| General The temporal
and spatial variation in ionospheric structures have often
frustrated the efforts of communications and radar system
operators who base their frequency management decisions on
monthly mean predictions of radio propagation in the high
frequency (short-wave) band. The University of Massachusetts
Lowell’s Center for Atmospheric Research (UMLCAR) has
produced a low power miniature version of its DigisondeTM
sounders, the DigisondeTM Portable Sounder (DPS),
capable of making measurements of the overhead ionosphere
and providing real-time on-site processing and analysis to
characterize radio signal propagation to support communications
or surveillance operations. The system
compensates for a low power transmitter (300 W vs. 10 kW for
previous systems) by employing intrapulse phase coding, digital
pulse compression and Doppler integration. The data acquisition,
control, signal processing, display, storage and automatic
data analysis functions have been condensed into a single
multi-tasking, multiple processor computer system, while the
analog circuitry has been condensed and simplified by the
use of reduced transmitter power, wide bandwidth devices,
and commercially available PC expansion boards. The DPS is
shown in the composite Figure 1-1 (with the integrated transceiver
package shown in Figure 1-1A,
and one of the four crossed magnetic dipole receive antennas
in Figure 1-1B).
Figure 1-1A DigisondeTM Portable Sounder |
Figure
1-1B Magnetic Loop Turnstile Antenna |
Noteworthy
new technology involved in this system includes:
-
Electronically
switched active crossed loop receiving antenna -
Commercially
sourced 10 MIPS TMS 320C25 digital signal processor
(DSP) -
4
million sample DSP buffer memory -
71
to 110 MHz digital synthesizer on a 4"x5"
card -
Compact
DC-DC converters allowing operation on one battery -
Four-channel
high speed (1 million 12-bit samples/sec) digitizer
board -
A
160 Mbits/sec parallel data bus between the digitizer
and the DSP -
A
proprietary multi-tasking operating system for remote
interaction via a modem connection without suspending
system operation
-
Direct
digital synthesized coherent oscillators -
21
dB signal processing gain from phase coded pulse compression -
21
dB additional signal processing gain from coherent
Doppler integration
-
Automatic
ionospheric layer identification and parameter scaling
by an embedded expert system
The
availability of a small low power ionosonde that could be
operated on-site wherever a high frequency (HF) radio or
radar was in use, would greatly increase the value of the
information produced by the instrument since it would become
available to the end user immediately.
One
of the chief applications for the real-time data currently
provided by digital ionospheric sounders is to manage the
operation of HF radio channels and networks. Since many
HF radios are operated at remote locations (i.e., aircraft,
boats, land vehicles of all sorts, and remote sites where
telephone service is unreliable) the major obstacle to making
practical use of the ionospheric sounder data and associated
computed propagation information is the dissemination of
this data to a data processing and analysis site. Since
HF is often used where no alternative communications link
exists, or is held in reserve in case primary communication
is lost, it is not practical to assume that a communications
link exists to make centrally tabulated real-time ionospheric
data available to the user. Furthermore, local measurements
are superior to measurements at sites of opportunity in
the user’s general region of the globe since extreme
variations in ionospheric properties are possible even over
short distances, especially at high latitudes [Buchau et
al., 1985; Buchau and Reinisch, 1991] or near the sunset
or sunrise terminator. However,
for most applications, the size, weight, power consumption
and cost of a conventional ionospheric sounder have made
local measurements impractical. Therefore the availability
of a small, low cost sounder is a major improvement in the
usefulness of ionospheric sounder data. Shrinking the conventional
1 to 50 kW pulse sounders to a portable, battery operated
100 to 500 W system requires the application of substantial
signal processing gain to compensate for the 20 dB reduction
in transmitter power. Furthermore, a compact portable package
requires the use of highly integrated control, data acquisition,
timing, data processing, display and storage hardware. The
objective of the DPS development project was to develop
a small vertical incidence (i.e., monostatic) ionospheric
sounder which could automatically collect and analyze ionospheric
measurements at remote operating sites for the purpose of
selecting optimum operating frequencies for obliquely propagated
communication or radar propagation paths. Intermediate objectives
assumed to be necessary to produce such a capability were
the development of optimally efficient waveforms and of
functionally dense signal generation, processing and ancillary
