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Agilent
Network Analysis Solutions
Advanced Filter Tuning Using
Time Domain Transforms
Application Note 1287-10
Introduction
The level of experience and The basic technique has been This application note reviews
expertise required to accurately comprehensively covered in these time domain tuning
tune coupled-resonator cavity Agilent application note 1287-8, techniques, and extends the
filters, cross-coupled filters, and which also describes how technique for use in tuning
duplexers effectively precludes coupled-resonator band-pass filters with cross-coupled
these devices from mass produc- filters can be easily and deter- resonators that produce
tion at high speed. Ironically, ministically tuned. To achieve transmission zeros near the
these same filters are increasing- the proper passband response, filter passband, as well as
ly needed in large quantities, as and to achieve good return loss duplexer filters that have a
a result of the spectral density and passband ripple, the center common (antenna) port, an
resulting from the runaway suc- frequency of each resonator is upper passband (transmit) port,
cess of wireless communications precisely tuned, and each and a lower passband (receive)
services. The time required to coupling between resonators port.
tune these filters accurately lim- precisely set. The method is
its manufacturers from increas- based on the time-domain Together with application note
ing their production volumes response of a filter's return 1287-8, this application note
and reducing manufacturing loss, in which the time-domain provides a comprehensive
cost. Fortunately, it is possible to response is obtained by a special compilation to filter tuning in the
dramatically reduce both the type of discrete inverse Fourier time domain, including theory,
time required to tune these transform of the frequency application, set-up, and tuning
types of filters, as well as the response. Readers are encouraged procedures.
experience and expertise to review the material contained
required. The method removes in application note 1287-8 for
filter tuning from the realm of information about the basic
art, and makes the process pre- technique and how it is applied
dictable and repeatable. Even to tuning coupled-resonator
relatively inexperienced filter cavity filters.
tuners can tune multiple-pole
filters with great success with a
minimal amount of training.
2
The technique defined
A five-pole coupled resonator fil-
ter with four coupling structures
will be used to illustrate the
basic tuning technique. A
schematic of the filter is shown
in figure 1, with the distributed
loss of the filter represented as
shunt resistance. To apply the
Figure 1.
tuning method, the network
analyzer's frequency sweep is
centered at the desired center
frequency of the bandpass filter.
The frequency span is set to two
to five times the expected filter
bandwidth. The bandpass mode
of time domain transform is
applied to the return loss trace.
Figure 2 shows the frequency
response and the bandpass
mode time response of the filter,
a fifth-order Chebyshev with
0.25 dB of passband ripple.
Each plot shows two traces. The
lighter (red) one is the filter
return loss response with ideal
values for all the components,
and the darker (blue) trace
shows the effect of mistuning
one of the resonator elements
(in this case, the second res-
onator). The upper plot is the
frequency response and the
lower plot is the time domain
response. Notice the
Figure 2.
distinctive dips in the time
response S11 of the filter The essence of the tuning tech- domain response with only the
(indicated by the triangles nique is that the dips in the time second resonator mistuned from
labeled 1-5). These are charac- domain response correspond its ideal (derived) value. In this
teristic nulls that occur if the exactly to each resonator in the case the capacitor CII was tuned
resonators are exactly tuned. If filter. When the resonator is to a few percent above its ideal
the center frequency of the tuned properly, the null is deep. value. It is clear that the dip has
measurement is changed even If the resonator is not tuned, the nearly disappeared. The dip will
slightly, the nulls start to disap- null starts to disappear. Though only be maximized when the
pear, indicating that the filter is it may seem remarkable that this capacitor is returned to its
no longer tuned. The peaks exact relationship exists, exten- ideal value. Note that mistuning
between the nulls relate to the sive testing with many different one resonator can affect the
coupling factors of the filter. kinds of filters, as well as simu- response from the other
This type of response holds true lations and direct mathematical "downstream" resonators.
