A Filter Network Arrangement
Field of the Invention
The invention relates to filtering of signals. In particular, the invention relates
to the equalising of attenuation and group delay across the passband of a filter
network arrangement.
Background of the Invention
Communication satellites typically process signals received and transmitted in
a number of communication channels. To separate and combine the
communication channels, the satellite may make use of a number of filters.
Filters involved in signal processing are designed to meet often strict
requirements on signal quality. Gain and group delay variation as a function of
frequency can cause signal degradation. It is therefore desirable that filters
involved in signal processing exhibit close to gain flatness and group delay
flatness where possible.
In non-minimum phase filters, cross-couplings within the filters have been
used to equalise the group delay of the filters. This technique requires a higher
order filter and there is a limit to the percentage of the bandwidth of the filter
which can be corrected.
It is also known to use external networks to equalise group delay across the
passband of a filter. For example, external one-port networks have been used to
add appropriate delays to signals in the passband.
Moreover, it is known that the quality factor (Q) of the resonators of a filter can
be adjusted in order to give some flattening of the passband. The Qof a
resonator is a measure of the strength of the damping of its oscillations. To
obtain a flat passband, very high Qfilter resonators are conventionally used,
which results in a filter with a relatively large size. To obtain a flat passband
with lower Qresonators, pre-distortion, the introduction of complex
transmission zeros or lossy cross couplings in the filter have been suggested.
The use of complex zeros or lossy cross couplings to flatten the passband means
that an increase in the complexity and the order of the filter is required.
The invention aims to improve on the prior art.
Summary of the Invention
According to the invention, there is provided a filter network arrangement
comprising a filter network; and one or more correction networks, wherein the
one or more correction networks is configured to substantially equalise the
passband gain and group delay of the filter network arrangement.
It will be realised that the passband gain and group delay of the filter network
arrangement will not be perfectly flat across the whole passband. However, the
one or more external correction networks are arranged to reduce the variation
in gain and group delay across the passband. The correction network is
designed to have a gain and group delay that equalise the overall gain and the
group delay to provide an approximately flat overall passband group delay and
gain. By group delay, it is meant the derivative of the phase with respect to
angular frequency.
The Qs of the filter network and the one or more correction network may be
r f 1 f 1
selected such that 0.4 <— < 2.5 , where r = — and rf = — , Q is the Q
rf bw Q bw Qf
of the one or more correction networks, Qf is the Qof the filter network and f0
and bw are the centre frequency and the bandwidth of the filter network
respectively. It has been found that if the Qs comply with the above
relationship, the group delay across the passband of the filter network
arrangement is substantially uniform when the gain across the passband is
uniform and a polynomial can be constructed for the one or more correction
networks that equalises both the gain and the group delay across the passband
of the filter network arrangement.
The Qs may be selected such that r is greater than r in order to achieve
flattening of the passband for a greater percentage of the filter bandwidth for a
given order of the one or more correction networks.
Alternatively, the Qof the or each of the one or more correction networks can be
chosen to be the same or substantially the same as the Qof the filter network.
The filter network and the one or more correction network may be realised in the
same or similar medium to ensure that the Qof the one or more correction
networks is the same as the Qof the filter network.
It has been found that if the one or more external correction networks has
substantially the same Q as the filter network, there exist solutions for the one
or more correction networks which will flatten the filter passband and also
equalise the group delay of the filter network arrangement irrespective of the
actual Q. Consequently, the precise Qof the filter network does not need to be
known as long as the one or more correction network has substantially the
same Q as the filter network, for example if the resonators of the one or more
correction networks are of the same type and realised in the same or similar
medium as the filter network. In other words, if the Qof the filter and the at
least one correction network are the same a solution for the at least one
correction network which flattens gain will also flatten group delay. The
relationship is not exact so flattening of the gain will not give precise flattening
of the group delay. A solution for the one or more correction networks may be
selected which gives the best gain flatness, thereby providing a substantially
flat group delay as well. In other words, a correction network that minimises
the variation in gain of the filter network arrangement can be chosen and, as a
consequence, the group delay variation would also be reduced.
The invention allows a filter network with low Qresonators to be used and the
size of the filter network can be reduced compared to filter networks that use
high Qresonators to flatten the passband. Since an external network is added,
the overall size of the filter network arrangement could of course be double the
size of the filter network. However, the overall size would still be smaller or
comparable to the filter network in which the passband of the filter was
flattened by increasing the Q and, according to the invention, the group delay
would also be equalised.
The one or more correction networks may comprise one or more one-port
networks and the filter network may comprise means for connecting the one or
more one-port networks to the filter. At least one one-port correction network
may be required depending on the arrangement of the circuit. In more detail,
the means for connecting the one or more correction networks to the filter
network may comprise a coupler or a circulator. If the connection means is a
coupler, two one-port correction networks would be required.
The order of the or each of the one or more correction network may be the same
or lower than the order of the filter network to minimise the size of the network
arrangement. Alternatively, the order of the correction network may be greater
that the order of the filter network.
The one or more correction networks may be synthesised from a polynomial
H(s), the coefficients of which may be determined through optimising an error
function for the gain and the group delay of the filter network arrangement.
The roots of the polynomial may be optimised by minimising the error function
for the gain on its own. The symmetry of the poles and zeros in the complex
plane may be maintained about the line s=r .
The filter network arrangement may be a low temperature co-fired ceramic filter
network arrangement. The filter network arrangement may be a multi-layer filter
network arrangement.
