3F3 – Digital Signal Processing (DSP), January 2009, lecture slides 6a, Dr Elena Punskaya, Cambridge University Engineering Department

1. Basics of Analogue Filters
Elena Punskaya
www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by
Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
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2. Analogue Filters
• Specified in a manner similar to digital filters (although
frequencies are specified in the Ω domain (in rad/s))
ωp – pass-band edge frequency
ωs – stop-band edge frequency
δp – pass-band ripple
ds – stop-band attenuation
• The pass-band magnitude response is usually required to be in
the range [1-dp, 1] – matter of convenience, can be adjusted to
make the pass-band ripple symmetrical with respect to
• Expressed as
Ap= -20log10(1-δp) dB
As= -20log10δs dB
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3. Analogue Filter Parameters
√( )
10 0.1Ap -1
• The discrimination factor 10 0.1As -1
The selectivity factor ωp/ωs
•
• The -3dB cutoff frequency – at which the magnitude response of
the filter is 1/√2 of its nominal value at the bass band
• The asymptotic attenuation at high frequencies
20(p-q) dB/decade
p,q – numerator and denominator degrees
(not defined for a digital filter as the frequency
of interest is in the range from [–π,π]
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4. Analogue Filter Prototypes
Analogue designs exist for all the standard filter types (lowpass,
highpass, bandpass, bandstop). The common approach is to define a
standard lowpass filter, and to use standard analogue-analogue
transformations from lowpass to the other types, prior to performing the
bilinear transform.
It is also possible to transform from lowpass to other filter types directly in
the digital domain, but we do not study these transformations here.
Important families of analogue filter (lowpass) responses are described in
this section, including:
• Butterworth
• Chebyshev
• Elliptic
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5. Butterworth Filter
Maximally flat frequency response near W=0
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6. Nth-order Butterworth Filter
An Nth-order lowpass Butterworth filter has transfer function H(s) satisfying
This has unit gain at zero frequency (s = j0), and a gain of -3dB ( = √0.5 ) at s = jΩc.
The poles of H(s)H(-s) are solutions of
N=3 N=4
Imag(s)= ω Imag(s)= ω
X
X
X X
ωc ωc X
X
i.e. at Re(s) Re(s)
X X
X
X
X
X X X
as illustrated on the right for N = 3 and N = 4:
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7. Butterworth Filter Poles
Clearly, if λi is a root of H(s), then - λi is a root of H(-s).
For a stable filter, the poles of H(s) must be those roots lying in the left half-plane,.
The frequency magnitude response is obtained as:
H ( jω )H (− jω ) = H ( jω ) =
1
2
1 + (ω ω C )
(*)
2N
Butterworth filters are known as quot;maximally flatquot; because the first 2N-1 derivatives of (*)
w.r.t. ω are 0 at ω = 0.
Matlab routine BUTTER designs digital Butterworth filters (using the bilinear transform):
[B,A] = BUTTER(N,Wn) designs an Nth order lowpass digital Butterworth filter and
returns the filter coefficients in length N+1 vectors B and A. The cut-off frequency Wn must
be 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate.
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8. Chebyshev Filter
Chebyshev – equiripple response in pass-band (up to ωc),
monotonically decreasing in stop-band
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9. Chebyshev Filter
Chebyshev filters are characterised by the frequency response:
where Tn(Ω) are so-called Chebyshev polynomials.
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10. Elliptic Filter
Equiripple in both pass-band and sto-band
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11. Elliptic Filter
Elliptic filters allow for equiripple in both pass and stop-bands. They are
governed by a similar form:
Where E(Ω) is a particular ratio of polynomials.
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12. Other Types of Analogue Filter
Other filter types include Bessel filters, which are almost linear
phase.
In general, there is a wide range of closed form analogue filters.
• Some are all-pole; others have zeros.
• Some have monotonic responses; some equiripple.
• Each involve different degrees of flexibility and trade-offs in
specifying transition bandwidth, ripple amplitude in pass-
band/stop-band and phase linearity.
For a given bandedge frequency, ripple specification, and filter
order, narrower transition bandwidth can be traded off against
worse phase linearity
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13. Thank you!
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