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LINEARIZATION OF FM-CW RADAR
SWEEP BY FEEDBACK
Steven Alfred Marshall
UUL li I’AHV
IMA . ' M F SUHOUfl
^iuII.l y. ia y-jwao
NAVAL POSTGRADUATE SCHOOL
Monterey, California
TH IESSS
LINEARIZATION OF FM-CW RADAR
SWEEP BY FEEDBACK
by
Steven Alfred Marshall
December 197^
Thesis Advisor: D.B. Hoisington
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Linearization of FM-CW Radar Sweep
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Master’s Thesis;
December 197^
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Steven Alfred Marshall
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Radar
FM-CW Radar
Frequency Modulation Linearization
20. ABSTRACT (Continua on ravaraa aida if na caaaary and idantify by block numbar)
A frequency-modulated radar is a radar in which a
continuous-wave transmission is frequency modulated in a known
manner in order to obtain range information. When the
frequency modulation is linear with time, the difference between
the frequency of the received signal and the transmitted signal
is directly proportional to the target range. The difference
frequency is given by (2R/c) (df/dt ) , R being the range to the
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target, c the velocity of electromagnetic waves and df/dt the
slope of the sawtooth frequency modulation. Therefore, the
difference frequency is proportional to both range and the
slope of the modulated waveform. In an ideal sawtooth
frequency-modulated waveform, the slope would be a constant;
however, the device to be used in the portable FM-CW radar,
a varactor-tuned Gunn oscillator, exhibits a non-linear
frequency sweep. If the frequency sweep is non-linear, the
return from a target at fixed range is frequency modulated.
As a result the bandwidth of the echo signal is increased, and
range resolution is impaired. Work on this thesis will be
directed towards linearization of this FM sweep and,
accordingly, to optimize range resolution.
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Linearization of FM-CW Radar
Sweep by Feedback
by
Steven Alfred Marshall
Lieutenant, United States Navy
B.S.E.E., University of Colorado, 1969
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
December 197^
7~/vej * J
1 * 13 $$^
a. /
L/bl " *
v ’ | ‘A«V
v H.JC
ABSTRACT
A frequency-modulated radar Is a radar In which a
continuous-wave transmission is frequency modulated in a
known manner in order to obtain range information. When
the frequency modulation is linear with time, the difference
between the frequency of the received signal and the
transmitted signal is directly proportional to the target
range. The difference frequency is given by (2R/c ) (df/dt ) ,
R being the range to the target, c the velocity of electro-
magnetic waves and df/dt the slope of the sawtooth frequency
modulation. Therefore, the difference frequency is
proportional to both range and the slope of the modulated
waveform. In an ideal sawtooth frequency-modulated waveform,
the slope would be a constant; however, the device to be
used in the portable FM-CW radar, a varactor-tuned Gunn
oscillator, exhibits a non-linear frequency sweep. If the
frequency sweep is non-linear, the return from a. target at
fixed range is frequency modulated. As - a result the bandwidth
of the echo signal is increased, and range resolution is
impaired. Work on this thesis will be directed towards
linearization of this FM sweep and, accordingly, to optimize
range resolution.
4
TABLE OF CONTENTS
I. FREQUENCY-MODULATED RADAR 10
A. FREQUENCY-MODULATED VERSUS PULSE RADAR — 10
B. FM-CW MARINE RADAR APPLICATIONS 11
C. RF SOURCE FOR FM-CW 13
1. FM Generation 13
2. Improving Sweep Linearity 13
II. GUNN-DIODE VOLTAGE-TUNABLE OSCILLATOR l4
A. CONCEPT OF LOW-POWER PORTABLE RADARS 14
B. GUNN-DIODE VOLTAGE-TUNABLE OSCILLATOR
CHARACTERISTICS l4
III. MODULATION LINEARIZATION BY FEEDBACK 17
IV. DELAY 19
A. TWO METHODS FOR CREATING DELAY 19
1. Delay Using Coaxial Cable 19
2. Delay Using MAD Line 21
V. SYSTEM PARAMETERS 24
A. MODULATION FREQUENCY AND PERIOD 25
B. STATIONARY TARGET AND RANGE
AMBIGUITY 26
C. DOPPLER FREQUENCY SHIFT AND RESULTING
RANGE ERROR 29
D. RANGE RESOLUTION 34
E. BEAT FREQUENCY DEPENDENCE ON FREQUENCY-
SWEEP BANDWIDTH 35
VI. DETERMINING LINEARITY 40
A. TUNED CAVITY AND PROBE METHOD 40
1. Cavity Linearity 41
2. Probe/Diode Detector Linearity 42
5
3. Limitations on Cavity Use 45
B. BEAT- FREQUENCY METHOD 48
1. Equipment Setup 48
2. Measuring Time Between Beat-
Frequency Peaks 48
3. Accuracy in Time Measurements 51
4. Meaning of Time Measurements 51
VII. FINAL EQUIPMENT SETUP 52
VIII. EFFECT OF NOISE IN CORRECTION CIRCUIT 58
IX. PHASE-LOCKED LOOP 60 '
X. DELAY AND FILTERING 64
XI. VOLTAGE RAMP RETRACE TIME 65
XII. OPTIMUM CORRECTING VOLTAGE 66
XIII. LINEARITY ANALYSIS 70
XIV. OTHER METHODS OF FM LINEARIZATION 74
A. SAMPLE AND HOLD METHOD 74
B. ANALOG/DIGITAL METHOD 75
XV. CONCLUSION 86
LIST OF REFERENCES 87
INITIAL DISTRIBUTION LIST 88
6
LIST OF FIGURES
1. Varactor-tuned Gunn oscillator change in
output frequency Af versus varactor voltage 16
2. Block diagram of basic feedback system 18
3. Microwave acoustic delay device with isolators
attached to input and output ports 23
4. Shape of frequency swept waveform 24
5. Transmitted and received signals from a
stationery target at less than maximum range
with resulting beat frequency shown below 27
6. Transmitted and received signals from a
stationary target at maximum range with
resulting beat frequency shown below 28
7. Transmitted and received signals from a
stationary target at greater than maximum
range with resulting beat frequency shown
below 30
8. Transmitted and received signals from a
stationary target and a moving closing target
at the same range showing difference in
beat frequency 31
9. Cavity and probe apparatus 42
10. Resonant peak showing f and Af 43
11. Diode-detector output voltage versus
varactor voltage 44
12. Diode-detector output voltage versus
diode-detector input voltage 46
13. Power output versus varactor voltage 47
14. Block diagram of equipment setup used for
measuring time between peaks of the
beat-frequency signal 49
15. Method used for measuring time between
beat frequency peaks 50
16. Block diagram of final equipment setup 53
7
17. Photograph, laboratory system assembly showing
delay device, mixer, cavity and probe and
circuitry view no. 1 54
18. Photograph, same as Figure 17 except
view no. 2 54
19. Correcting circuit, closeup view 55
20. External amplifier, closeup view 56
21. Schematic diagram of correcting circuitry 57
22. Block diagram of phase-locked loop 6l
23. Correcting signal (AC) 63
24. Corrected ramp (AC) shown with correcting
signal (AC) for one. cycle 67
25. Corrected ramp (AC) shown with correcting
signal (AC) for two cycles 67
26. Uncorrected output frequency waveform as
obtained from cavity and probe 68
27. Corrected output frequency waveform as
obtained from cavity and probe 68
28. Beat frequency (uncorrected) as recorded at
input to PLL 71
29. Beat frequency (corrected) as recorded at
input to PLL 71
30. Deviation of ramp slope, df/dt ,
from average 73
31. Sample and hold method No. 1 block diagram 76
32. Sample and hold method No. 1 circuit
waveforms (Part I) 77
33. Sample and' hold method No. 1 circuit
waveforms (Part II) 78
34 . Sample and hold method No. 2 block diagram 79
35. Sample and hold method No. 2 circuit waveforms — 80
36. Block diagram of analog/digital linearization
method 82
37. Analog to digital error conversion 83
8
ACKNOWLEDGEMENTS
The author wishes to thank Professor David B. Hoisington
for his help and encouragement throughout the course of this
thesis. Appreciation is expressed to Mr. Ernie Kirchner and
to Teledyne MEC for the generous loan of a microwave acoustic
delay line, without which this thesis would not have been
possible. A great deal of thanks goes to Lieutenant Glenn
Ewing whose helpful suggestions aided in the successful
completion of this thesis. And most importantly, a million
thanks to my wife, Linda, and my children for their patience
and understanding during the last hectic weeks of this
thesis.