Alternatively, the filter network arrangement could be implemented as a
microstrip.
According to the invention, there is also provided a processing arrangement for a
communication satellite comprising a filter network arrangement as set out
above.
Furthermore, according to the invention, there is provided a method of obtaining
a filter network arrangement comprising a filter network and one or more
correction networks for equalising the gain and the group delay across the
passband of the filter network arrangement, the method comprising:
determining a polynomial and a Qfor a filter network; choosing a Qfor the one
or more correction networks; selecting a starting polynomial for the one or more
correction networks and adjusting the coefficients of the polynomial to
substantially equalise gain and group delay across the passband of the filter
network; and synthesising the filter network and the one or more correction
networks from the polynomial for the filter network and the polynomial for the
one or more correction networks.
The Qof the one or more correction networks may be selected based on the Qof
the filter network. The Qs of the filter network and the one or more correction
r f 1
networks may be selected such that 0.4 <—< 2.5 , where rc = — and
rf bw Q
r f , Q is the Qof the one or more correction networks, Qf is the Qof
bw Qf
the filter network and f0 and bw are the centre frequency and the bandwidth of
the filter network respectively.
Adjusting the coefficients of the polynomial may comprise adjusting the
coefficients of the polynomial in dependence of the Qs of the filter network and
the correction networks. Determining the coefficients of the polynomial may
comprise minimising the error function
E ( k , ) = fE ( k , k ) + gE ( k , ) , where f, are weighting constants,
and
*=1 ((s * - rc + - k )
where + i are the roots of the polynomial for the correction network,
sm + id)m are the roots of the numerator pr
olynomial for the filter network and
s p + id) p are the roots of the denominator polynomial for the filter network.
When the Qs of the filter network and the one or more correction networks are
the same, the precise values of r and r may be not required for the
determination of the coefficients of the polynomial and only an estimate may
be used.
Minimising the error function may comprise first minimising E ( k , i) k ) to find
approximate coefficients for the correction network polynomial and then
minimising Ei on its own to optimise the roots of the correction network
polynomial.
Finding the coefficients of the polynomial H(s) may comprise finding
coefficients that minimise said error functions while still maintaining the
symmetry of the correction network polynomial about the line s=r .
Brief Description of the Drawings
Embodiments of the invention will now be described, by way of example, with
reference to Figures l to 10 of the accompanying drawings, in which:
Figure l is a schematic block diagram of a system in which a filter network
arrangement is used;
Figure 2 illustrates the phase variation with frequency of a real and an ideal
filter network;
Figure 3 illustrates the gain variation with frequency of a lossy and a lossless
filter network;
Figure 4 is a schematic diagram of a filter network arrangement according to a
first embodiment of the invention;
Figure 5 illustrates a method for designing a correction network according to
an aspect of the invention;
Figure 6a and 6b illustrate a prototype correction network and an example of a
correction network at a later stage in the correction network synthesis process;
Figure 7a and 7b illustrate the loss and the group delay respectively of one
example of a correction network;
Figure 8 illustrates a microstrip layout for one example of a filter network
arrangement;
Figure 9a and Figure 9b illustrate the loss and group delay respectively of the
complete circuit of Figure 8 compared to the loss and group delay of the filter
of the circuit on its own; and
Figure 10 is a schematic diagram of a filter network arrangement according to a
second embodiment of the invention.
Detailed Description
With reference to Figure 1, a high-level diagram of a communication system 1is
shown comprising receiver 2 for receiving signals, a signal processing
arrangement 3 for processing signals and a transmitter 4 for transmitting
signals. The system may be a satellite communication system. The processing
arrangement 3 may comprise one or more filter network arrangements 5. The
filter network arrangement may, for example, comprise a bandpass filter
network and may, for example, be used in a demultiplexer for demultiplexing
received signals into a number of frequency channels. Alternatively, the filter
network arrangements may be used in an up- or down-converter. It should be
realised that a filter network arrangement could alternatively be used for any
other suitable purpose.
The filter network arrangement is configured to let through signals of a specific
frequency range and stop signals with frequencies outside the frequency range.
The filter network arrangement will have some effect on the amplitude and the
phase of the signals passed. For example, the loss of a real filter causes
rounding of the passband amplitude and the amplitude selectivity of the filter
causes a varying group delay.
Group delay is a measure of how long it takes for a signal of a particular
frequency to traverse a network. The group delay is conventionally considered
the derivative of the phase response of the network with respect to the angular
frequency, and, accordingly, the derivative of the phase response with
do)
respect to angular frequency will hereinafter be referred to as the "group
delay".
The dashed line of Figure 2 shows the relationship between the phase and the
frequency for an ideal filter and the solid line of Figure 2 shows the
relationship between the phase and the frequency for a real filter. As indicated
by the dashed line, there is a linear relationship between the phase and the
frequency for all signals that pass through an ideal filter network. In other
words, the group delay for the passband of the filter network is the same
irrespective of the frequency of the signal. However, a real filter will not have a
constant group delay. Instead, the signals at the edges of the filter bandwidth
will undergo a larger phase adjustment in a real filter than in an ideal filter, as
indicated by the solid line of Figure 2.
The dashed line of Figure 3 shows the amplitude response of a lossless filter
and the solid line of Figure 3 shows the amplitude response of a lossy filter. For
equi-ripple filter functions, an nth order lossless filter will have n frequency
points of perfect transmission, between which a small amount of power will be
reflected. As shown in Figure 3, a lossy filter attenuates the amplitude
significantly more than a lossless filter. The solid line of Figure 3 also shows
rounding of the passbands for the lossy filter, caused by the finite quality factor
(Q) of the filter. Loss is generally not a problem as long as it is uniform.