circuitry. Since the need for an embedded general purpose
computer was a given imperative, real-time control software
was developed to incorporate as many functions as was feasible
into this computer rather than having to provide additional
circuitry and components to perform these functions. The
DPS duplicates all of the functions of its predecessor the
DigisondeTM 256 [Bibl et al., 1981] and [Reinisch,
1987] in a much smaller, low power package. These include
the simultaneous measurement of seven observable parameters
of reflected (or in oblique incidence, refracted) signals
received from the ionosphere:
1)
Frequency
2) Range (or height for vertical incidence measurements)
3) Amplitude
4) Phase
5) Doppler Shift and Spread
6) Angle of Arrival
7) Wave Polarization Because
the physical parameters of the ionospheric plasma affect
the way radio waves reflect from or pass through the ionosphere,
it is possible by measuring all of these observable parameters
at a number of discrete heights and discrete frequencies
to map out and characterize the structure of the plasma
in the ionosphere. Both the height and frequency dimensions
of this measurement require hundreds of individual measurements
to approximate the underlying continuous functions. The
resulting measurement is called an ionogram and comprises
a seven dimensional measurement of signal amplitude vs.
frequency and vs. height as shown in Figure 1-2 (due to
the limitations of current software only five may be displayed
at a time). Figure 1-2 is a five-dimensional display, with
sounding frequency as the abscissa, virtual reflection height
(simple conversion of time delay to range assuming propagation
at 3x108 m/sec) as the ordinate, signal amplitude
as the spot (or pixel) intensity, Doppler shift as the color
shade and wave polarization as the color group (the blue-green-grey
scale or "cool" colors showing extraordinary polarization,
the red-yellow-white scale or "hot" colors showing
ordinary polarization). 
Figure
1-2 Five-Dimensional Ionogram
Another
objective of the DPS development was to store the data created
by the system in an easily accessible format (e.g., DOS
formatted personal computer files), while maintaining compatibility
with the existing base of DigisondeTM sounder
analysis software in use at the UMLCAR and at over 40 research
institutes around the world. This objective often competed
with the additional objective of providing an easily accessible
and simply understood standard data format to facilitate
the development of novel post-processing analysis and display
programs.
Ionospheric
Propagation of Electromagnetic Waves back
to top An
ionospheric sounder uses basic radar techniques to detect
the electron density (equal to the ion density since the
bulk plasma is neutral) of ionospheric plasma as a function
of height. The ionospheric plasma is created by energy from
the sun transferred by particles in the solar wind as well
as direct radiation (especially ultra-violet and x-rays).
Each component of the solar emissions tends to be deposited
at a particular altitude or range of altitudes and therefore
creates a horizontally stratified medium where each layer
has a peak density and to some degree, a definable width,
or profile. The shape of the ionized layer is often referred
to as a Chapman function [Davies, 1989] which is a roughly
parabolic shape somewhat elongated on the top side. The
peaks of these layers usually form between 70 and 300 km
altitude and are identified by the letters D, E, F1 and
F2, in order of their altitude. By
scanning the transmitted frequency from 1 MHz to as high
as 40 MHz and measuring the time delay of any echoes (i.e.,
apparent or virtual height of the reflecting medium) a vertically
transmitting sounder can provide a profile of electron density
vs. height. This is possible because the relative refractive
index of the ionospheric plasma is dependent on the density
of the free electrons (Ne), as shown in Equation
1-1 (neglecting the geomagnetic field):
m2(h)= 1 – k (Ne/f2)
(1–1) where k
= 80.5, Ne is electrons/m3, and f
is in Hz [Davies, 1989; Chen, 1987]. The
behavior of the plasma changes significantly in the presence
of the Earth’s magnetic field. An exhaustive derivation
of m [Davies, 1989] results in the Appleton Equation for
the refractive index, which is one of the fundamental equations
used in the field of ionospheric propagation. This equation
clearly shows that there are two values for refractive index,
resulting in the splitting of a linearly polarized wave
incident upon the ionosphere, into two components, known
as the ordinary and extraordinary waves. These propagate
with a different wave velocity and therefore appear as two
distinct echoes. They also exhibit two distinct polarizations,
approximately right hand circular and left hand circular,
which aid in distinguishing the two waves. When
the transmitted frequency is sufficient to drive the plasma
at its resonant frequency there is a total internal reflection.
