Microwave recursive filter

A broadband microwave recursive filter that provides sharp transitions in the frequency domain between adjacent stop and passbands comprising a signal input node; a signal output node; a filter circuit connected between the signal input node and the signal output node for providing a signal flow therebetween which has a predetermined frequency bandwidth characteristic; a microwave transistor circuit, with the microwave transistor circuit being band-limited to provide gain in only a restricted window of frequencies within the predetermined frequency bandwidth and connected for providing amplification to signals flowing in the filter circuit between the signal input node and the signal output node while suppressing out-of-window signals resulting from design approximations. The filter circuit includes a plurality of distributed feedback loop circuits, with each of the feedback loop circuits including the microwave transistor circuit therein, and wherein each of the feedback loop circuits has a different electrical length in relation to the other of the plurality of feedback loop circuits. Finally, the present filter design includes a passive filter connected in common to all of the plurality of distributed feedback loop circuits in the filter circuit for circuit for providing filter zeros on both sides of the restricted window of frequencies. In a preferred embodiment, the microwave transistor circuit includes an FET which is impedance-matched to obtain an approximately flat gain response across the window. It is preferred that the plurality of feedback loops all have amplitude weighting factors of the same sign.

BACKGROUND OF THE INVENTION 
The present invention is directed generally to broadband microwave filters, 
and in particular, to broadband recursive microwave filters that provide 
sharp transitions in the frequency domain between adjacent stopband 
passbands. 
There is a need for microwave filters which have sharp transitions in the 
frequency domain between adjacent stopbands and passbands. This need is 
especially acute for very wideband signal systems which operate by 
chopping the wideband signal up into a series of contiguous smaller 
bandwidths, and then processing these individual smaller bandwidths. This 
type of wideband signal chopping into smaller contiguous bandwidths is 
required, for instance, in monolithic full-band millimeter-wave receivers 
for processing the ultra-wideband microwave I.F. signals that result. 
Typically, this signal chopping leads to individual smaller bandwidths of 
octave (or near-octave) bandwidths. However, if the transition between 
individual contiguous octave filters is not sharp in the frequency domain, 
then spurious frequency responses will be generated in a particular octave 
filter by signals adjacent to but not within the particular octave filter 
band. Such spurious signals are sometimes referred to by the term 
frequency aliasing. 
At very low frequencies, active filters are commonly utilized in order to 
obtain the higher order transfer functions needed to provide sharp 
transitions between stopbands and passbands. Such active filters are 
advantageous at these low frequencies because they have the ability not 
only to compensate for parasitic losses affiliated with passive circuit 
elements, but also to provide overall amplification. However, the direct 
transposition of low-frequency design principles to the microwave range is 
impeded by the lack of appropriate broadband, high-gain devices to perform 
operational amplifier functions. Additionally, a principal limiting factor 
of current microwave active devices is that such active devices have an 
intrinsic time delay. Accordingly, most of the interest in microwave 
active filters has concentrated on alternate approaches in which 
individual reactances and resonators are replaced with microwave active 
substitutes that yield higher-Q performance. 
OBJECTS OF THE INVENTION 
Accordingly, it is an object of the present invention to provide a 
broadband microwave filter that has a sharp transition in the frequency 
domain between adjacent stopband passbands. 
It is a further object of the present invention to achieve wideband 
filtering of up to an octave, in conjunction with sharp cut-off 
characteristics at the band edges. 
It is still a further object of the present invention to provide a 
broadband microwave filter that has a sharp cut-off characteristic at the 
band edges while utilizing a minimum number of active devices. 
It is yet a further object of the present invention to provide a wideband 
microwave filter with improved circuit Q and reduced size. 
It is still a further object of the present invention to provide a 
broadband microwave filter with sharp transitions in the frequency domain 
between adjacent stopbands and passbands utilizing transistors, while 
avoiding the gain and time delay limitations normally associated with the 
use of such transistors. 
Other objects, advantages, and novel features of the present invention will 
become apparent from the detailed description of the invention, which 
follows the summary. 