Ideally, the gain across the passband should be constant.
Afilter network arrangement according to embodiments of the invention
exhibits approximately uniform attenuation of signals across the frequencies in
the passband. It will be shown that the filter network arrangement according to
the invention will also have a flattened group delay. In reality, the gain and the
group delay will not be exactly flat. An approximation to flatness in the form of
a ripple function is considered acceptable. The key is that the gain and the
group delay will be significantly flatter for the overall filter network
arrangement than for the filter network on its own. In other words, the
correction network improves the gain and group delay characteristics compared
to the filter network on its own.
With reference to Figure 4, one embodiment of a filter network arrangement
according to the invention is shown. The filter network arrangement 5
comprises a first port 6, a filter network 7, a coupler 8, two correction networks
9a, 9b and a second port 10. The correction networks 9a, 9b are provided to
correct for the varying loss of the filter network 7 across the passband such that
the signals passing through the filter network arrangement 5 are attenuated to
approximately the same extent across the whole bandwidth of the filter. The
correction networks also equalise the group delay. The two correction networks
would effectively be identical.
The filter network 7 of the filter network arrangement 5 may be a microwave
filter. For example, it could be a combline filter, interdigital filter or a
waveguide filter. However, it will be realised that the filter network 7 is not
limited to a microwave filter and it could be any frequency filter. The filter
network 7 could be implemented in ceramics. For example, it may be a low
temperature co-fired (LTCC) ceramic filter. The coupler shown in Figure 4 is a
3dB coupler. According to some embodiments of the invention, the two
correction networks 9a, 9b are both one-port correction networks. In the
example of Figure 4, a signal would be subject to the correction network
whether it is passed from the first 6 to the second port 10 or the second port 10
to the first port 6.
According to some embodiments of the invention, given the transfer function of
the filter network 7, an appropriate correction network may be implemented by
determining a correction network polynomial that will flatten the passband and
synthesising the correction network from the polynomial. It is realised that if
the correction network equalises the gain of the overall filter network
arrangement it will also flatten the group delay, provided the Q of the
correction network is close to the Q of the filter network 7. It is recognised,
according to the invention, that the set of polynomials H(s) that give
approximately flat group delay in the desired range includes polynomials that
also flatten the gain. It is realised that as long as the correction network and
the filter network have similar Qs, by selecting a polynomial that flattens the
gain of the passband the group delay is also flattened. However, if the Qs are
different, an appropriate correction network polynomial can still be found that
substantially equalises both the gain and the group delay of the filter network
arrangement. In other words, a correction network can be synthesised that
simultaneously corrects both the group delay and the gain of the filter network
arrangement. As will be described in more detail below, in the design of the
filter network arrangement, the focus is on flattening the gain and the
flattening of the group delay then follows.
It will now be shown mathematically that the group delay is equalised when the
gain across the passband is equalised, provided that the respective Qs of the
filter network and correction network have suitable values. It will then be
shown how the relationship between the group delay and the loss can be used
to design the correction network. Scattering parameters, also referred to as Sparameters,
will be used to describe the electrical behaviour of the network. In
the discussion below, the filter and correction network are described in the
lowpass domain. The units of angular frequency are in radians/seconds, the
units of the group delay are in seconds and the units of the gain are in decibles.
S of a lossless correction network can be written as
K
Y { k + i ( - k ) )
- s + i w - w ) )
k=l
where H(s) is a Hurwitz stable polynomial representing the transfer function of
the correction network, w is angular frequency, + i are the roots of H(s)
and H(s) have Kroots.
For a lossless purely reactive one-port network, there is symmetry of the poles
and zeros of S around ¾ = 0 and the gain of the correction network can be
written as
where 511 is the complex conjugate of 511 .
If w ad some loss to the correction network we break the unity condition and
where f(s) < 1. According to embodiments of the invention, f(s) of the
correction network is controlled in order to compensate for the rounding of the
passband of the filter network due to loss in the filter network. In words,
additional loss is added to flatten the gain across the passband. Loss in a filter
is generally not a problem as long as it is uniform. The filter network
arrangement would typically be provided next to a low noise amplifier (LNA)
and the extra loss in the filter network arrangement could be compensated for
by increasing the LNA gain.
Mathematically, adding loss corresponds to shifting the poles and zeros of the
polynomial representing the transfer function to the left in the complex plane.
Put a different way, making a network lossy is equivalent to adding a negative
real part to the roots of the network polynomials. The poles and zeros are
shifted to the left in the complex plane by shifting the real part by a
constant amount r , where r is related to the Q of the resonators used in the
final realisation of the correction network as shown below:
where f0 is the centre frequency and bw the bandwidth of the correction
network. The bandwidth of the correction network is the same as the
bandwidth of the filter network. The symmetry of S is maintained for a lossy
network but the line of symmetry is shifted to s = r .
The group delay of a correction network can be written as
The Taylor series for the group delay and the insertion loss with respect to - s
(the real part of s) can be examined to see the effect of adding loss to lossless
networks. As will be shown below, a relationship between the group delay and
the insertion loss which is considered when designing the filter network
arrangement according to the invention can then be found.