for any all-pole filter, regardless derivation, confirm this relation-
of filter type. ship. Figure 2 shows the time
3
Basic tuning method
The basic time domain tuning This first adjustment will exactly 3. Adjust the next coupling from
method for simple all-pole center the filter and provide the input and output to match
filters, is to measure the time optimum tuning for the given the associated peak in the
domain response of S11 and S22 coupling factors. Many filters template response. Readjust
of the filter. The filter resonators have adjustable coupling factors the resonators adjacent to this
are adjusted with the following that must be tuned to generate coupling to restore the nulls
steps: the desired filter response, par- to be as deep as possible.
ticularly bandwidth and return
1. Starting with the first and last loss. The coupling adjustment 4. Continue in this manner until
resonators, tune to create a can be accomplished with the all couplings have been
deep null in the time responses following steps: adjusted to match the peaks
of the S11 and S22 measure- of the filter template, and all
ments respectively (the nulls 1. Create a filter template by resonators have their
will be at approximately t=0). measuring an existing tuned associated nulls as deep as
filter or from a filter simula- possible.
2. The next resonator from the tion, and load it into the net
input and output are then work analyzer's memory Note that adjusting one coupling
tuned for deep nulls (which traces for S11 and S22. will affect all couplings that
will appear approximately at follow, so it is important to start
t=1/BW where BW is the filter 2. After the initial resonator with the couplings at the input
bandwidth). Tuning the tuning described above, adjust and output and work toward the
second resonator will slightly the input and output coupling center.
pull the first, since they are to match the amplitude of the
coupled. first peak of the target S11
and S22 filter response.
3. The previous resonators (first Readjust the first and last
and last, in this case) are resonator to restore the first
readjusted to restore the null S11 and S22 nulls to make
in the time domain trace to them as deep as possible.
make it as deep as possible.
4. Continue in this manner,
working in toward the center,
until all the resonators have
been adjusted for a deep null.
4
Tuning filters with
cross-coupled
resonators
For many communication appli-
cations, it is necessary to make a
filter skirt response steeper than
normally obtained by all-pole
type filters. Discrete transmis-
sion zeros (where the S21 goes Figure 3.
to zero) can be obtained in the
filter stopband by adding cross-
coupling (coupling between res-
onators other than nearest
neighbors). The number of res-
onators that the coupling "skips
over" will determine the charac-
teristics of the transmission
zeros. Skipping over an odd
number of resonators, as seen in
figure 3, results in an asymmet-
ric frequency response, with a
zero on only one side of the
passband. Skipping over two res-
onators results in transmission
zeros on both sides of the pass-
band. The time domain response
of these filters differs from the
all-pole filters, in that tuning the
characteristic nulls to be as deep
as possible does not result in the
Figure 4.
filter being properly tuned.
Figure 4 shows the frequency The filter was optimized for
and time response of the four- return loss in the passband and
pole filter with asymmetric cross rejection in the upper stop band.
coupling from figure 3. The filter, Notice from the time response
in this case, had coupling adjust- that the nulls are not deep for
ments for only the input, output, many of the resonators. The
and cross-coupling. The coupling design methods for simple,
between resonators was fixed. all-pole filters help illustrate
why this is so, and how to tune
these filters.
5
All-pole filters
All-pole filters are designed by
starting with a low-pass proto-
type filter, then applying a trans-
form to shift it up in frequency
from "DC-centered" to the
desired center frequency. The
essence of the design process is
that the coupling values are
derived only from the low-pass
prototype component values.
The resonator values are derived
by making the resonant frequen-
cy of the node (which includes Figure. 5. The time domain response separates
the input and output coupling) the response from each node.
