Hioki Memory HiCorder MR6000, Инструкция по эксплуатации, Страница: 378 / 392

372
FFT Definitions
The above relationships are represented on the following complex plane.
Imaginary part
Real part
F ( k )
ø ( k )
│
F
(
k
)
│
Linear time-invariant system
Consider a linear time-invariant (LTI) system that produces a response y ( n ) to a discrete time-
domain signal
x
(
n
) .
The linear time-invariant system (hereafter referred to as LTI system) is a system in which the
following expression holds for any integer
Ai
where the equation 𝑦𝑦
𝑖𝑖
(𝑛𝑛) = 𝐿𝐿[𝑥𝑥
𝑖𝑖
(𝑛𝑛)] is the response to
x
i
( n )
.
L [ A
1
x
1
( n ) + A
2
x
2
( n )] = A
1
y
1
( n ) + A
2
y
2
( n ) .............................................(6)
Let h ( n ) denote a system function of the LTI system, the relationship between an input and output is
given by the following expression.
𝑦𝑦(𝑛𝑛) = ∑ ℎ( 𝑚𝑚 )𝑥𝑥(𝑛𝑛 − 𝑚𝑚) =
∞
𝑚𝑚𝑚𝑚
∑ ℎ(𝑛𝑛 − 𝑚𝑚)𝑥𝑥(𝑚𝑚)
∞
𝑚𝑚𝑚𝑚∞
....................(7)
Assign a unit impulse δ ( n ) (1 for n = 0, 0 for other numbers) to x ( n ) , and the following expression
holds.
y ( n ) = h ( n ) ......................................................................................(8)
Thus, applying the unit impulse to the input signal allows the LTI system characteristics to
absolutely outputs. The response waveform of a system to a unit impulse is called the impulse
response.
On the other hand, where the discrete Fourier transforms of
x ( n )
,
y ( n )
, and
h ( n )
are
X ( k )
,
Y ( k )
, and
H ( k )
, respectively, expression (7) can be written as follows:
Y
(
k
) =
X
(
k
)
H
(
k
) ...............................................................................(9)
The function
H
(
k
) is also called the transfer function, which is 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 instrument calculates an impulse response and a transfer function using the relationships of
expression (9).
Input
(Calculation channel 1)
LTI system
Output
(Calculation channel 2)
h ( n )
H ( k )
x ( n )
X ( k )
y ( n )
Y ( k )
HIOKI MR6000A966-03