The Taylor expansion of the group delay is
gd i = gd + Vgd .r + 0(r 2 )
(6)
where gd is the gradient of the group delay with respect to - s (the real part of
s), and gd is the group delay of the lossy correction network. Due to the
symmetry of the poles and zeros of S of the lossless, purely reactive correction
network, V i = 0 and thus
gd gd , (7)
provided that the higher order terms in the Taylor expansion are small.
Consequently, it can be seen from Equations 6 and 7 that the group delay of the
correction network is affected very little by the addition of a small amount of
loss. Some of the higher order terms in the Taylor series are non-zero and
so gd is not exactly equal to gd .
The Taylor expansion of the gain of Equation 2 gives
V (5i i).r + 0 (r ) (8)
where 511; is 511 of the lossy network and V(511511) is the gradient of 511511
with respect to —s and V2 (51 1511) is the r adient of V(511511) with respect to
- s , etc. It can be found that V(511511) = .
which can be written as V(511511) = -2.51 151 l.gd . It can further be found that
V2 (511511) = 4511511.^ + 2511511.V^ . Considering that Vgd = 0 , it can be
( (2.gd .r )2 (2.gd .r
seen that 5 11,51 = 511511 l - 2.gd r + + . . .
2 ! 3 !
¥ ( m
Since 511511 = 1, it follows that 511,511, = 1+ (2. .r ) , that is
m=l ml
1511 =cxp(-2gd .r ) (9)
Equation 9 shows that the loss of the correction network is a function of its
group delay.
A similar analysis can be carried out for a filter network. S21 for a lossless filter
network can be written as
where w is angular frequency, m + i m are the roots of the numerator
polynomial and s +ί ) are the roots of the denominator polynomial for the
filter network. The numerator polynomial has Mroots and the denominator
polynomial has P roots.
Consequently, the gain of the filter network can be written as
and the group delay of the filter network can be written as
To take into account the loss and therefore the finite Qof the filter resonators
(Qf), the poles and zeros of the filter transfer function must be shifted to the
left in the complex plane by a constant amount r , where r/ is related to Qf by
where f0 is the centre frequency and bw the bandwidth of the filter network 7.
As before, the Taylor series for the group delay and the insertion loss with
respect to - s (the real part) can be examined to see the effect of adding loss to
the lossless filter networks. The Taylor series expansion of the group delay
becomes
gd f = gd f + Vgd f .r (14)
where Vgdf the gradient of the group delay with respect to - s , and gdf is
the group delay of the lossy filter network. For the filter network, there is no
symmetry for the poles and zeros and so Vgdf ¹ 0
If Equation 14 is expanded, it becomes
For a small change r in sr and m, the gain of the filter network becomes
1
S2l,S2l, =521521 + V(521521).r +-V (521521).r/ + 0 (r ) (16)
where 521; is 521 of the lossy filter network and V(521521) is the gradient of
521521 with respect to - s . Now V(521521) = -2.gd f .521521 and thus
521,521, = 521521 -2.(521521.^ ).r - —V(S2AS21.gdf ).r 2 - - 2(S2AS 21.gdf ).r ...(17)
where V(S2\S21.g ) is the gradient of S2\S2\.gd f with respect to - s , etc.
This time, since dgf ¹ 0, there will be terms dgf , dg f etc. However, by
expanding Equation 17 it can be seen that the dominant terms can be written
as 521521.2m.gd f
m.rf
m. Hence, it can be found that
f f
521,521, 521521.exp(-2.g).rf =K for - l £ ¾ £ l (21)
and Kis a constant.
If the filter network and the correction network have very similar or the same
Qs, r =rf and Equation 2 1 can be rewritten as follows:
gd ( ) + gd ( ) = K' for - 1£ ¾ < 1 (22)
For different Qs, we write
gd ( ) +— gd f )) = K" for
rf
In practice, the gain and group delay will not be exactly flat and it is acceptable
if the gain and group delay across the passband is a ripple function.
Consequently, equations 20, 2 2 and 23 can be written as
gd ( )+ gd f (w) = (w) for - 1£ ¾ <1 (24)
gd ( )) + gd f { ) ) = g { for - 1£ ¾ < 1 (25)
r
gd ) ) +— gd f ) ) =h G)) for - l £ £ l (26)
r f
where ( ¾) , g ( ) and h(a) are ripple functions. It is of course desired that
the ripples are small to minimise the variation in gain and group delay and
provide a gain and a group delay that is approximately uniform across the
passband.
From Equation 7, it is known that gd i ~ gd c and from Equation 15 it is known
that gd f (of) equals gd f (of) plus some extra higher order terms in the Taylor
series. Consequently, from Equations 24 and 25 it can be seen that if the filter
and correction network have the same or substantially the same Q, the
conditions for a flat group delay and passband are the same and when the
passband is flat the group delay is also flat and vice versa, provided that the
extra terms in the Taylor series are small enough. In reality, the extra terms are
small but still significant and so there has to be a slight trade off between the
group delay equalisation and amplitude equalisation.
If the filter network and the correction network have very similar but not
exactly the same Qs, an acceptable approximation to group delay and gain
flatness is still provided if the group delay of the filter network and the
correction network exhibit the relationship of Equation 25.
Moreover, it has been found, that if the Qs are different, a polynomial for the
correction network can still be found that satisfies both Equations 24 and 26.
In more detail, it has been found that as long as
0.4 <- <2.5 (27)
r f
the group delay is approximately flat when the amplitude is approximately flat.
