Normalized Lowpass Filters

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Normalized Lowpass Filters
All-pole lowpass filters, such as Butterworth
and Chebyshev filters, have transfer functions of
the form
Where N is the filter order.
Filter functions are tabulated in normalized
form
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Normalized Lowpass Filters
Normalized form means the tabulated functions are
for filter prototypes with
If
we can write
so
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Normalized Lowpass Filters
Unity gain means that in the transfer function
An Nth order filter has N poles.
If N is odd, one pole is purely real (Its
imaginary part is zero, so it lies on the real
axis.
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Normalized Lowpass Filters
If N is even, no pole lies on the real axis.
If a pole is not on the real axis (its imaginary
part is not zero) then its complex conjugate is
also a pole.
If N is even, the transfer function may be
factored into
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Normalized Lowpass Filters
If N is even, the transfer function may be
factored into
For Butterworth filters, e 0
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Normalized Lowpass Filters
If N is odd, the transfer function may be
factored into
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Normalized Lowpass Filters
Second order
1
1
-1
-1
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Normalized Lowpass Filters
Third order
jw
x
1
1
-1
x
s
x
-1
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Normalized Lowpass Filters
The pole locations are tabulated for Butterworth
filters of other filter orders, and for Chebyshev
filters of orders up to 8 and various ripple
factors, in the textbook. Specialized filter
references contain far more extensive tabulations
for these and other filter types (Bessel,
elliptic, etc.)
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Lowpass to Lowpass Transformation
Denormalizing the normalized filter
We will denormalize a prototype lowpass filter
(Wc 1) by scaling it so its cutoff frequency
is wc. Take the normalized transfer function
H(s), and replace s with
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Lowpass to Lowpass Transformation
Denormalizing the normalized filter
For a second-order Butterworth, the normalized
prototype is
If were designing a filter with
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Lowpass to Lowpass Transformation
Denormalizing the normalized filter
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Lowpass to Lowpass Transformation
Denormalizing the normalized filter
This illustrates how we can denormalize a
complex-conjugate pole pair, or second-order
section.
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Problems

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