SUMMARY OF THE INVENTION 
Briefly, the present invention comprises a broadband microwave recursive 
filter that provides sharp transitions in the frequency domain between 
adjacent stopbands and passbands, and comprises a signal input node; a 
signal output node; a filter circuit connected between the signal input 
node and the signal output node for providing a signal flow therebetween 
which has a predetermined frequency bandwidth characteristic, the filter 
circuit including a plurality of distributed feedback loop circuits; a 
two-port microwave transistor circuit which is band-limited for providing 
gain in only a restricted window of frequencies within the filter 
circuit's predetermined frequency bandwidth and for providing 
amplification to signals flowing between the signal input node and the 
signal output node while suppressing out-of-window signals resulting from 
design approximations; wherein each of the feedback loop circuits include 
the microwave transistor circuit therein, and wherein each of the feedback 
loop circuits has a different electrical length in relation to the other 
of the plurality of feedback loop circuits to provide the filter circuit's 
predetermined frequency characteristic; and one or more passive filters 
included in the feedback loop circuits for providing transmission zeros at 
each end of the restricted window of frequencies. 
In a preferred embodiment, the plurality of feedback loops in the filter 
circuit all have amplitude weighting factors of the same sign and the 
microwave transistor is an FET impedance-matched to obtain an 
approximately flat gain response across the window. It is also preferred 
that the one or more passive filters be common to all of the plurality of 
distributed feedback loops.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
A generalized schematic of the recursive filter of the present invention is 
shown in FIG. 1. This schematic embodiment is composed of transmission 
elements labeled with the designation "T" and feedback elements labeled 
with the designation "F", with the labels having various identifying 
indices associated with them. In the present design, any one of these 
transmission elements "T" may contain one or more of the following: 
(a) simple transmission lines, including the degenerate case of zero line 
length; 
(b) passive or active filter structures; and 
(c) active devices, such as transistors, with appropriate impedance 
matching elements. The feedback elements may be any two-terminal elements, 
but are typically composed of Rs, Ls, and Cs. 
The basic recursive portion of the filter of the present invention 
comprises the transmission elements T.sub.k, k=1, . . . , N, and the 
feedback elements F.sub.k, k=1, . . . , N. The transmission elements 
T.sub.in and T.sub.out provide additional flexibility and can be used to 
implement window functions and/or achieve supplementary amplification and 
filtering. 
A recursive filter typically may be regarded as a tandem combination of a 
transversal section and a feedback-only section. The presence of the 
feedback constitutes a very powerful extension relative to a strictly 
transversal process, permitting poles of transmission of arbitrarily high 
Q factors to be realized. Introduction of such poles (where in the limit, 
the transfer function of the filter may go to infinity, i.e., the filter 
begins to oscillate) permits the generation of significant gain. However, 
introduction of such poles in order to obtain significant gain makes the 
filter sensitive and introduces stability concerns. 
The recursive filter characteristics are the result of the interaction 
between time-delayed signal components. Under the assumption that 
particular time delays are multiples of some common minimum delay 
.tau.=2.pi./.omega..sub.s, where 107 .sub.s is the frequency at which the 
periodic transfer function repeats itself, the recursive transfer function 
of a structure with M+1 feedforward branches and N feedback branches can 
be written in the form 
##EQU1## 
In this equation, .omega. is the angular frequency, .alpha..sub.m 
represents the transversal amplitude weighting factors, and .beta..sub.n 
represents the recursive amplitude weighting factors. Note that in this 
equation, the numerator represents the sum of the various transversal 
terms, while the denominator represents the sum of the various recursive 
or feedback terms. In the present design, for ease of explanation, it is 
assumed that there are no transversal terms, such that the numerator of 
the equation is a constant. Accordingly, the basic design task is to 
determine the coefficient values .beta..sub.n, n=1, 2, . . . , N, that 
yield the best approximation to a prescribed target response function. A 
number of methods may be utilized to determine the .beta..sub.n 
coefficients. One such method is the impulse response method. However, 
this method gives rise to various disturbing aliasing effects. An 
alternate method for determining the coefficients .beta..sub.n is to use 
the bilinear transformation method. This method is used to derive the 
recursive filter response from an appropriate lumped-element prototype 
response. Using this method, a bilinear-type transformation is defined 
which establishes correspondance between the frequency variable j.OMEGA. 
in the lumped-element domain and the frequency variable j.omega. in the 
distributed domain, according to some chosen substitution formula 
j.OMEGA.=F(j.omega.). Two examples of such F(j.OMEGA.) transformations are 
listed in equations 2 and 3 below. 
