Tel Aviv University
Raymond and Beverly Sackler
Faculty of Exact Sciences
Staggered PRI and Random Frequency Radar Waveform
Submitted as part of the requirements towards an M.Sc. degree in Physics
School of Physics and Astronomy, Tel Aviv University
By:
Yossi Magrisso
The work has been carried out under the supervision of
Prof. Nadav Levanon and Dr. Roy Beck-Barkay
And with the assistance of Dr. Aharon Levi
2
1. Table of contents
1. Table of contents………………..…………………………………………..……………..………….……………...2
2. Abbreviations………………………………………………...……………………………………………………….4
3. Abstract………………………………………………………………………………………………………………..5
4. Introduction………………………………………………………………………………..…………...……………..6
4.1. Definitions……………………………………………………..…………………………………………………9
4.2. Wide-Band Radar signals……..………………………..…………………….………………………...…...…..12
4.3. Generalized Ambiguity function………………...…………………………………….…………………..…....16
4.4. Pulse – Doppler train waveform Ambiguity Function………………………….….……….………........……..21
4.5. Linear Stepped-frequency waveform Ambiguity Function…………...…………………....…………………..26
4.6. Random Frequency Ambiguity Function……………………………...………………………….…...………..28
4.7. Staggered PRI based waveform Ambiguity Function……………….…………………..……...…...………….32
5. Staggered PRI and Random frequency Based Waveform…………………….………………………………..35
5.1. Description……………………………………………………………………………………………....…...….35
5.2. Optimization for minimum sidelobes, Normalized PSLR…………………………………………….…….…..39
5.3. Integration loss in the first range ambiguity zone using staggered PRI waveform………………….…………41
5.4. Processing Staggered PRI waveform using perfect Reconstruction……….…….……………………………..43
5.5. Processing Staggered PRI with random frequency waveform………………….………………………………47
5.5.1. Processing……………………………………………………………………..……………………………47
5.5.2. Implementation complexity.………………………………………………………………….………………49
5.6. Simulation results……………………………………………………………………………………………….55
5.6.1. Staggered PRI waveform...………………………………………………….......……….………………….57
5.6.2. Single target (noise free and no clutter)....………...…………………………………….…………………….60
5.6.3. Single target in the presence of noise and clutter…………...…………………………………...…………….60
3
5.6.4. Two targets in the presence of noise…...………………………………………….……….………...……….62
5.7. Experimental results…………………………………………………………………………………………….65
5.8. Signal-to-Clutter Ratio considerations…………...…………………..………....………………………………70
5.9. Advantages and Disadvantages ………………………………..……………………………………….………72
5.9.1. Advantages……………………………………………....……………………………………………….…72
5.9.2. Disadvantages…………………………………………………………………………...…………………..74
6. Conclusions…………………………………………...……………………………………………………………..75
Appendix A – Code Review….………….………………………………………………..……………………………77
Bibliography….……………………………………………………………..……………………………………………82
4
2. Abbreviations
Radar RAdio Detection And Ranging
WF Wave-Form
PRI Pulse Repetition Interval
PRF Pulse Repetition Frequency
SNR Signal-to-Noise Ratio
SCR Signal-to-Clutter Ratio
RF Radio Frequency
IF Intermediate Frequency
PSLR Peak-to-Sidelobes Level Ratio
NPSLR Normalized PSLR
RCS Radar Cross Section
CPI Coherent Processing Interval
FFT Fast Fourier Transform
DFT Discrete Fourier Transform
NUFFT Non-Uniform Fast Fourier Transform
RDM Range-Doppler Map
RVM Range-Velocity Map
ECM Electronic Counter Measures
ECCM Electronic Counter-Counter Measures
RMS Root Mean Square
STD Standard Deviation
CW Constant Wave
BW Band Width
SF Stepped-Frequency
LPI Low Probability of Intercept
SAR Synthetic Aperture Radar
ISAR Inverse Synthetic-Aperture Radar
PDF Probability Distribution Function
CFAR Constant False Alarm Rate
5
3. Abstract
Radar systems are electromagnetic sensors that are in nowadays one the most important remote
sensing tools in civilian and military uses. One of the main aspects of Radar systems design is the
Radar waveform, which defines the modulation of the signal the Radar transmits. The Radar
waveform directly affects the performance of all types of Radar systems, from detection capability
and accuracy of surveillance Radars, to image quality and resolution of imaging Radars. Two of the
main issues discussed in the literature of Radar waveform development are the sidelobes and
recurrent lobes in the ambiguity function. High sidelobes and recurrent lobes in the ambiguity
function usually lead to degradation and even limitations to Radar systems performance, so the main
goal of the research in the area is to find waveforms having an ambiguity function that aspires the
perfect "thumbtack" ambiguity function, which has no sidelobes and no recurrent lobes.
In this work a new type of Radar waveform is proposed, one that utilizes a staggered Pulse
Repetition Interval (PRI) and random frequency pulse train, and its performances are examined. The
examination includes introduction of a new generalization of the ambiguity function – one that can
represent wide-band physical signals, detailed simulation results for different scenarios including
analysis to evaluate the performance and implementation feasibility of the waveform, and
experimental data analysis. Advantages and disadvantages of the waveform are presented, including
implementation consequences in possible Radar application.
6
4. Introduction
A Radar is an electromagnetic device aimed to sense objects from a distance by transmitting
electromagnetic waves towards an object and measuring the reflections scattered from it. By precise
time difference measurement between the time of transmission and time of arrival of the scattered
echo, one can measure with high accuracy the range to the object, given that the speed of light in the
medium is known.
One of the main challenges in remote sensing using electromagnetic waves is that the radiation
power attenuates proportionally to in each direction (marking the range from the transmitter
to the object), giving a total attenuation proportional to between the transmitted power and the
power of the reflected radiation by the object received at the location of the transmitter [1]. This
drastic attenuation limits the maximal range in which a Radar can sense objects, since the reflected
radiation always exists in an environment of noises added to it from different sources (thermal, solar,
other man made transmitters, etc.), and from some range it will weaken enough to be masked by
them. In order to increase this maximal detectable range, estimation methods are usually used to
filter the noises as much as possible. The common method used is the Matched-Filter aimed to
maximize the Signal-to-Noise Ratio (SNR) defined by:
| |
{| | }
meaning the ratio between the power of the reflection and the RMS of the noise added to it (E{X}
symbols the average of the random variable X). By maximizing the SNR we can reach optimization
in the sense of maximum detection probability given a defined false detection probability (known
also as the "Neyman–Pearson lemma" [1,2,3]).
The matched-filtering action is actually an optimally weighted integration of the power in time, and
can be done in a coherent manner or in a non-coherent one. If the transmitter is capable of
transmitting a coherent radiation and the measurement instrumentation is able to measure the phase
of the returning reflections - than an optimal complex matched-filter can be applied, yielding a full
complex integration of the wanted signal and averaging the noise to the minimum. Coherent
7
transmission and reception capabilities also enabled the measurement of the Doppler frequency shift
of the reflected radiation, enabling the measurement of the radial velocity of the object relative to the
transmitter. The Doppler effect on the Radar measurements will be further discussed in the following
chapters 4.1.2 and 4.2.
Since a coherent Radar can measure the range and velocity to an object, the following challenges are:
1. Measuring them at a good precision. 2. Providing the Radar with the ability to differentiate
between different objects in the medium (placed at different ranges and/or moving at different
velocities), by being able to distinguish between each object's reflection.
In order for the Radar to have these capabilities, we aspire to maximize the Radar range and velocity
resolutions. In order to achieve a higher range and velocity resolutions, one has to consider the
waveform of the transmitted signal - meaning its amplitude and frequency modulation. High range
resolution, meaning high time compression resolution, can only be achieved if the waveform
modulation has a wide enough bandwidth. If the bandwidth of the modulated signal is much smaller
than the carrier frequency of the signal, the signal is said to be a "narrow-band signal". However if
the bandwidth is in the scope of the carrier frequency, the signal is said to be a "wide-band signal".
We will further elaborate on wide-band Radar signals in the following chapter 4.2.
Since the first Radar invention in the late 19th
century, different Radar applications have evolved and
today include a wide span of implementations, amongst them: search and detection, targeting,
triggering, weather sensing, navigation, mapping and imaging. Even though the different
applications can have a very different purpose and function, they all share a common need for high
SNR and high resolution.
Unfortunately, there are some unwanted artifacts in the outputs of Radar systems utilizing matched-
filter in their signal processing – the existence of sidelobes and recurrent-lobes (ambiguities) added
to the main-lobe peak, that can lead to false detection or false reflectors in the Radar image [2]. The
reason for the existence of the side-lobes and recurrent-lobes in the processed signal, usually has to
do with the finite time-frame in which the data is collected and analyzed. The finite time frame is, in
most cases, a natural limitation regarding the physics of the scenario the Radar has to operate in. For
example, a search Radar meant to detect and locate a flying airplane, has only a few seconds or even
less to do so before the airplane will fly out of its detectable region, meaning that the maximal time
frame for the Radar's operation in this case can be only a few seconds at best. This whole time frame
8
has to also be divided into even smaller time frames in order to collect enough data needed to acquire
the wanted range and velocity resolutions and unambiguous spans.
An instrument used to analyze the properties of the Radar waveform, including resolution, sidelobes
and recurrent-lobes is the Ambiguity Function - as defined in equation (25) in chapter 4.3. An ideal
ambiguity function is a single spike, with no sidelobes or recurrent lobes, centered in the range-
Doppler domain. It is referred to in the Radar literature as “thumbtack ambiguity function”. Its
physical realization would yield superior target-resolution and clutter-rejection capabilities for the
Radar. Finding Radar waveforms that can produce ambiguity functions having characteristics close
to that of the ideal thumbtack ambiguity function is a major research subject in the field, and is also
one of the goals of this work. Chapter 4.3 will be dedicated to the subject of the ambiguity function,
also expanding it to wide-banded signals case.
Another important feature that has to be taken into account in designing modern day military Radars
and Radar waveforms, is the Radar's ability to deal with ECM (Electronic Counter Measures) meant
to disrupt and confuse it, leading it to malfunction. Some ECM systems operate by detecting and
studying the Radar's waveform, then transmitting matched counter signals that are received and
analyzed in the Radar as a false target or even many false targets [4]. In order to prevent these ECM
systems from disrupting the Radar's operation, some Radars are equipped with ECCM (Electronic
Counter-Counter Measures) capabilities, using the Radar's waveform as major tool for that [5]. Two
of the ways to provide the Radar waveform with ECCM capability are: 1. Helping it to avoid
detection by Low Probability of Intercept (LPI) techniques. 2. By being unpredictable, and making it
very hard for the ECM to synthesize disruptive signals. One of the ways proposed to do both these
things is by using a random noise-like Radar waveform [6,7]. In addition to its ECCM capabilities,
the noise-like waveforms also possess a good trait of uniformly distributed noise-like sidelobes in
their ambiguity function, that can reduce the probability of false detections. The random
characteristics of the waveform proposed in this work will enable the Radar using it to have similar
ECCM and noise-like sidelobes properties.
