This document discusses different types of filter responses including low-pass, high-pass, band-pass, and band-stop filters. It defines key terms like passband, cutoff frequency, and quality factor. It also covers active filters, filter order, and different approximations for filter design including Butterworth and Chebyshev approximations. The Butterworth filter has a maximally flat passband while the Chebyshev filter trades off ripples in the passband for a steeper roll-off in the transition region.
2. INTRODUCTION
• The frequency response of a filter is the graph of its
voltage gain versus frequency.
• A filter is a circuit that passes certain frequencies and
attenuates or rejects all other frequencies.
• The passband of a filter is the range of frequencies
that are allowed to pass through the filter with
minimum attenuation. (usually defined as less than
– 3 dB of attenuation).
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3. • The critical frequency, fc ,(also called the cutoff
frequency) is defined as the end of the passband
and is normally specified at the point where the
response drops – 3 dB (70.7%) from the
passband response.
• Following the passband is a region called the
transition region that leads into a region called
the stopband.
• There is no precise point between the transition
region and the stopband.
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4. Low-Pass Filter Response
• A low-pass filter is one that passes frequencies
from dc to fc , and significantly attenuates all
other frequencies.
• The passband of the ideal low-pass filter is shown
in the blue shaded area of Figure 15–1(a);
• The bandwidth of an ideal low-pass filter is equal
to fc
BW = fc
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5. A decade of frequency change is a ten-times change
(increase or decrease).
15–1(a);
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7. Poles
• A single RC circuit is called a pole.(In filter terminology).
• Actual filter responses depend on the number of poles.
• The -20 dB/decade roll-off rate for the gain of a basic RC
filter means that at a frequency of 10fc,the output will be
-20 dB (10%) of the input.
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8. • The critical frequency of a low-pass RC filter
occurs when XC = R, where
fc =1/2πRC
• The output at the critical frequency is 70.7%
of the input.
• This response is equivalent to an attenuation
of -3 dB
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10. Active filters
• Filters that include one or more op-amps in the
design are called active filters.
• by combining an op-amp with frequency-selective
feedback circuits, these filters can optimize:
1. the roll-off rate or
2. phase response
• In general, the more poles the filter uses, the
steeper its transition region will be
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11. Order of Filter
• The order of a passive filter (symbolized by n)
equals the number of inductors and capacitors in
the filter.
• If a passive filter has two inductors and two
capacitors, n=4.
• The order of an active filter depends on the
number of RC circuits (called poles) it contains.
n ≈ the number of capacitors
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14. Band-Pass Filter Response
• A band-pass filter passes all signals lying within a band
between a lower-frequency limit and an upper-
frequency limit and essentially rejects all other
frequencies that are outside this specified band.
Bandwidth BW = fc2 - fc1
center frequency
F0 =
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15. Quality Factor (Q)
• The frequency about which the passband is
centered is called the center frequency,
f0, defined as the geometric mean of the
critical frequencies.
• Quality Factor (Q) of a band-pass filter is the
ratio of the center frequency to the bandwidth.
• Q=f0/BW
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16. • The value of Q is an indication of the selectivity of
a band-pass filter.
• The higher the value of Q, the narrower the
bandwidth and the better the selectivity for a
given value of f0.
• Band-pass filters are sometimes classified as
narrow-band (Q > 10)or wide-band(Q < 10).
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17. EXAMPLE 15–1
• A certain band-pass filter has a center frequency of 15
kHz and a bandwidth of 1 kHz. Determine Q and classify
the filter as narrow-band or wide-band
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19. All-Pass Filter
• The all-pass filter is useful when we want to produce a
certain amount of phase shift for the signal being filtered
without changing its amplitude.
• The phase response of a filter is defined as the graph of
phase shift versus frequency.
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21. Approximate Responses
• Attenuation
• Attenuation refers to a loss of signal.
• With a constant input voltage, attenuation is
defined as the output voltage at any frequency
divided by the output voltage in the midband:
Attenuation =vout/vout(mid)
• if the output voltage is 1 V and the output voltage in the midband
is 2 V, then:
• Attenuation = 1 V/ 2 V = 0.5
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22. • Attenuation is normally expressed in decibels using
this equation:
• Decibel attenuation = -20 log attenuation
• For an attenuation of 0.5, the decibel attenuation is:
Decibel attenuation = -20 log 0.5 = 6 dB
• We will use the term attenuation to mean decibel
attenuation.
an attenuation of 3 dB means that the output voltage
is 0.707 of its midband value.
An attenuation of 6 dB means that the output voltage
is 0.5 of its midband value.
An attenuation of 12 dB means that the output
voltage is 0.25 of its midband value.
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23. Passband and Stopband Attenuation
• In filter analysis and design, the low-pass filter is a
prototype,
• Typically, any filter problem is converted into an
equivalent low-pass filter problem and solved as a
low-pass filter problem;
• the solution is converted back to the original filter
type.
• For this reason, our discussion will focus on the
low-pass filter and extend the discussion to other
filters.
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24. • Zero attenuation in the passband, infinite
attenuation in the stopband, and a vertical
transition are unrealistic.
• To build a practical low-pass filter, the three regions
are approximated as shown in Fig. 19-6.
• The passband is the set of frequencies between 0
and fc.
• The stopband is all the frequencies above fs.
• The transition region is between fc and fs.
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25. Five approximations
• The five approximations we are about to discuss are trade-
offs between the characteristics of the passband,
stopband, and transition region.
• The approximations may optimize the flatness of the
passband, or the roll-off rate, or the phase shift
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26. ??????????
• In some fi lters, the attenuation
• at the edge frequency is less than 3 dB. For
this reason, we will use f3dB for
• the frequency when the attenuation is down 3
dB and fc for the edge frequency,
• which may have a different attenuation.
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27. Order of Filter
• The order of a passive filter (symbolized by n)
equals the number of inductors and capacitors in
the filter.
• If a passive filter has two inductors and two
capacitors, n=4.
• The order of an active filter depends on the
number of RC circuits (called poles) it contains.
n ≈ the number of capacitors
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28. 1- Butterworth Approximation
• Also called the maximally flat approximation
because the passband attenuation is zero through
most of the passband and decreases gradually to Ap
at the edge of the passband.
• Well above the edge frequency, the response rolls
off at a rate of approximately 20n dB per decade,
where n is the order of the filter:
• Roll-off = 20n dB/decade
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29. • For example, a first-order Butterworth filter rolls
off at a rate of 20 dB decade,
• a fourth-order filter rolls off at a rate of 80 dB per
decade,
• The major advantage of a Butterworth filter is the
flatness of the passband response.
• The major disadvantage is the relatively slow roll-
off rate compared with the other approximations
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31. 2-Chebyshev Approximation
• It rolls off faster in the transition region than a
Butterworth filter.
• The price paid for this faster roll-off is that
ripples appear in the passband of the frequency
response.
• # Ripples =n/2
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