##EQU2## 
In these equations, the terms 106 .sub.c and .omega..sub.c are the 
designated cut-off frequencies in the respective domains, while the term 
.omega..sub.s is the symmetry frequency at which the periodic transfer 
function repeats itself, where .omega..sub.s =2.pi./.tau.. Utilizing these 
transformations results in a nonlinear distortion of the prototype 
frequency axis such that no aliasing effects occur. 
In order to employ the above bilinear transformation method, the desired 
target response for the recursive transfer function must be translated 
into the j.OMEGA. domain. The magnitude of the resulting prototype target 
response is arrived at through application of the inverse frequency 
variable transformation to the corresponding j.omega.-domain response by 
way of the substitution j.omega.=F.sup.-1 (j.OMEGA.). Standard synthesis 
techniques are then utilized to find a rational function in terms of this 
prototype-domain frequency variable that represents the prototype target 
response along the j.OMEGA. axis. Once such a function has been 
established, it may then be frequency transformed back to the j.omega. 
domain to yield the recursive filter response. 
This response function assumes the form of a rational function in various 
powers of e.sup.-j2.pi..omega./.omega..sbsp.s. These powers reflect signal 
contributions of specific time delays K.multidot.2.pi./.omega..sub.s, K=0, 
1, . . . The denominator terms describe the recursive part of the transfer 
function. Corresponding coefficients are the .beta.-coefficients. (The 
numerator terms describe the transversal part of the function with its 
.alpha.-coefficients). 
The .beta. coefficients will, in general, comprise a mix of positive and 
negative coefficients, with corresponding signal contributions summed to 
provide the desired filter signal response. These terms will tend to add 
in-phase across the band of frequencies where the response is to be a 
linear passband, and will add out-of-phase at the edges of the passband in 
order to provide the signal response zeros. However, the combination of 
both positive and negative coefficients requires the capability in the 
circuit for reversing the signal phases over wide microwave frequency 
ranges. 
Signals with positive and negative coefficients can be obtained in a 
microwave environment by splitting an incident signal into two signals of 
opposite phase and then utilizing separate delay lines to derive the 
required signal component with the appropriate delay and sign. However, 
this approach requires a disproportionate amount of circuit board area. 
Alternately, a single feed line could be used and the sign requirements 
could be satisfied by using microwave active devices in both inverting and 
non-inverting configurations. However, this approach sacrifices design 
uniformity by typically requiring amplifier sections with different 
reflection, isolation, and delay characteristics. 
In order to avoid opposite sign terms in the present design, an 
approximation approach is utilized to approximate negative .beta. 
coefficient terms with positive .beta. coefficient terms. These 
approximate substitution terms may be determined, by way of example, 
according to the following equation: 
EQU .beta..sub.n .multidot.e.sup.-j2.pi.n.omega./.omega..sbsp.s 
.apprxeq.-.beta..sub.n 
Y.multidot.[e.sup.-j2.pi.(n-1).omega./.omega..sbsp.s 
+e.sup.-j2.pi.(m+1).omega./.omega..sbsp.s ], (4) 
for .beta..sub.n /.beta..sub.n/2 &lt;0, and n&gt;0. For more information on this 
substitution process, see the paper by C. Rauscher, "Microwave Active 
Filters Based on Transversal and Recursive Principles" IEEE Trans. 
Microwave Theory Techn. Typically, the empirical parameter .gamma. will be 
in the vicinity of 0.6 in order to render the approximation useful over 
bandwidths of on the order of an octave. Furthermore, if the contribution 
by a given .beta. term is judged to be negligible, it may be convenient to 
delete it all together. 
The above-described substitutions and deletions of the various .beta. terms 
result in some passband deviations from the original desired response. 
These passband deviations can be obviated by means of computer 
optimization to adjust the pertinent parameters and bring the filter 
response back in line with the desired filter response. This computer 
optimization is also used to compensate for the microwave circuit 
parasitic effects. There are a variety of computer optimization programs 
well known in the art for accomplishing the above described optimization. 
Typically, these computer optimization programs function by generating an 
error function, such as the least-squares function. By way of example, and 
not by way of limitation, various IMSL subroutine library programs or the 
program COMT may be utilized for optimization purposes. 