9
4.1. Definitions
Velocity
In this work we will analyze the effects of a search Radar waveform on the ability to detect a moving
target, and on the target's range and range-rate measurements. The range-rate of a target (also
referred to as the 'radial velocity'), is directly related to its three dimensional Cartesian velocity
relative to the position of the Radar system, in the following way:
( )
| |
| | is the distance from the target to the Radar (where denotes the three dimensional Cartesian
relative position of the target), and will simply be referred to as the "target's range".
is the target's range-rate relative to the Radar, and is not necessarily identical to the target's
relative velocity | |. A target moving toward the Radar will have a negative range-rate, whereas
a target moving away from the Radar will have a positive one.
For convenience, however, throughout this work the range-rate of the target will be denoted simply
as the "target's velocity". Wherever the word "velocity" is mentioned, the interpretation should
always be of "range-rate", regardless of the context.
01
Range and Velocity Profiles
The Range-Doppler Map (RDM) or Range-Velocity Map (RVM) are, in many cases, the final output
of a Radar signal processing flow. They represent the amplitude of the filtered signal as function of
the range and velocity. A reflecting target placed at a certain range and moving in a certain velocity
will yield a peak in a range-velocity cell in the map (see Figure 1a).
Throughout this work, RDMs or RVMs will be presented as images. In some cases it is constructive
to examine the range or velocity profiles of the map for some range (showing all the amplitude
values for the different velocities in that range), or for some velocity (showing all the amplitude
values for the different ranges in that velocity). These will be referred to as the "Velocity Profile"
and as the "Range Profile", respectively. The range and velocity profiles are shown in Figure 1b.
Figure 1a – An example of a Range-Velocity Map (RVM). This RVM is a simulation output of a target placed at 250 m range and moving at a velocity of -30 m/sec, in the presence of noise and clutter (in this case, many stationary strong reflectors places at different ranges). The dotted-red line marks the velocity "slice" of all ranges in velocity -30 m/sec (Range Profile), whereas the dashed green line marks the range "slice" of all velocities in range 250 m (Velocity Profile).
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-60
-50
-40
-30
-20
-10
0
-30
-25
-20
-15
-10
-5
0
[dB]
00
Figure 1b – Range and velocity profiles of the marked lines in Figure 1a. The Range-Profile is the amplitude in the map for velocity -30 m/sec and for all the ranges, and the Velocity-Profile is the amplitude in the map for range 250 m and for all the velocities.
0 100 200 300 400 500 600-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
-70-60-50-40-30-20-100-40
-20
0
20
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
02
4.2. Wide-Band Radar signals
A wide-band signal is a signal containing a frequency-span not much smaller than the carrier
frequency itself. The use of wide-band signals in Radar applications can be attractive because they
can produce very high range resolution outputs. Besides its obvious advantage in yielding high
measurement precision and better scatterers separation, high range resolution enables adding to
Radars systems some enhanced capabilities, such as object classification and target recognition [8,9].
However, the Doppler behavior of wide-band electromagnetic signals can be quite different than that
of narrow-band ones, and this has to be taken into account. An electromagnetic signal transmitted
from a moving object relative to a receiver, has a frequency shift known as a Doppler shift. If the
moving object transmits the real part of the complex signal:
The complex signal received at a stationary receiver will be:
( ) ( ( ) )
Where is the time delay caused by the wave propagation time. In free space, assuming the
transmitter moves in a constant velocity (denoting the speed of light) and that the frequency
is constant in time, the time delay is:
thus:
(
)
And the received signal will be:
03
( (
)
)
( ( )
)
(
)
denoting
The received signal has a Doppler frequency shift:
where is the carrier wavelength, and is the attenuation of the signal as a result of the
wave's propagation and the system losses. The approximation in equations (9-11) assume
narrowband signal in which the frequency is the carrier frequency of the transmitted signal.
In the case of Radar signals, reflected from an object moving at a constant velocity and received at
the same stationary point of the transmitter, the two way delay is:
The received signal will be:
( (
)
)
( (
)
)
(
)
denoting the object's complex Radar Cross Section (RCS), and the two-way Doppler frequency
shift.
04
The calculations presented above are true for a narrow band signal (relative to the carrier frequency)
reflected from a slow moving object (relative to the velocity of light). However, if the signal is a
wide-banded signal, or alternatively the object moves at a high velocity, the assumptions leading to
equation (13) are no longer valid. In the general case, we have to recalculate the received signal as
function of a more accurate time-dependent delay (equation (6)).
Using a time symmetry property, assuming the object moves at a constant velocity and that the signal
propagation delay is equal for both directions (to and from the target) [10,11], the delay fulfills the
relation:
(
)
( (
))
(
)
It is interesting to notice that using only the symmetry property of the time delay, the wide-band time
shift is consistent with the one predicted by the Special Relativity theory. The one-way relativistic
time dilation transformation is given by:
√
If is the time period of the waves emitted from the Radar, then the moving object sees waves
hitting it with a period of . The object then reflects the waves having the same period, and these
reflections are received back in the Radar with a secondary period shift of [12]:
√
05
Finally, a wide-band signal reflected by a moving object and received at the same spatial point as the
transmitter's location, is given by [13,14,15]:
√
√
(
)
√
(
)
( (
) )
where denotes:
The frequency can be also a function of time, so the time dependency of the received frequency ( )
can be different than the time dependency of the transmitted frequency ( or simply ):
(
)
In that case the received signal will be:
√
√
(
)
( (
) (
) )
06
4.3. Generalized Ambiguity function
The standard ambiguity function defined as [2]:
| |∫
Denoting the complex conjugate signal with a time shift, and the normalization factor:
∫| |
is a tool meant to help a Radar waveform developer. If we inspect the correlation function between
the signal and the signal , we get:
| |∫
| | ∫
| | ∫
| | | |
So we can see that the amplitude of ambiguity function (that part of the ambiguity function that
mostly interest us) is actually equal to the amplitude of the correlation function between the signal
and the signal . As we know, the signal is the matched filter of the
signal .
07
If the following conditions are fulfilled:
a. Matched filtering is used to detect the signal
b. The target's velocity is small relative to the speed of light
c. The signal is narrow-banded
it can be argued that the ambiguity function represents well enough the matched-filter output of zero-
range, zero-velocity object reflections with different range-velocity hypothesis filters. However, this
is not necessarily true if one or more of the conditions above are not fulfilled. In the case one of them
is not fulfilled, we have to find and work with a generalization of the ambiguity function. Several
such functions have been proposed in the past [16,17], and here we develop a generalization
corresponding to the wide-band signal reflections.
As we've seen in chapter 4.2, the reflections received from a point target are (equation (24)). In
this case the correct matched–filtering output of a zero-range, zero-velocity object reflections using
different range-velocity hypothesis filters, is:
| |∫
| |∫
(
)
| |∫
(
)
( (
) (
) )
| |∫
(
)
(
) (
)
Where is the envelope's complex conjugate.
In order to simplify the last expression in equation (30) we assume the signal envelope changes
relatively slowly in time, therefore we can use the approximation:
08
(Here is constant in time)
(
) (
)
In this approximation we define the Generalized Ambiguity function:
| |∫
(
)
that is, in fact, the correlation between and the function:
(
)
The difference between the standard ambiguity functions and the Generalized ambiguity function in
the case that one or more of the three conditions mentioned above are not complied, can be seen in
the following example (in this case – the signal has a wide-band, namely condition c. is not fulfilled).
The waveform's parameters are:
Waveform Stepped-frequency, 3 pulses per batch
Number of pulses 10
Carrier frequency 400 kHz
Bandwidth* 50 kHz
Pulse Repetition Interval 325 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 30,000 m/sec
* The Bandwidth represents the whole frequency span of the different frequency steps.
In this case the transmitted carrier frequencies (in kHz) are [400, 425, 450, 400, 425,
450, 400, 425, 450,400], spanning over a total bandwidth of 50 kHz.
09
Figure 2a shows the pulse modulation and frequency as function of time, and Figures 2b,2c the
waveform's standard ambiguity function and Generalized ambiguity function respectfully. The poor
velocity resolution in this examples stems from the low carrier frequency (400 kHz)
Figure 2a – A stepped-frequency waveform, 3 pulses in batch.
Figure 2b – Standard Ambiguity function of a wide-band stepped-frequency modulated waveform.
0 500 1000 1500 2000 2500 3000 3500-1
-0.5
0
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 500 1000 1500 2000 2500 3000 35000
2
4
6x 10
4
Time [sec]
Base B
and f
requency [
Hz]
f *
CP
I
t / PRI
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-25
-20
-15
-10
-5
0
5
10
15
20
250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
21
Figure 2c - Generalized Ambiguity function of a wide-band stepped-frequency modulated waveform. This figure demonstrates the difference between the Generalized Ambiguity function and the standard ambiguity function shown in Figure 2b, regarding the behavior of both the sidelobes and the recurrent lobes.
Simulation code for calculating the Generalized ambiguity function presented here is reviewed in
Appendix A.
Sibul and Titlebaum have discussed the wide-band ambiguity function volume properties [18], and
showed that they can be quite different than those we know of the ordinary ambiguity function [2].
For example, they showed that the integrated volume of the wide-band ambiguity function can in fact
be larger than 1, as opposed to the volume of the ordinary ambiguity function that is equal to 1.
Velo
city [
m/s
ec]
t / PRI
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
-3
-2
-1
0
1
2
3
x 106
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20
4.4. Pulse – Doppler train waveform Ambiguity
Function
A well-studied ambiguity function is of the Pulse-Doppler Waveform. The Pulse-Doppler waveform
is a very common Radar waveform, used in a wide span of Radar applications. The way a pulsed
Radar generally works is by transmitting a short time-span high energy pulses, and then shutting the
transmission down and listening to the returning echos. A major advantage of the pulsed waveform
over a Constant-Wave (CW) waveform, is that Radars using CW waveforms usually suffer from
high transmission leaks to the receiver channel (producing unwanted "self-noise"), reducing the SNR
of the received signals. Radars using pulsed waveforms usually suffer less from this problem since
they usually do not transmit anything while listening. If several pulses are transmitted coherently one
after another, than the waveform is actually a pulse-train waveform.
A solution that can helps get some intuition as to how a pulse train ambiguity function may look like
is the general pulses train periodic ambiguity function given by [2]:
| | | | |
|
where denotes the PRI and denotes the ambiguity function of a single pulse. However,
the periodic ambiguity function assumes matched filtering an infinite train of identical pulses with a
finite pulses train, therefore equation (35) can serve only as a simple approximation to the case in
which the matched filtering is between two finite pulse trains.
In Figure 3a, the amplitude and base-band frequency (frequency shift relative to the RF frequency)
of a single pulse waveform are drawn as function of time. In this example we see the transmission of
the pulse lasts for 40 µsec and then shuts down. The pulsed signal does not have any frequency shift,
hence the base-band frequency is a constant zero. The single pulse waveform Generalized ambiguity
function is shown in Figure 3b. Time and frequency profiles of the Generalized ambiguity function
at the zero time zero frequency point are shown in Figure 3c.
22
Figure 3a – An example of a single rectangular pulse waveform. The top figure shows the amplitude of the signal, and the bottom figure
shows the base-band frequency shift relative to the RF frequency (in this case – constant zero shift), as function of time
Figure 3b – The Generalized ambiguity function of the single pulse waveform. Time and frequency profiles of the function are shown in Figure 3c.
