Hioki 8847 - Linear Time-Invariant Systems

Appendix 5 FFT Definitions
A16
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).
)(kF
)(k
φ
)
(
k
F
実数部
虚数部
Imaginary component
Real component
)()()]()([
22112211
nyAnyAnxAnxAL +=+
(6)
(7)

∞
−∞=
∞
=
−=−=
mm
mxmnhmnxnhny )()()()()(
0
(8)
)()( nhny =
(9)
)()()( kHkXkY =
Input
(Analysis channel 1)
Output
(Analysis channel 2)
x(n)
X(k)
y(n)
Y(k)
h(n)
H(k)
LTI System