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ESD-TDR-63-232 TM-3506
TAPPED DELAY LINE REALIZATIONS OF FREQUENCY
PERIODIC FILTERS AND THEIR APPLICATION TO LINEAR
FM PULSE COMPRESSION
TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-63-232
May 1963
R. Manasse
Prepared for
DIRECTORATE OF RADAR AND OPTICS
ELECTRONIC SYSTEMS DIVISION
AIR FORCE SYSTEMS COMMAND
UNITED STATES AIR FORCE
L. G. Hanscom Field, Bedford, Massachusetts
Prepared by
D D C
JUNl < j;:;
TISIA A
THE MITRE CORPORATION
Bedford, Massachusetts
Contract AF33(600)-39852 Project 750
When US Government drawings, specifications, or
other data are used for any purpose other than a
definitely related government procurement oper¬
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nished, or in any way supplied the said drawings,
specifications, or other data is not to be regarded
by implication or otherwise, as in any manner
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ration, or conveying any rights or permission to
manufacture, use, or sell any patented invention
that may in any way be related thereto.
Do not return this copy. Retain or destroy.
(T)
TM-3506
Qx
s **
/.japped delay line realizations of frequency
PERIODIC FILTERS AND THEIR APPLICATION TO LINEAR
FM PULSE COMPRESSION
l f '/• 0 1 k' C t T >
TECHNICAL DOCUMENTARY REPORT NO. ESD-TDR-63-232
(v ' ' ' f
V
' (?)
& L May 1#63y
/^j l-yj R. Manasse /
Prepared for
DIRECTORATE OF RADAR AND OPTICS
ELECTRONIC SYSTEMS DIVISION
AIR FORCE SYSTEMS COMMAND
UNITED STATES AIR FORCE
L.G. Hanscom Field, Bedford, Massachusetts
Prepared by
THE MITRE CORPORATION
Bedford*, Massachusetts
Contract AF33|600|-39852jProj«<l
750
I ABSTRACT
It is shown that a linear network having an amplitude and
phase response which is a periodic function of frequency can
be synthesized with a tapped delay line with amplitude and
phase weightings on each tap. The theory of this technique
for the realization of frequency periodic filters is devel¬
oped. The example which motivates the discussion of this
problem is the use of a single frequency periodic filter to
replace a bank of complex dispersive subpulse networks em¬
ployed in a large time -bandwidth product linear FM pulse
compression network. The availability of high quality tapped
quartz delay lines and the ease with which amplitude and
phase adjustments can be made on each tap appear to make
this technique attractive for a number of future applications.
7
i
iii
PREFACE
The material for this paper was prepared approximately a year
and one half ago with the expectation that it would eventually form one
section of a considerably larger report on linear FH pulse compression.
Since that time the material for the larger report has grown and evolved
into several papers, two of which were presented at the recent Pulse
Compression Symposium at RADC. It is more than timely therefore to
publish this material at this time and, except for minor editorial
revisions, this TM reproduces the draft version of this paper prepared
earlier.
TM-3506
1.
TAPPED DELAY LINE REALIZATIONS OF FREQUENCY PERIODIC FILTERS
AND THEIR APPLICATION TO LINEAR FM PULSE COMPRESSION
In this paper we shall see how the desired periodic filter response
can be realized with the aid of a high quality tapped delay line having
amplitude and phase weightings on the output of each tap. Such a filter
has a frequency response which is periodic in frequency with a period
equal to the reciprocal of the delay line tap spacing.
In our case, we are Interested in synthesizing a set of dispersive
networks which are all Identical except for a center frequency displacement.
If this tapped delay line simulates the desired frequency response over
a frequency band, then the repetitive character of the network in frequency
enables us to use the filter at a number of frequencies simultaneously.
Thus we are able to replace a whole set of dispersive networks by one
tapped delay line.
Before considering the reasoning which leads us to the use of a
tapped delay line, we will review briefly the basic properties of the
complex notation which is used to characterize the response of linear
networks .*
Consider a linear time- invariant network which is characterized by
an Impulse response h(t). We let x(t) be the real waveform input to the
filter, and y(t) be the real waveform at the output of the filter. See
Figure 1.
For a discussion of the complex representation of real waveforms, see
P. M. Woodward, "Probability and Information Theory, with Applications to
Radar" (McGraw-Hill, 1953), and D. Gabor, Journal Institute of Elect.
Engineers (Pt. Ill), 93, p. 429, 1946. See also, J. Dugundji, "Envelopes
and Pre-Envelopes of Real Waveforms", Vol. IT-4, PGIT, March, 1958.
TM-3506
2.
