This is the first in a series of articles describing applications of the LTC1562 connected as a lowpass, highpass or bandpass filter with added stopband notches to increase selectivity. Part 1 covers lowpass filters.

Lowpass filters with stopband notches are useful in applications seeking steep attenuation in the vicinity of the cutoff frequency. When compared to classical all-pole realizations (such as Butterworth or Chebyshev) they are more “efficient”; that is, they meet a given attenuation requirement with the least number of poles.

Lowpass filters with stopband
notches (broadly referred to as
“elliptics” or “Cauers”) can be designed
with the aid of some literature or with
commercially available software. The
new FilterCAD^{™} for Windows^{®} program,
supplied free of charge by Linear
Technology Corporation, is an excellent
example.

For instance, a 100kHz lowpass filter with a 0.1dB passband ripple and 40dB attenuation at 200kHz can be realized with a 6th order Chebyshev or a 4th order textbook elliptic. Curves A and B of Figure 1 illustrate the respective amplitude responses.

When considering the practical implementation of the filters of Figure 1 (curves A and B), in the author’s experience, it is easier to implement the higher order all-pole filter (curve A), rather than the 4th order version with the stopband notches. The realization of deep stopband notches may result in hardware complexity. This is especially true if a discrete R-C active implementation is chosen and if a single 5V supply and a wide input dynamic range are required.

Nevertheless, curve C of Figure 1 is of particular interest because of its rather simple hardware implementation. Curve C is derived from the classical elliptic response, curve B, where the high frequency notch is “pushed” to infinity and the highest Q pole pair is readjusted to maintain passband flattness. The penalty is the slight gain roll-off at the cutoff frequency, which, for many applications, is acceptable. For sake of simplicity the amplitude response of the filter of Figure 1, curve C, is called a “p-e” (pseudo-elliptic) response.

Figure 2 illustrates the group delay responses of the three filters of Figure 1, with the same curve letter designations. The group delay of curve C is the flattest.

Hardware Implementation

High order filter realizations were a subject of passionate interest in the 1960s and ’70s. One very popular method, which stems from the simplicity of its hardware implementation, consists of decomposing a high order filter polynomial into cascaded second and first order polynomials. Each polynomial is then implemented with commercially available active and passive components. The major drawback of the “cascaded” method is the relatively high Q of at least one of the 2nd order sections and the resulting requirement for precision components for its realization.

Using the “cascading” principle outlined above, the “p-e” response of curve c, Figure 1, can be treated as a self-contained 4th order block and two ( or more) of these blocks can be cascaded to form an 8th order (or higher) lowpass filter with two (or more) stop-band notches. This interesting novelty is driven by the simplicity of its hardware realization; it requires, however, the transformation of an 8th order classical elliptic lowpass response into two cascaded 4th order “p-e” responses.

Figure 3 shows a compact hardware
implementation of the 4th order “p-e”
filter using one half of the LTC1562
quad Operational Filter IC, which was
introduced in the February 1998 issue
of *Linear Technology* magazine.^{1} Two
2nd order sections form the 4th order
filter function. A phase-shifting external
capacitor, C_{IN1}, and a feedforward
path through resistor R_{FF2}, create the
desired notch.

To make the circuit technique of Figure 3 intuitively obvious, consider the following:

A signal of a given frequency can be notched if it is phase shifted by 180 degrees and then summed with itself. If the summation is governed by equal gains, a complete signal cancellation occurs and the notch depth, at least in theory, is infinite.

A phase shift of 180 degrees at a
single frequency, f_{O}, is easily provided
by a second order inverting
bandpass filter; hence, in Figure 3, if
C_{IN1} equals zero, a notch is formed as
the bandpass output (pin 2) is
summed with the input via (R_{IN2}, R_{FF2}).
Moreover, if the summation has equal
gains (1), the notch should, in theory,
have infinite depth.

In Figure 3, an external capacitor,
C_{IN1}, is added to provide additional
phase lead, so that the frequency of
the notch is higher than the center
frequency, f_{O1}, of the second order
section used to create it.

The notch frequency, f_{n1}, is directly
proportional to the center frequency,
f_{O1}, and indirectly proportional to the
time constant (R_{IN1} • C_{IN1}) divided by
the (R_{Q1} • C) (C is an internal capacitor
of 159pF); therefore:

A step-by-step algorithm for building compact “p-e” lowpass filters with the new LTC1562 quad Operational Filter building block is outlined below:

Start with a set of two (lowpass) pole pairs and one finite stopband notch. Arrange the pole pairs in ascending order of Q values.

Example 1:

1. Calculate the frequency-setting resistor, R21:

2. Calculate the Q-setting resistor,
R_{Q1}:

Note: The calculations for R21 and
R_{Q1} are from the LTC1562 Data Sheet;
they are applicable to any 2nd order
section using the LTC1562 proprietary
architecture.

3. Calculate the input resistor, R_{IN1},
from the following expression:

Make sure that R_{IN1} > R21; If not,
make R_{IN1} = R21 and proceed to Step
4a.

Note: R21/R_{IN1} is the DC gain from
the input to the lowpass output of the
first building block, pin 3. The expression
for R_{IN1} ensures optimum
dynamic behavior of all nodes of the
LTC1562.

4a. Use the value of R_{IN1}, calculated
above, and calculate the value of
the input capacitor C_{IN1} from the
notch equation (2).

Use a commercially available NPO-type
0402 surface mount capacitor with the value nearest the ideal value
of C_{IN1} calculated above. For instance,
if C_{IN1 (ideal)} is 60.14pF, choose an off-the-shelf
56pF standard value.