equal to the center frequency of
the filter. For example, in the fil- To illustrate this point, consider time zero, the reflection from
ter in figure 1, the resonant fre- the response of a filter to an node 2 will look as though both
quency of the second node is impulse, as shown in figure 5. C12 and C23 are grounded. The
defined by the elements L2 in As the impulse proceeds though delay between these pulses will
parallel with CII plus C12 and each node of the filter, part is be due to the coupling, so less
C23 (the coupling elements), and reflected and most is transmit- coupling (which results in a
it exactly equals the filter center ted. If the filter is uncharged narrower filter) will have more
frequency. This is true for all the before the pulse arrives, the delay. This is the same relation-
nodes, including the first and reflection from the first node ship used to design the all-pole
last, which have only one cou- will look as though the coupling filter. It then becomes clear why
pling added. capacitance, C12, is grounded on tuning for deep nulls with the
the far side. That is, the time network analyzer tuned to the
The time domain response of a domain reflection will be the filter center frequency succeeds:
filter node has a deep null when- same as a circuit that is tuned to The response from each individ-
ever the frequency sweep of the the "node frequency" consisting ual node is centered on the same
network analyzer is exactly cen- of C12 + C1 in parallel with L1. frequency.
tered on the resonant frequency Since the pulse goes to zero after
for that node. Further, the time
domain response shows the
response of the filter nodes
separated in time. This separation
is caused by the delay through
each filter section, which Fano
showed to be inversely propor-
tional to the filter bandwidth.
The time domain response will
have sufficient resolution if the
frequency sweep is at least twice
as wide as the filter bandwidth.
6
Effects of cross coupling
With cross-coupling added to the
filter, the time domain response
no longer has the simple
relationship to filter tuning.
Further, especially in filters with
asymmetric transmission zeros, VNA center frequency tuned
for deepest null on resonator 2
tuning of the filter is not optimum
when each node frequency is VNA center frequency =
tuned to the filter center filter center frequency
frequency. Recall that the node Time ( ns)
frequency is defined to be the
resonant frequency of the node Figure 6. Change in time response when the
with all connected couplings, VNA center frequency is tuned.
including cross coupling,
grounded. The resonators are The argument still holds for the Figure 6 illustrates the time
often "pulled" to compensate for time response of any particular response of the filter tuned at
the effect on the pass-band of node of a filter having a deep the filter center frequency, and
the transmission zeros in the null when the node frequency is then tuned to a frequency that
stopband, thus achieving the exactly centered on the network maximizes the null associated
desired passband return loss analyzer frequency. The difficul- with resonator 2 (one of the res-
specification. This results in an ty with these complex filters is onators with cross coupling).
asymmetric shape to the return that the node frequencies are no
loss, as demonstrated in figure 4. longer easy to determine. But This process is repeated for each
Tuning for deep nulls results in the network analyzer itself can of the filter's resonators, adjust-
a filter that does not meet the be used, on a properly tuned or ing the VNA center frequency
return loss specifications. "golden" filter, or on a simulated until each null is maximized. For
However, the discussion about filter, to discover the individual best sensitivity, the frequency
figure 5 points to a method that node frequencies. This is done span is reduced to just two times
will allow tuning filters with by setting up the vector network the bandwidth. Table 1 gives the
cross coupling in the time analyzer (VNA) in dual-channel node frequencies determined for
domain. mode, with one channel on fre- each resonator for the filter from
quency domain and one on time figure 4. Armed with this infor-
domain. The center frequency of mation, and using the measure-
the VNA is adjusted while look- ment from figure 4 as the tuning
ing at the null associated with a template, a filter tuning process
particular resonator. When the for complex filters can be
null is maximized, that frequen- defined.
cy is recorded as the node
frequency for that resonator.
Table 1. Node frequency for each resonator
Resonator no. Node frequency
1 836.25 MHz
2 833.85 MHz
3 834.55 MHz
4 836.45 MHz
7
Tuning of complex filters
The filter from figure 4, with all 2. Adjust the coupling to align 4. Finally, to get the resonators
four resonators, the input and the time domain response tuned to their correct final
output coupling, and the cross peaks with those of the target values, set the VNA center
coupling detuned, is used to filter, remembering to readjust frequency to that listed in
demonstrate this process. the resonators to get deep table 1 for each resonator,
nulls. Figure 8 shows the and tune that resonator for
1. Assuming that the input and result of coupling adjustment. maximum null. After a first
output coupling is sufficient pass, go back again and retune
to produce an approximate 3. Adjust the cross coupling to each resonator to account for
filter shape, start by tuning set the zero frequency to the pulling effect of tuning the
the filter as though it were an match the S21 frequency other resonators. Figure 10
all-pole filter. Figure 7 shows response target, as shown in shows the final result of
the frequency response before figure 9. tuning this filter. It is clear
any tuning, and after the res- that the final response is
onators (but not coupling) nearly identical. Remember
have been adjusted for that the return loss tuning
maximum nulls. was done entirely in the time
domain.