0 20 40 60 80 100 120 1400
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 20 40 60 80 100 120 140-1
-0.5
0
0.5
1
Time [sec]
Base B
and f
requency [
Hz]
f *
CP
I
t / PRI
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-100
-50
0
50
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
23
Figure 3c – Time and frequency profiles of the Generalized ambiguity function of the single pulse waveform at the zero-time and zero-frequency point. If we
look at the time profile for f = 0 we will see the "correlation triangle" which is the product of a correlation of a rectangular time-window with itself. If we look at the frequency profile for t = 0 we will see a Dirichlet periodic sinc pattern formed by a Discrete Fourier Transform of the same rectangular time-window.
The next example shows the Generalized Ambiguity function of a 10 pulses train waveform with the
parameters:
Waveform Pulse - Doppler
Number of pulses 10
Carrier frequency 5 GHz
Bandwidth 0 MHz
Pulse Repetition Interval 200 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 2 m/sec
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80
0.5
1
Tim
e-P
rofile
for
f =
0
t / PRI
-150 -100 -50 0 50 100 1500
0.5
1
Fre
quency-P
rofile
fot
t =
0
f * CPI
24
In Figure 4a, the amplitude and base-band frequency of the 10 pulses train waveform are drawn as
function of time. Here again the pulse train signal does not have any frequency shift, hence again the
base-band frequency is a constant zero. The 10 pulses waveform Generalized ambiguity function is
shown in Figure 4b.
Figure 4a – 10 Pulses train waveform with uniform PRI
Figure 4b – The Generalized Ambiguity function of the 10 pulses train Pulse-Doppler waveform. The figure on the right shows a zoom on the zero-time mainlobe. In this case in addition to the recurrent lobes in the velocity axis, recurrent lobes in the range axis also appear.
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.5
0
0.5
1
Time [sec]
Base B
and f
requency [
Hz]
Velo
city [
m/s
ec]
t / PRI
-8 -6 -4 -2 0 2 4 6 8
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
25
In this case not only do the sidelobes and recurrent lobes exist in the velocity axis, but now because
of partial correlations between the train and the time shifted train – time recurrent lobes also arise.
Sidelobes and recurrent lobes could lead to false detections or distortions in a Radar image if not
treated properly.
26
4.5. Linear Stepped-frequency waveform Ambiguity
Function
The linear stepped-frequency waveform is a pulse train, having each pulse in the train modulated
by linearly increasing frequencies (see Figure 5a). Stepped–frequency waveforms are generally
used when a large bandwidth is required in order to achieve high range resolution, but it is
impossible to increase the single pulse bandwidth using inter-pulse modulation. The linear
stepped-frequency waveform and it's properties are discussed in detail by Levanon in [2,19].
Combining a linear SF with linearly increasing pulse intervals waveform was proposed in order
to reduce ambiguity levels [20].
The next example shows the generalized ambiguity function of a waveform with the parameters:
Waveform Stepped-frequency
Number of pulses 10
Carrier frequency 5 GHz
Bandwidth 50 kHz
Pulse Repetition Interval 200 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 2 m/sec
Looking at the generalized ambiguity function of this waveform (Figure 5b) and comparing it to
the generalized ambiguity function of the single frequency pulse train, we can see some of the
next features:
a. Improvement in the time resolution due to the use of a wider bandwidth.
b. Appearance of additional time/velocity recurrent lobes, closer to the mainlobe.
c. Some reduction in the amplitude of the far time recurrent lobes, due to the miss correlation
between pulses carrying different frequencies.
d. Appearance of additional sidelobes in the time/velocity domain.
27
Figure 5a – 10 pulses train waveform with a linear stepped-frequency modulation.
The waveform consists of 2 batches of 5 pulses per batch, having
a 10 kHz frequency step between each two consecutive pulses
in the batch. This creates a batch with a total bandwidth of 50 kHz.
Figure 5b – The Generalized Ambiguity function of the 10 pulses train with stepped-frequency modulated waveform. The figure on the right shows a zoom on the zero-time mainlobe. When comparing it to the ambiguity function of a pulse-Doppler train (shown in Figure 4b), we can see that the range resolution has improved due to the wider bandwidth, but at the price high and close recurrent lobes.
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.5
0
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2
4
6x 10
4
Time [sec]
Base B
and f
requency [
Hz]
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
28
4.6. Random Frequency Ambiguity Function
In order to minimize the velocity and range sidelobes and ambiguities of the linear stepped-
frequency waveform, several non-linear frequency series were proposed. Costas proposed a
theoretical optimal series [19,21]. The use of random stepped-frequency series was also proposed
[22,23].
The next example shows the generalized ambiguity function of a random stepped-frequency
waveform with the parameters:
Waveform Randomized Stepped-frequency
Number of pulses 10
Carrier frequency 5 GHz
Bandwidth 50 kHz
Pulse Repetition Interval 200 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 2 m/sec
As shown in Figure 6a, in this case a total bandwidth of 50 kHz is also achieved but now by having
each pulse in the train carrying a different frequency, and sorted in a random non-linear fashion. The
waveform's general ambiguity function is shown in Figure 6b.
Figure 6a – Random frequency 10 pulses train waveform.
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.5
0
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2
4
6x 10
4
Time [sec]
Base B
and f
requency [
Hz]
29
Figure 6b – The Generalized Ambiguity function of the random frequency 10 pulses train waveform. The figure on the right shows a zoom on the zero-time mainlobe.
Comparing the general ambiguity functions shown in Figure 6b and Figure 5b we can see that once
we randomizing the frequency, significant decorrelation occurs and that the energy of the time
recurrent lobes existing in the linear stepped-frequency waveform are now spread randomly all over
the velocity axis, and suppressed in their amplitude. The reason that the decorrelation does not
happen in the linear stepped-frequency case can be better explained by the following calculation:
The total number of pulses in the Coherent Processing Interval (CPI) is:
Marking the PRI as , and the frequency step as , each pulse in the linear stepped-frequency batch
has a phase component of:
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
31
Correlating the whole batch with a reference signal (matched filter) will yield the integration output:
∑
Denoting as the index of the stepped-frequency batch in the CPI ( =0,1,2,…, ),
and it's corresponding coherent phase. The meaning of equation (38) is full coherent integration
of the pulses of the batch.
Correlating the reference with a similar signal, only delayed by exactly one pulse interval, will yield
the output:
∑
∑
The result is that the partial correlation between the shifted pulses and the reference signal yields a
high integration result, smaller than main-lobe only by the factor of
The random SF waveform, however, does not have this partial correlation and therefore its' time
recurrent lobes are low and spread all over the velocity axis.
As can be seen in Figure 6b, because of the uniform PRF sampling and the low bandwidth - the
velocity recurrent lobes in the zero-time vicinity remains high. If a gradually higher and higher
bandwidth will be used, the velocity ambiguities will also gradually disappear as each reflected pulse
in the train will have an increasingly different Doppler shifts and the train will not be integrated
properly at non-zero velocities (see Figure 6c).
30
The random SF waveform was recently suggested for different applications, amongst them Synthetic
Aperture Radar (SAR) [24,25] and Inverse SAR (ISAR) [26] imaging for improved point-spread
function, and also for improved multiple target detection by using an iterative maximum-likelihood
based algorithm [27].
Figure 6c – Different Generalized Ambiguity function of the random frequency 10 pulses train waveform, zooming on the zero-range main-lobe, for different bandwidth-to-transmission frequency ratios. In this case different ratios are achieved by reducing the carrier frequency while keeping the bandwidth constant.
Velo
city [
m/s
ec]
t / PRI
BW/f0 = 1e-05
-0.4 -0.2 0 0.2 0.4
-200
-100
0
100
200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
BW/f0 = 0.03
-0.4 -0.2 0 0.2 0.4
-5
0
5
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
BW/f0 = 0.09
-0.4 -0.2 0 0.2 0.4
-1
0
1
x 106
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
BW/f0 = 0.33333
-0.4 -0.2 0 0.2 0.4
-4
-2
0
2
4
x 106
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
32
4.7. Staggered PRI based Waveform Ambiguity
Function
In order to reduce the range and velocity ambiguities in the ambiguity function, a non-uniform, intra
cycle staggered PRI was proposed [23]. When using this waveform, a spreading of the velocity
ambiguities all over the volume of the general ambiguity function is caused by: 1. The non-uniform
sampling of the Doppler phase that prevents the aliasing phenomenon. 2. The reduction of the range
ambiguities is caused by the non-constant range wraparound of the far distance objects reflections –
preventing their proper integration. This waveform type was suggested to enable Radars suppress
clutter [28,29] and interferences [30]. Lately the use of a staggered PRI waveform was also proposed
for SAR applications in order increase the imaging coverage using high resolution, without the need
for a long antenna to do so [31].
The next example shows the generalized ambiguity function of a waveform with the parameters:
Waveform Pulse – Doppler, Non-uniform PRI
Number of pulses 10
Carrier frequency 5 GHz
Bandwidth 0 kHz
Pulse Repetition Interval 200 μsec + ΔT
ΔT ~ U[ -TPRI/2 , TPRI/2]*
TPRI* 120 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 2 m/sec
*ΔT is a random time shift in each of the pulse intervals from the average PRI of 200 μsec,
distributed uniformly between the values [ -TPRI/2 , TPRI/2] (~U[a,b] symbols uniform
distribution between the real values a and b). TPRI defines the time frame length
around the average PRI of the waveform, in which each pulse interval is randomized.
33
In Figure 7a the waveform is shown, having different random time delays between each two
sequential pulses, and carrying the same frequency. Figure 7b shows the generalized ambiguity
function of the waveform, and in it we can see a significant reduction of the both the time and
velocity recurrent lobes.
Figure 7a – 10 pulses train with a staggered PRI waveform.
Figure 7b – The Generalized Ambiguity function of the 10 pulses train with a staggered PRI waveform.
0 500 1000 1500 2000 25000
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 500 1000 1500 2000 2500-1
-0.5
0
0.5
1
Time [sec]
Base B
and f
requency [
Hz]
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
34
To emphasize the effects, Figure 7c shows the generalized ambiguity function of the same
waveforms, but with a length of 40 pulses. In it we can see the spreading of the time and velocity
recurrent lobes all over the volume, but that the time recurrent lobes are still relatively high in the
zero velocity line due to some partial correlation in the time domain.
Figure 7c – The Generalized Ambiguity function of the 40 pulses train having a staggered PRI waveform.
One of the main challenges in implementation of the staggered PRI waveform is its signal
processing. Li and Chen [32] propose to use the Non-Uniform Fast Fourier Transform (NUFFT)
algorithm for the processing. In chapter 5.4 we propose a method of processing it using perfect
reconstruction.
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
35
5. Staggered PRI and Random frequency Based Waveform
5.1. Description
In order to gain from both worlds in terms of ambiguity rejection, we propose a waveform based on
the combination of a staggered PRI, and the random frequency shift between sequential pulses in the
train.