The plasma resonance frequency (fp) is defined
by several constants, e – the charge of an electron,
m – the mass of an electron, eo
– the permittivity of free space, but only one variable,
Ne – electron density in electrons/m3
[Chen, 1987]:
fp2 = (Ne e2/4peom) = kNe (1–2)
A
typical number for the F-region (200 to 400 km altitude)
is 1012 electrons/m3, so the plasma
resonance frequency would be 9 MHz. The value of m
in Equation 1–1 approaches 0 as the operating frequency,
f, approaches the plasma frequency. The group velocity of
a propagating wave is proportional to m,
so
m
= 0 implies that the wave slows down to zero which is obviously
required at some point in the process of reflection since
the propagation velocity reverses. The
total internal reflection from the ionosphere is similar
to reflection of radio frequency (RF) energy from a metal
surface in that the re-radiation of the incident energy
is caused by the free electrons in the medium. In both cases
the wave penetrates to some depth. In a plasma the skin
depth (the depth into the medium at which the electric field
is 36.8% of its incident amplitude) is defined by:
l0/2p
d = —————————— (1–3)
where l0 is
the free space wavelength. The
major difference between ionospheric reflection and reflection
from a metallic surface is that the latter has a uniform
electron density while the ionospheric density increases
roughly parabolically with altitude, with densities starting
at essentially zero at stratospheric altitudes and rising
to a peak at about 200 to 400 km. In the case of a metal
there is no region where the wave propagates below the resonance
frequency, while in the ionosphere the refractive index
and therefore the wave velocity change with altitude until
the plasma resonance frequency is reached. Of course if
the RF frequency is above the maximum plasma resonance frequency
the wave is never reflected and can penetrate the ionosphere
and propagate into outer space. Otherwise what happens on
a microscopic scale at the surface of a metal and on a macroscopic
scale at the plasma resonance in the ionosphere is very
similar in that energy is re-radiated by electrons which
are responding to the incident electric field. Coherent Integration
back to top During
the 1960’s and 1970’s several variations in sounding
techniques started moving significantly beyond the basic
pulse techniques developed in the 1930’s. First was
the coherent integration of several pulses transmitted at
the same frequency. Two signals are coherent if, having
a phase and amplitude, they are able to be added together
(e.g., one radar pulse echo received from a target added
to the next pulse echo received from the same target, thousandths
of a second later) in such a way that the sum may be zero
(if the two signals are exactly out of phase with each other)
or double the amplitude (if they are exactly in phase).
Coherent integration of N signals can provide a factor of
N improvement in power. This technique was first used in
the DigisondeTM 128 [Bibl and Reinisch, 1975].
In
ionospheric sounding, the motion of the ionosphere often
makes it impossible to integrate by simple coherent summation
for longer that a fraction of a second, although it is not
rare to receive coherent echoes for tens of seconds. However,
with the application of spectral integration (which is a
byproduct of the Fourier transform used to create a Doppler
spectrum) it is possible to coherently integrate pulse echoes
for tens of seconds under nearly all ionospheric conditions
[Bibl and Reinisch, 1978]. The integration may progress
for as long a time as the rate of change of phase
remains constant (i.e., there is a constant Doppler shift,
Df).