9
I. FREQUENCY-MODULATED RADAR
A frequency-nodulated radar is a radar in which a
continuous-wave transmission is frequency modulated in a
known manner in order to obtain range information. The
range information is obtained by comparing the received
signal with the transmitted signal such that a difference
frequency is obtained. This difference frequency is directly
proportional to the distance to the reflecting object that
caused the echo.
A. FREQUENCY-MODULATED VERSUS PULSE RADAR
In contrast, pulse radars send out a series of short
bursts or pulses of radio- frequency energy and receive the
delayed echos in the silent intervals between transmitted
pulses .
Since frequency-modulated radar both transmits and
receives signals simultaneously, this prevents the time-
division antenna duplexing used in pulse radar. Typically
separate transmitting and receiving antennas are utilized
when the transmitted power exceeds about one watt.
One of the advantages of FM-CW radar is that ranges of
only a few feet may be measured. This explains the develop-
ment of frequency-modulated radar for the measurement of
altitude of aircraft. In World War II, all major combatants
utilized frequency-modulated altimeters [1].
10
Frequency-modulated and pulse radar systems having the
same average power, frequency of operation, antenna gain,
signal bandwidth, signal Integration times, and so on, have
maximum ranges of the same order of magnitude. One advantage
of FM-CW over pulse radar is that it is not required to
handle the large peak power levels associated with pulse
radars.
Range resolution is dependent upon the radio-frequency
bandwidth (range resolution is discussed in detail in
Section V). In frequency-modulated radar range resolution
can be increased simply by increasing the modulation
bandwidth. In conventional pulse radar increasing range
resolution is accomplished by making the transmitted pulse
narrower. This means that in order to obtain a given average
power out of the radar, the peak power must be increased,
and the system must be capable of handling the higher peak
power level. Moreover, receiver bandwidth must be changed
when the pulse width is changed or maximum range capability
suffers. Therefore, high range resolution is more easily
attained in a frequency-modulated radar, but the theoretical
range resolution can only be achieved if there is a high
degree of modulation linearity. The linearity must be
present over the entire range of the frequency sweep.
B. FM-CW MARINE RADAR APPLICATIONS
Frequency-modulated marine radars are designed
principally for obtaining range information. These radars
11
would be at a great disadvantage when compared to pulse
radars for application against high-speed targets. The
greater the relative velocity between the radar platform
and the target, the greater the doppler shift of the
reflected signal; therefore, the difference frequency would
be altered by this doppler shift, causing a range error.
The effect of doppler-frequency shift caused by a moving
target is discussed further in Section V. The pulse radar,
unlike the frequency-modulated radar, does not depend upon
a frequency shift to provide range information, but rather
it depends only upon the time it takes for a narrow pulse
of RF energy to travel from the radar, strike the target,
and return to the radar. The total time of travel is
directly proportional to the range to the target and is
totally independent of the doppler frequency shift.
The doppler shift is not utilized in the standard pulse
radar; however, there are certain pulse radars which do make
use of the doppler shift to detect moving targets in clutter.
These radars are known as MTI (Moving Target Indication)
radars when the duty cycle is low, or pulse-doppler radars
when the duty cycle is high.
The FM-CW radar can be very effective in providing
accurate range information on stationary targets or targets
moving with a low relative velocity where the doppler shift
encountered is small, see Section V for doppler calculations.
12
C. RF SOURCE FOR FM-CW
The portable FM-CW radar being designed will utilize
a varactor-tuned Gunn oscillator as an RF source for
reasons of practicality. The device is practical for
portable applications since it is capable of operating from
low voltage (typically 10 VDC) supplies and is easily
frequency-modulated .
1. FM Generation
The frequency modulation will be generated by placing
a ramp voltage on the varactor portion of a varactor-tuned
Gunn oscillator. To optimize range resolution, it is
necessary to obtain as linear a frequency sweep as possible.
This is a problem when utilizing the Gunn diode source ,
as the frequency sweep obtained by placing a linear ramp
voltage on the varactor is inherently non-linear over its
entire range. The degree of non-linearity is small when
frequency sweep is restricted to a small bandwidth, but the
non-linearity increases as the sweep amplitude increases.
This effect can be seen on the change in output frequency
versus varactor voltage (time) curve. Figure 1.
2 . Improving Sweep Linearity
As one possible method of improving the sweep
linearity, a feedback circuit, utilizing an acoustic delay
device, a microwave mixer, and a phase-locked loop as major
components was used to create- a correcting signal. The
correcting signal was applied to the varactor in addition to
the linear- volt age ramp. The combination of the two signals
greatly reduced the non-linearity in the sweep.
13
II. GUNN-DIODE VOLTAGE-TUNABLE OSCILLATOR
A. CONCEPT OF LOW-POWER PORTABLE RADARS
The concept of low-power portable radars operating from
low-voltage battery supplies became more appropriate with
the advent of the Gunn-diode oscillator. The low-voltage
supply used with the Gunn oscillators, typically 10 VDC,
is less expensive and more practical than the high-voltage
supply necessary for magnetron or klystron operation. Gunn
oscillator development has reached the point of obtaining
from approximately 5 watts continuous-wave power output at
C band, to 0.5 watts continuous-wave power output at X band
[ 2 ].
Police hand-held doppler radars utilizing stable CW
Gunn oscillators as RF sources are capable of detecting
doppler frequency shifts and converting these shifts into
corresponding relative vehicle speeds. These simple doppler
radars are capable of very accurate speed measurement, but
they do not have range finding capabilities. By frequency
modulating the CW signal, range information can be extracted
from comparison of the instantaneous frequencies of
transmitted and echo signals.
B. GUNN-DIODE VOLTAGE-TUNABLE OSCILLATOR CHARACTERISTICS
The typical Gunn-diode voltage-tunable oscillator
consists of a Gunn-diode oscillator with an integral varactor
1H
diode. In the 9 GHz frequency range the apparatus typically
is mechanically tunable over a bandwidth of 1-2 GHz, and
voltage tunable over a wide range, typically 100 MHz, with
a tuning voltage of 0-35 VDC. Figure 1 is a curve showing
variation with varactor voltage for the device used in this
work.
The varactor tuning provides an ideal method for generat-
ing an FM sweep since sweeping the dc voltage on the varactor
causes a frequency shift proportional to the amount of
voltage applied. This frequency modulation of the RF
signal provides the necessary reference point to obtain
range information.
The varactor-tuned Gunn oscillator is extremely sturdy.