The following example shows the Generalized Ambiguity function of a waveform with the
parameters:
Waveform Randomized Stepped–frequency,
staggered PRI
Number of pulses 10
Carrier frequency 5 GHz
Bandwidth 50 kHz
Pulse Repetition Interval 200 μsec + ΔT
ΔT ~ U[ -TPRI/2 , TPRI/2] *
TPRI* 120 μsec
Pulse width 40 μsec
Sampling rate 400 kHz
Velocity bin size 2 m/sec
*ΔT is a random time shift in each of the pulse intervals from the average PRI of 200 μsec,
distributed uniformly between the values [ -TPRI/2 , TPRI/2] (~U[a,b] symbols uniform
distribution between the real values a and b). TPRI defines the time frame length
around the average PRI of the waveform, in which each pulse interval is randomized.
In Figure 8a the waveform is shown, having both different random time delays between each two
sequential pulses, with each one carrying different frequency in a random order. Figure 8b shows
36
the generalized ambiguity function of the waveform, and in it we can see again a significant
reduction of the both the time and velocity recurrent lobes.
Figure 8a – 10 pulses train with staggered PRI and random frequency waveform.
Figure 8b – The Generalized Ambiguity function of the 10 pulses train with staggered PRI and random frequency waveform. The figure on the right shows a zoom on the zero-time mainlobe.
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.5
0
0.5
1
Time [sec]
Am
plit
ude (
Real part
)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2
4
6x 10
4
Time [sec]
Base B
and f
requency [
Hz]
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
37
Figure 8c – The Generalized Ambiguity function of the 40 pulses train with staggered PRI and random frequency waveform. The figure on the right shows a zoom on the zero-time mainlobe.
To emphasize the effects, Figure 8c shows the generalized ambiguity function of the same
waveforms, but with 40 pulses train. In it we can see the spreading of the recurrent lobes all over the
range and velocity axes. In this example it is also apparent that the far velocity recurrent lobes (in
this example located at and ) are spread locally, but not entirely all
over the velocity axis. The reason is that the PRI, although not uniform, is still localizes around 200
μsec for each pulse in the CPI. If we use a wider stagger in the PRI (by increasing TPRI) the spreading
will increase, but as a consequence we might also affect other parameters in the system (caused by a
decrease in the minimal pulse interval and increase in the maximal pulse interval in the CPI). Figure
8d shows the effects of different staggers on the velocity recurrent lobes in the generalized ambiguity
function.
Velo
city [
m/s
ec]
t / PRI
-4 -3 -2 -1 0 1 2 3 4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Velo
city [
m/s
ec]
t / PRI
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-200
-150
-100
-50
0
50
100
150
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
38
Figure 8d – Four Generalized Ambiguity functions of the 40 pulses train of staggered PRI and random frequency waveform, with different PRI staggers (zooming on the zero-range main-lobe). In this case the mean pulse interval is 200 μsec.
Velo
city [
m/s
ec]
t / PRI
TPRI
= 200 sec
-0.4 -0.2 0 0.2 0.4
-200
-100
0
100
200
Velo
city [
m/s
ec]
t / PRI
TPRI
= 100 sec
-0.4 -0.2 0 0.2 0.4
-200
-100
0
100
200
Velo
city [
m/s
ec]
t / PRI
TPRI
= 50 sec
-0.4 -0.2 0 0.2 0.4
-200
-100
0
100
200
Velo
city [
m/s
ec]
t / PRI
TPRI
= 0 sec
-0.4 -0.2 0 0.2 0.4
-200
-100
0
100
200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
39
5.2. Optimization for minimum sidelobes, Normalized
PSLR
In the case strong targets or clutter are present, the ambiguities or side lobes of the ambiguity
function might unwantedly be detected as false targets. Therefore one of the main objectives in
finding a good waveform is by reducing these unwanted artifacts down to the minimum.
When using random series for the pulse intervals and frequencies, an obvious question rises:
Are there optimal series in the sense of minimal ambiguities / side lobes in the ambiguity
function?
A useful quality criterion for the "goodness" of the side lobes level is the main-lobe to Peak Side-
Lobe Ratio (PSLR), in other words the ratio between the amplitude of the main-lobe and the
amplitude of highest side lobe. Here we will treat the recurrent lobes (or ambiguities) as unwanted
sidelobes.
Using a waveform that spreads the ambiguities all over the ambiguity function, the PSLR will be
increased as we increase the length of the pulse train. The reason is that the main-lobe is the product
of coherent integration of all the samples of the target reflections in the pulse train (generating an
integration power proportional to , where is the number of pulses in the CPI), as opposed to the
sidelobes that are also integrated, but in a non-coherent manner (generating an average integration
power proportional to ). This produces a PSLR proportional to
in power and √ in
amplitude. Hence, a better criterion for the "goodness" of the waveform regardless of the length of
the pulse train can be a Normalized PSLR (NPSLR), which we defined as:
√
The higher the NPSLR, the better. Figure 9 shows histograms of 40, 100 and 200 pulses train with
different PRI staggers ranges, TPRI. In it we can see that for large TPRIs the NPSLR depends only
mildly on the pulse train length.
41
Exercising an exhaustive search in order to find good series for PRIs and frequencies (located on the
histogram's tail at high NPSLRs), can help us find the best NPSLR achievable as function of the PRI
stagger range for different train lengths, as well as the good series itself.
Figure 9 – NPSLR Histograms (proportional to the Probability Distribution functions, or PDFs) of different pulse-train lengths, of the staggered PRI (showing different PRI staggers) and random frequency waveform.
As can be seen in Figure 9 the maximal NPSLR found in the exhaustive search was ~0.7, and in
most cases represents the maximal achievable suppression of the ambiguities. The maximal
theoretical statistical value of the NPSLR is of course 1.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810
0
102
104
PD
F
40 pulses
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810
0
102
104
PD
F
100 pulses
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810
0
101
102
103
PD
F
NPSLR
200 pulses
TPRI
= 20 sec
TPRI
= 70 sec
TPRI
= 130 sec
TPRI
= 320 sec
40
5.3. Integration loss in the first range ambiguity zone
using staggered PRI waveform
In many cases Radars work only in the first range-ambiguity zone, meaning they receive and process
reflections of a transmitted pulse from targets located at close distances, and filter the reflections of
the same pulse returning from farther objects - after the following pulse was transmitted. The
filtering can be achieved by different methods such as transmitting in large frequency shift between
two consecutive pulses, then applying a filter matched to the second pulse that rejects reflections of
the first pulse returning from farther objects. In that case, the maximal range from which we receive
reflections and integrate them coherently (and without loss) will be restricted by the pulse interval
duration. Here we also assume, of course, that there is no reception while transmitting. The relation
between the PRI and the maximal first ambiguity-zone range (in the case of uniform PRI waveform)
is given by:
The assumption is that any reflection returning from a farther distance than will be filtered. In
the case of using a staggered PRI, different pulse intervals will dictate different maximum ranges,
although the smallest pulse interval will not necessarily be the constraint to the maximal detectable
range:
1. Integration level of reflections returning at delays smaller than the smallest pulse interval
in the cycle batch will not be affected (therefore there will be no additional losses).
2. Some losses will be inflicted to reflections returning at times between the minimal and
the maximal pulse intervals.
3. Reflections returning at later times than the maximal pulse interval in the series will not
be detected, since filtering of signals returning from ambiguity zones is assumed.
42
In Figure 10 the integration loss is shown as function of the time delay of the Radar echo, for
different PRI staggers. It shows that for a waveform with no stagger at all (TPRI = 0), the loss will be
zero for all times between the pulse width and the PRI, but will be infinite for time larger than the
PRI. When using a stagger in the PRI, however, some loss will occur at times smaller than the mean
PRI because of the narrower pulse intervals in the train, but because of the wider pulse intervals in
the train – the loss at times larger than the mean PRI will not be infinite. This means for example that
although suffering from significant loss, a target can be detected in that area.
Figure 10 – Integration loss as function of the time delay (proportional to the target's range) for different PRI staggers. The average pulse interval used in this example is 200 μsec, and the average duty cycle is 0.2 (the pulse width is 40 μsec).
The integration loss is an additional parameter that has to be taken into account when optimizing the
waveform. Large PRI stagger range can increase the NPSLR - reducing probability for a false
detection, but will also increase the integration loss - reducing the probability of detection.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
5
10
15
20
25
30
35
t / mean PRI
Inte
gra
tion loss [
dB
]
TPRI
= 200 sec
TPRI
= 100 sec
TPRI
= 50 sec
TPRI
= 0 sec
43
5.4. Processing Staggered PRI waveform using perfect
Reconstruction
One of the main challenges in implementation of the staggered PRI waveform is its signal
processing. Different methods were proposed, amongst them direct Discrete Fourier Transform
(DFT) processing [33] and the more efficient Non Uniform FFT (NUFFT) [32] . The NUFFT
algorithm is described in detail at [34]. In this chapter we propose and analyze another method using
non-uniform to uniform sampling interpolation followed by FFT.
Perfect reconstruction of a periodic signal band limited to , from
non-uniformly spaced samples ( ), sampled at times , is given by [35]:
∑ ( )
where:
{
∏
( ( ) )
( ( ) )
( ( ) ) ∏ ( ( ) )
( ( ) )
Non-uniform to uniform sampling interpolation is given by resampling of the perfect reconstruction:
∑ ( )
{
∏
( ( ) )
( ( ) )
( ( ) ) ∏ ( ( ) )
( ( ) )
44
In matrix representation the interpolation transformation is:
In our case ( ) represents the different pulses in the CPI. Once ( ) is resampled uniformly as
, the Doppler frequency (and the velocity) of the target can be estimated by an FFT just like in
the uniform PRI case.
Because DFT is also a linear operator that is applied on the samples, we can combine the two matrix
multiplications, and therefore reduce the amount of calculations:
{ }
Notice in this analysis that we assume all the pulses in the CPI are transmitted at the same carrier
frequency. A waveform that contains both non-uniformity in its PRI and frequency shifts between its
pulses, as described in chapter 5.1, cannot be simply processed by non-uniform to uniform
interpolation and DFT as described here. The reason is that the frequency shifts dictates a nonlinear
phase shifts from pulse to pulse, that cannot be compensated by simple Doppler processing.
Therefore - when also using random frequency we need to take a different approach. Figures 11a-c
show simulation results for three different cases:
a. A single target
b. Three targets having different frequencies and amplitudes (amplitudes of 1 at 50Hz, 2 at 100
Hz, and 0.5 at 140 Hz).
c. Same three targets in the presence of noise. The noise Standard Deviation (STD) is 2,
meaning that the stronger target has an SNR of 0 dB before integration (the weakest target
has an SNR of ~ -12 dB before integration).
In all cases good reconstruction and Doppler estimation is demonstrated.
45
Figure 11a – Doppler estimation of single target with non-uniform PRI waveform by interpolation to uniformly sampled signal and DFT.