In fact, it has been found that using a slightly lower Qfor the correction
network than for the filter network allows flattening of the passband for a
greater percentage of the filter bandwidth for a given order of correction
network. In other words, when extra loss is added to the correction network to
give an r that is higher than rf, a greater percentage of the filter passband can
be flattened for a given order of correction network. As a specific example, it
has been found that a Qfor the correction network that gives r =2rf is suitable
for increasing the bandwidth over which the passband is flattened but also
providing a substantially uniform group delay. In fact, as long as the Qof the
correction network is selected such that
l <- <2.5 (28)
r f
a satisfactorily uniform group delay may be obtained while, at the same time,
the percentage of the bandwidth over which the passband is flattened is
increased.
According to embodiments of the invention, the polynomial of the correction
network is chosen to flatten the passband amplitude and the group delay for
the selected Q factors. It will now be shown, with reference to Figure 5, how an
appropriate correction network polynomial that gives an approximately flat
passband amplitude and group delay for the whole filter network arrangement
can be found, according to the invention, and how a correction network can be
synthesised based on the polynomial.
The polynomial for the desired filter network is first determined (step S5.1).
The skilled person would know how to determine a polynomial for a given filter
network and the process will not be described in detail herein. The Q of the
filter network is then chosen at step S5.2. The Q of the filter network depends
on the type of resonators of the filter network and the medium in which they
are implemented. Choosing the Q of the filter network may involve determining
the technology to use and then determining the resulting Q. Alternatively,
choosing the Q of the filter network may involve selecting a desired Q and then
determining the technology to be used to provide the selected Q.
An appropriate Q for the correction network is then determined based on the Q
of the filter network (step S5.3). If the resonators of the correction network are
selected to be of the same type as the resonators of the filter network and
further selected to be implemented in the same medium, the Q of the correction
network will be the same as the Q of the filter network. The Qs of the networks
can also be chosen to be different. As will be described in more detail below, if
the Qs of the correction network and the filter network are the same or
substantially similar, the exact Q does not have to be known for the design and
synthesis of the correction network and step S5.3 of choosing the Q of the
correction network may involve choosing to make the Q of the correction
network the same as the filter network. For example, choosing the Q of the
correction network may involve choosing to manufacture the correction
network and the filter network in the same material and with the same type of
resonators. If different Qs are instead used for the filter network and the
correction network, step S5.3 may instead involve choosing a specific value for
the Qfor the correction network. The value of the Qfor the correction network
may be chosen to be sufficiently close to the value of Qfor the filter network to
provide a value of r that complies with Equation 28 or at least Equation 27.
Apolynomial for representing the desired transfer function of the correction
network then has to be determined. For example, a Generalised Reverse
Coefficient Bessel Polynomial may be used as the starting point for finding a
suitable polynomial H(s) for the correction network. The Generalised Reverse
Coefficient Bessel Polynomial is a solution to the 2nd order differential
equation:
sQn - {2n - 2+a+bs).e +b.n£ = 0 (29)
where a and b are complex values and n is a positive integer.
The solutions to Equation 29 are of the form:
n =å f . -k where
k=l
If a Bessel polynomial is used, it can be seen from Equation 30 that in the
optimisation process to find the correction network polynomial, no matter what
its order, there are only two variables, namely a, b. The insertion loss of the
correction network, 5 1, = F(0n(a,b),Q) , is a function of Qof the correction
network and of q , and therefore of a and b.
Of course the order of the correction network has a bearing on the degree of
equalisation achievable. Generally, a correction network of order equal to that
of the filter network is sufficient. However a lower order network can be used
with some compromise on the flatness of the gain and the group delay. Alower
order correction network may be desirable in order to reduce the size of the
overall filter network.
Once a suitable polynomial as a starting point for the correction network has
been chosen, the coefficients of the polynomials need to be optimised (step
S5.4). To find the coefficients of the final polynomial, an error function is first
constructed.
A suitable error function for minimising the gain is given by
E (0),
(31)
where
K M
+ + a - a m - rf + ( - ( m
k=l
(32)
where k + i are the roots of the polynomial for the correction network,
s m + id)m are the roots of the numerator pr
olynomial for the filter network and
s P
+ i
P are the roots of the denominator polynomial for the filter network.
From Equations 7, 15 and 22, and assuming that the higher order terms in the
Taylor expansion are zero such that g d f (of) equals g d f (of) in Equation 15, it
is realised that the group delay is flattened when the gain is flattened as long as
the Qfactors for the filter network and the correction network are the same and
Equation 3 1 would then be the only error function that needs to be minimised
to flatten both the gain and the group delay. However, the higher order terms
in the Taylor expansion are in fact not zero and slightly different Qfactors may
be appropriate and it may therefore also be desired to consider an error
function for minimising the group delay.
Asuitable error function for minimising the group delay is given by
K ( - r ).(ffl - fflt )
E , ) = 2å
k=l (( - r ) + ( >- w )
P (s r - rf ) .( - p ) ( - ) .( - w9 )
p=l ((s r - rf + (w - wr ) 2 )2
?=i (( - ) 2 + («- « ) 2 )
(33)
It should be noted that the error function for the group delay is found by first
differentiating Equation 20 with respect to , which gives
gd ( ) + gd f ( ) = for - 1< w < 1 (34)
where gd ( ) is the derivative of gd ( )) with respect to 0 and gd f ( ) is the
derivative of gd f i with respect to ¾ . In reality, the sum of the derivatives
of the group delays of the correction network and the filter network will not be
exactly zero but should be minimised to flatten the overall group delay of the
filter network arrangement. The error function for the group delay can be
found by differentiating expressions for group delay based on Equations 5 and
12 for a lossy network with respect to and replacing the derivatives of the
group delay in Equation 34 with the differentiated expressions.