With a little caution exercised \\ T hen applying maximum bias
conditions, the device should give many thousands of
maintenance-free hours of operation. Solid-state sources
have life expectancies in excess of 100,000 hours, but with
the attached varactor, mean life time between failure
varies between 2000 hours and 40,000 hours of operation for
the varactor-tuned Gunn oscillator [ 3 ] •
The long lifetime along with the low-voltage supply
used with the Gunn oscillator makes it possible to design
a low-power FM-CW radar that is small in size and weight
while requiring a minimum of maintenance.
15
CHANGE IN OUTPUT FREQUENCY Af (MHz)
FIGURE 1. Varactor tuned Gunn Oscillator
change in output frequency Af
versus varactor voltage
l6
III. MODULATION-LINEARIZATION BY FEEDBACK
The basic idea for linearizing the frequency sweep of
the varactor tuned Gunn oscillator is shown in Figure 2.
A portion of the RF output from the Gunn oscillator is
delayed then mixed with an undelayed portion of the RF
output. The resulting difference frequency out of the
mixer is a function of the rate of change of frequency of
the sweep waveform. For a linear ramp of frequency sweep,
i.e., constant slope, the difference frequency fg, out of
the mixer will be a constant. The value of fg is determined
by the rate of change of frequency and the time delay
introduced by the delay device.
f„ = (slope) (time delay)
iD
- <§>
However, the slope of the output frequency versus modulating
voltage is not linear for practical devices; therefore, fg
will be variable over some range. It is this variable
frequency which will be used to generate a correcting
signal, which when summed with the ramp voltage and applied
to the varactor, produces a more linear frequency sweep.
This scheme is a negative feedback system, but it must be
realized that instabilities are present throughout the system
which could cause the system to go into oscillation under
various conditions.
FIGURE 2. 'Block diagram of basic feedback system
18
IV. DELAY
The first problem encountered in setting up the system
was how to obtain the necessary time delay. The delay must
be sufficiently long such that the resulting difference
frequency is high in comparison to the frequency of the
modulation sweep. This is a necessary condition which
allows for an ample number of corrections to be made during
the period of the modulation sweep. The delay must also be
constant over the entire bandwidth of operation.
A. TWO METHODS FOR CREATING DELAY
Two possible methods were investigated which could
create the delay: 1) coaxial cable, and 2) a microwave
acoustic delay device. A third possibility, creating the
delay by propagation through a waveguide, was entirely ruled
out as being infinitely impractical for this application
because of its bulk and high cost.
1 . Delay Using Coaxial Cable
For coaxial cable, the velocity of propagation is
given by the expression:
1 _ 1 c
v = = -
\/ « \J VoVo \l v r Z T
19
where: c = speed of light, approximately
3 xlO 8 m/sec
y r = relative permeability, approximately 1.0
y Q = permeability constant
= 4 ttx 10 - ^ weber/amp-m
E r - relative dielectric constant
e Q = permittivity constant = 8.85*1 x 10 -12 F/m
c
therefore, v = — • -
\P^
When the above expression is applied to the solid
dielectric coax such as RG-213, with a relative dielectric
constant of 2.3, and loss equal to 47 dB per 100 feet at
10 GKz, the following results:
RG-213
DELAY CABLE LENGTH LOSS
0.1 ysec 19*7 meters 30*6 dB
1.0 ysec 197 meters 306 dB
Since a delay in the order of 1.0 ysec is necessary
to give an adequate difference frequency, it is evident that
the RG-213 coaxial cable will not be suitable because of its
high loss. This loss is much too high for use in a system
designed for operation in the milliwatt power range.
20
Even the lowest-loss cable gives unacceptably high
attenuation for a 1.0 usee delay. The lowest loss cable
listed In [4] Is RG-263 with a solid polytetraflouroethylene
dielectric, relative dielectric constant of 1.5 and loss
equal to 9 dB per 100 feet. Loss versus length at 10 GHz
Is as follows :
RG-263
DELAY CABLE LENGTH LOSS
0.1 sec 24 meters 7.1 dB
1.0 sec 240 meters 71 dB
The only acceptable case is the RG -263 with a
short delay.
The idea of using coaxial cable to obtain a delay
at microwave frequencies is only intended for experimental
purposes. It would be illogical to design a miniature radar
requiring several hundred feet of bulky coaxial cable for
the sole purpose of creating a delay.
2 . Delay Using MAD Line
The development in microwave acoustic devices has
made possible the miniaturization of microwave delay lines.
This miniaturization allows the delay line to become an
integral part of the radar, making it in this respect ideal
for use in the small portable type radar under study. The
major problem in using microwave acoustic delay (MAD) lines
is their very high attenuation loss, typically 55 dB for a
21
1.0 usee delay at X band. Yet, the high loss associated
with the MAD lines is still lower than that from low-loss
coaxial cable for the same delay. Since the MAD line offers
small size and relatively low loss, it was chosen as the
means of creating the delay.
a. Description of MAD Device Used
The device used had a delay of 1.0 ysec, constant
over a 2 GHz bandwidth, and produced an attenuation loss of
approximately 5^ dB at the system operating frequency of
9.5 GHz. The device was very conveniently packaged as can
be seen from a photograph of the device with accompanying
isolators. Figure 3. The isolators were matched to the device
by Teledyne MEC for the purpose of minimizing reflection
problems. Configuration of the MAD device as received from
Teledyne made its application within the system very easy.
The device performed exactly to specifications, producing a
1.0 ysec delay over the required frequency bandwidth. Since
the delay device was capable of handling 200 mW of average
power, no problems were encountered in this respect.
22
23
FIGURE 3. Microwave acoustic delay device with isolators attached to input
and output ports. Note scale for actual size
V. SYSTEM PARAMETERS
The portable radar was designed around a center frequency,
V a PP rox i m ately 9.5 GHz, a maximum range R equal to
in 3.x
5 miles and a ramp (sawtooth) frequency swept waveform as
shown in Figure 4.
FIGURE . Shape of frequency swept waveform
A. MODULATION FREQUENCY AND PERIOD
Restrictions on the value of the ramp period, x , can
be found as follows:
Velocity
distance
time
or
v
d
t
Utilizing the above equation with velocity equal to the
velocity of electromagnetic propagation c, the distance equal
to twice the maximum range 2R to allow the necessary time
max
for electromagnetic energy to transit the R max distance and
return, and time equal to one-half of the modulation period
x m (the reason for using the factor one-half x m is explained
later in this section), the following results for a range of
5 miles:
, 2R^ ov (2) (5 miles) (1600 m/mile)
_L max _
2 T m c / 0 I T r
(3x10 m/sec)
x = 107 ysec
m
f m < ~ = 9.35 KHz
m
As a comparison, if R max were chosen to be 3 miles, x m
would be equal to 6*4 ysec and f m would be less than 15.6 KHz.
For a system using the sawtooth frequency sweep described
above, a brief background on theoretical frequency-modulation
performance follows.
25
B. STATIONARY TARGET AND RANGE AMBIGUITY
Figure 5 depicts a transmitted signal and a received
signal which are mixed together at the radar to obtain the
beat frequency as shown in the lower part of the figure.
From the figure:
f* = f + ft where f = , t . = —
d dt 3 d c
f* = f + (~)f
f* - f = (^)f
DESIGNATE: Range Frequency, f„ = (— )f
Thus, for a stationary target, the beat frequency or range
frequency, is directly proportional to target range [5].
The beat frequency f^ as shown in Figure 5, is the
frequency which corresponds to the actual target range,
while frequency ?2 corresponds to false range information.