*(t)
h(t)
FIGURE I
y(t) Is related to x(t) and h(t) by a convolution Integral.
00
y(t) - J h(t-T)x(T)dT
• 00
All realizable filters satisfy h(t) « 0 for t < 0, so that the convolu
tion can equally well be written
t
y(t) - J h(t-T)x(T)dT
Convolution in the time domain corresponds to multiplication in the
frequency domain, and therefore
Y(f) - H(f)X(f) [or Y(cu) - H(uj)X(co), at - 2nf ]
where Y(f), H(f) and X(f) are Fourier transforms of y(t), h(t) and x(t),
respectively.
Usually, in circuit theory, the response of an electrical network is
characterised by its effect on complex time waveforms (e.g., e^*, J - /^T)
rather than real time waveforms (e.g., cosU)t)a The reason for this, of
course, is that any real time waveform must have a Fourier transform which
TM-3506
3.
Is conjugate symmetric about zero frequency and therefore any real wave¬
form is completely characterized by its frequency function for positive
frequencies alone: The frequency function for negative frequencies can
be mapped to zero without destroying any information about the time wave¬
form, and this is exactly the procedure used to obtain the complex
representation which proves to be notationally convenient and leads to
algebraic simplifications. Xc(f), the complex (frequency) representation
of X(f), is defined simply as
xc(f) -
Yc(f) is similarly defined. The frequency response of the network is then
written
Y (f) - H(f)X (f)
c c
The complex time representation of x(t), written xc(t) , is the
Inverse Fourier transform of Xc(f). The real waveform can always be obtained
from its complex representation by taking twice the real part. This complex
representation sets up a one-to-one correspondence between real and complex
waveforms. In the frequency domain the correspondence is obtained by mapping
the frequency function to zero for negative frequencies, and leaving the
frequency function for positive frequencies unperturbed. In the time domain
the correspondence between real and complex waveforms is set up using Hilbert
transforms. xc(t) , the complex representation of x(t) , is given by
xc(t) - -J x(t) + J $fx(t)
where Vx(t) s Hilbert transform of x(t). We find, taking the Fourier
transform of both sides, that must satisfy
TM-3506
4.
Xc(f) .|x(f) +i j p|tfx(t)] ,
(F denotes Fourier transform)
or
pjVx(t)]
/
■ JX(f) > f > 0
JX(f), f<0
It is seen that 7/ introduces a multiplication by -j for positive
frequencies and multiplication by +j for negative frequencies. In the
time domain this operation corresponds to convolution with the time
function 1/nt, so
which is the usual formula for the Hilbett transform. It is as though
x(t) had been passed through a linear filter with impulse response 1/nt.
However, this filter is not realizable because the impulse response is not
zero for t less than zero, can be viewed in the frequency domain as an
inf inite-bandwidth 90° phase shifter. If we allow arbitrarily long time
delays in the impulse response, it should be possible to realize 2*/ to any
desired degree of accuracy. Fortunately in the synthesis of filters, time
delays are usually of no consequence.
hǤ a number of useful properties, but we shall need only a few of
them.
14 (cos out) ■ sin uut (u> - 2tt£)
Hi sin out) ■ - cos out
Ma *(o ■ ‘ *(*>
»
^4 (real function) ■ real function
TM-3506
5.
The first two relations are easily proved by mapping cosuut and slwut to
their complex representations, multiplying by -J, and taking twice the
real part. The third relation results from, the fact that 14 applied twice
converts X(f) Into -X(f). The fourth relation Is true because V
preserves conjugate symmetry of the frequency function.
There Is another important relationship which we will need. Consider
the following function
x(t) * u(t) cosuut
where u(t) has a frequency function which vanishes outside the Interval
(-uu ,uu). We can obtain the Hilbert transform of x(t) by replacing the right-
hand side of the equation by its complex representation, multiplying by -j,
and taking twice the real part.
The complex representation of x(t) is given by
The Fourier transform of this function, which can be expressed as a
convolution of the respective frequency functions, is zero for negative
frequencies. Twice the real part of this function equals x(t). Therefore
the function must indeed be the complex representation of x(t).
Multiplying the complex representation by -J, and taking twice the
real part, we have
^ £u(t)cosuutJ ■ 2fte u(t)e^U)tJ ■ u(t)sinU>t
Similarly
^ J\i(t)sin'J)tJ * -u(t) cosuut
Let us now proceed to synthesize, using the above relationships, q
function of time whose frequency function is a repetitive version of some
TM-3506
6.
desired (non-periodic) function. Consider s network characterized by
an impulse response h(t) and a frequency transfer function H(f) with
the desired amplitude and phase response over a band of frequencies Wy.
