„Signal Theory” Zdzisław Papir
Signal Filtering
„Signal Theory” Zdzisław Papir
Signal Filtering
Signal Filtering
• Signal filtering – Fourier series
• Signal filtering - examples
• Signal filtering – Fourier transform
• Frequency characteristics of a signal after filtration
• Signal filtering – example
„Signal Theory” Zdzisław Papir
)
(s
H
)
(t
x
st
e
s
H
n
t
jn
n
e
X
t
x
o
)
(
n
t
jn
n
n
t
jn
Y
n
e
Y
e
jn
H
X
t
y
n
o
o
o
)
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The Fourier series of the output signal y(t)
Signal filtering – Fourier series
st
e
t
y
„Signal Theory” Zdzisław Papir
Sawtooth input signal x(t)
0
0.5
1
1.5
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2.5
3
0
0.1
0.2
0.3
0.4
0.5
0.6
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0.8
0.9
1
Sawtooth signal (period T)
time t/T
x(t)
t
n
n
e
n
j
t
x
n
t
jn
n
n
o
1
2
0
sin
1
1
2
1
1
2
2
1
Signal filtering - examples
„Signal Theory” Zdzisław Papir
Lowpass filter
RC
T
T
s
Ts
Cs
R
Cs
s
H
,
1
1
1
1
1
1
1
g
g
R
C
„Signal Theory” Zdzisław Papir
Lowpass
filter
1
,
log
20
1
,
0
log
20
1
1
,
1
1
g
g
g
2
g
g
H
H
j
j
H
„Signal Theory” Zdzisław Papir
10
-1
10
0
10
1
10
2
10
-4
10
-3
10
-2
10
-1
10
0
g
dec
dB
H
log-log amplitude characteristics of the LPF (1st order)
Input/output signals – Fourier
series
t
jn
n
n
e
jn
n
j
t
y
o
g
o
o
1
1
1
2
2
1
t
n
n
e
n
j
t
x
n
t
jn
n
n
o
1
o
sin
1
1
2
1
1
2
2
1
o
g
1
1
j
j
H
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 9)
0
5
10
15
20
25
30
35
40
45
50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lowpass filter and sawtooth signal – amplitude spectra
n
f
o
Lowpass filter
Sawtooth signal
f
g
/f
o
= 9
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 9)
0
0.5
1
1.5
2
2.5
3
3.5
0
0.2
0.4
0.6
0.8
1
Lowpass filter response
time t/T
f
g
/f
o
= 9
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 3)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Lowpass filter and sawtooth signal – amplitude spectra
Lowpass filter
Sawtooth signal
f
g
/f
o
= 3
n
f
o
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 3)
0
0.5
1
1.5
2
2.5
3
3.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lowpass filter response
time t/T
f
g
/f
o
= 3
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 1)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Lowpass filter and sawtooth signal – amplitude spectra
Lowpass filter
Sawtooth signal
f
g
/f
o
= 1
n
f
o
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 1)
0
0.5
1
1.5
2
2.5
3
3.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lowpass filter response
time t/T
f
g
/f
o
= 1
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 1/3)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Lowpass filter and sawtooth signal – amplitude spectra
f
g
/f
o
= 1/3
n
f
o
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 1/3)
0
0.5
1
1.5
2
2.5
3
3.5
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Lowpass filter response
time t/T
f
g
/f
o
= 1/3
„Signal Theory” Zdzisław Papir
Highpass filter
RC
T
T
Ts
Ts
Cs
R
R
s
H
g
,
1
1
1
R
C
„Signal Theory” Zdzisław Papir
Highpass
filter
„Signal Theory” Zdzisław Papir
10
-1
10
0
10
1
10
2
10
-1
10
0
g
dec
dB
H
log-log amplitude characteristics of the HPF (1st order)
1
,
0
1
,
log
20
1
,
1
g
g
g
2
g
g
g
g
H
H
j
j
j
H
„Signal Theory”
Zdzisław Papir
o
g
o
g
o
g
o
g
o
o
o
o
1
2
1
1
1
2
n
n
t
jn
t
jn
n
n
e
jn
e
jn
jn
n
j
t
y
t
n
n
e
n
j
t
x
n
t
jn
n
n
o
1
o
sin
1
1
2
1
1
2
2
1
o
g
g
1
j
j
j
H
Input/output signals – Fourier
series
Amplitude spectra (f
g
/f
o
= 9)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Highpass filter and sawtooth signal – amplitude spectra