Equations 3 1 and 33 can be combined to give a combined error function:
E ( k , )k ) = E ( k , )k ) + E ( k , )k ) (35)
where , g are weighting constants defining the importance of group delay or
amplitude flatness. E >) can be minimised for a number of points in the
interval - 1 < ( < 1, using one or many appropriate optimisation techniques, to
find the coefficients of the Generalized Reverse Coefficient Bessel Polynomial
or other suitable polynomial chosen.
The relationships between the group delay and the gain shown above are not
exact and it is has been found that sometimes it is better to flatten the
amplitude and accept the group delay that follows. In other words, a
polynomial may be chosen that only minimises the error function for the gain
El. Having a nearly perfectly flat passband is sometimes more important than
having a perfectly constant group delay. In some embodiments, the
determination of the coefficients of the correction network polynomial can be
considered to be carried out in two parts. First, equation 35 is minimised to
find approximate coefficients. The roots of the correction polynomial are then
optimised by minimising Ei from Equation 31only. For example, it is
contemplated that when the Qs are the same or substantially the same, only the
error function for the gain is minimised to find the coefficients of the
polynomial since when the Qs are the same or similar the group delay is
substantially flat when the gain is substantially flat. Only minimising the error
function for the gain El can be considered to correspond to minimising the
error function E of Equation 35 but with a zero value assigned to the weighting
coefficient g for E2. Moreover, it is contemplated that when the Qs are
different, both the error functions for the gain and the group delay are
minimised to find the coefficients of the polynomial. Values may be selected for
the weighting constants of Equation 35 that give more weight to either the
error function for the gain or the error function for the group delay. Of course,
both error functions may be considered even when the Qs are the same or
similar.
Suitable values of a and b of the Generalised Reverse Coefficient Bessel
Polynomial to minimise Ei and/or a combination of Ei and E2 are determined
using an appropriate optimisation technique. The error function may be
minimised by an interactive process. It should be noted that, when the Qs are
the same or substantially the same, the exact value of Q is not needed to find
suitable values of a and b.Arough estimate of the Qis sufficient and the design
will still be valid for a Qof, for example, half the design value. For a low Q
filter, a fixed value of r of 0.1has been found suitable.
If different Qs are used, the bandwidth over which both the passband and the
group delay are flat could be slightly increased. However, the precise values of
Qs for both networks are then needed for the optimisation.
Accordingly, when the Qs are the same or substantially the same an estimate of
r and would be used in the error functions El and E2. When the Qs are
different, more precise values of r and would instead be used in the error
functions El and E2.
When finding the coefficients of the polynomial to minimise the error function,
the symmetry of the correction network polynomial is maintained about the
line s=r . In other words, the coefficients are optimised to provide a correction
network in which all resonators have the same Q. A property of this symmetry
is that when the network is synthesised there will be a constant resistive
residue associated with each resonator.
It will be appreciated that although one specific example of a gain error
function and a group delay error function has been given above, alternative
error functions can be constructed and the optimisation of the correction
network is not limited to the use of the error functions described herein.
Moreover, although the polynomial has been described to be obtained from a
Bessel polynomial as a starting point, it should be realised that other suitable
polynomials can be used. The polynomial does not have to be a Bessel
polynomial. If a Generalised Reverse Coefficient Bessel polynomial is used as a
starting point, the final polynomial may not be a Bessel polynomial. In a first
stage of the minimisation of the error function, the constraints that maintain
the polynomial as a Bessel polynomial may be kept but in a second stage, when
the error function is close to being minimised, the constraints are removed and
the final polynomial for the correction network may therefore only be close to a
Bessel polynomial.
The determined polynomials for the filter network and the correction network
can then be used to synthesise the filter network and the correction networks
(step S5.5). An example of how to synthesise a correction network from the
lossless H(s) is described below. However, it should be realised that a lossy
H(s) could also be used. Aconstant resistive element would be extracted with
each resonator.
To start the network synthesis process, it can be considered that the admittance
Y(s) for the one- ort network is iven b
Equation 36 can be rewritten as
real[H (s)]+imaginary[He s )]
Y(s) = (37)
real[H (s)]+imaginary[H s )]
where H (s) is the odd part of H(s) and He(s) is the even part of H(s).
For the special case of pole-zero complex conjugate symmetry this reduces to
H s)
Y s) (38)
He (s)
The synthesis of the one-port network can be performed by removing elements
from the admittance function of Equations 37 or 38. When H(s) has complex
conjugate symmetry, elements can be extracted from Equation 38 by continued
fraction expansion. However, for the general case, H(s) has complex
coefficients and elements must be extracted from the admittance function of
Equation 37. It is then useful to consider that Equation 37 can be written as
C s)
Y s) (39)
A(s)
where A(s) and C(s) are chain matrix parameters.
Shunt capacitors and frequency invariant susceptances followed by unit
inverters can then repeatedly be extracted by multiplying the chain matrix from
the polynomials by that of the negative of the element to be extracted, to give a
network with the topology of Figure 6a, as will be described in more detail
below. Equations for removing elements are provided as Equations 40 to 42
below.
For the removal of a capacitor Ck, the following equation can be used.