Since H(f) is to be used only over the band Wp, we can set H(f) equal to
zero outside this band, as shown in Figure 2.
H(f) is the Fourier transform of a real function, and therefore it must
be conjugate synnetric. If it were not for the image component of H(f)
at -f , we could form the Rep*. H(f) in order to obtain a periodic function
of frequency! The desired result can be obtained by carrying out the
following steps:
^ For discussion of the Rep and Comb operations, see P. H. Woodward,
loc. cit., p. 28.
TM-3506
7.
1. Replace h(t) by hc(t), so that the frequency transfer function
is zero for f<0.
h(t) - hc(t) - \ h(t) + ~ j#h(t)
2. Make the resulting frequency function periodic by performing
Repy^ on Hc(f). As Woodward shows, the operation Rejty in the
frequency domain corresponds to the operation (1/Wp) Comb^y^ in
the time domain,
proportionality,
The result is, omitting constants of
CombT [h(t)] + j CombT ^h(t)] (Tj
i/wF)
This impulse response as it stands is not satisfactory because
it is not real. We must make the frequency function of this
waveform conjugate symmetric by mapping the frequency spectrum
to zero for negative frequencies and taking the real part of
the resulting time waveform.
3. We map the negative frequency part of the frequency function of
the above time waveform to zero. For any time waveform (real or
complex) this result can be obtained by adding to it J times the
Hilbert transform of the time waveform.
The result is
CombT [h(t)J + j CombT j?/ h(t)J
+ .#/combT [h(t)^| -^CombT h(t)J
F F
4. The above waveform has a frequency function with the desired
repetitive character for positive frequencies. Because the
frequency function is zero for negative frequencies, the wave¬
form is the complex representation of a real waveform which is
obtained by taking twice the real part. Ignoring constants of
proportionality, as before, we have the desired impulse response
h*(t)
TM-3506
8.
h'(t) - CombT [h(t)J - ?/CombT |?{h(t)~)
F F
In order to see how this impulse response can be realized with a
tapped delay line, let us consider the Comb operation. The Comb opera¬
tion, as defined by Woodward, multiplies the time waveform by an infinite
series of uniformly spaced delta functions.
CombT [h(t)J - h(t) £ 6(t-kTF)
F
«. on
GO
- £ h(krF) 6 (t-KTF)
«• oo
where the summations are over the index k. This impulse response consists
of a weighted sequence of 6- functions and it can be realized with a
tapped delay line where the output of the kth tap is weighted with amplitude
h(kT_). Writing out the expression for h1, we have
F
oo oo
h'(t) “Ihk6(t'kV "^[IV(t*krF)]
• OD - 00
where we have abbreviated h^ * h(kT^) and h^ * ?/h(kTp).
Regarding the problem of realizability, the first term in the expression
for h*(t) will be zero for negative t, but the second term in the expression
for h*(t) is not necessarily zero for negative t. The reason for this is
that the Hilbert transform of a function which is zero for negative t yields
a function of time which is not, in general, zero for negative t. We have
seen earlier that the Hilbert transform operation is equivalent to passing
the waveform through a linear filter with non-reallzable Impulse response
1/nt and so this result is not surprising. However, as mentioned earlier,
the Hilbert transform can be closely approximated by a realizable filter if
sufficiently large time delays are allowed. For example, consider a filter
with an impulse response ^(t-T)^"1 for t>0 , and zero for t<0. This
function is plotted in Figure 3.
FIGURE 3
TM-3506
10.
This Impulse response is a truncated version of a Hilbert transformer In
series with a time delay T, the effects of the truncation becoming smaller
as T becomes larger. We assume in the discussion which follows that this
type of approximation is used for the Hilbert transform wherever it occurs
and that sufficient time delay is introduced where necessary to maintain
realizability of the required impulse responses. We assume, in particular,
that h(t) has been delayed sufficiently so that both h^ and are zero
for k < 0.
The tapped delay line realization of h'(t) is shown in Figure 4.
As we have pointed out earlier, the operation gives a broad-band
+ 90° phase shift. A somewhat more practical scheme may be to use at the
outputs of the adder channels two broadband + 45° phase shifters as shown
in Figure 5. Here a sufficient (but equal) time delay is assumed to be
included in both phase shifters so that they can be accurately realized
over the frequency band of interest.
In effect, we can view the operation on the output of each delay line
tap as an amplitude weighting a^ and a phase shift 0^, where
hk * \ cos K
^k - ak 8in 0k
K - tan'x<Vhk>
It Is useful to work out the expression for the frequency response of the
tapped delay line, H'(f), in terms of the coefficients a^ and 0^. H'(f)
is found by taking the Fourier transform of the expression obtained earlier
for h'(t) .