Figure 11b – Doppler estimation of 3 targets at frequencies 50 Hz, 100 Hz and 140 Hz.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-1
-0.5
0
0.5
1
t [sec]
Am
plit
ude (
Real part
)
0 20 40 60 80 100 120 140 160 180 200-50
-40
-30
-20
-10
0
frequency [Hz]
Norm
aliz
ed A
mplit
ude [
dB
]
True signal
Non-uniformly sampled signal
Interpolated signal
FFT of True signal
Spectral estimation of Non-uniformly sampled signal
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
-6
-4
-2
0
2
4
6
t [sec]
Am
plit
ude (
Real part
)
0 20 40 60 80 100 120 140 160 180 200-40
-30
-20
-10
0
frequency [Hz]
Norm
aliz
ed A
mplit
ude [
dB
]
True signal
Non-uniformly sampled signal
Interpolated signal
FFT of True signal
Spectral estimation of Non-uniformly sampled signal
46
Figure 11c – Doppler estimation of 3 targets at frequencies 50 Hz, 100 Hz and 140 Hz
in the presence of noise.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
-6
-4
-2
0
2
4
6
t [sec]
Am
plit
ude (
Real part
)
0 20 40 60 80 100 120 140 160 180 200-40
-30
-20
-10
0
frequency [Hz]
Norm
aliz
ed A
mplit
ude [
dB
]
True signal
Non-uniformly sampled signal
Interpolated signal
FFT of True signal
Spectral estimation of Non-uniformly sampled signal
47
5.5. Processing Staggered PRI with random frequency
waveform
5.5.1. Processing
We take a waveform composed of pulses. Between each 2 pulses, pulse and pulse , there is
a time interval (PRI) of ( ). In this notation . Each pulse is modulated by a
different random frequency . Given that the first pulse is transmitted at , the timeline
between the different pulses is:
∑
We also mark the sample time relative to beginning of each pulse transmission as , where is the
sample index. The global sampling time is therefore:
∑
is the number of samples in each pulse interval, and depends on the sampling
frequency.
As was shown in chapter 4.2 (equation (24)), the relative phase between the transmitted and received
signal is given by:
(
) (
) ( )
(
)
( )
48
The RF signal is usually down-converted to base-band in order to sample and process it digitally.
The demodulated received signal is:
{
( )}
{
} {
}
The received signal amplitude, (
), is also a function of the time delay
.
Assuming all the transmitted pulses have the same amplitude, we can treat each pulse as independent
and regardless of :
(
) (
) (
)
and get:
(
) {
} {
}
If we also use the "Stop and Hop" approximation, neglecting the phase dependency on the intra-pulse
time, , meaning:
we get:
49
(
) {
} {
}
(
) {
}
Matched filtering of the signal is applying the filter on the received signal ( ).
The product is the Range-Velocity Map (RVM, as opposed to the traditional Range-Doppler Map,
RDM):
∑ ∑ (
)
{
}
∑ {
} ∑ (
)
The inner sum in the last expression of equation (60) represents a temporal short-time matched filter.
The outer sum represents the "generalized" Doppler processing (now actually Range-Velocity
processing), that carries out the coherent integration.
Using digital signal-processing, the range and velocity hypotheses can be quantized in order to
receive a finite data size. In addition, the grids of and do not have to be uniform, and can vary
in different manners as needed by the application.
Implementation of the waveform's signal processing in MATLAB code is reviewed in Appendix A.
5.5.2. Implementation complexity
One of the most important aspects in the implementation of a Radar waveform is its signal
processing computational complexity and memory usage. We will now compare between the
processing complexity of an ordinary modulated pulse-Doppler waveform, and that of the staggered
PRI random frequency waveform.
51
The ideal signal processing in terms of maximal output SNR, is matching the received signal with all
the possible range-velocity hypotheses:
∫
Where is the expected received signal transformed by the propagation from objects at
different ranges and velocities. As was shown in chapter 4.3 (equation (30)), the correct matched-
filter response should be:
(
)
(
) (
)
Processing complexity of a basic pulse-Doppler waveform
If we use a simple modulated waveform, keeping all of the limiting conditions presented at chapter
4.3 fulfilled, then we can reduce the reference signal to a simpler form:
(
)
( )
Usually, in order to reduce the dependency of the signal on the high carrier frequency the signal is
coherently down-converted to base-band, so the reference signal will actually have to be:
(
)
The matched-filter output then has the form:
∫
(
)
50
If the signal is sampled uniformly and the waveform is a pulse train with a uniform PRI, the
matched-filter becomes a sum instead of an integral:
∑ (
)
Here is the uniform PRI, is the number of pulses transmitted in the cycle, and
is the number of samples sampled at each pulse interval reception window. Here again marks the
sampling time relative to beginning of each pulse transmission, and is the sample index.
Two more assumptions are usually made:
a. All the pulses in the train are identical to each other.
Using this assumption the reference signal's amplitude modulation, (
), does not depend
on the pulse number .
The sum then becomes:
∑
∑ (
)
b. The phase change within a transmitted pulse reflection due to the Doppler shift is negligible
(again, this is the "Stop and Hop" approximation) :
In this case we can assume the additional phase term is constant:
52
and it will not have a major effect on (taking into account that the absolute phase does not
interest us, only the relative phases). Finally we have:
∑ ∑ (
)
The signal processing computations of equation (70) includes the following:
1. Performing times (for each pulse interval) a matched filter on a sample set at a size of at least
samples. We will mark the final output size of this process as - the number of range bins.
The reason the number of range bins ( ) might be different than the number of sample at each
pulse interval ( ) is that we might want to use a finer sampling by interpolating the data, or
alternatively might not need all of the samples.
2. Performing FFTs at the size of - the number of velocity bins. Here might be different
than for similar reasons as in the case of and .
The complexity will be in the order of:
However, if we are only interested in a subset of Doppler frequencies and not in all of them, digital
filtering followed by decimation can be used, reducing the total necessary FFT size. If digital
filtering is applied, its complexity has to be also taken in account and it will add FIR
calculations. Assuming decimation factor of (in this case the number of velocity bins will be ),
the computational complexity will be in order of:
(
)
53
In terms of space in both cases, memory will be needed in order to remember all the
sampled data before performing the FFT (and filtering).
For example, assuming that and and that no decimation is applied,
the computational complexity will roughly be ( ) , and
memory units will be needed.
Processing complexity of a staggered PRI random frequency waveform
As shown in equation (60), the staggered PRI random frequency waveform's signal processing is
given by:
∑ {
} ∑
(
)
The signal processing computations includes the following:
1. Performing times (for each pulse interval) a matched filter on a sample set at a size of at least
samples. We again mark the final output size of the process as - the number of range bins.
2. Performing DFTs of size , marking and the number of range bins and number of
velocity bins (respectfully) that are inspected. Here and can be any number, because the
ranges and velocities that are inspected can be arbitrary.
In this case the complexity will be in the order of:
In terms of space, of memory will be needed in order to remember all the
sampled data before performing the DFT, plus all of the required DFT coefficients.
54
Using the previews example, assuming that and , and
that no decimation is applied, the computational complexity will roughly be
, and memory units will be needed.
If we compare between the computational complexities of the two types of processing in this
example, it seem that the second is much more complex in both terms of number of calculations and
memory usage (about ~100 fold more calculations and ~1000 fold more memory units). However,
this is true only if and . In reality we might not need to do all of the calculations for
the entire range and velocity spans, so it is possible to reduce the amount of calculations and memory
usage in a controlled way by reducing , or both.
55
5.6. Simulation results
A simulation was written in order to evaluate the performance of the staggered PRI random
frequency waveform and its signal processing. The simulation includes the transmitted signals with
the wanted frequencies and pulse intervals, propagation of the waves in free space reflected from
moving objects, clutter and in the presence of noise, and the reception process including RF to IF
conversion, filtering and sampling, signal processing and analysis. Figure 12a shows a block-
scheme of the simulation. Figure 12b demonstrate the simulation output for a single target – the
Range Velocity Map (RVM). Figure 12c shows the range and velocity profiles of the target in the
RVM.
Physical
medium
Signal
generation
Clutter
Targets
IF Reciever
and sampling
Simulation
parameters
Filtering and
Decimation
Convertion to
Base-Band
Signal
processing
Display and
analysis
Noise
Figure 12a – A scheme of the simulation's modules and data-flow
56
Figure 12b – Range-Velocity Map by simulation of single target at 250 m range and -31 m/sec radial velocity. The waveform includes integration of 1000 pulses.
Figure 12c – Range and velocity profiles of the target in the Range-Velocity Map shown in Figure 12b. The meaning of the range and velocity profiles is explained in chapter 4.1.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-60
-50
-40
-30
-20
-10
0
-30
-25
-20
-15
-10
-5
0
180 200 220 240 260 280 300-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
-70-60-50-40-30-20-100-60
-40
-20
0
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
[dB]
57
5.6.1. Staggered PRI waveform
We first demonstrate the cycle coherent integration using the method described in chapter 5.4 - non-
uniform to uniform interpolation of staggered PRI sampled waveform (Figures 13a, 13b).
Unfortunately the method cannot be applied in the case of using frequency hopping, so we show the
results using only staggered PRI with a constant frequency modulated waveform. In this case of
course we lose range resolution, and the effective range resolution is determined only by the pulse
width. We compare the results to the general processing (equation 60) described in chapter 5.5
(Figures 13c, 13d).
Figure 13a – RVM of single target using staggered PRI with single frequency waveform, created by non-uniform to uniform interpolation and DFT. The waveform includes a 100 pulses.
Figure 13b – Range and velocity profiles of the target in Figure 13a.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400
-160
-140
-120
-100
-80
-60
-40
-20
0
-35
-30
-25
-20
-15
-10
-5
0
160 180 200 220 240 260 280 300 320-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
-180-160-140-120-100-80-60-40-200-80
-60
-40
-20
0
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
[dB]
58
Figure 13c – RVM of single target using staggered PRI with single frequency waveform (no frequency hopping), created by the general signal processing. The waveform includes a 100 pulses.
Figure 13d – Range and velocity profiles of the target in Figure 13c.
It appears that processing the signals using non-uniform to uniform interpolation may yield better
results, but in fact it seems to be very sensitive to the exact PRI series, and demands greater
computational resources than the general processing method. An example for a simulated PRI series
for which the calculation did not yield the correct result is shown in Figure 13e.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-150
-100
-50
0
-35
-30
-25
-20
-15
-10
-5
0
180 200 220 240 260 280 300 320-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
-160-140-120-100-80-60-40-200-60
-40
-20
0
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
[dB]
59
Figure 13e – A different result for a similar case as was shown in Figure 13d for a different PRI series, using non-uniform to uniform interpolation and DFT. The waveform includes a 100 pulses. For this PRI series the calculation did not yield the correct result.
The failure to reconstruct the signal properly in some of the cases tested in the full simulation is due
to the sensitivity of the reconstruction solution to the bandwidth requirement, presented at chapter
5.4. In reality, it is difficult to constrain signals to a strictly confined bandwidth with no leaks none
so ever to higher frequencies. The perfect reconstruction method seems to be very sensitive to such
frequency leaks, leading in some cases to a wrong signal reconstruction and making it a non-practical
solution.
180 200 220 240 260 280 300 320-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]Range [m]
-180-160-140-120-100-80-60-40-200-60
-40
-20
0
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
61
5.6.2. Single target (noise free and no clutter)
In this scenario we use the simulation to simulate a single target at different ranges and velocities.