4b. Recalculate the value of R_{IN1}
after a C_{IN1} of 56pF is chosen.

5. Calculate the frequency- and Q-setting
resistors R22, R_{Q2}, as
done in steps 1 and 2, above.
Choose the closest 1% standard
resistor values.

6. Calculate the feedforward
resistor, R_{FF2}:

7. Calculate the input resistor R_{IN2},
to satisfy the gain condition for
the notch (1).

Make the practical value of R_{IN2} as
close as possible to the value calculated
above; otherwise, the stopband
notch depth will be affected.

An Example Using FilterCAD

The following is a comprehensive example of how to synthesize and realize a complex lowpass filter using two “p-e” 4th order sections in cascade. FilterCAD for Windows will be used to synthesize the filter.

A classical 8th order, 100kHz lowpass
elliptic filter with theoretical
passband ripple, A_{MAX}, of 0.005dB,
and a minimum stopband attenuation,
A_{MIN}, of 85dB at twice cutoff, can
be synthesized by cascading four biquadratic
2nd order sections, as
shown in Table 1. Each biquadratic
section comprises a complex pole pair
of center frequency f_{O}, and an imaginary
zero pair of notch frequency f_{n}.
The amplitude response is shown in
Figure 4, curve A. The filter above is easily transformed into two cascadable
4th order “p-e” sections by
performing the following steps.

- Set the two highest notch frequencies to infinity and expect a decrease in stopband attenuation as well as gain peaking in the vicinity of the cutoff frequency (Figure 4, curve B).
- Use the interactive capability of FilterCAD to increase the frequency of the right hand notch (Figure 5 curve C), until the stopband ripple has equal peaks.
- Use the interactive capability of FilterCAD to flatten the passband by lowering the Qs. Start with the highest Q, then proceed with the second highest, then the third.

f_{O} |
Q | f_{n} |

61.8049e3 | 0.5471 | 957.9224e3 |

81.2817e3 | 0.9230 | 343.0259e3 |

99.9948.e3 | 1.9047 | 235.4796e3 |

109.8890e3 | 6.4428 | 203.3896e3 |

Table 2 illustrates the parameters of the transformed filter. Compared to Table 1, two notch frequencies are set to infinity, one notch frequency has been increased and the three highest Qs have been reduced. Figure 5, curve C, illustrates the amplitude response of the transformed filter. The original filter shown in Figure 4 is also shown in Figure 5, curve A, for comparison. The main difference between curves A and C is the theoretical stopband attenuation. Curve C, with its lower Q, will also exhibit improved transient behavior.

f_{O} |
Q | f_{n} |

61.8000e3 | 0.5471 | ∞ |

81.2800e3 | 0.9046 | ∞ |

99.9900.e3 | 1.7555 | 250.6400e3 |

109.8800e3 | 5.874 | 203.3900e3 |

A Practical Case

The high Qs of the previous synthesized filters ensure, at least in theory, passband flatness all the way up to the cutoff frequency. In practice, errors occur in the vicinity of the filter cutoff. They are most often manifested as gain peaking and they are caused by the tolerances of the passive components and the finite bandwidth of the active circuitry. The gain peaking at the filter cutoff can be addressed by predistorting the high Q section, that is, by intentionally lowering the Q so that the theoretical response will show some gain rolloff at the cutoff frequency.

The synthesized filter of Table 2
can be efficiently realized by two cascaded
“p-e” 4th order sections, as
illustrated in the block diagram, Figure
6. Note the arrangement of the
pole-zero pairs of Figure 6 and compare
it with Table 2. In Table 2, the
sections appear in order of increasing
f_{O} and Q. In Figure 6, within each 4th
order “p-e” filter, the 2nd order section
with the highest Q is placed first;
the 4th order “p-e” filter containing
the highest Q is cascaded last. The
notches (f_{n1} and f_{n2}) are so arranged
that the highest frequency notch is
formed from the pole pair whose center
frequency (f_{O}) is closest to the filter
cutoff frequency. For example, the
250kHz notch is placed with the
99.99kHz pole pair. This nonobvious
arrangement allows for a stopband
attenuation approaching the theoretical
values. The highest Q of 5.87 is reduced to 3.97 for reasons mentioned
above and for improving the transient
response of the circuit. See Figure 7
for the amplitude response; note the
slight rolloff at the cutoff frequency.
Figure 8 shows the complete hardware
realization using all four sections
of an LTC1562 continuous-time quad
Operational Filter IC. The algorithm
outlined above was followed to calculate
the values of the external passive
components. The circuit occupies as
much real estate as a U.S. dime. This
is quite significant considering the
cumbersome alternative of a fully discrete
realization with op amps, Rs
and Cs.

Experimental Results

Figure 7, curve A, shows the amplitude
response of the filter hardware
illustrated in Figure 8. No attempt
was made to adjust any component.
Both notches are fully resolved, but
due to the tolerances of the components
and the finite bandwidth of the
active circuitry, the stopband attenuation,
although impressive, is 2dB
above the theoretical value. Subsequently,
the value of R_{Q1} was lowered
to 16.2k (curve B) to better define the
first notch. The filter reaches attenuation
levels beyond 85dB all the way
up to 0.5MHz input frequencies. The
measured attenuation at 1MHz was
still better than 78dB. The dynamic
range of the circuit is quite impressive:
the measured wideband noise
was 40µV_{RMS} and the THD for 1V_{RMS}
and 50kHz input signal was better
than –80dB.

^{1} Hauser, Max. “Universal Continuous-Time Filter Challenges Discrete Designs.” *Linear Technology* VIII:1 (February 1998), p.1.