If the radar is designed to distinguish between the two
frequencies by tracking the lower of the two, then the false
signal will not cause errors for the case depicted. But,
if the received signal is from a target farther away, f^ is
increased and decreased. When the echo signal returns
to the radar in a time x m /2, f-^ and fj are equal. Figure 6.
Any target at a greater range will cause f, to be larger
26
BEAT FREQUENCY FREQUENCY
FIGURE 5. Transmitted and received signals from a
stationary target at less than maximum
range with resulting beat frequency
shown below
27
BEAT FREQUENCY FREQUENCY
FIGURE 6. Transmitted and received signals from a
stationary target at maximum range with
resulting beat frequency shown below
28
than f 2 as shown in Figure 7. If this is the case, the
radar would then be presenting false range information.
From the above discussion, range ambiguities are possible
when the echo from a target arrives after a time greater than
t_/ 2 since it was transmitted. Thus, R is chosen as that
range from which an echo return would be received in time
x m /2. Ambiguities resulting from echo returns from targets
beyond designed maximum range could be eliminated by
accepting the range frequency only if at the corresponding •
time, the transmitted frequency is greater than the received
frequency. This insures that the echo return is from a
target within the designed range capabilities of the radar.
A system utilizing this concept would be unacceptably complex
for simple portable radar application. It would be much more
reasonable to simply accept the ambiguity in the few cases
it became applicable.
C. DOPPLER FREQUENCY SHIFT AND RESULTING RANGE ERROR
V/hen there is relative motion between a target and a
radar, it can easily be shown that [6]:
2v
f = — — f where v = relative velocity between
d c 0 r radar and target
f = RF frequency
c = velocity of electromagnetic
propagation
Figure 8 shows a transmitted signal, a received signal
from a stationary target and a received signal from a moving
target .
29
BEAT FREQUENCY FREQUENCY
FIGURE 7. Transmitted and received signals from a
stationary target at greater than maximum
range with resulting beat frequency
shown below
30
FREQUENCY
received
(stationary target) -7- .
FIGURE 8. Transmitted and received signals from a
stationary target and a moving closing
target at the same range showing
difference in beat frequency
The beat frequency caused by the moving target is:
f
B
f '
- f"
= [f + (^)f] - [f +
c c
(2£) f -
2v
(— f
c
c
range
speed
frequency
frequency
DESIGNATE: Speed frequency, fg
2 v
±(— )f
31
It should be noted that the sign associated with the
speed frequency is (+) if target and radar are moving apart
relatively and (-) if target and radar are closing relatively.
It is evident that the beat frequency obtained depends
both on the range to the target and the relative speed
between target and radar platform. A determination of the
range error associated with the doppler- frequency shift
follows :
f R -
f
or R =
2f
f
d
f
o
The FM radar range to a moving target at a distance R
consists of the range associated with a stationary target
at the same distance R, plus or minus a range error AR,
associated with the doppler-frequency shift.
R + AR
cf R + ^d
2f 2 f
If the
expression
expression for f d is substituted into the above
the results are:
C *R f o
R + AR = — £ ± v„ —
2f r f
Therefore, the range error AR - v
32
To determine the extent of the range error caused by
the doppler-frequency shift, a hypothetical case where the
relative speed between two craft was 120 miles/hour was
considered. This is an extreme case for marine radar
application and should represent a maximum effect in range
error.
120
miles
hour
52.8
m
sec
a ... f o _ (52.8 m/sec) (9. 5 GHz)
K V r J, 20 MHz
lo7 ysec
AR = 2.68 meters
This small value of range error indicates that the
doppler-frequency shift will have negligible effect on the
accurate range measuring capability of the system. The
percentage effect varies with the range to the target
involved. For the example above, a relative velocity of
120 miles/hour, and a range of 1000 meters, gives a
percentage change as follows:
2.68 ( 100 )
1000
. 268 $
With an error in measured range of less than 1$ in
300 meters even under the extreme case depicted above, it
was appropriate to neglect the effect of the doppler-
frequency shift in the present system.
33
D. RANGE RESOLUTION
f R = = ( ^r } ¥■
K C C T
m
T f R = — Af
m R c
T m^R = avera S e number of cycles of the beat frequency
occurring in the time of one complete
modulation cycle (t )
Let :
AN = t f D
m R
AN
(^)Af
or R
CAN
2Af
6R
c6 ( AN)
2Af
[ 7 ]
There exists an indefinite borderline for target resolution
at 6 (AN) approximately equal to one complete beat cycle over
the modulation period. That is to say if there is
approximately one complete cycle variation over the modula-
tion period T m between the beat frequencies produced by two
different targets, then they will be resolved by the radar
as two separate targets.
Let: 6 (AN) = 1.0
c
2Af
3*1
6R =
The last equation shows that range resolution 6R is
dependent on the frequency sweep width Af. The larger the
frequency sweep width, the better (lower value for 6R) the
range resolution. Values for 6R are tabulated below for
frequency sweep widths varying from 10 MHz -100 MHz.
Af 6R
10 MHz 15 meters
20 MHz 7*5 meters
30 MHz 5.0 meters
60 MHz 2.5 meters
100 MHz 1.5 meters
A Af of 20 MHz was selected for the present system
because it offered a range resolution capable of distinguishing
navigational aids, and yet, still represented less than
one-fourth of the available Af of the varactor-tuned Gunn
oscillator.
E. BEAT FREQUENCY DEPENDENCE ON FREQUENCY-SWEEP BANDWIDTH
When using the delay-line discriminator, the delay must
be constant over the entire operating range of the system.
The beat frequency fg is a function of both the delay time
H f*
t d and the rate of change of frequency [8],
f = (— )t = (— )t
*B Mt ;t d K xJ d
m
35
In order to obtain a suitable beat frequency, a proper
combination of Af, t m and t d is required. Since the modula-
tion period was calculated to be 107 ysec on the basis
of maximum range requirements, and the delay device selected
produced a fixed delay time t^ of 1.0 ysec, then from the
above expression, the beat frequency was directly dependent
on the frequency-sweep bandwidth Af.
The choice of a proper Af was based on the following
analysis: With the receiver blanked for the first half of
the transmitted sweep, the received signal would be present
for a time x m /2 in the receiver. This time was designated
tp and is approximately equal to 1/BW, where BW is the 3dB
bandwidth of the signal in the i-f amplifier. The 3dB signal
bandwidth determines the range discrimination since the
difference frequency for a target is directly proportional
to its range. Anything that increases the signal bandwidth
from a target causes a loss in range discrimination capability.
Non-linear frequency modulation of the transmitted signal is
one effect leading to this increased bandwidth. The
approximation that the increased bandwidth created by errors
in the frequency modulation of the transmitted signal is
approximately the same as the corresponding increase in beat
frequency was used in the analysis.
In the analysis of an FM-CW ranging system [9], it was
stated that the separation between beat frequencies of two
targets must exceed — — — Hz (t is the sweep duration and
T m T
t = 2R/c ) before the positions of the individual targets
36
become clearly defined. This expression with t = —
where R max Is 5 miles, results In the approximation of
t R = T m /2 used above.
The concept of the above discussion Is developed as
follows :
B
= (— )(t )
v dt Mr d ;
,df\ 2R
Mt } c
Af B
or
c
2R
Af B
A €>
(df)
dt avg
2R Af B
(M)
m
(10058)
The above equation expresses the percentage deviation of
the slope of the output frequency sweep as compared to the
average slope (Af/x ) . The permissible deviation of df/dt
from the average Is range dependent with a larger range
requiring a smaller deviation of the slope.