FIGURE 4
TM-3506
12
FIGURE 5
oo
H'(f) - J h,(t)e'J2nft dt
- 00
I
V
-j2TTfkrF
+ J
■j2nfierF
-I-
j0k -j2TTfwrF
e e
As expected, this function is periodic in frequency with period
Noting the orthogonality relation
f + *
o
;
J2TTf (m-k)Tp
2
d£
iAF
TM-3506
13.
where m and k are integers, the expression for H'(f) can be used to find
a^ and 0^ in terms of H 1 ( f ) .
f0* V wt
- J H(£)e)2"£klF «
V 4 "p
The prime has been left off the H(f) because H*(f) * H(f) over the region
of integration (again ignoring constants of proportionality). Taking the
real and imaginary parts of both sides, we obtain
j0lt p j;mrKi-F
h^ ■ Re (a^e ) ■ Be J H(f)e df
j 2rrf kTF
*0“ ^ Wf
\ - ti»n(akeJ<}k) = <£»t[ H( f)1
fo+ * WF
J2nfkTp
df
V*wr
Thus it is seen that the requf/?.d tap weightings, h^ and can be
easily computed from the required H(f).
The tapped line has a frequency response which is equal to the
desired frequency response H(f) in the band of interest, and is periodic
in frequency with period equal to the reciprocal of the delay line tap
spacing. The behavior of the network frequency response over the band
of interest depends only on these tap weightings. Since these tap
weightings can be obtained from a set of potentiometer adjustments, the
response of the line can be changed easily by readjusting these pots.
With the aid of a high quality quarts delay line having many taps, one
should be able to synthesize a very complicated network characteristic.
TM-3506
14.
It should be noted that this tapped delay line method ot filter synthesis
is perfectly general; Its use is not restricted to the synthesis of
dispersive networks only.
Suppose we wish to utilize the characteristics of this filter over
only one of these frequency periods. The situation Is shown schematically
in Figure 6. Hfl(f) denotes the frequency response of a band selection filter.
I M* (f)l
In the ebove plot only the amplitude of the frequency response has been
shown. A similar plot could be shown indicating the repetitive phase shift
characteristics of the network. That portion of the frequency response
which we desire to use can be extracted by means of the band selection
filter H#(f) with a linear phase and a constant amplitude characteristic
over the region in which H'(f) differs appreciably from aero, but which
TM-3506
15.
drops off sufficiently rapidly to exclude responses from neighboring
bands •
More generally, we can consider the desired H(f) to be decomposed
into the product H'(f)*H (f), where H*(f) is realized with a tapped delay
s
line and Hs(f) represents the band selection filter. In this manner
the non ideal characteristics of H (f) can be compensated somewhat by
s
adjustment of H*(f). Thus the burden of accurately approximating H(f)
can be shared between both H*(f) and H (f), though most of the burden
s
will probably still fall on H*(f) because of its ease of adjustment.
The number of taps required on the line depends on the nature of
the filter we are attempting to synthesize. If the filter response is
identically zero outside of some band, then an infinite number of taps
are required, in principle. However, if the filter response is smoothly
tapered over this band, then only a finite number of these tap weightings
will differ appreciably from zero and a good approximation to the desired
filter response can be had by utilizing a delay line with only a finite
number of taps. The filter impulse response, including the band selection
filter, can occupy at most a band of width W^, in frequency and approximately
NT„ * N/VL, in time. The time -bandwidth product of the filter impulse
r r
response can at most be equal to the product of these quantities which is
simply N, the number of taps on the line. The practical design problems
in obtaining a band selection filter, and the approximation problem associated
with truncating the number of taps on the delay line will result in a filter
impulse response whose time -bandwidth product is somewhat less than N.
The tapped delay line realization of the repetitive filter is valid
at all frequencies from zero to infinity. In practice, the device would
have a bandwidth limited by the frequency response of the delay line employed.
If a band selection filter were used, it would probably be centered on the
center frequency of the bandpass tapped delay line.
