MAE 334 – LAB 5
2
The low pass filter drawn in Figure 1 can be modeled as a first order differential
equation of the form:
F
y
dt
dy
RC
=
+
If
F(t)
is the input signal and
y(t)
the output. This equation has the following general
solution:
∫
∞
−
∆
−
−
∆
⎥
⎦
⎤
⎢
⎣
⎡
∆
=
t
RC
t
d
e
RC
F
t
y
/
)
(
1
)
(
)
(
(4)
following equations (1), (2) and (3) we conclude
/
1
0
()
00
tRC
et
ht
RC
t
−
⎧
≥
⎪
=
⎨
⎪
<
⎩
…
…
(5)
You should recognize this as a standard first order system response function of the
same form as used to describe the thermocouple in lab 2 and the calorimeter in lab 3. In
this case the time constant,
τ
, would be
RC
, the resistance and capacitance values.
The Impulse Response of the Low Pass Filter
Recall that the delta
or impulse function
,
δ
(t)
, is zero at all times except
t
= 0, where it
is infinite and that it is infinite in such a way that integral for any time that includes
t=0
is
1.
∫
+
−
=
ε
δ
1
)
(
dt
t
Therefore
ε
can be as small as we please. If we subject our filter to an impulse function,
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 Spring '09
 Digital Signal Processing, Derivative, Impulse response, Lowpass filter

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