It should be noted that the approximations obtained from equation 4 in 
order to allow the use of .beta. weighting coefficients of only one sign 
cause a signal response breakdown in the stopband. At points where zeros 
of transmission should occur in the desired response, the recursive filter 
components actually add up to produce maximums of transmission. 
In order to control this stopband behavior as well as perform the functions 
of the disregarded .alpha.-coefficients, one or more window functions must 
be added to the design. In a preferred embodiment, a microwave active 
device which is intrinsically band-limited, in combination with passive 
matching circuitry and a separate passive filter segment are added to the 
filter design. The microwave active device with its matching elements 
should provide appreciable gain roll-off beyond the passband edges, but 
offer flat amplification within the passband, so as not to interfere with 
the main recursive bandpass operation. 
A preferred embodiment of the present design is shown in FIG. 2. This 
filter was designed to achieve a 9-15 GHz triple-hump bandpass response, 
with stopbands spanning the frequency ranges from 5-8 GHz and 16-19 GHz. 
In essence, the passband characteristic is centered around .omega..sub.s 
/4.pi.=12 GHz. The recursive solution that resulted from the use of the 
high-pass-type bilinear transformation, and one-sign-only .beta. 
approximations, together with the deletions of less significant terms, 
resulted in two feedback loops with affiliated feedback resistors, in 
combination with a windowing amplifier section and a passive filter 
segment to assist in defining the bandpass characteristics. 
The basic recursive filter embodiment shown in FIG. 2 comprises an RF input 
node 10, and RF output node 12, and a filter circuit with a basic signal 
path 14 and including a first feedback loop 16 and a second feedback loop 
18. The circuit further includes a microwave amplifier section comprising 
active microwave device 20 with matching circuitry for providing 
amplification, delay, and windowing to the filter circuit. Finally, this 
circuit includes a passive filter 22 which, in the present embodiment, is 
comprised of two parallel connected transmission lines 24 and 26. 
Referring first to the microwave amplifier section, this section is 
comprised in FIG. 2 of the FET 20 with its attendant matching circuits. In 
essence, the amplifier section encompasses all circuit elements to the 
right of a set of blocking capacitors 30 and 32. These blocking capacitors 
30 and 32 are necessary in order to prevent any of the dc biasing current 
for the FET from undesirably propagating into other parts of the filter. 
The standard FET 20 has a gain function which decreases approximately as a 
straight line from lower frequencies to higher frequencies. In order to 
obtain a flat gain characteristic at the bandwidth of interest, the gain 
at the lower end of the frequency band must be reduced. Additionally, the 
FET has certain parasitic capacitances which must be compensated. 
The transmission line elements 34, 36, 38, 42, 46, 48, 50, 52, 58, and 60, 
the resistors 40 and 56, the inductors 39 and 54, and the capacitor 44 
comprise the matching circuit and the biasing circuit for the FET 20. More 
specifically, the transmission lines 36 and 38 and the resistor 40 operate 
to reduce the lower frequency gain response of the FET 20. The 
transmission lines 36 and 38 are used to impedance match the resistor 40 
to the FET gate. In essence, the resistor 40 is shunted between the gate 
and the source terminals of the FET 20 at low frequencies. This shunting 
operation of the resistor 40 occurs because at low frequencies, the 
impedance of the FET 20 is very high, thereby shunting the signal into the 
resistor 40. However, at higher frequencies, the impedance of the FET 20 
drops such that the effects of resistor 40 become less noticeable in the 
circuit. 
A bias voltage is applied to the gate by means of a terminal 41, an 
inductor 39, and the transmission line 36. The inductor 39 operates as an 
RF choke to prevent the RF signal from being dissipated in the biasing 
circuitry. 
The transmission lines 42 and 46 are specifically chosen in length to 
provide a shunt inductance compensation to the gate of the FET 20 in order 
to compensate for the parasitic capacitance of the FET gate. 
The output impedance of the FET 20 is also capacitive. In order to 
compensate for this capacitive output of the FET, a transmission line 48 
is connected to the drain of the FET 20. At the other end of the 
transmission line 48 are connected the transmission lines 50 and 52 and 
the resistor 56 and the transmission line 58. The combination of the 
transmission lines 48, 50, 52, 58, and the resistor 56 help tune out the 
capacitive affects at the output of the FET. 
