Numerator>0.10534</Numerator> <Numerator>0.063304</Numerator> <Numerator>0.0229931</Numerator> <Numerator>-0.00949755</Numerator> <Numerator>-0.030137</Numerator> <Numerator>-0.0376226</Numerator> <Numerator>-0.0333958</Numerator> <Numerator>-0.0210541</Numerator> <Numerator>-0.0053287</Numerator> <Numerator>0.00910094</Numerator> <Numerator>0.0186664</Numerator> <Numerator>0.0215935</Numerator> <Numerator>0.018085</Numerator> <Numerator>0.010004</Numerator> <Numerator>0.000173827</Numerator> <Numerator>-0.00847457</Numerator> <Numerator>-0.0136934</Numerator> <Numerator>-0.0144376</Numerator> <Numerator>-0.01101</Numerator> <Numerator>-0.00483219</Numerator> <Numerator>0.00206594</Numerator> <Numerator>0.00766437</Numerator> <Numerator>0.0105175</Numerator> <Numerator>0.010091</Numerator> <Numerator>0.00682721</Numerator> <Numerator>0.00192309</Numerator> <Numerator>-0.00307194</Numerator> <Numerator>-0.00672831</Numerator> <Numerator>-0.00812791</Numerator> <Numerator>-0.00707596</Numerator> <Numerator>-0.00409193</Numerator> <Numerator>-0.000199801</Numerator> <Numerator>0.00340861</Numerator> <Numerator>0.0057273</Numerator> <Numerator>0.00620583</Numerator> <Numerator>0.00486516</Numerator> <Numerator>0.00224477</Numerator> <Numerator>-0.000793211</Numerator> <Numerator>-0.00334667</Numerator> <Numerator>-0.0047233</Numerator> <Numerator>-0.00463139</Numerator> <Numerator>-0.0032208</Numerator> <Numerator>-0.001009</Numerator> <Numerator>0.0013037</Numerator> <Numerator>0.00304597</Numerator> <Numerator>0.00376825</Numerator> <Numerator>0.00335166</Numerator> <Numerator>0.002014</Numerator> <Numerator>0.000215785</Numerator> <Numerator>-0.00148752</Numerator> <Numerator>-0.0026174</Numerator> <Numerator>-0.00290311</Numerator> <Numerator>-0.00233568</Numerator> <Numerator>-0.00115385</Numerator> <Numerator>0.00025075</Numerator> <Numerator>0.00145855</Numerator> <Numerator>0.00214287</Numerator> <Numerator>0.00215685</Numerator> <Numerator>0.00155645</Numerator> <Numerator>0.000571431</Numerator> <Numerator>-0.000484518</Numerator> <Numerator>-0.0013037</Numerator> <Numerator>-0.00167733</Numerator> <Numerator>-0.00154047</Numerator> <Numerator>-0.000983021</Numerator> <Numerator>-0.000201799</Numerator> <Numerator>0.000559443</Numerator> <Numerator>0.00108792</Numerator> <Numerator>0.00125675</Numerator> <Numerator>0.00105695</Numerator> <Numerator>0.000581421</Numerator> <Numerator>-7.99164e-06</Numerator> <Numerator>-0.000533469</Numerator> <Numerator>-0.00085415</Numerator> <Numerator>-0.000903101</Numerator> <Numerator>-0.000694309</Numerator> <Numerator>-0.000315686</Numerator> <Numerator>0.000108891</Numerator> <Numerator>0.000456546</Numerator> <Numerator>0.000638365</Numerator> <Numerator>0.000623379</Numerator> <Numerator>0.000438563</Numerator> <Numerator>0.000155845</Numerator> <Numerator>-0.000135864</Numerator> <Numerator>-0.000353648</Numerator> <Numerator>-0.000448554</Numerator> <Numerator>-0.000408593</Numerator> <Numerator>-0.00026074</Numerator> <Numerator>-6.09391e-05</Numerator> <Numerator>0.00013087</Numerator> <Numerator>0.000263737</Numerator> <Numerator>0.000310691</Numerator> <Numerator>0.000268733</Numerator> <Numerator>0.000163837</Numerator> <Numerator>3.29668e-05</Numerator> <Numerator>-8.59142e-05</Numerator> <Numerator>-0.00016084</Numerator> <Numerator>-0.000178822</Numerator> <Numerator>-0.00014086</Numerator> <Numerator>-6.49352e-05</Numerator> <Numerator>2.49751e-05</Numerator> <Numerator>0.000104895</Numerator> <Numerator>0.000157843</Numerator> <Numerator>0.000174826</Numerator> <Numerator>0.000158842</Numerator> <Numerator>0.000118881</Numerator> <Numerator>6.7932e-05</Numerator> <Numerator>1.8981e-05</Numerator> <Numerator>-1.8981e-05</Numerator> <Numerator>-4.09589e-05</Numerator> <Numerator>-4.79522e-05</Numerator> <Numerator>-4.39561e-05</Numerator> <Numerator>-3.29668e-05</Numerator> <Numerator>-1.99798e-05</Numerator> <Numerator>-1.09889e-05</Numerator> </Coefficients> <Decimation> <InputSampleRate unit="HERTZ">200.0</InputSampleRate> <Factor>5</Factor> <Offset>0</Offset> <Delay>0.585</Delay> <Correction>0.585</Correction> </Decimation> <StageGain> <Value>1.0</Value> <Frequency>0.05</Frequency> </StageGain> </Stage> </Response> </Channel> </Station> </Network> </FDSNStationXML>Extensometer
Pressure Transducer