A s)
(40)
C s) - s.C .A s)
C(s) where C = S —
sA(s)
For the removal of a frequency invariant susceptance bk, the following equation
can be used.
_-
where b =-^- | k i.A(s)
For the removal of a unit admittance inverter, the following equation can be
used
(42)
Capacitor C and frequency invariant susceptance b k pairs are removed
separated by unit admittance inverters, as is known in the art. Each stage
reduces the order of the polynomials A(s) and C(s) by l .
Once the correction network has been synthesised, various network
transformations are carried out to arrive at a network approximation of the
correction network in the form it would be manufactured as would be well
known by the person skilled in the art. The correction network can then be
manufactured. It will be appreciated that the correction network and the filter
network would be manufactured using the technology determined to provide
the Qs determined at steps S5.2 and S5.3. If a similar or the same Q factor is
required in the filter network and the correction network, the correction
network can be manufactured in the same or in a similar medium to the filter.
An example of finding a suitable correction network for a specific filter and of
incorporating the correction network into a filter network arrangement will
now be described. As an example, a 4th order generalised Tchbeyshev filter
with transmission zeros at -1.741 and 1.41 and poles (for a lossless filter) at
-0.9640-0.64061, -0.1589+1.17671, -0.2419-1.22571, -0.8810+0.79381 can be
considered. The Q of the filter is approximately 100 with rf=o.i2.
The filter may be realised as a folded edge-coupled microstrip filter with centre
frequency 1.345GHz and bandwidth 110MHz. A substrate with a relative
permittivity x of 9.8, and a thickness 0.635mm was considered.
Using the above described method for finding the coefficients of the correction
network polynomial, a 2nd order correction network polynomial can be
obtained. The Q of the correction network is also chosen to be approximately
loo with r =o.i2. The poles of the obtained suitable lossless polynomial are
determined to be at -0.8582+0.46151 and -0.8856-0.52451. The zeros would be
at 0.8582+0.46151 and 0.8856-0.52451. The lossy poles and zeros would of
course be shifted to the left in the complex plane by -r .
The one-port correction network may be synthesised from the polynomial using
the method described above. A prototype one-port network as shown in Figure
6a was obtained, with values 01=0.5734, bi=o.0225, 02=1.7385 and
b2=o.04i3. The prototype network was then normalised, the shunt capacitors
and frequency invariant susceptances were transformed to series half
wavelength resonators and transformers were introduced into the inverters to
scale the resonator admittance to unity, giving the network shown in Figure 6b.
For the network shown Figure 6 , =— ==— , J 2 = and 0 is the
Y Y
centre frequency of the filter network and the correction network and given a
lower and upper band edge frequencies of 1.29GHz and 1.40GHz respectively
we have for a 1Ohm system ¥=7.8056, = #-0.005 , f = -0.003, Ji=o. 4727
and J2=o.i283.
This network was then scaled to a 50 Ohm system and transformed to a coupled
line microstrip circuit. The insertion loss and the group delay of the microstrip
correction network obtained from an Agilent ADS circuit simulation are shown
in Figures 7a and 7b respectively. The graph of Figure 7a shows the gain of the
correction network, in decibels, across the passband and the graph of Figure 7b
shows the group delay, in seconds, of the correction network across the
passband.
Two copies of the correction network were incorporated with a branch line
coupler and the microstrip filter was added to this circuit to give a final circuit
as shown in Figure 8. The filter network 7, the coupler 8 and the correction
networks 9a, 9b are indicated in the circuit of Figure 8.
Figure 9a shows the insertion loss of the final microstrip circuit compared to
the insertion loss of the filter, obtained from an Agilent Momentum simulation.
Figure 9b shows the group delay of the final microstrip circuit compared to the
group delay of the filter, obtained from an Agilent Momentum simulation. The
gain is shown in decibels and the group delay is shown in seconds in the
graphs. It is clear that both the gain and the group delay of the filter network
arrangement comprising the correction network is flatter than the gain and the
group delay of the filter network on its own. It can be seen from Figure 9a that
the variation in gain across the passband of the filter network arrangement is
only approximately 30% of the variation in gain of the filter on its own.
Correspondingly, it can be seen from Figure 9b that the variation in group
delay of the filter network arrangement is only approximately 40% of the
variation in group delay of the filter on its own. The variation in gain and group
delay can be reduced further by increasing the order of the correction
networks. Consequently, the example of Figures 6a, 6b, 7a, 7b, 8, 9a and 9b
show that a correction network can be provided, according to the invention,
that flattens the attenuation and the group delay of a filter across the passband
of the filter. Accordingly, the invention provides a filter network arrangement
with improved group delay and gain characteristics compared to the filter
network on its own.
It should be realised that the filter network and the correction networks
discussed with respect to Figures 6a, 6b, 7a, 7b, 8, 9a and 9b are just one
specific example and the correction network design technique, according to the
invention, can be used to obtain a suitable correction network polynomial, and
corresponding synthesised correction network, for compensating for the gain
and group delay variation of any filter network.