A dc bias is applied to the drain of the FET 20 via a contact point 55. An 
inductor 54 is connected between the contact point 55 and the drain to act 
as a choke inductor to prevent high frequency signals from leaking out 
through the bias path at the contact point 55. 
The FET 20 in combination with its matching network of transmission lines, 
capacitors, inductors, and resistors, constitutes an 8-to-16 GHz broadband 
amplifier, and serves as part of the lowest order transmission module. By 
way of example, the FET 20 may be implemented by an Avantek M126 
sub-half-micron device biased at 1/2I.sub.DSS and at a drainsource voltage 
of V.sub.DS=+ 3.0 V. The transfer response of this FET amplifier section 
is shown in FIG. 3 by the dotted line curve 100. The passband for this 
amplifier FET 20 is intentionally designed to be wider than the actual 
filter response so as to reduce sensitivity of the critical filter skirts 
to transistor-related tolerance effects. 
As noted previously, an additional passive filter 22 is included in the 
circuit in order to aid the shaping of the filter characteristic by 
introducing zeros of transmission at 6, 8, 16, and 18 GHz. In the 
embodiment shown in FIG. 2, this passive filter 22 was comprised of the 
parallel connection of two 100 ohm transmission lines with lengths chosen 
such that the signals propagating therethrough are added in-phase in the 
center of the passband of interest, but are added 180.degree. out-of-phase 
to cancel at the desired zero points in the response function. In this 
particular embodiment, the transmission line lengths for the transmission 
lines 24 and 26 were chosen to be equivalent to a quarter wave and a five 
quarter wave length, respectively, at the band center of 12 GHz. 
Accordingly, at band center, there is a difference of 360.degree. in the 
phases of the two signals when they are added together after propagation 
through the passive filter 22, thus ensuring that signals around the band 
center will be passed. 
As noted previously, the present recursive filter includes two feedback 
loops 16 and 18. Each of these circuit loops requires a certain amount of 
feedback loop time delay. A portion of this time delay is inherently 
provided by the FET 20 and its attendant matching circuitry. Additional 
feedback loop delay is provided by the passive filtering section 22. The 
transmission line elements 70, 72, 74, and 76 provide the remaining time 
delays to implement the entire desired delays for the feedback loops. The 
transmission line elements 72 and 74 are common to both of the feedback 
loops 16 and 18. However, the feedback loop 18 includes the delay element 
76 with a first and second ends which is not common to any of the other of 
the feedback loops in the circuit and is connected at one end to the RF 
output node 12. This feedback loop 18 further includes a resistive element 
78 which is not common to any of the other of the feedback loops in the 
circuit and which is connected between the RF output node 12 and the RF 
input node 10. Finally, in the embodiment of FIG. 2, the feedback loop 18 
includes the delay line 70 which is connected at a first end to the RF 
input node 10 and at a second end thereof indirectly to the gate of the 
FET 20. The other feedback loop 16 includes a resistive element 80 which 
is not common to any of the other of the feedback loop circuits and is 
connected at one end thereof to the second end of the delay line 76 and is 
connected at the other end thereof to the second end of the delay line 70. 
The calculated frequency response characteristic for this filter is shown 
by the dashed line curve 102 in FIG. 3. The actual measured filter 
response characteristic for the recursive filter of FIG. 2 is shown by the 
solid line curve 104 in FIG. 3. 
By way of example and not by way of limitation, the following specific 
circuit element values may be utilized to implement the embodiment of the 
present invention shown in FIG. 2. Note that all electrical lengths 
.theta. are for the midband frequency of 12 GHz. 
______________________________________ 
TL.sub.70 : 
Z.sub.o = 50.OMEGA. 
.theta. = 210.degree. 
TL.sub.50: 
Z.sub.o = 75.OMEGA. 
.theta. = 84.degree. 
TL.sub.72 : 
Z.sub.o = 50.OMEGA. 
.theta. = 50.degree. 
TL.sub.52 : 
Z.sub.o = 50.OMEGA. 
.theta. = 90.degree. 
TL.sub.34 : 
Z.sub.o = 35.OMEGA. 