If the bandwidth was changed by errors In the frequency
modulation of the transmitted signal by an amount for example
of 7 KHz, about one-third of the i-f signal bandwidth, then
Afg = 7 KHz, and the expression for the percentage deviation
of the slope becomes:
37
164 x 10^ yds/sec
lb4 yds
) (7 KHz) (100*)
avg
where: R = R m j_ n = 82 yds for convenience
^ = 107
c = 1 64 x 10° yds/sec
Afg= 7 KHz
avg
75 MHz
Af
Since Af was selected as 20 MHz, the permissible percentage
deviation of the slope is 3 . 75 %.
Maximum deviation of the slope from the average was
determined in order to calculate the required number of
corrections that must be applied to maintain a deviation
from linearity less than a required amount. Prom measure-
ments on an uncorrected frequency sweep curve, it was found
that the slope at the beginning of the 20 MHz sweep deviated
from the average by 37 . 5 % and the slope at the end of the
sweep deviated by 17.5# from the average. The average slope
being that which would be obtained providing the frequency
sweep was perfectly linear. The number of corrections which
must be made to the system depends on the worse-case deviation
(i.e., 37 . 5 % deviation over half of the modulation period,
53-5 ysec) and the deviation desired. If for example, the
38
slope is allowed to deviate by as much as 5% at any instant
of time, then this would require approximately 8 or more
corrections over the 50 usee time period (i.e.,
37*5#/# cor. = 5% -*■ if cor. _> 7.5). This results in a
beat frequency requirement of greater than 160 KHz.
For the system parameters Af = 20 MHz, t d = 1.0 usee,
T m - 107 usee and fg = 187 KHz, the following results:
f B = 187 KHz
T.
m
T B ff cor.
# cor. =
m
B
or t b = 5 . 35 usee
53.5 usee _
5.35 ysec
37.5%
10 cor.
3-75% deviation maximum
Thus, the 10 corrections in x /2 or 20 corrections in
resulting from a beat frequency of 187 KHz, provide
a sufficient number of corrections to the frequency sweep
to reduce the linearization errors to approximately .
The above is true providing the correcting circuit works
perfectly as designed.
39
VI. DETERMINING LINEARITY
Some method Is required for measuring the frequency
modulation linearity. A point-by-point plot of output,
frequency versus varactor voltage (time) can be obtained
by applying specific voltages on the varactor and monitoring
the corresponding output frequencies. A plot of this type
is shown in Figure 1 for the device used in this work.
However, the point-by-point plot does not give either a
continuous nor a dynamic representation.
Two different methods were used to determine dynamically
the linearity of frequency-modulation: 1) the tuned cavity
and probe method, and 2) the beat-frequency method.
A. TUNED CAVITY AND PROBE METHOD
Some method for observing the output in the form of a
continuous plot of frequency versus time on an oscilloscope
is desirable. This method offers the advantages of being
able to view and to record easily the output waveshape
(i.e., output frequency versus time), and to detect changes
in waveshape while varying parameters. To obtain an oscil-
loscope display, a frequency discriminator is required.
A discriminator was constructed using a tuned cavity with
probe and crystal detector. The basic idea is to use the
variation of impedance with frequency on a resonant peak
near the physical center of the cavity [10]. The cavity
section is coupled to the main guide through an inductive
40
iris. Its resonant frequency can be varied by means of an
adjustable short. Figure 9 illustrates this apparatus. The
probe depth is varied to obtain the maximum coupled signal
on the oscilloscope while the sliding short is varied to
\
obtain an output voltage proportional to frequency.
1. Cavity Linearity
The cavity utilized was a standard waveguide section
cut to length for resonance in the TE^q^ mode at X-band.
The resonance frequency was made alterable by means of the
adjustable short which was tuned until a good linear portion
of the slope of a resonant peak was obtained.
Since the frequency sweep is only 20 MHz while the
center frequency is approximately 9.5 GHz, the frequency
sweep represents a very small change in frequency when
compared to the center frequency, i.e.,
(20 MHz/9.5 GHz) (100) * 0.258.
Cavity Q can be calculated by the expression
Q = f /Af, where f Q is the center resonant frequency of
the cavity and Af is the 3 dB frequency bandwidth, see
Figure 10. For f equal to 9.5 GHz and Af approximately
40 MHz, the cavity Q is around 240 . Then it is a reasonable
assumption that cavities with Q's lower than 200 would
exhibit linear regions over a 20 MHz frequency band on the
slope of the resonant peak. It is reasoned that a lower
Q cavity would be more desirable since a larger linear
FIGURE 9. Cavity and probe apparatus
portion could be utilized. The ability to utilize the large
linear portion of a lower Q cavity reduces the possibility
of waveform distortion caused by operation off of the
linear portion of the resonant slope.
2 . Probe/Diode Detector Linearity
A questionable area Involving linearity when using
this method is whether or not the diode detector is actually
linear over the region of interest. To determine whether
or not the diode detector is linear, the probe was first
inserted in the cavity and data was taken to determine the
/
FIGURE 10. Resonant peak showing f and Af
range of output voltages as the varactor voltage level was
changed through its entire range. Then the diode detector
was removed from the assembly, and a modulated source was
fed through an attenuator directly to the input of the
detector. The input level w as changed through a range which
caused the output voltage from the diode detector to change
through the full range of interest.
Figure 11 is a plot of diode detector output voltage
versus the varactor voltage. This establishes the range of
output voltage over which the diode detector linearity is
^3
FIGURE 11. Diode-detector output voltage versus
varactor voltage
f
in question. Figure 12 shows the output voltage of the
diode detector as the input was varied over a range
sufficiently large to establish the same range in output
voltage as obtained from Figure 11. Form a comparative
check of the two figures, diode detector response was
essentially linear from the high end 140 MV, down to approxi-
mately 15 MV of output voltage. This linear range corresponds
to that obtainable by changing the varactor voltage through
its entire range. Thus, diode detector linearity was established
over the entire operating range capability of the system.
3. Limitations on Cavity Use
The use of the cavity to measure modulation linearity
as outlined above presents problems when it becomes necessary
to check the linearity very closely. First, the cavity is a
power sensitive device which means the voltage output picked
up at the probe changes as the power level of the varactor-
tuned Gunn oscillator changes (Figure 13 shows the power
level changes in the Gunn oscillator as varactor voltage is
changed). Therefore, output waveform representations must
be compensated for power fluctuations at the source. Also,
there exists the problem of being able to determine with
any accuracy the degree of linearity of the output wave-
shape from its presentation on the oscilloscope.
47
FIGURE 13. Power output versus varactor voltage
Gunn bias = 10 volts
B. BEAT-FREQUENCY METHOD
Because of the above problems with the tuned cavity
and probe method, another method for determining the degree
of linearity was also utilized. In this second method, the
delay line and mixer which are part of the linearizing
circuit were used as a discriminator. The time between the
peak of each cycle of the beat frequency was determined
through the use of an oscilloscope, a pulse generator and
a frequency counter.
1. Equipment Setup
The beat frequency was fed into one channel of a
dual-trace oscilloscope while the pulse generator was fed
into the other channel. The frequency counter was synchro-
nized with the pulse generator, and consequent] y , it
displayed the pulse repetition rate selected on the pulse
generator. Figure 14 shows a block diagram of this equipment
setup .
