文档介绍：Fundamentals of Measurement Technology(6)
Prof. Wang Boxiong
For dynamic measurement, the measuring system must be a linear one.
We can only process linear systems mathematically.
It is rather difficult to perform nonlinear corrections in situations of dynamic measurement.
Practical systems may be considered as linear systems within a certain range of operation and permissible error limits.
It is of general significance to study linear systems.
Dynamic characteristics of measuring systems
The input-output relationship of a linear system:
where
x(t)= input of the system
y(t)= output of the system
an, a1, a0, and bm, b1, b0 are system’s parameters.
A linear constant-coefficient system or linear time-invariant (LTI) system: the parameters are constants.
Mathematical representation of linear systems
()
Properties:
Superposition property (superposability):
If for
then
Proportionality
If
then
Where a is a constant.
Mathematical representation of linear systems
()
()
Differentiation
If
then
Integration
If
and for a zero initial condition of the system, then
Mathematical representation of linear systems
()
()
Frequency preservability
If

and for
then the output
Mathematical representation of linear systems
Proof:
According to the proportionality property
According to the differentiation property
Since
Mathematical representation of linear systems
()
()
()
Letting the left-hand side of Eq. () be zero,
then the right-hand side of Eq. () must also be zero,
Solving the equation yields:
where φ is the phase shift.
Mathematical representation of linear systems
Transfer function
Definition:
For t0, y(t)=0, the Laplace transform Y(s) of y(t) is defined as
where s is the Laplace operator: s=a+jb for a>0.
Representation of system’s characteristics in terms of transfer function or frequency response
()
If all the system’s