S21 &
S21 and S11 S11
after first tuning Untuned
S11 Target
Freq .
MHz Freq . MHz
S11 before tuning
S11 after first tuning
S11 Target S11 Target
Time ( ns ) Time ( ns )
Figure 7. Figure 8.
S21 Target
Freq . MHz
Fr eq . MHz
Figure 9.
Time ( ns )
8 Figure 10.
Duplex filter tuning
Duplex filters (sometimes called
duplexers), as seen from the
antenna port, have two paths
that contribute to the return loss
response, each with its own
delays and responses. The task
for the filter tuner, and the focus
of this section, is to separate
these responses so that each
side of the filter can be deter-
ministically tuned.
Figure 11.
Duplex filters are used primarily
to separate the transmission
channel (Tx) from the receive
channel (Rx) in a wireless com-
munications base station.
Because the Tx and Rx are near-
ly adjacent, the filters tend to be
very asymmetric to create sharp
cutoffs for each band. Figure 11
shows the schematic of such a
duplexer. Note that a single
cross-coupling is used in each
side, but that the cross-coupling
is capacitive in one side and
inductive in the other. This gives
a lower transmission zero for the
Rx band (Rx is upper in this
case) and an upper transmission
zero in the Tx band as shown in
figure 12. Figure 12.
Duplexers that have more than resonator responses at the
a bandwidth of separation common port can come from
between the Tx and Rx bands either the Tx side or the Rx side.
are easily tuned with the method
noted above for tuning filters In figure 11, the duplexer uses
with cross-coupling. That is quarter-wave transformers to
because the network analyzer isolate each side of the duplexer
can be centered on the Tx band, (the input impedance of the Tx
with the span at greater than side is a short circuit at the Rx
two bandwidths, and still not frequency). Other topologies
have the Rx band interfere with couple the common port to a
the input or output reflection broader-band common resonator,
response. However, most duplex- which is in turn coupled to the
ers have substantially less than first resonator on both the Tx
one bandwidth between the and Rx sides. With this configu-
edges of the Tx and Rx bands ration, the common resonator
(a typical filter might have an clearly cannot be centered on
80 MHz bandwidth with 20 MHz either the Tx or Rx passbands,
of separation). These types of instead it is centered somewhere
duplexers make time-domain in between.
tuning difficult, because
9
Time domain response of
duplexers
The time domain response of
duplexers is complicated by the
fact that at the common port,
reflections from both the Tx side
and Rx side will cause some
nulls in the time domain. Figure
13 shows the time domain and
frequency response of a real
duplexer. To view the time-
domain response in a way that
makes sense, it is necessary to
set the network analyzer center
frequency to the frequency
between the Rx and Tx pass-
bands. The span of the analyzer
must be set to at least two times
the overall bandwidth of the Tx Figure 13. Top half of display: Upper trace = antenna common
and Rx bands. The following Lower trace = Rx
example of tuning a real duplex Lower half of display: Upper trace = antenna common
Lower trace = Rx
filter uses a duplexer which has
the common port coupled to a
common resonator, which in
turn is coupled to both the
last (5th) Tx resonator and the
last (6th) Rx resonator.
Setting up the tuning process
Just as with the complex filter of
figure 4, the tuning process for a
duplexer requires a properly
tuned prototype filter to allow
the node frequencies and target
couplings to be determined.
However, the nodes will be more
difficult to associate with indi-
vidual resonators, especially
from the common port.
10
Identifying the resonator
The upper half of figure 13
shows that there are more nulls
in the time domain response of
the reflection from the common
port than there are from the Tx
port. The first null is associated
with the common resonator
(figure 14). The second null asso-
ciation is found by changing the
tuning slightly on the last Tx res-
onator, and in the same manner
the last Rx resonator can be
associated with the third null
from the common port (figure 15).
