Analog Lowpass Filter Specifications
• Typical magnitude response |Ha(j)| of analog lowpass filter:
• In the passband, defined by 0 p , we require
1 p |Ha(j)| 1+ p, || p , i.e., |Ha(j)| approximates unity within
an error off ² p
• In the stopband, defined by s < f, we require
|Ha(j)| s, s || < f , i.e.,
i e |Ha(j)| approximates zero within an error
of s
4-1-30
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Analog Lowpass Filter Specifications
• p - passband edge frequency
• s - stopband edge frequency
• p - peak ripple value in the
passband
• s - peak ripple value in the
stopband
t b d
• Peak passband ripple
p = 20log
20l 10(1 p )
• Minimum stopband attenuation
s = 20log10(
( s)
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4-1-31
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Analog Lowpass Filter Specifications
• Magnitude specifications may alternately be given in a
normalized
li d fform as iindicated
di t d b
below
l
We will use this form
more frequently.
Maximum Passband Attenuation
Minimum Stopband Attenuation
• Here, the maximum value of the magnitude in the passband
assumed to be unityy
•
- Maximum passband deviation, given by the
minimum value of the magnitude in the passband
• 1/A -Maximum stopband magnitude
4-1-32
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Analog Lowpass Filter Design
• Two additional parameters are defined:
1) Transition ratio k = p/s
for a lowpass filter k < 1
2) Discrimination parameter
usually
ll k1 << 1
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Butterworth Approximation (1/5)
• The magnitude-square response of an N-th order analog
l
lowpass
B tt
Butterworth
th filt
filter is
i given
i
b
by
• First 2N 1 derivatives of |Ha(j)|2 at = 0 are equal to
zero
• The Butterworth lowpass filter thus is said to have a
maximally flat magnitude at = 0
maximally-flat
• Gain in dB is G() = 10log10|Ha(j)|2
• As G(0) = 0 and G(c) =10log10(0.5)
(0 5) = 3.0103
3 0103 ؆ 3dB
3dB
• c is called the 3-dB cutoff frequency
4-1-34
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Butterworth Approximation (2/5)
• Typical magnitude responses with c = 1
• Two parameters completely characterizing a Butterworth
lowpass filter are c and N
• These are determined from the specified banedges p and
s, and minimum passband magnitude
, and
maximum
i
stopband
t b d ripple
i l 1/A
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Butterworth Approximation (3/5)
• c and N are thus determined from
You can also use log
instead of log10 for both
• Solving the above we get
• Since order N must be an integer, value obtained is
rounded up to the next highest integer
• N is used next to determine c by satisfying either the
stopband edge or the passband edge spec exactly
• If the stopband edge spec is satisfied, then the passband
edge spec is exceeded providing a safety margin
4-1-36
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Butterworth Approximation (4/5)
• Transfer function of an analog Butterworth lowpass filter is
given
i
b
by
where
• Denominator DN(s) is known as the Butterworth
polynomial of order N
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Butterworth Approximation (5/5)
• Example- Determine the lowest order of a Butterworth
l
lowpass
filt with
filter
ith a 1-dB
1 dB cutoff
t ff frequency
f
att 1 kHz
kH and
da
minimum attenuation of 40 dB at 5 kHz
• Now
No
which
hi h yields
i ld 2 = 0.25895,
0 25895 and
d 10l
10log10(1/A
10(1/A2) = 40
40
Which yields A2 = 10,000
and
• Therefore
Hence
• We choose N = 4
4-1-38
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Introduction to Digital Signal Processing
Comparison of Magnitude Responses of
Different Analog Filters
2009/4/23
H D sˆ
H LP s
|
s F
s
HIntroduction
H D Signal
sˆ |Processing
LP s to Digital
ˆ 1
s F sˆ
45
46
Steps involved in the design process:
Step 1 – Develop specifications of a prototype analog
lowpass filter HLP(s) from specifications of desired analog
filter HD(s) using a frequency transformation
Step 2 – Design the prototype analog lowpass filter
Step 3 – Determine the transfer function HD(s) of desired
filter by applying inverse frequency transformation to HLP(s)
Let s denote the Laplace transform variable of prototype
analog lowpass filter HLP(s) and denote the Laplace
transform variable dž of desired analog filter HD(dž)
s F sˆ
Then
Note that there is a tradeoff between width of the transition band
and the smoothness of the filter.
For the same set of design requirements, the lowest order filter
can be achieved using an elliptic filter. But the elleptic filter has
ripples in the passband and the stopband. A butterworth filter will
need the highest order of all types to satisfy the same
Design of Analog Filters
requirements. But it is smooth in both bands.
„
„
„
2009/4/23
23
Design of Analog Highpass,
Bandpass, and Bandstop Filters
• Steps involved in the design process:
Step 1 – Develop specifications of a prototype analog
lowpass filter HLP(s) from specifications of desired analog
filter HD(s) using a frequency transformation
Step
p 2 – Design
g the p
prototype
yp analog
g lowpass
p
filter
Step 3 – Determine the transfer function HD(s) of desired
filter by applying inverse frequency transformation to HLP(s)
• Let s denote the Laplace transform variable of prototype
analog lowpass filter HLP(s) and denote the Laplace
transform variable dž of desired analog filter HD(dž)
• Then
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Analog Highpass Filter Design (1/2)
• Spectral Transformation:
where p is the passband edge frequency of HLP(s) and
is the passband edge frequency of HHP(dž)
g
y axis the transformation is
• On the imaginary
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Analog Highpass Filter Design (2/2)
• Example - Determine the lowest order of an Butterworth
lowpass filter with the specifications:
• Choose p = 1, then
• Analog lowpass filter specifications:
p = 1,, s = 1,, p = 0.1 dB,, s = 40 dB,,
• Code fragments used
[ Wn]] = buttord(1,
[N,
( 4, 0.1, 40, ‘s’);
)
[B, A] = butter(N, Wn, ‘s’);
[num, den] = lp2hp(B, A, 2*pi*4000);
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4-1-47
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