Another embodiment of the invention is shown in Figure 10. Like reference
numerals indicate like components. The filter network arrangement 5 of Figure
10 comprises a first port 11, a circulator 12, an external correction network 9, a
filter network 7, and a second port 13 . Consequently, instead of the coupler, a
circulator is used. The correction network 9 is identical to the correction
networks 9a and 9b of Figure 4, apart from that the correction network would
have to be optimised to match the circulator instead of the couplers. As before,
the filter network may be a microwave filter. The filter network may be
implemented in ceramics. The path taken by a signal received at the first port
11 is shown by solid arrows. A signal received at the first port 11 is passed by
the circulator 12 to the one-port correction network 9 and from there to the
filter network 7 to be filtered and output at the second port 13. Consequently,
the external network 9 will correct the group delay and gain of a signal entering
at the first port 11 and leaving at the second port 13. The path taken by a signal
received at the second port 13 is shown by dashed arrows. Due to the nature of
the circulator 12, a signal received at the second port 13 is passed directly from
the filter network 7 to the first port 11, by-passing the correction network 9.
Since a circulator is used instead of a coupler in the embodiment of Figure 10,
only a single correction network is required instead of two correction networks
as shown in Figure 4. However, for some frequencies 3dB couplers can be made
much smaller than circulators and consequently the filter network arrangement
of Figure 10 could be bigger and heavier than the filter network arrangement of
Figure 4.
It is contemplated that the filter network arrangements of Figure 4 and 10 may
be implemented as low temperature co-fired ceramic (LTCC) filter network
arrangements. To provide a small and compact structure, a filter network as
shown in Figure 4 may be provided as a multi-layer filter network arrangement.
As a specific example, it may be a three layer LTCC structure with the filter
network and the coupler on the middle layer and the correction networks on the
top and bottom layer respectively.
Whilst specific examples of the invention have been described, the scope of the
invention is defined by the appended claims and not limited to the examples.
The invention could therefore be implemented in other ways, as would be
appreciated by those skilled in the art.
For example, it should also be realised that although the correction network
has been described in the two embodiments above as a one-port network, the
correction network may be any suitable network. It may, for example, be a twoport
network in which case no coupler or circulator would be needed. The
described technique for synthesising the correction network would also be
applicable for the two-port network. The polynomials discussed above give rise
to a simple ladder one-port network. The equivalent two-port network would be
more complex but could be used if appropriate for the application.
Moreover, although the filter network and the correction network have been
described to be connected by a coupler or a circulator, it should be realised that
if the correction network allows, for example if the correction network is a twoport
network, the correction network may be directly connected to the filter
network.
It should also be understood that although the embodiments have been
described with respect to a microstrip or an LTCC network, any type of suitable
integrated circuit can be used.
Additionally, it should be realised that the correction network can be used with
any type of suitable filter for which it is desired to equalise the group delay and
gain. The filter network arrangement does not have to be used in a
communication satellite.

Claims
1. An analogue filter network arrangement comprising:
a filter network (7); and
one or more correction networks (9a, 9b), wherein the one or more
correction networks is arranged to substantially equalise the passband gain and
group delay of the filter network arrangement and wherein group delay is the
derivative of phase with respect to angular frequency wherein the Qs of the filter
network and the one or more correction networks are selected such that
r f f
1<—< 2.5 , where r =— and rf =— , Q is the Q of the one or more
rf bw Qc bw Qf
correction networks, Qf is the Q of the filter network and f0 and bw are the
centre frequency and the bandwidth of the filter network respectively.
2. A filter network arrangement according to any one of the preceding
claims, wherein the one or more correction networks comprises one or more oneport
networks.
3. Afilter network arrangement according to claim 2, wherein the one or
more correction networks comprise two identical correction networks and the
filter network arrangement comprises a coupler for connecting the correction
networks and the filter network.
4. Afilter network arrangement according to claim 2, wherein the one or
more correction networks comprises a single correction network and the filter
network arrangement comprises a circulator (8) for connecting the correction
network to the filter network.
5. Afilter network arrangement according to any one of the preceding
claims, wherein the order of the one or more correction networks is the same or
lower than the order of the filter network.
6. Alow temperature co-fired ceramic (LTCC) structure comprising a filter
network arrangement according to any one of the preceding claims.
7. A processing arrangement for a communication satellite comprising a
filter network arrangement according to any one of claims 1 to 5 or a LTCC
structure according to claim 6.
8. A method of equalising the gain and the group delay across the passband
of an analogue filter network arrangement, the method comprising:
determining a polynomial and Q for a filter network;
choosing a Q for the one or more correction networks;
selecting a starting polynomial for the one or more correction networks
and adjusting the coefficients of the polynomial to substantially equalise gain
and group delay across the passband of the filter network; and
synthesising the filter network and the one or more correction networks
from the polynomial for the filter network and the polynomial for the one or
more correction networks, wherein the Qs of the filter network and the one or
r f 1
more correction networks are selected such that 1<— < 2.5 , where rc =—
rf bw Q
and r f 1 f =— , Q is the Q of the one or more correction network, Qf is the Q
bw Qf
of the filter network and f0 and bw are the centre frequency and the bandwidth
of the filter network respectively.
9. A method according to claim 8, wherein determining the coefficients of
the polynomial comprises minimising the error function
E )k ) = E ( , )k ) + g ( , )k ) where , g are weighting constants,
where
+ + - wί ) s - + w- w )
|511, («, ,« ) |2.|521, («)| 2 =l
( - rc + ( - ( ) ) ( - rf ) 2 + w- w ) )
k=l p=l
and
where + i are the roots of the polynomial for the correction network,
s m + id)m are the roots of the numerator pr
olynomial for the filter network and
s P
+ id)
P are the roots of the denominator polynomial for the filter network.
10. Amethod according to claim 9, wherein minimising the error function
comprises first minimising E ) to find approximate coefficients for the
correction network polynomial and then minimising E on its own to optimise
the roots of the correction network polynomial.