Highpass filter
Sawtooth signal
f
g
/f
o
= 9
n
f
o
„Signal Theory”
Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 9)
0
0.5
1
1.5
2
2.5
3
3.5
-25
-20
-15
-10
-5
0
5
10
Highpass filter response
time t/T
f
g
/f
o
= 9
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 3)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Highpass filter and sawtooth signal – amplitude spectra
Highpass filter
Sawtooth signal
n
f
o
f
g
/f
o
= 3
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 3)
0
0.5
1
1.5
2
2.5
3
3.5
-6
-5
-4
-3
-2
-1
0
1
2
Highpass filter response
time t/T
f
g
/f
o
= 3
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 1)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Highpass filter and sawtooth signal – amplitude spectra
Highpass filter
Sawtooth signal
n
f
o
f
g
/f
o
= 1
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 1)
0
0.5
1
1.5
2
2.5
3
3.5
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
High pass filter
time t/T
f
g
/f
o
= 1
„Signal Theory” Zdzisław Papir
Amplitude spectra (f
g
/f
o
= 1/3)
0
10
20
30
40
50
0
0.2
0.4
0.6
0.8
1
Highpass filter and sawtooth signal – amplitude spectra
Highpass filter
Sawtooth signal
n
f
o
f
g
/f
o
= 0,3
„Signal Theory” Zdzisław Papir
Output signal y(t) (f
g
/f
o
= 1/3)
Highpass filter response
0
0.5
1
1.5
2
2.5
3
3.5
-0.06
-0.04
-0.02
0
0.02
0.04
time t/T
f
g
/f
o
= 0,3
„Signal Theory” Zdzisław Papir
„Teoria sygnałów” Zdzisław Papir
d
e
j
X
t
x
t
j
2
1
)
(
j
X
j
H
j
Y
d
e
j
X
j
H
t
y
t
j
2
1
)
(
Transformata Fouriera sygnału wyjściowego y(t)
)
(s
H
)
(t
x
t
j
e
j
X
j
H
2
1
t
j
e
j
X
2
1
t
y
Signal filtering - Fourier transform
j
X
j
H
j
Y
j
Y
t
y
t
t
x
t
h
t
y
t
x
t
h
j
X
j
H
j
Y
Impulse response of the filter
Signal filtering - Fourier transform
)
(s
H
j
X
t
x
)
(
)
(t
h
)
(t
h
t
y
t
t
x
)
(
Impulse response of a filter is its output signal when
input of a filter is excited by the Dirac delta impulse
(t).
„Signal Theory”
Zdzisław Papir
Frequency characteristics
of a signal after filtration
j
X
j
H
j
Y
j
Y
t
y
)
(s
H
j
X
t
x
)
(
j
j
j
e
j
H
j
X
j
Y
e
j
H
j
H
e
j
X
j
X
Filtration changes both:
• amplitude spectrum
• phase spectrum
of an input signal.
„Signal Theory” Zdzisław Papir
Signal filtering - example
W
j
H
j
t
t
x
2
1
1
Wt
d
d
t
d
t
j
t
j
d
e
j
d
e
j
t
y
Wt
W
W
W
W
W
t
j
t
j
W
W
Si
1
2
1
sin
1
2
1
sin
1
2
1
sin
cos
2
1
2
1
2
1
2
1
1
2
1
0
0
x
d
x
0
sin
Si
„Signal Theory”
Zdzisław Papir
Sinus integral
properties
x
d
x
0
sin
Si
„Signal Theory” Zdzisław Papir
1. Sinus integral is an odd function
x
du
u
u
du
d
u
d
x
x
x
Si
sin
sin
Si
0
0
2. Sinus integral in zero vicinity (x 0)
0
sin
0
Si
0
0
d
0
sin
,
0
Si
0
x
d
x
x
Sinus integral
properties
x
d
x
0
sin
Si
„Signal Theory”
Zdzisław Papir
3. Horizontal asymptote (x )
2
sin
Si
lim
0
x
d
x
0
0
sin
2
1
sin
2
1
sin
F
d
d
4. Local extrema
0
,
0
sin
Si
k
k
x
x
x
dx
x
d
Rising time of a filter output
is inversely proportional
to its bandwidth.
Signal filtering - example
„Signal Theory” Zdzisław Papir
1
0
Wt
t
y
Si
1
2
1
+
/W
-
/W
t
r
= 2
/W =
1/B
W
f
W
y
Si
1
2
1
The peak output value
does not depend on
the filter bandwidth.
Summary
• The output signal of the filter excited by a periodic signal
is a periodic signal as well; in most cases its Fourier series
is not summable to a closed form expression.
•
The Fourier transform of the output signal is a product of the
filter transfer function and input signal Fourier transform.
• The impulse response of the filter is its output signal when
the filter is driven by the Dirac delta
(t).
•
The filter output signal is a convolution of the filter
impulse response and the input signal.
„Signal Theory”
Zdzisław Papir