.theta. = 78.degree. 
TL.sub.58 : 
Z.sub.o = 65.OMEGA. 
.theta. = 180.degree. 
TL.sub.36 : 
Z.sub.o = 85.OMEGA. 
.theta. = 35.degree. 
TL.sub.60 : 
Z.sub.o = 50.OMEGA. 
.theta. = 30.degree. 
TL.sub.38 : 
Z.sub.o = 85.OMEGA. 
.theta. = 100.degree. 
TL.sub.26 : 
Z.sub.o = 100.OMEGA. 
.theta. = 90.degree. 
TL.sub.42 : 
Z.sub.o = 40.OMEGA. 
.theta. = 12.degree. 
TL.sub.24 : 
Z.sub.o = 100.OMEGA. 
.theta. = 450.degree. 
TL.sub.46 : 
Z.sub.o = .theta. = 32.degree. 
TL.sub.74 : 
Z.sub.o = 50.OMEGA. 
.theta. = 10.degree. 
TL.sub.48 : 
Z.sub.o = 95.OMEGA. 
.theta. = 34.degree. 
TL.sub.76 : 
Z.sub.o = 50.OMEGA. 
.theta. = 330.degree. 
C.sub.30 = C.sub.32 = 10 pF 
C.sub.44 = 20 pF 
L.sub.39 = L.sub.54 = 15 nH 
R.sub.40 = 100.OMEGA. 
R.sub.56 = 25.OMEGA. 
R.sub.80 = 820.OMEGA. 
R.sub.78 = 680.OMEGA. 
______________________________________ 
FET.sub.20 : Avantek M126? 
It should be noted that a number of additional external window functions 
may be added to the present design. 
The present circuit was constructed on a 0.25 mm thick 
fiberglass-reinforced Teflon substrate, with coaxial 50 ohm adapters at 
the input and the output. Noise figures were measured at 1 GHz intervals 
within the 9-to-15 GHz passband. The values of the noise decreased from a 
maximum reading of 7.8 dB at the lower band edge to a minimum of 5.2 dB at 
14 GHz, before rising to 5.9 dB at the upper band limit. These noise 
numbers indicate a noise advantage for the recursive filter in comparison 
to previous designs including the transversaltype designs. 
It should be noted that the present design does not require the location of 
the amplifier section to be common to all or a plurality of the feedback 
loops in the filter. In this regard, an active amplifier section could be 
located in a portion of each feedback loop which is not common to the 
other feedback loops. Likewise, the desired windowing effect of the active 
amplifier section may be obtained by inserting the amplifier either before 
the filter section or after the filter section. However, in order to 
minimize size and elements and, in particular, the number of transistors 
used, it was deemed to be desirable to provide the active amplifier 
windowing function by means of a single amplifier which is common to all 
of the feedback loops in the filter circuit. The above comments also hold 
for the passive filter section 22. This passive filter section may be 
disposed at the input to the main microwave filter or at the output of the 
main microwave filter, or a separate passive filter section may be 
disposed in each feedback loop in an uncommon portion of that feedback 
loop. However, in order to reduce circuit size, the passive filter section 
was disposed to be common to all of the feedback loops in the main 
microwave filter. Note that the amplifier section with the FET 20 performs 
the multiple functions of providing gain for the circuit loops, providing 
windowing for the desired filter frequency response, and providing a 
portion of the time delay required in the feedback loops 16 and 18. Note 
that the elements of the passive filter section 22 perform the dual 
function of providing the passive filtering while also contributing to the 
time delay for the individual feedback loops 16 and 18. 
In order to impose a sense of direction on each of the feedback loops 16 
and 18, and thereby allow the feedback scheme to operate as intended, the 
levels of the individual signals fed back through the nondirectional 
impedance elements 78 and 80 should exceed, by a comfortable margin, the 
levels of parasitic contributions fed forward through the same elements. 
This is accomplished through the assignment of gain functions to the loops 
16 and 18 in order to introduce an appropriate differential between the 
output and the input relative signal levels. This gain function is 
implemented in the circuit of FIG. 2 simply by the amplifier section with 
the FET 20. Note that this appropriate differential in the output and the 
input relative signal levels permits the use of high resistance values for 
the feedback resistors 78 and 80 to thereby minimize perturbations in the 
input and the output lines. 