The range resolution of the simulated Radar is 32 m, and the velocity resolution is 1 m/sec. As
shown in Figure 14, the target signal is coherently integrated regardless of the target range and
velocity.
Figure 14 – Range-Velocity Maps of single target at different ranges and velocities.
5.6.3. Single target in the presence of noise and clutter
In this scenario we simulate a single target in the presence of thermal noise and clutter. The clutter is
simulated by many strong point reflectors at zero velocity and at different ranges. In Figures 15a,
15b simulation results are shown with the target standing out on the background of the clutter and
noise after the signal processing.
Velo
city [
m/s
ec]
Range [m]
Range: 180 m, Velocity: -1e-08 m/sec
0 200 400
-60
-40
-20
0
Velo
city [
m/s
ec]
Range [m]
Range: 250 m, Velocity: -30 m/sec
0 200 400
-60
-40
-20
0
Velo
city [
m/s
ec]
Range [m]
Range: 70 m, Velocity: -20 m/sec
0 200 400
-60
-40
-20
0
Velo
city [
m/s
ec]
Range [m]
Range: 430 m, Velocity: -50 m/sec
0 200 400
-60
-40
-20
0
-30
-25
-20
-15
-10
-5
0
[dB]
60
Figure 15a – Simulated Range-Velocity Map of single target at 250 m
range and -30 m/sec radial velocity, in the presence of thermal noise and clutter.
Figure 15b – Range and velocity profiles of the target in the Range-Velocity Map. Even though the target is approximately 11 dB weaker than the clutter located in its range, after integration it is approximately 18 dB stronger than the clutter's velocity sidelobes.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-60
-50
-40
-30
-20
-10
0
-30
-25
-20
-15
-10
-5
0
0 100 200 300 400 500 600-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
-70-60-50-40-30-20-100-40
-20
0
20
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
[dB]
62
5.6.4. Two targets in the presence of noise
In this scenario we simulate two targets with the same RCS at:
1. Different ranges but same velocity (Figure 16a)
2. Different velocities but same range (Figure 16c)
in the presence of thermal noise. As shown in their range and velocity profiles in Figure 16b and
Figure 16d, the two targets are still separable although they are close to each other (relative to the
resolution) in both dimensions.
Figure 16a – Simulated Range-Velocity Map of two close targets at 250 m and 280 m
ranges, and each at -30 m/sec radial velocity, in the presence of thermal noise.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-60
-50
-40
-30
-20
-10
0
-30
-25
-20
-15
-10
-5
0
[dB]
63
Figure 16b – Range profile of the targets in the Range-Velocity Map shown in Figure 16a.
Figure 16c – Simulated Range-Velocity Map of two close targets at 250 m
range, and at -30 m/sec and -32.5 m/sec radial velocity, in the presence of thermal noise.
0 100 200 300 400 500 600-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Range-P
rofile
Am
plit
ude [
dB
]
Range [m]
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300 350 400 450 500
-60
-50
-40
-30
-20
-10
0
-30
-25
-20
-15
-10
-5
0
[dB]
64
Figure 16d – Velocity profile of the targets in the Range-Velocity Map shown in Figure 16c.
-70 -60 -50 -40 -30 -20 -10 0-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Velo
city-P
rofile
Am
plit
ude [
dB
]
Velocity [m/sec]
65
5.7. Experimental results
In order to prove the implementation feasibility of a staggered PRI random frequency waveform,
experimental data including real linear stepped-frequency Radar raw samples were manipulated to
produce an effective random waveform, and a proper signal processing was applied to produce the
required Range-Velocity maps. The original data processed was of a waveform similar to the one
shown schematically in Figure 17a, made of a train of some 3500 pulses. Processing the data using
equation (60) produces the RVM shown in Figure 17b, in it we can see a real target located at 115 m
range and moving at -34 m/sec velocity, and also strong static clutter (at zero velocity) located at all
ranges. The range profiles of the target and clutter are shown in Figure 17c, and the velocity profile
of the target is shown in Figure 17d.
Figure 17a – A full stepped-frequency waveform data (scheme)
0 5 10 15 20 250
1
2
3
4
5
6
Fre
qu
en
cy [M
Hz]
PRI index
66
Figure 17b – Range-Velocity Map created by processing of a full stepped-
frequency train. In this example an target appears at 115 m range and at -34 m/sec velocity, in the presence of strong clutter (located at the zero velocity). The number of integrated pulses is ~3500.
Figure 17c – Range profiles of the target (-34 m/sec velocity) and the
clutter (zero velocity) in the Range-Velocity Map, created by processing of a full stepped-frequency train.
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300
-50
-40
-30
-20
-10
0
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
0 50 100 150 200 250 300 350-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude o
f C
lutt
er
and T
arg
et
[dB
]
Range [m]
Target
Clutter
[dB]
67
Figure 17d – Velocity profile of the target (at 115 m range) in the Range-Velocity
map, created by processing of a full stepped-frequency train.
An effective random waveform was created from the original data by selecting a random series of
pulses from an entire linear stepped-frequency cycle. Because the selection of the PRI series is
random, the pulse interval and the frequency difference between each two consequent pulses in the
series are also random (although quantized by the basic original PRI and frequency step). The diluted
data contains about 1500 pulses of the original 3500. This method is illustrated in Figure 18a. The
RVM produced by processing the diluted data is shown in Figure 18b. In it we can still see the
clutter at the zero velocity, but now additional velocity sidelobes of the strong clutter also appear. In
fact, the clutter sidelobes level is so high (about -35 dB under the clutter level, as expected – see
Figure 18c), that they reach the target's level and mask it. The range profiles of the target and clutter
are shown in Figure 18d. Looking carefully at the range profile of the target we can still see it rising
a little above the sidelobes level at 115 m range. If we were to use a larger pulse-train, or if the target
was stronger enough, then the target's profile would have been more prominent on the background of
the clutter sidelobes. We can also see that both the range profile of the clutter and its velocity
resolution did not change significantly. This gives good indication that the random waveform is
feasible for implementation.
It is important to mention here that pulse cancelling technique, meant to reduce the clutter level,
cannot be applied in this case due to the different frequencies of the consecutive pulses and also due
to the different time interval between the pulses (creating changing phase differences between them).
-60-50-40-30-20-100-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Velo
city-P
rofile
Am
plit
ude o
f T
arg
et
[dB
]
Velocity [m/sec]
68
Figure 18a – An effective staggered PRI and "random" (disordered)
frequency waveform achieved by random dilution of a stepped- frequency train data.
Figure 18b – Range-Velocity Map created by processing of a randomly diluted stepped-frequency train data. In this example the clutter's velocity sidelobes are so high, that they mask the target and it seems to disappear.
0 5 10 15 20 250
2
4
6Before dilution
Fre
quency [
MH
z]
0 5 10 15 20 250
2
4
6After dilution
PRI index
Fre
quency [
MH
z]
Velo
city [
m/s
ec]
Range [m]
0 50 100 150 200 250 300
-50
-40
-30
-20
-10
0
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
[dB]
69
Figure 18c – Velocity profile of the target in the Range-Velocity Map, created by processing of a randomly diluted stepped-frequency train data. The average velocity sidelobes level of ~-35 dB is concurrent with the number of ~1500 pulses integrated in the sub-series.
Figure 18d – Range profiles of the target and the clutter in the Range-Velocity Map, created by processing of a randomly diluted stepped-frequency train data. The similarity between the clutter range-profiles in this example and in the case of full stepped-frequency data processing (shown in Figure 17c) indicates that range and velocity compressions are achieved properly by this waveform. The target is hardly seen because it has approximately the same amplitude as the clutter's velocity sidelobes.
-60-50-40-30-20-100-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Velo
city-P
rofile
Am
plit
ude o
f T
arg
et
[dB
]
Velocity [m/sec]
0 50 100 150 200 250 300 350-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Range-P
rofile
Am
plit
ude o
f C
lutt
er
and T
arg
et
[dB
]
Range [m]
Target
Clutter
71
5.8. Signal-to-Clutter Ratio considerations
In order to detect a target in search Radars, Constant False Alarm Rate (CFAR) methods are usually
used [1]. These methods usually include estimation of the noise by averaging the amplitude of
resolution cells close to an inspected cell (not including it), and by comparing the amplitude of the
inspected cell to that average. If the amplitude of the inspected cell is stronger than the estimated
noise by a factor of (called the CFAR threshold), than the cell is pronounced as a detection.
However, if a strong clutter is located at the range vicinity of the inspected cell, it's velocity
sidelobes might be stronger than the noise level and by that might reduce the probability of detecting
the target. If the target's amplitude will not be stronger than the clutter's sidelobes level by a factor of
- the target will not be detected. Unlike in the case of the standard pulse-Doppler waveform for
which the Doppler weighting window can determine the level of velocity side lobes, the sidelobes
level in the staggered PRI random frequency waveform case will be directly affected by the number
of pulses integrated in the CPI. If we mark the amplitude of the target as , and the amplitude of the
clutter as , than the Signal-to-Clutter Ratio (SCR) is defined as:
(
)
As we have seen in chapters 5.2 and 5.7, when using the staggered PRI random frequency waveform
the ratio between the clutter amplitude and the mean clutter sidelobes' amplitude is √ ( marking
the number of pulses in the train). If the clutter's sidelobes are stronger than other noises in the
system, it will be the dominant factor in determining the detection threshold, making the average
estimated noise level to be:
√
The estimated SNR of the target will then be:
70
In order for a detection to be declared, the estimated SNR must pass the threshold , giving:
so the detection would be possible only if :
When designing a Radar system, this can be a crucial consideration. The most appropriate
applications to use the waveform would therefore be ones that operate in high SCR environments.
Figure 21 shows an example of the SCR environment in which Radar system with a staggered PRI
random frequency waveform can work, being able to detect a target at detection
threshold (the target has to be at least 10 dB stronger than the clutter's average velocity side lobes
surrounding it).
Figure 19 – Signal-to-Clutter Ratio (SCR) lower limit needed for the ability to detect a target surrounded by the clutter velocity side lobes in the case of staggered PRI random frequency waveform, using a 10 dB detection threshold, as function of the total number of pulses in the CPI. For example – in order to be able to detect a target in an SCR environment of
-30 dB using the waveform, one would need to integrate at least 104 pulses which would yield a -40 dB velocity side lobes level. Because a 10 dB threshold is used for detection, the target could be detected on the background of the side lobes.
100
102
104
106
108
1010
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
Number of integrated pulses in the CPI
Sig
nal-to
-Clu
tter
Ratio [
dB
]
72
5.9. Advantages and Disadvantages
As was shown in the previous chapters, the staggered PRI random frequency waveform has some
advantages and some disadvantages which have to be taken into account when considering it's
implementation in a Radar application.
5.9.1. Advantages
Ambiguity rejection
Implementation of a Radar system which, for example, utilizes a uniform PRI pulse-Doppler
waveform usually brings about the problem of ambiguities in the target's measured range-velocity
state.
A range ambiguity can be created if a reflection of a transmitted pulse by a far target is received after
the transmission of the following pulse. The received signal can then be interpreted as the second
pulse reflected by a closer target.