2 . Measuring Time Between Beat-Frequency Peaks
To measure the time between consecutive peaks of
the beat frequency it was only necessary to adjust the pulse
generator repetition rate such that the second pulse
coincided exactly with the first peak of the beat frequency
signal, the frequency was read on the counter and repeated
for the remaining peaks of the beat frequency. Since the
first pulse occurs at the beginning of the beat-frequency
signal and remains there throughout all the measurements.
OSCILLOSCOPE
I
DIFFERENCE FREQUENCY SCOPE ON INTERNAL SYNC.
FIGURE 14. Block diagram of equipment setup used for
measuring time between peaks of the beat
frequency signal
it was strictly utilized as a reference point. This meant
that the reciprocal of the frequency read from the counter
was representative of the time from the reference point to
each of the individual peaks of the beat-frequency signal.
To obtain the time between consecutive peaks, it was simply
a matter of taking the difference between the time from the
reference point to each of the consecutive peaks. This
concept is shown diagrammed in Figure 15.
49
BEAT
FREQUENCY
SIGNAL
PULSE-GENERATOR
ADJUSTED FOR
MEASURING TIME
TO FIRST PEAK
OF f B
PULSE-GENERATOR
ADJUSTED FOR
MEASURING TIME
TO SECOND PEAK
OF f B
PULSE-GENERATOR
ADJUSTED FOR
MEASURING TIME
TO THIRD PEAK
0F . f B
time between 1 and 2
time between 2 and 3
t = t^ - tg
FIGURE 15. Method used for measuring time between
beat frequency peaks
50
3. Accuracy in Time Measurements
In the method outlined above, the counter was used
to measure the frequency of the pulse generator. It would
also be appropriate to use the counter to directly measure
the period of the pulse train and make calculations more
simple. The determining factor for which to. use was governed
by the accuracy obtainable. The counter used had resolution
of 0.1 ysec on the period scale while it had 10 Hz resolution
when reading frequency in the KHz range. It was found when
this method was attempted, that the best accuracy obtainable
was governed not by the counter but rather by the system,
and it was on the order of 0.1 ysec. This made the use of
the period measuring function of the counter more attractive
since this eliminated converting frequency measurements to
period measurements.
4 . Meaning of Time Measurements
If the frequency sweep is perfectly linear, the time
between consecutive peak values of the beat frequency is a
constant. Any deviation from linearity will be evident from
the varying times noted between the peak values of the beat
frequency. This deviation can be represented as a percent
deviation from linearity; therefore, this method offers a
more accurate approach to the determination of linearity of
the frequency sweep.
51
VII . FINAL EQUIPMENT SETUP
A block diagram of the final equipment setup is shown
in Figure 16. The 1 mA ammeter was used to monitor the
mixing current which was controlled to approximately 1 mA
for proper mixing. A Wavetek model l42 HF generator was
utilized as a ramp generator. The ramp retrace time
associated with this piece of equipment was excessive and
produced problems in the correcting system as discussed in
Section XI. To minimize adverse effects on the correcting
system because of this condition, a simple special purpose
ramp generator with a fast retrace could be constructed
using a few active devices as described in [11]. Figures 17
and 18 are photographs of the system showing the delay
device, mixer, cavity and probe, milliammeter and circuitry.
Figures 19 and 20 show closeup views of the correcting
circuit and external amplifier, respectively. Figure 21
gives ’a detailed schematic diagram of the entire correcting
circuit between the output of the mixer and the input to
the varactor.
52
SPECTRUM
ANALYZER
53
FIGURE 16. Block diagram of final equipment setup
FIGURE 17. Laboratory system assembly showing delay
device, mixer, cavity and probe and
circuitry. View no. 1.
FIGURE 18. Laboratory system assembly showing delay
device, mixer, cavity and probe and
circuitry. View no. 2.
5 1 *
FIGURE 19. Correcting circuit, closeup view
55
56
FIGURE 20. External amplifiers, closeup view
57
FIGURE 21. Schematic diagram of correcting circuitry
VIII. EFFECT OF NOISE IN CORRECTION CIRCUIT
The first amplifier following the mixer in the final
setup is the video amplifier, 501, as shown in Figure 21.
At this point in the system, the desired signal is at its
minimum value. The following is a brief analysis on the
possible effects of noise on the correcting circuitry.
N
out
kTB GF
n
where :
k = Boltzman's constant = 1.38xl0 -23 Joules-°K -1
T = Room temperature « 300°K
B n = Noise bandwidth » 20 MHz
G = Gain of video amplifier 501: 25 dB = 300
F o = Noise figure of video amplifier 501: 5 dB = 3-16
N Qut = (1.38x10 -23 Wat S- - SeC -)(300 o K ) (20xl0 6 /sec) ( 300 ) ( 3 ■ 16 )
= 7.87x10~' L0 watts
The signal level at the input to the delay device was
approximately 15 mW. After traveling through the delay
device (55 dB attenuation) and the mixer (5 dB attenuation),
the signal at the input to the video amplifier was down
60 dB from 15 mW. The amplifier gain of 25 dB made the
signal at the output of the video amplifier 35 dB down from
58
is far
15 mW, or ^.7^x10"^ watts. Therefore, noise power
less than signal power and should not be a problem in
circuit operation.
59
IX. PHASE-LOCKED LOOP
Phase-locked loops are a class of circuits based on
frequency-feedback technology. A phase-locked loop (PLL)
consists of a phase detector followed by a low-pass filter
and a voltage-controlled oscillator. If the Incoming
frequency Is changing, the phase detector output voltage
changes just enough to keep a nearly constant phase difference
between the oscillator signal and the Incoming signal. In
this manner, the PLL has the ability to track or stay locked
onto a changing incoming frequency on a cycle for cycle
basis. The average "error" voltage applied to the oscillator
is a function of the incoming frequency. The low-pass filter
voltage is the demodulated output when the signal at the
input is frequency modulated. If the oscillator frequency is
a linear function of the control voltage, the phase-locked
loop can be used as a linear discriminator, with variation in
output voltage proportional to change in frequency.
Phase-locked loops can have good noise immunity, easily
adjusted center frequency, bandwidth adjustment, high
selectivity, high-frequency operation capability and center-
frequency tuning by means of a single external component [12].
From the block diagram of a phase-locked loop, Figure 22,
with no signal input the error voltage , v error ,j ls equal to
zero. Under this condition, the voltage-controlled oscillator
VCO, operates at its free-running frequency, f .
60
INPUT
SIGNAL
OUTPUT
SIGNAL
FIGURE 22. Block diagram of phase-locked loop
The free-running frequency is determined by the choice of
components used. When an input signal is applied to the PLL,
the phase comparator compares the frequency and phase of the
input frequency to that of the VCO and generates an error
voltage, V ^ , which is related to the phase difference
error
between the signals. The error voltage is filtered, amplified
and applied as a control voltage on the VCO. The error
voltage forces the VCO to change frequency in the direction
that will minimize the phase difference between oscillator
and input signal.
The range of frequencies over which the PLL can acquire
lock with an incoming signal is called the "capture-range",
and the range of frequencies which the PLL can maintain
track with an incoming signal is called the "lock-range".
6l
Once the PLL is in lock, the VCO frequency is the same
as the incoming signal frequency and the error voltage is dc.
The effects of the low-pass filter are: 1) the capture
process becomes slower with increased filtering, 2) the
capture range decreases with increased filtering, and
3) the transient response of the loop becomes underdamped
with increased filtering [13].