It should be noted that the same basic concept utilized in the present 
recursive filter may also be utilized in designing transversal filters. In 
this regard, a transversal filter composed of a plurality of feedforward 
circuit loops may be utilized to obtain a desired filter response. In 
designing this transversal filter, the various coefficients for the 
different feedforward signal paths may be set so that there are no 
opposite sign terms, i.e., so that there are no negative coefficients. 
This may be accomplished by utilizing an approximation equation, such as 
the one set forth in equation 4 of the present disclosure. Additionally, 
the insignificant terms may be dropped from this feedforward filter 
design. In order to delete the undesired out-of-band response of this 
filter caused by the above-described approximations, a series of window 
function circuits may be utilized in combination with this transversal 
filter design. More specifically, an active amplifier section may be 
utilized which is inherently band-limited to thereby provide the windowing 
function and the desired gain for the circuit. Additionally, a passive 
filter section may be included to provide zeros at specific locations in 
the filter response. As noted above for the recursive design, the 
transversal filter design may utilize the amplifier section and the 
passive filter section either prior to the transversal feedforward loops 
or at the output of the transversal filter, or an individual passive 
filter and amplifying section may be utilized in each separate feedforward 
path, or in a portion of the transversal filter which is common to all of 
the feedforward loops. 
Referring again to the recursive filter design of the present invention, 
one of the principle attractions of the present approach is that the 
active filter segment and the passive filter segment are incorporated into 
the overall filter design in such a manner that their associated time 
delays contribute to the delays already required for the realization of 
the recursive process. This design feature leads to very compact filter 
circuits. Accordingly, the present design is especially advantageous in 
microwave applications where space is at a premium. 
It should also be noted that in the present design, the time delays do not 
have to be multiples of some prescribed lowest-order time delay, as is 
otherwise generally the case in digital recursive filters. This feature 
provides extra degrees of design freedom. 
It should also be noted that opposite-sign feedback signals have been 
approximated by using a section of transmission line to achieve the 
180.degree. phase shift at midband. This permits the recursive filter 
design to include only one type of active element configuration, such as a 
common source configuration in the case of FET active elements. 
In essence, the present design including the combination of the recursive 
filter feedback loops, with the passive filter structure and the amplifier 
section both integrated into the recursive filter feedback loops, provides 
an extremely compact structure with a minimum of circuit elements and, in 
particular, a minimum of active circuit elements. 
It should be noted that the present invention is not limited to the use of 
FETs in the active amplifier portion of the design. In this regard, any 
type of transistor or active two-port device may be utilized. In addition, 
it should be noted that various feedforward paths may be incorporated into 
the present recursive structure in order to provide additional design 
freedom. 
It should be noted that the recursive approach to obtaining a particular 
frequency response characteristic enjoys a significant advantage over 
similar transversal approaches through the availability of powerful 
feedback options. This recursive approach also tends to be more receptive 
to the efficient integration of active devices and passive filter segments 
into the recursive filter structure. This integration permits the time 
delays associated with the active devices and the passive filter segments 
to be used constructively, thereby allowing circuit dimensions to be 
minimized. Moreover, the present recursive filter design does not require 
the space-consuming junction elements as does a typical transversal filter 
design. Although the recursive filter structure with the poles in its 
transfer function gives the appearance of being more suseptible to 
instability than transversal-type circuits, parasitic feedback within 
typical microwave active 2-port devices tends to disallow any meaningful 
distinction in practical design situations. 
The principle function of the feedforward terms in the numerator is to 
establish the zeros of transmission for the recursive filter. The implied 
reliance on transversal principles to achieve the zeros quite often 
represents, however, an inefficient use of resources when compared to 
conventional passive filter alternatives. Accordingly, to implement this 
function, the numerator of the recursive filter response expression is 
split off, with the zeros provided by the passive filter section noted 
previously. 
From the above, it can be seen that a broadband microwave filter has been 
realized that has sharp transitions in its frequency domain between 
adjacent stopband and passbands. 
Obviously, many modifications and variations of the present invention are 
possible in light of the above teachings. It is therefore to be understood 
that within the scope of the appended claims, the invention may be 
practiced otherwise than as specifically described.