Depending upon the filter, it may
also be possible to identify other ANT
resonators in the Tx or Rx filter,
but soon the nulls become con-
Figure 14. Tuning the common ANT resonator shows a response change primarily in
fusing, with the tuning of one the first null. In this way the first node resonator is determined and the first node
resonator affecting two nulls. frequency can be found by changing the VNA frequency to find the deepest null
Rx 6
Figure 15. Tuning the Rx 6 resonator shows the primary effect at the second null. By
looking for the frequency of the VNA, which makes the null deepest, we know this
node frequency. Note: the next null also shows some effect from tuning
11
Finding node frequencies Table 2. Node frequency for tuned duplexer
Once the association of nulls Common port Tx port Rx port
with resonators has been done Node Freq. Node Freq. Node Freq.
from the common port for the Com 1800 TX1 1747 RX1 1848
RX6 1800 TX2 1749 RX2 1848
last Tx and Rx resonators, the
TX5 1796 TX3 1750 RX3 1851
individual node frequency for
RX5 1805 TX4 1760 RX4 1841
each resonator is found by tun- TX4 1788
ing the analyzer's center fre- RX4 1810
quency until the associated null
is deepest. This frequency is also
recorded for each null while
measuring reflection from the Tx
and Rx ports, and for the first
several nulls from the common
port. These frequencies (in MHz)
are shown in table 2.
Separating Tx and Rx
Responses
These node frequencies will be
used for the final tuning of the
duplexer, but experimental
research shows that it is not
practical to try to tune the
duplexer directly to these fre-
quencies. This is because there Rx 6
is so much interaction from the
Rx 1
Rx side on the Tx response,
especially at the common port,
that the resonators cannot be Figure 16. The upper trace shows the Rx path frequency response with a VNA center
sufficiently isolated unless they frequency selected to obtain the deepest null for the respective resonators. These
frequencies are recorded for the Rx filter tuned and the Tx first resonator tuned low.
are already very close to their
correct values. The solution for
initial tuning is to mistune one Table 3. Node frequency for duplexer with sides isolated
side (say the Tx side) and then Common Port Tx Port * Rx Port **
recharacterize the filter for the Node Freq. Node Freq. Node Freq.
Rx side node frequencies. Figure Com 1803** TX1 1746 RX1 1848
16 shows the response of the 1793*
duplexer with Tx5 (the one RX6** 1829 TX2 1749 RX2 1848
closest to the common port TX5* 1762 TX3 1749 RX3 1850
resonator) mistuned. RX5** 1848 TX4 1787 RX4 1850
TX4* 1738
In figure 16, the VNA center fre- RX4** 1860
quency is changed such that the *Rx untuned; **Tx untuned
null associated with the Rx 6
resonator measured at the com-
mon port is deep (time domain, After the Rx frequencies are Note that from the Tx and Rx
upper trace). This frequency is determined, the Rx 6 resonator ports, the node frequencies are
recorded in table 3 as Rx 6 fre- is set high, and the Tx resonator nearly unchanged, indicating
quency. But with the same filter frequencies are determined in a that these are very nearly isolat-
measured at the Rx port, with an similar way. The precise node ed from their respective other
analyzer center frequency of frequency for each node was sides even in a tuned duplexer.
1850 MHz (time domain, lower recorded in table 3.
trace), each Rx node is nearly a
null.
12
Tuning a filter
A duplexer tuning process
proceeds as follows:
1. Start with resonator RX6
tuned high in frequency. Tune
the Tx side of the filter, and
common port according to the
starred (*) frequencies in
table 3. Tune coupling and
cross-coupling as described in
application note 1287-8.