In the application of the PLL for detecting changes in
the difference frequency, the input signal to the PLL was
the difference frequency while the error voltage, V error ,
was sensed and utilized as a correcting signal for the
varactor tuned Gunn oscillator. Optimum filtering was found
to be a problem. A lack of filtering caused the error
voltage to have a large ripple on it. This ripple voltage
frequency modulated the signal by a corresponding amount,
with a resulting distorted output waveform. Excessive
filtering distorted the correcting signal to the point where
it was no longer a "correcting signal", but once optimum
filtering was accomplished, the error voltage obtained was
clean and practically distortion-free. A photograph of the
optimum correcting signal obtained is shown in Figure 23.
The limited distortion evident in the correcting signal was
because of the delaying effects of filtering and problems
involved with the retrace time between successive voltage
ramps. These problems are discussed further in Sections X
and XI.
62
FIGURE 23. Correcting signal (AC) as viewed at
input to differential amplifier (536)
Scale: 0.1 volt/division
63
X. DELAY AND FILTERING
The basic idea of the system was to detect a nonlinearity
in the output 'frequency sweep, develop a signal to correct
for the nonlinearity and apply the correcting signal onto the
varactor. This would alter the output frequency in such a
manner that the resulting frequency sweep was more linear
than before.
An inherent problem in the system was the fact that
instantaneous correcting was not possible. A finite delay
time existed between the time of detection of the fault and
the actual application of the correcting signal. This finite
delay time was increased greatly by the addition of filters
in the correction loop.
Since the output of the phase-locked loop was somewhat
noisy, considerable filtering was required before application
of the correcting voltage on the varactor. The filtering
was necessary because any ripple voltage riding on the
correcting signal caused the output frequency to be modulated
by a corresponding amount. Thus, it was essential that a
certain amount of filtering be utilized within the closed-
loop system.
In contrast, too much filtering distorted the correcting-
voltage waveform so badly that rather than correct the
nonlinearity, it caused a further deviation from linearity.
Optimum filtering was determined by laboratory testing.
6H
XI. VOLTAGE RAMP RETRACE TIME
The FM-CW radar herein described depends upon a constant
rate of change of frequency for its operation. It is there-
fore necessary to gate off the transmitter or receiver
during the frequency retrace time. It is desirable to reduce
the retrace time to an absolute minimum in order to maximize
the useful average power.
By minimizing the retrace time, not only is the useful
average power maximized, but also the time available for
correcting the non-linearity in the sweep. This is discussed
further in Section XIII.
65
XII. OPTIMUM CORRECTING VOLTAGE
The correcting signal which produced optimum linearization
of the output-frequency sweep is shown in Figure 23. This
signal is shown again in Figures 24 and 25 with the AC
portion of the corrected ramp signal to the varactor for
one and two cycles of modulation frequency, respectively.
In each figure, the distortion in the ramp introduced by the
correcting circuit in order to linearize the frequency sweep
is clearly evident.
To obtain the optimum correcting voltage, many hours of
laboratory measurements were made. A possible optimum signal
was selected, then the output-frequency waveform was observed
using the cavity and probe technique of Section VI, and the
beat-frequency signal was monitored for linearity as also
outlined in Section VI. When the optimum signal was found,
it was recorded by taking photographs of the output-
frequency waveform Figure 27, beat-frequency signal. Figure
29, and the correcting signal. Figure 23, and recording data
for plotting the percent deviation from linearity versus
time during the modulation sweep Figure 30.
Figure 26 is a photograph of the uncorrected output-
frequency waveshape and is shown above that of the corrected
output-frequency waveshape Figure 27 for comparison. Upon
comparing these two figures, the corrected output-frequency
waveshape appears to be much more linear than that of the
66
r
FIGURE 24. Corrected ramp (AC) with correcting
signal (AC) shown below.
Scale: Ramp 1 v/div. cor. sig. .2v/div
Figure 25 . Corrected ramp (AC) with correcting
signal (AC) shown below for two cycles.
Scale: Ramp 1 v/div. cor. sig. .2v/div
67
FIGURE 26. Uncorrected output frequency waveform
as obtained from cavity and probe
FIGURE 27.
Corrected output frequency waveform
as obtained from cavity and probe
68
uncorrected. This comparison even though limited when
viewed with the restrictions discussed in Section VI, still
represents a significant linearization of the output-
frequency sweep. Analyzing the degree of linearization
present was difficult using this method. All numerical
calculations on the sweep deviation from linearity were
based on the beat-frequency method and are presented in
Section XIII.
69
XIII. LINEARITY ANALYSIS
Figure 28 shows a photograph of the uncorrected beat
frequency as recorded at the output of the external
amplifier. The peaks of the beat-frequency signal are quite
apparent, and the time between successive peaks became
larger when proceeding from left to right. This corresponds
to a frequency sweep starting with a high slope and ending
with a low slope.
Figure 29 shows the beat frequency with the correcting
signal applied to the varactor. This photograph shows the
much smaller deviation of the period between the peaks of
the beat frequency, and represents a more linear frequency
sweep.
The beat frequency associated with a perfectly linear
slope was 187 KHz for this system. This frequency is based
on the average slope of the output-frequency waveform and
the delay provided by the delay device. In order to present
the recorded data in an acceptable manner, consider the
following:
f B = (slope) (time delay)
H “P
= (:rr) (At) where At is a constant 1.0 usee
dt
70
FIGURE 28. Beat frequency (uncorrected) as recorded
at input to PLL
FIGURE 29. Beat frequency (corrected) as recorded
at input to PLL
71
where Af.
B
(beat frequency measured
- 187 KHz)
Af B (100$)
The above expression gives the percentage deviation from
linearity for the output-frequency slope.
The beat frequency was monitored as outlined in the
beat-frequency method of Section VI. The time between each
successive peak of the beat frequency was measured and
converted to a corresponding frequency. The difference
between these frequencies and 187 KHz was recorded. Three
sets of data are plotted in Figure 30 showing percentage
deviation from linearity of the output-frequency slope versus
time. Each of these represents a slightly different bias
on the varactor. On the same figure is also plotted the
uncorrected deviation.
In analyzing the corrected curves of Figure 30, the
deviation oscillated around the desirable. The worst case
deviation above was approximately 7 % while the worst case
deviation below was approximately 9 %. However, the uncorrected
curve deviated above by 14$ and below by 18$. Thus,
indications are that linearity errors have been reduced
by approximately 50$ for the worst case deviations.
72
20. OH
73
FIGURE 30. Deviation of ramp slope, df/dt, from average
XIV. OTHER METHODS OF FM LINEARIZATION
Two other methods capable of linearizing the FM sweep
were postulated, but time limitations prohibited the actual
construction and testing of each. The following few para-
graphs will offer a brief description of these methods and
their ability to linearize the FM sweep.
A. SAMPLE- AND-HOLD METHOD
This method utilizes a linear discriminator which
produces a voltage proportional to the period of the beat-
frequency signal by means of a sample-and-hold circuit, a
ramp generator and gating pulses. The process is simply to
sample a gated ramp signal once each cycle of the beat
frequency and apply the sampled value of voltage on the
varactor.
The ramp generator is in the form of an integrator
with a constant voltage on the input and is to be gated on
as the beat frequency starts a cycle. The ramp signal is
placed on the input to a sample-and-hold circuit. Prior to
reinitiating the ramp generator at the beginning of the
next cycle of the beat frequency, the sample-and-hold circuit
should be gated on in order to sample the ramp at the input.
If the beat frequency is high, the ramp will not reach
a high value because the period of the beat frequency for
that cycle is short and ramp reinitiation will occur in a
7H
shorter time. Conversely, if the beat frequency is low,
the ramp has a larger time before reinitiation and will
reach a higher value.
