Hioki MR8740, Instruction Manual

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Appendix 4 FFT Definitions
A
14
Representing the above relationship on a complex flat surface produces the fol-
lowing figure.
Linear Time-Invariant Systems __________________________________
Consider a linear time-invariant (LTI) system y ( n ) that is a response to discrete
time-domain signal
x
(
n
).
In such an LTI system, the following expression applies to any integer A
i
when
the response to x
i
( n ) is y
i
( n ) = L [ x
i
( n )].
If the system function of an LTI system is h ( n ), the input/output relationship can
be obtained by the next expression.
Therefore, when a unit impulse δ (
n
) (which is 1 when
n
= 0, and 0 when
n
≠ 0) is
applied to x ( n ), the input/output relationship is:
This means that when the input signal is given as a unit impulse, the output is
the LTI system characteristic itself.
The response waveform of a system to a unit impulse is called the
impulse
response .
On the other hand, when the discrete Fourier transforms of x ( n ), y ( n ) and h ( n ) are
X ( k ), Y ( k ) and H ( k ), respectively, expression (7) gives the following:
H ( k ) is also called the transfer function, calculated from X ( k ) and Y ( k ). Also, the
inverse discrete Fourier transform function of
H
(
k
) is the unit impulse response
h ( n ) of the LTI system. The impulse response and transfer function of this instru-
ment are calculated using the relationships of expression (9).
) ( k F
) ( k
φ
)
(
k
F
実数部
虚数
部
Imaginary component
Real component
) ( ) ( )] ( ) ( [
2 2 1 1 2 2 1 1
n y A n y A n x A n x A L
+ = +
(6)
(7)
∑ ∑
∞
−∞ =
∞
=
− = − =
m m
m x m n h m n x n h n y ) ( ) ( ) ( ) ( ) (
0
(8) ) ( ) ( n h n y
=
(9)
) ( ) ( ) ( k H k X k Y
=
Input
(Analysis channel 1)
Output
(Analysis channel 2)
x ( n )
X
(
k
)
y ( n )
Y
(
k
)
h ( n )
H ( k )
LTI System