A velocity ambiguity can be created because the uniform PRI used by the waveform is, in fact, the
sampling of the Doppler phase. Because the sampling rate is finite, a fast target's reflection having a
Doppler frequency higher than the PRF, can alias onto a lower frequency and be interpreted as an
echo returning from a slower target.
On the one hand - in order to avoid range ambiguities one would wish to use a long PRI (meaning a
low PRF) in order have the second-time around reflections return from as farther distances as
possible. On the other hand - the same one would wish to use a high PRF (meaning a short PRI) in
order have the Doppler aliasing happen for higher velocities as possible.
In many cases the conflict between the two types of ambiguities enforces the Radar implementer to
make some compromise regarding the PRI selection (of course, there are many other considerations
that has to be taken into account when selecting the optimal PRI for the system, and controlling the
range-velocity ambiguities is only one of them).
73
However, using a staggered PRI random frequency waveform can profoundly diminish the range and
velocity ambiguities (even make them vanish completely), and by that enables the Radar
implementer more degrees of freedom in choosing the optimal PRI series for the system. The reason
for the vanishing of the range ambiguities is the frequency mismatch between the corresponding
transmitted pulses. The reason for the vanishing of velocity ambiguities is the non-uniform phase
sampling created by the staggering of the PRI, that prevents the aliasing phenomenon from
happening.
The range and velocity ambiguities can be visualized in the Generalized Ambiguity function. In
Figure 4b (the pulse-Doppler waveform case) we can see the recurrent lobes that exist alongside the
zero-time, zero-velocity main-lobe. The main-lobe represents the true target range-velocity state that
will appear on the Range Doppler Map, and the recurrent lobes represent the ambiguities that might
also appear in it as false targets. In Figure 8b (the staggered PRI random frequency case) we can see
that there are no prominent recurrent lobes – indicating that there will be no prominent ambiguities in
the Range-Velocity Map.
ECCM (Electronic Counter-Counter Measures)
One of challenges a Radar developer has to face is Electronic Counter Measures (ECM) that attempts
to disrupt the functionality of the Radar and confuse it. Some ECM techniques will attempt to study
some repeating behaviors of the Radar and use them in order to send disruptive counter signals in
real time. Transmitting a train of identical pulses with a uniform time-interval between them
(uniform PRI) is an example of such a repeating behavior.
Using a staggered PRI and a random frequency based waveform significantly decreases the ability of
the ECM to predict the behavior of the Radar system, therefore making it harder for it to disrupt the
Radar's functionality.
74
5.9.2. Disadvantages
Implementation complexity
As we have seen in chapter 5.5.2, the computational complexity of signal processing the staggered
PRI random frequency waveform is greater than in the case of other equivalent waveforms – both in
time (CPU / FPGA) resources and in space (memory) resources. This might make a Radar system
using the waveform more costly and less attractive for implementation.
High velocity sidelobes
The implications of high velocity side lobes presented in chapters 5.7.3 and 5.8 indicates to what
may be a major disadvantage of the staggered PRI random frequency waveform. While an ordinary
pulse-Doppler Radar may have velocity ambiguities due to its uniform PRI, its velocity side lobes
level can be controlled by applying an appropriate weighting window onto the velocity axis (before
performing the FFT), no matter how many pulses are integrated in the CPI. In the case of using a
staggered PRI random frequency waveform, however, the velocity side lobes level can be
determined only by the number of pulses integrated in the CPI and there is no way to reduced them
by applying such an equivalent weighting window (or, to be precise, the ability of applying such an
equivalent weighting window is not known to the author while writing this paper).
This poses a serious challenge for the Radar when it works in an environment containing strong and
weak reflectors at the same time (for example - a weak target in an environment of strong clutter).
The high side lobes level of the strong reflector might exceed the target's level and prevent it from
being detected, as demonstrated in Figure 18b.
However, the velocity side lobes level are determined by the factor (where is the number of
coherent pulses transmitted in the CPI), and therefore this problem can be reduced by increasing if
possible. In addition – the problem will be less significant if the application works in an environment
of weak clutter (such as in the case of airborne Radars for example).
75
6. Conclusions
The Radar waveform is one of the most important parameters in the Radar system design. Usually, in
order to inspect the waveform characteristics, the Ambiguity function tool is used. Unfortunately, the
ambiguity function gives a good indication of how point targets will appear on the Radar Range-
Doppler Map only if some limiting conditions are fulfilled. In the general case, when one or more of
the limiting conditions are not fulfilled, we will need to start working with a more accurate tool
which is the Generalized Ambiguity function.
Several of the common waveforms used in Radar systems were presented with their Generalized
Ambiguity functions. A new waveform is proposed, one that is based on a train of coherent pulses,
with random frequency shifts between each two consecutive pulses in the train, and also a stagger in
the pulse interval between them. Evaluation of the staggered PRI random frequency waveform's
Generalized Ambiguity function shows a response inspiring a wished "thumbtack" response, with
mean PSLR of √ in amplitude ( in power) where is the number of pulses in the train. A method
of finding "good" PRI and frequency series for the pulse train in the sense of minimal maximum
PSLR was proposed, and the integration loss in the first ambiguity zone due to the PRI stagger was
analyzed.
The signal processing of the staggered PRI random frequency waveform was shown to be more
computationally complex than the regular pulse-Doppler train's signal processing. A method of
processing a waveform containing only PRI stagger with no frequency hopping using perfect
reconstruction was shown to work in general, but not in a robust way.
A detailed simulation was written in order to evaluate the waveform's properties and performance,
and its results prove the feasibility of implementation of the waveform. Different scenarios including
some with several targets in the presence of noise and clutter were simulated and analyzed in terms
of resolution, mean PSLR and SCR.
Experimental data from a Radar system using linear stepped-frequency waveform was manipulated
to create an effective staggered PRI random frequency waveform data. Implementation of the
76
appropriate signal processing on the manipulated data indeed yields the expected results, with a good
fit to the detailed simulation results.
The overall work shows that the staggered PRI random frequency waveform can in fact be
implemented in a Radar systems, having the advantages of range and velocity ambiguity rejections
and ECCM capabilities, but at the cost of greater computational complexity, and also velocity side
lobes level that cannot be decreased below 1/√ in amplitude ( in power) relative to the main-
lobe.
77
Appendix A – Code Review
In this appendix we review some of the code used in simulations and signal processing throughout
the whole work. The simulations and analysis were written using the MATLAB®
application.
Generalized Ambiguity Function
The following function receives the waveform's parameters in the structure AF_in and outputs the
standard and Generalized ambiguity functions in the structure AF_out.
function AF_out = AF_calc(AF_in)
Light_velocity = 3e8; %m/sec
AF_out.mean_PRI = AF_in.small_PRI + AF_in.rand_PRI_Amp/2;
T_ps = AF_in.T_p*ones(AF_in.NumOfPRIsInCycle,1); % sec
PRIs = AF_out.mean_PRI*ones(AF_in.NumOfPRIsInCycle,1); % sec
PRI_ind = 0:1:AF_in.NumOfPRIsInCycle;
PRI_ind = PRI_ind(randperm(AF_in.NumOfPRIsInCycle+1));
PRI_ind = PRI_ind(1:AF_in.NumOfPRIsInCycle);
% Random PRI:
if (AF_in.WF_type == 4) || (AF_in.WF_type == 5)
PRIs = PRIs + round(AF_in.rand_PRI_Amp./(AF_in.NumOfPRIsInCycle)*...
(PRI_ind-(AF_in.NumOfPRIsInCycle)/2).'.*...
AF_in.f_sampling)./AF_in.f_sampling;
end
% Pulse - Doppler:
BB_freqs = 0e4*ones(1,AF_in.NumOfPRIsInCycle);
% Stepped frequency:
if (AF_in.WF_type == 2)
freq_ind = 0;
for PRI_ind = 1:AF_in.NumOfPRIsInCycle
if freq_ind>=AF_in.NumOfPRIsInBatch
freq_ind = freq_ind-AF_in.NumOfPRIsInBatch;
end
BB_freqs(PRI_ind) = freq_ind.*AF_in.BW./(AF_in.NumOfPRIsInBatch-1);
freq_ind = freq_ind + 1;
end
end
% Random Frequency:
if (AF_in.WF_type == 3) || (AF_in.WF_type == 5)
BB_freqs = AF_in.BW/(AF_in.NumOfPRIsInCycle-1).*(0:1:(AF_in.NumOfPRIsInCycle-1));
BB_freqs = BB_freqs(randperm(AF_in.NumOfPRIsInCycle));
end
NumOfSamplesInPRI = floor(AF_in.f_sampling.*PRIs);
78
Cycle_NumOfSamples = sum(NumOfSamplesInPRI);
Velocity_Bin_Size = AF_in.Velocity_Bin_Size;
NumOfVelocityBins = AF_in.NumOfVelocityBins;
Velocity_vec = (-NumOfVelocityBins:NumOfVelocityBins).*Velocity_Bin_Size;
Slow_time_freq = 2*Velocity_vec/Light_velocity*AF_in.RF_freq;
time_vec = (0:1:(Cycle_NumOfSamples-1)).'./AF_in.f_sampling;
CPI = Cycle_NumOfSamples / AF_in.f_sampling;
AF_out.mean_CPI = AF_in.NumOfPRIsInCycle.*AF_out.mean_PRI;
transmission_Start_times = 0;
PRI_times = cumsum(PRIs);
for PRI_ind = 2:AF_in.NumOfPRIsInCycle
transmission_Start_times = [transmission_Start_times; PRI_times(PRI_ind-1)];
end
transmission_End_times = transmission_Start_times + T_ps;
Delay = 0;
PRI_Start_sample = 1;
PRI_End_sample = floor(AF_in.f_sampling*PRIs(1));
Envelope = zeros(Cycle_NumOfSamples,1,'single');
Wave_length_Env = zeros(Cycle_NumOfSamples,1,'single');
Signal = single(Envelope);
AF_out.IntegrationGain = zeros(ceil(2*AF_out.mean_PRI*AF_in.f_sampling),1);
AF_out.Integration_Times = (0:1:(length(AF_out.IntegrationGain)-1)).'./AF_in.f_sampling;
for PRI_ind = 1:AF_in.NumOfPRIsInCycle;
Curr_time_vec = (time_vec(PRI_Start_sample:PRI_End_sample)-Delay);
Envelope(PRI_Start_sample:PRI_End_sample) = ...
(Curr_time_vec>=transmission_Start_times(PRI_ind)).*...
(Curr_time_vec<=transmission_End_times(PRI_ind));
temp_IntegrationGain = (1-Envelope(PRI_Start_sample:PRI_End_sample));
temp_IntegrationGain = temp_IntegrationGain.*(Curr_time_vec-T_ps(PRI_ind)-...