Prom the above, it is seen that the higher the beat
frequency, the lower the signal from the sample-and-hold
circuit. The reference point for the output from the sample-
and-hold circuit may be chosen as that value produced by the
desired beat frequency component. Then, portions of the
beat frequency signal where frequency is above that desired
will cause a decrease while those below that desired will
cause an increase in the voltage levels placed on the
varactor. The net result is a linearizing of the output-
frequency waveform.
Two possible methods of implementing the sample and hold
technique are outlined in Figures 31 * 32 and 33 } and Figures
34 and 35 .
B. ANALOG/DIGITAL METHOD
All methods discussed so far for linearizing the FM
sweep have utilized analog means. The following discussion
outlines a method for achieving the same results with a
large portion of the correcting circuit being digital. In
the following method the voltage ramp placed on the varactor
would be generated digitally with subsequent error signals
added to it in a digital adder.
75
FIGURE 31. Sample and hold method no. 1
block diagram
76
VIDEO AMPLIFIER
OUTPUT
n
_ n
, ,n
_ T n
CLIPPER # 2
- , OT FTPITT 1
U
u
u
1 "i ■ (JUli UI
□_
a
_n_
n
DIODE
OUTPUT
FIGURE 32. Sample and hold method ’no. 1
circuit waveforms (Part I)
DIFF. §2
OUTPUT
77
AT
SAMPLE
GATE
7
7
AT
RAMP RESET
GATE
RAMP
INPUT
RAMP OUTPUT/
SAMPLE AND
HOLD INPUT
SAMPLE AND
HOLD
OUTPUT
FIGURE 33. Sample and hold method no. 1
circuit waveforms (Part II)
78
FIGURE 3^1. Sample and hold method no. 2
block diagram
79
VIDEO
AMPLIFIER
OUTPUT
INVERTED
VIDEO
AMPLIFIER
OUTPUT
MULTIVIBRATOR # 1 /
RAMP RESET
MULTIVIBRATOR #2/
SAMPLE GATE
RAMP
INPUT
RAMP OUTPUT/
SAMPLE AND
HOLD INPUT
r SAMPLE AND
HOLD OUTPUT
FIGURE 35. Sample and hold method no. 2
circuit waveforms
80
Figure 36 illustrates a block diagram of the proposed
correcting method. The major components include a ramp
counter, a bank of parallel shift registers, a digital-to-
analog converter, two digital adders and a clock. The
input marked "error signal" on the block diagram represents
an error signal from the system in digital form. A possible
method for achieving this is discussed later.
The ramp is generated by simply clocking the ramp
counter through its entire range. The clock must be set
at such a rate that the counter reaches maximum count in
the time necessary for one modulation sweep (modulation
period) of the radar system. For example, if the counter
was an 8-bit device, it must be clocked from 0 through 256
counts during one modulation period. If the period were
100 psec, then the clock must count to 256 in 100 psec or
256 counts per 100 psec which corresponds to a 2.56 MHz
clock rate. Thus the number of bits in the ramp counter
establishes how small each step in the ramp staircase will
be, and the clock rate sets up the timing for the whole
system.
The required size of the shift register depends on the
maximum deviation of the slope from linearity. If this
deviation is small, the shift register could be small.
A possible method for generation of the digital error
signal is shown in block diagram form in Figure 37. The idea
uses the method described in this thesis for generating an
analog error signal out of the phase-locked loop. From the
81
MIXER
82
FIGURE 36. Block diagram of analog/digital linearizing method
GUNN
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s
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Ph
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X
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o
o
o
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83
FIGURE 37. Analog to digital error conversion
output of the PLL, the analog error would be compared to a
reference value established for the zero-error state. The
output from the comparator through steering gates causes an
up/down counter to be clocked either up or down. The
resulting output from the counter Is a digital signal
proportional to the error sensed. The clock in this
arrangement could be the same clock as used in Figure 36.
An advantage of this method is that the total signal
sent to the varactor is not only a combination of the present
error signal and the ramp voltage, but it also includes the
error which was sent to the varactor at precisely the same
correcting instant one modulation cycle before. In this
manner, the correcting system should reach a steady-state
such that the error signal to the varactor is based primarily
on past information with a resulting minimum deviation from
linearity in the system.
In a certain manner, the correcting system is predicting
what the error should be at a given instant from past
information, comparing the predicted error with the actual
instantaneous error of the system to arrive at a new error
signal, summing this error signal with the ramp for applica-
tion on the varactor and storing this same value of error
signal for use as the predicted error signal at the same
time instant the next modulation cycle.
The ramp retrace time would probably be limited by the
settling time of the D/A converter which should be less than
84
1.0 ysec. With retrace time this short, the low end correction
problems as discussed in Section XI and receiver/transmitter
blanking time would be reduced.
85
XV. CONCLUSION
Varactor-tuned Gunn oscillators have operating
characteristics which make them desirable for application
as RF sources in portable marine FM-CW radars. When the
Gunn device is used in this application, it is very important
that the output-frequency sweep vary linearly with time.
Any deviation from linearity throughout the output-frequency
sweep, will cause a loss of range resolution in the system.
The output-frequency sweep of the typical varactor-tuned
Gunn oscillator is inherently non-linear, with the degree
of non-linearity increasing with increasing sweep widths.
Therefore, in order to achieve frequency-modulation linearity
while using large sweep widths, it becomes necessary to
linearize the frequency sweep. One method of linearizing
the sweep was developed in detail, while several others were
outlined, in this thesis. The approach to linearization used
in this thesis was successful, but it was limited by both the
inability of the system to correct errors instantly and the
lack of quality correcting during the ramp retrace. Better
results would have been possible if the retrace time had
been shorter and the beat frequency had been longer. The
method requiring digital generation of the ramp and error
signal or some variation of it, is predicted to be the most
effective means for linearizing the frequency sweep.
86
LIST OF REFERENCES
1. Luck, D.G.C., Frequency Modulated Radar, p. 3,
McGraw-Hill, 19W.
2. Sobol, H. and Sterzer, F. , "Microwave Power Sources,"
I.E.E.E. Spectrum , p. 29, April 1972.
3. Bushnell, T.R., and Isaacs, A.T. , "Wideband Varactor-
Tuned Solid-State Sources to 20 GHz," The Microwave
Journal , p. 45-48, June 1973.
4. Times Wire and Cable Company, RF Transmission Line
Catalog and Handbook , No. TL3, 1970.
5. Luck, D.G.C., Frequency Modulated Radar, p. 7,
McGraw-Hill, lJW.
6. Skolnik, M.I., Introduction to Radar Systems, p. 72,
McGraw-Hill, 19W.
7. Luck, D.G.C., Frequency Modulated Radar, p. 27-28,
McGraw-Hill, 19W.
8. Rodrigue, G.P., "Microwave Solid-State Delay Lines,"
Proceedings of the I.E.E.E., Volume 53, No. 10,
p. 1428-1437, October 1965 .
9. Hymans, A.J., and Lait , J., "Analysis of a Frequency-
Modulated Continuous Wave Ranging System," I.E.E.
Paper No. 3264E, July i 960 .
10. Ginzton, E.L., Microwave Measurements, p. 405-407,
McGraw-Hill, 1957.
11. Millman, J. and Taub , H. , Pulse, Digital , and Switching
Waveforms, p. 514-569, McGraw-Hill , 1965 . '
12. Signetics Corporation, Applications Handbook, p. 6-1,
1974.
13. Signetics Corporation, Applications Handbook, p. 6-6,
1974.
87
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