Sometimes it is desirable to use a tapped delay line whose center
frequency is zero, i.e. a lowpass line. A number of commercially available
TM-3506
16,
distributed parameter electrical delay lines are of this type* In order
to see how the desired response can be realized with tapped lowpass delay
lines, let us write the impulse response h(t) in the form
h(t) * u(t) cos WQt - v(t) sin uu^t
If h(t) is assumed to be bandlimited to the region (-2uuo, 2uT), then
u(t) and v(t) will have a finite spectrum only in the frequency range
(-uuo> uoq) . Noting the relations proved earlier,
^ [u(t) cos V] « u(t) sin uu^t
9f[ v(t) sin WQtJ * • v(t) cos uu^t
the desired impulse response h'(t) can be written
h'(t) - CombT [h(t)J - ^Comhj j^h(t)]
F F
* CombT [u(t) cos u^t - v(t) sin uUQt J
F
-VComb^ ^(u(t) cos U)Qt - v(t) sin U)Qt/]
* cos uu^t Comb^ u(t) -
F
-^£aln ouQt Combj, u(t)
sin (JU^t Combtj, v(t)
+ cos u> t
o
Comb^, v(t)J
Now, If Comb^ u(t) and Comb^ v(t) were functions whose spectra were
confined to the frequency range (*ouQ, ouq) , the could be brought Into the
bracket in the last term according to the rule. The Comb operator, of course,
TM-3506
17.
gives an infinite bandwidth, but suppose we consider a Comb operator which
has been filtered so as to eliminate all but the very low frequency com¬
ponents. We shall denote this operator by Comb^ . As a practical matter,
the Comb operation which can be realized with a lapped delay line is
limited in this way because of the finite bandwidth of the delay line.
Recalling that the Comb operation involves multiplication in the time
domain, the bandwidth occupied by Comb^ u(t) is equal to the sum of the
bandwldths occupied by CombJl and u(t).F Provided that the carrier
frequency U)o is chosen to be larger than the resulting bandwidth, W
can be brought into the bracket according to the rule. Combining
coefficients of cos 'JU^t and sin (JUQt , we have the simple result, ignoring
constants of proportionality as usual,
h*(t) * cos w t ComUl u(t) - sin U) t Comb* v(t)
0 F 0 F
Here the desired impulse response is repetitive in frequency, but this
repetition extends only over the bandwidth of the delay line. As before,
the Comb operations can be realized with the aid of tapped lines, where
the tap weightings are equal to sampled values of the function in question.
We denote uk = u(kTp) and v^ = v(kTp) . A method for realizing h’(t) using
tapped delay lines is shown in Figure 7. It can be easily shown that
/
all time-varying terms in the impulse response of this system are zero.
The configuration shown requires two lowpass delay lines, but no Hilbert
transformers are necessary. If the band over which the frequency response
should be repetitive has width W, the delay lines should have a bandwidth
Let us consider now how the tapped delay line network can be used in
the synthesis of a linear FM pulse compression network with large TW product.
The required group delay versus frequency characteristic is repeated in
Figure 8. This characteristic can be resolved into the sum of the two time
delay versus frequency characteristics shown in Figure 9.
FIGURE 7
TM-3506
20.
The sawtooth function is periodic in f end hence could be realized
with the aid of a tapped line. The staircase function also suggests
the use of a tapped delay line with uniformly spaced taps spaced Tq
apart and with a band selection filter of width Wq at the output of
each tap. Each of theso networks would have to be all-pass, l.e. have
uniform amplitude characteristic across the band, and the desired group
delay characteristic would be obtained by cascading these two networks
In series.
As a practical matter It Is not possible to construct band selection
filters which are perfectly rectangular and a repetitive filter which Is
a perfect sawtooth. One method which might prove satisfactory would be
to use a set of amplitude selection filters, somewhat overlapping, to
partition the signal band Into a number of frequency bands, as shown In
Figure 10. The output of each filter could be translated In frequency
SET OF AMPLITUDE SELECTION FILTERS
FIGURE 10
•o as to obtain a set of equally spaced but non overlapping bands as shown
in Figure 11. Each band can be delayed an amount proportional to Its
L A/VAAA/lAAAyi ,
FIGURE II
TM-3506
21.
frequency , the results added and passed through a repetitive network
with the desired dispersive characteristic for each band. The result
can be resolved back down into its separate frequency components with
a set of band-selection filters, and the resulting bands can be
translated so they once again occupy adjacent positions in frequency,
as shown in Figure 10.
If the amplitude selection filters are reasonably tapered and
reasonably closely spaced, the overall amplitude characteristic should
be almost flat. The repetitive network need only have its nearly saw¬
tooth character within the bands where substantial signal exists and
therefore the requirements imposed upon the repetitive network can be
relaxed considerably.
Other arrangements can be used employing one or several repetitive
networks which will allow us to obtain the desired dispersive
characteristics over several frequency bands simultaneously. The above
arrangement is meant to serve only as an example. The application of
repetitive networks to the realization of this type of pulse compression
system will require much further study.
Acknowledgement
The author wishes to acknowledge helpful discussions relating to
the subject material of this report with Mr. E. L. Key and Mr. R. W. Jacobus.
Roger Manas 8 e
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