Curr_time_vec(1))./T_ps(PRI_ind).* (Curr_time_vec<=AF_out.mean_CPI);
temp_IntegrationGain(temp_IntegrationGain>1) = 1;
temp_IntegrationGain(temp_IntegrationGain<0) = 0;
AF_out.IntegrationGain(1:(PRI_End_sample-PRI_Start_sample+1)) =
AF_out.IntegrationGain(1:(PRI_End_sample-PRI_Start_sample+1)) + ...
temp_IntegrationGain;
Freq_Env(PRI_Start_sample:PRI_End_sample) =
Envelope(PRI_Start_sample:PRI_End_sample).*BB_freqs(PRI_ind);
RF_Freq_Env = AF_in.RF_freq + Freq_Env(PRI_Start_sample:PRI_End_sample);
Wave_length_Env(PRI_Start_sample:PRI_End_sample) = single(Light_velocity./RF_Freq_Env);
PRI_Signal = single(exp(2j*pi*BB_freqs(PRI_ind).*(Curr_time_vec-time_vec(PRI_Start_sample))));
Signal(PRI_Start_sample:PRI_End_sample) = Envelope(PRI_Start_sample:PRI_End_sample).*PRI_Signal;
PRI_Start_sample = PRI_Start_sample+NumOfSamplesInPRI(PRI_ind);
if PRI_ind<AF_in.NumOfPRIsInCycle
PRI_End_sample = PRI_End_sample+NumOfSamplesInPRI(PRI_ind+1);
else
PRI_End_sample = Cycle_NumOfSamples;
end
end
RF_Freq_Env_full = (double(Freq_Env) + double(AF_in.RF_freq)).';
AF_out.IntegrationGain = AF_out.IntegrationGain./AF_in.NumOfPRIsInCycle;
AF_out.IntegrationLoss = min(1./AF_out.IntegrationGain,1e2);
Expanded_WF = single(repmat(Signal,1,1));
Expanded_time_vec = single(repmat(time_vec,1,1));
79
Expanded_time_vec((Cycle_NumOfSamples+1):(2*Cycle_NumOfSamples-1)) =
single(Expanded_time_vec((1:(Cycle_NumOfSamples-1)))+(Cycle_NumOfSamples)/AF_in.f_sampling);
Expanded_time_vec = Expanded_time_vec-(Cycle_NumOfSamples-1)/AF_in.f_sampling;
Expanded_WF_length = 2*Cycle_NumOfSamples-1;
Ambiguity_Function = zeros(Expanded_WF_length,NumOfVelocityBins,'single');
Ambiguity_Function_gen= zeros(Expanded_WF_length,NumOfVelocityBins,'single');
Beta = Velocity_vec./Light_velocity;
for V_ind = 1:(2*NumOfVelocityBins+1)
v = Velocity_vec(V_ind);
Transformed_time_vec = (Light_velocity-v)/(Light_velocity+v)*time_vec;
sig = Expanded_WF.*exp(2j*pi*Slow_time_freq(V_ind).*time_vec);
sig_gen = Expanded_WF.*single(exp(-2j*pi*(double(RF_Freq_Env_full).*...
double(Transformed_time_vec) - double(RF_Freq_Env_full).*double(time_vec))));
Ambiguity_Function(:,V_ind) = xcorr(sig,Expanded_WF);
Ambiguity_Function_gen(:,V_ind) = xcorr(sig_gen,Expanded_WF);
end
Ambiguity_Function = Ambiguity_Function./max(abs(Ambiguity_Function(:)));
Ambiguity_Function_gen = Ambiguity_Function_gen./max(abs(Ambiguity_Function_gen(:)));
AF_out.Ambiguity_Function = Ambiguity_Function;
AF_out.Ambiguity_Function_gen = Ambiguity_Function_gen;
AF_out.time_vec = time_vec;
AF_out.Signal = Signal;
AF_out.Freq_Env = Freq_Env;
AF_out.Expanded_time_vec = Expanded_time_vec;
AF_out.Velocity_vec = Velocity_vec;
AF_out.CPI = CPI;
AF_out.Slow_time_freq = Slow_time_freq;
Mean_Wave_length = mean(Wave_length_Env);
Velocity_Resolution = 0.5*Mean_Wave_length./CPI;
if AF_in.BW>0
Time_Resolution = 1/AF_in.BW;
else
Time_Resolution = 1/AF_in.f_sampling;
end
AF_out.Velocity_Frame_size = 2*Velocity_Resolution/AF_in.Velocity_Bin_Size;
AF_out.Time_Frame_size = 2*Time_Resolution*AF_in.f_sampling;
% ----------------------------------------------------------------
% Find PSLR and NPSLR:
AF_size1 = size(Ambiguity_Function_gen,1);
AF_size2 = size(Ambiguity_Function_gen,2);
y = round((AF_size1-1)/2);
x = round((AF_size2-1)/2);
Center_Point = [x ; y];
x_min = round(Center_Point(1)-AF_out.Velocity_Frame_size/2+1);
x_max = round(Center_Point(1)+AF_out.Velocity_Frame_size/2+1);
y_min = round(Center_Point(2)-AF_out.Time_Frame_size/2+1);
y_max = round(Center_Point(2)+AF_out.Time_Frame_size/2+1);
Max_PSL_Ambiguity_Function_gen = Ambiguity_Function_gen;
Max_PSL_Ambiguity_Function_gen(y_min:y_max,x_min:x_max) = 0;
Max_PSL = max(abs(Max_PSL_Ambiguity_Function_gen(:)));
AF_out.PSLR = 1/Max_PSL;
AF_out.NPSLR = AF_out.PSLR/sqrt(AF_in.NumOfPRIsInCycle);
81
Signal Processing
The following function receives as an input the simulated or experimental parameters of the
staggered PRI random frequency waveform in the structure params, and the received and filtered
samples in the array th_Data.Decimated, and outputs the RVM processed by both the methods
described at chapters 5.4 and 5.5.1.
% Processing:
t_n = cumsum([params.Tp params.PRIs(1:(end-1))]);
R_index_max = ceil((params.R_max-params.R_min)/params.R_step);
V_index_min = floor((params.V_min-params.V_max)/params.V_step);
R0 = (0:1:R_index_max).*params.R_step + params.R_min;
v = (0:-1:V_index_min).*params.V_step + params.V_max;
th_Data.RVM = zeros(length(v),length(R0));
for Range_ind = 1:length(R0);
RangeGate = th_Get_RangeGate_From_Range(R0(Range_ind),params);
Sample_Data = th_Data.Decimated(:,RangeGate);
for Velocity_ind = 1:length(v);
ref_phases = 4*pi*params.RFFreqsTx./(params.LightVelocity+v(Velocity_ind)).*...
(v(Velocity_ind).*t_n + R0(Range_ind));
th_Data.RVM(Velocity_ind,Range_ind) = sum(Sample_Data.*exp(1j.*ref_phases.'));
end
end
th_Data.R_axis = R0;
th_Data.v_axis = v;
% Perfect reconstruction:
sample_size = params.LightVelocity/(2*params.fs); %m
MaxRange = min(params.LightVelocity/(2*min(params.PRIs)),params.R_max);
NumOfRangeSamplesInMap = ceil(MaxRange/sample_size);
mean_PRI = mean(params.PRIs);
t_PRI = t_n;
T = params.CPI;
N = 2.5*ceil(T/mean_PRI);
delta_t = (t_n(end)-t_n(1))/(N-1);
reconstruction_times = t_n(1)+(0:1:(N-1)).*delta_t;
pi_div_T = pi/T;
th_Data.PerfRecon_RDM = zeros(N,NumOfRangeSamplesInMap);
h = ones(N,params.NumOfPRIs);
for p = 1:params.NumOfPRIs
disp(['Run ' num2str(p)]);
t_p = t_PRI(p);
for k = 1:length(reconstruction_times)
for q = 1:params.NumOfPRIs
if q==p
continue
end
80
dem_sin = sin(pi_div_T*(t_p-t_PRI(q)));
if dem_sin == 0
sinc_factor = 1;
else
sinc_factor = sin(pi_div_T*(reconstruction_times(k)-t_PRI(q))) / dem_sin;
end
h(k,p) = h(k,p) * sinc_factor;
end
if ~mod(params.NumOfPRIs,2) % for even N
h(k,p) = h(k,p) * cos(pi_div_T*(reconstruction_times(k)-t_p));
end
end
end
max_V = 0.5*params.AverageWaveLength/delta_t;
W = exp(-2j*pi.*((1:length(reconstruction_times)).'-1)*((1:length(reconstruction_times))-1)./...
length(reconstruction_times));
Wh = W*h;
for sample_ind = 1:NumOfRangeSamplesInMap
th_Data.PerfRecon_RDM(:,sample_ind) = Wh*th_Data.Decimated(:,sample_ind);%fft(c);
end
th_Data.R_axis_PerfRecon = params.Range0ForGate+((0:(NumOfRangeSamplesInMap-1))./params.fs).*...
params.LightVelocity/2;
th_Data.v_axis_PerfRecon = -(0:1:(N-1)).*(max_V/(N-1));
end
82
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תקציר
מרחק( הינם חיישנים אלקטרומגנטיים המהווים נדבך משמעותי ביכולות החישה -מערכות מכ"מ )מגלה כיוון
אחד ההיבטים המרכזיים במימוש מערכות מכ"מ הוא צורת הגל )אפנון( מרחוק בשימושים אזרחיים וצבאיים.
, החל מיכולת ות המכ"מישירה ומשמעותית בכל מערכ לצורת הגל המכ"מית יש השפעה .המשודר האות המכ"מי
מרכזיים היבטיםהגילוי ודיוקי מכ"מי חיפוש וכלה באיכות התמונה ויכולת ההפרדה של מכ"מי הדמאה. שני
משמעות -המופיעים בפונקציית הרב ,הנידונים בספרות המחקר על צורות גל מכ"מיות הם אונות הצד והקיפולים
(Ambiguity Functionשל צורת הגל. אונות צד גב )משמעות מובילים לרוב לפגיעה -והות וקיפולים בפונקציית הרב
צורת גל מציאתואף למגבלות בביצועי מערכות מכ"מ, כך שאחת ממטרות המחקר המרכזיות בנושא היא
נטולת אונות צד וקיפולים. –משמעות שלה שואפת לפונקציית הרב משמעות האידאלית -שפונקציית הרב
של צורת גל לאות מכ"מי, המכיל רכבת פעימות )פולסים( במרווחי זמן לא קבועים, וכן עבודה זו מציעה סוג חדש
משמעות-שמשודרים בתדרים אקראיים. ביצועי צורת גל זו נבדקים על ידי בחינה של הכללה של פונקציית הרב
צועים והוכחת כולל ניתוח בי , בחינת תוצאות סימולציה מפורטת בתרחישים שוניםרחבי סרטפיסיקליים לאותות
, כולל , וניתוח תוצאות ניסויים. לבסוף מוצגים יתרונות וחסרונות של שימוש בצורת האותהיתכנות מימוש
. משמעויות למימושים במערכות שונות
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אוניברסיטת תל אביב
הפקולטה למדעים מדויקים
ע"ש ריימונד ובברלי סאקלר
צורת גל אקראית למכ"מ
(Staggered PRI and Random Frequency Radar Waveform)
עבודה זו הוגשה כחלק מהדרישות לקבלת התואר
באוניברסיטת תל אביב .M.Sc –"מוסמך אוניברסיטה"
החוג לפיסיקה
על ידי:
יוסי מגריסו
ברקאי )מנחה מלווה(-רועי בק ד"רהעבודה הוכנה בהנחייתם של פרופ' נדב לבנון ו
ובהדרכתו של ד"ר אהרון לוי