2. Tune resonator TX5 as low
as possible. Tune the Rx side
of the filter using the double
starred (**) frequencies in
Figure 17. The Rx side of the filter is being tuned here. The upper plot shows
table 3. Figure 17 shows the S11 and S22 of the filter; each set to a different center frequency appro-
tuning starting with Rx1 and priate for the first and last resonator. The lowest plot shows the null from
common. The result of tuning each resonator.
all Rx resonators is shown in
figure 18. The VNA is set to
the common resonator
frequency (about 1800 MHz)
so the Rx nulls don't appear
deep. Here, Tx 5 resonator is
not yet tuned.
3. Final tune TX5 and TX6 to
the frequency in table 2.
Final tune all resonators to
table 2 values. Results shown
in figure 19.
Figure 18.
Figure 19.
13
More complex
filter tuning
Dealing with multiple or
strong cross couplings
In the example filter shown in This is similar to the method pass band response. To do
figure 3, the coupling value for used to isolate the Tx side of the this, one may short out the res-
the cross coupling was much less duplexer from the Rx side. The onators beyond the cross cou-
than the main coupling. For resulting filter has only one path pling, essentially making a new
such cases, the cross coupling for coupling, and can be charac- filter with the cross coupling
does not have a strong effect on terized as an all pole filter. A being the main path through the
the time domain response. "golden" trace of this filter can filter. A "golden" trace may be
However, some filters have very be captured without the cross taken with a filter thus modified,
strong cross coupling (coupling coupling. and the value of the cross cou-
of the same order as the main pling in the time domain may be
coupling), or multiple cross cou- When tuning an untuned filter, it recorded. When tuning an
plings. In these cases, it may be can be set to have the same untuned filter, the process is
necessary to take a different response as the "golden" filter, reversed. The resonators beyond
approach to tuning the filters. with the cross coupling removed. the cross coupling are shorted
All that remains is to set the and the cross coupling is set in
One approach that has been cross coupling back to verify the the time domain. The shorting of
effective is to remove the cross final filter response. the resonators is removed, and
couplings (either by tuning them the filter is tuned as described
to a very low value, or shorting Another option for filters with above. This may be effective in
out cross couplings if they are adjustable cross couplings is to dealing with cross coupling that
not adjustable). This will result set the cross coupling first, is used for linearizing group
in a filter that does not have the before tuning the rest of the fil- delay in filters.
desired shape, but does have ter. This method may be effective
the correct settings for the res- for filters where the cross cou-
onators and main coupling. pling has a strong effect on the
14
Conclusions Other resources
In this application note, we have Tuning coupled resonator
shown ways to extend the time
domain tuning techniques to
cavity filters
1. Joel Dunsmore, "Simplify
more complex filters. These fil-
Filter Tuning Using Time
ters may contain complex trans-
Domain Transforms",
mission responses, with cross
Microwaves & RF, March 1999.
couplings. These filters may also
2. Joel Dunsmore, "Tuning Band
contain multiple paths, such as
Pass Filters in the Time
in duplexers, or even multiplex-
Domain, Digest of 1999 IEEE
ers. While good progress has
MTTS Int. Microwave Sym.,
been made on extending these
pp. 1351-1354.
techniques, there remain many
3. "Simplified Filter Tuning
opportunities for enhancements
Using Time Domain,"
to these methods, and many fil-
Application note 1287-8,
ter types that require further
literature number 5968-5328E
investigation. Agilent
Technologies is continuing
research into the area of filter Tuning cross-coupled
tuning, and will continue to pro- resonator filters
vide state-of-the-art tuning tech- 4. Joel Dunsmore, "Advanced
niques and applications to Filter Tuning in the Time
support innovation in the area of Domain,"Conference
coupled resonator filter design. Proceedings of the 29th
European Microwave
Conference, Vol. 2, pp. 72-75.
Tuning duplexer filters
5. Joel Dunsmore, "Duplex
Filter Tuning Using Time
Domain Transformers,"
Conference Proceedings of
the 30th European Microwave
Conference, Vol. 2, pp. 158-161.
Filter design
6. Zverev, "Handbook of Filter
Synthesis," John Wiley and
Sons, 1967.
7. Williams and Taylor,
"Electronic Filter Design
Handbook, 2nd Edition,"
McGraw Hill Publishers,
Chapter 5, 1988.
15
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