The DigisondeTM 128PS, and all subsequent versions
perform this spectral integration. Additional
detail on this topic is contained in Chapter 2 in this section. Coded Pulses
to Facilitate Pulse Compression Radar Techniques back
to top
A
third general technique to improve on the simple pulse sounder
is to stretch out the pulse by a factor of N, thus increasing
the duty cycle so the pulse contains more energy without
requiring a higher power transmitter (power x time = energy).
However, to maintain the higher range resolution of the
simple short pulse the pulse can be bi-phase, or phase reversal
modulated with a phase code to enable the receiver to create
a synthetic pulse with the original (i.e., that of the short
pulse) range resolution. A network of sounders using a 13-bit
Barker Code were operated by the U.S. Navy in the 1960’s. The
critical factor in the use of pulse compression waveforms
for any radar type measurement is the correlation properties
of the internal phase code. Phase codes proposed and experimented
with included the Barker Code [Barker, 1953], Huffman Sequences
[Huffman 1962], Convoluted Codes [Coll, 1961], Maximal Length
Sequence Shift Register Codes (M-codes) [Sarwate and Pursley,
1980], or Golay’s Complementary Sequences [Golay, 1961],
which have been implemented in the VHF mesospheric sounding
radar at Ohio State University [Schmidt et al., 1979] and
in the DPS. The internal phase code alternative has just
recently become economically feasible with the availability
of very fast microprocessor and signal processor IC’s.
Barker Coded pulses have been implemented in several ionospheric
sounders to date, but until the DPS was developed there
have been no other successful implementations of Complementary
Series phase codes in ionospheric sounders. The
European Incoherent Scatter radar in Tromso, Norway (VanEiken,
1991 and 1993) and an over-the-horizon (OTH) HF radar used
the Complementary Series codes. However most major radar
systems including all currently active OTH radars opted
for the FM/CW chirp technique, due to its resistance to
Doppler induced leakage and its compatibility with analog
pulse compression processing techniques. Basically, the
chirp waveform avoids the need for extremely fast digital
processing capabilities, since only the final stage is performed
digitally, while the pulse compression is best performed
entirely digitally. Even at the modest bandwidths used for
ionospheric sounding, this digital capability was until
recently, much more expensive and cumbersome than the special
synthesizers required for chirpsounding.
Another
new development in the 1970’s was the coherent multiple
receiver array [Bibl and Reinisch, 1978] which allows angle
of arrival (incidence angle) to be deduced from phase differences
between antennas by standard interferometer techniques.
Given a known operating frequency, and known antenna spacing,
by measuring the phase or phase difference on a number of
antennas, the angle of arrival of a plane wave can be deduced.
This interferometry solution is invalid, however, if there
are multiple sources contributing to the received signal
(i.e., the received wave therefore does not have a planar
phase front). This problem can be overcome in over 90% of
the cases as was first shown with the DigisondeTM
256 [Reinisch et al., 1987] by first isolating or discriminating
the multiple sources in range, then in the Doppler domain
(i.e., isolating a plane wavefront) before applying the
interferometry relationships.
Except
for the FM/CW chirpsounder which operates well on transmitter
power levels of 10 to 100 W (peak power) the above techniques
and cited references typically employ a 2 to 30 kW peak
power pulse transmitter. This power is needed to get sufficient
signal strength to overcome an atmospheric noise environment
which is typically 20 to 50 dB (CCIR Noise Tables) above
thermal noise (defined as kTB, the theoretical minimum noise
due to thermal motion, where k = Boltzman’s constant,
T = temperature in °
K, and
B = system bandwidth in Hz). More importantly, however,
since ionogram measurements require scanning of the entire
propagating band of frequencies in the 0.5 to 20 MHz RF
band (up to 45 MHz for oblique measurements), the sounder
receiver will encounter broadcast stations, ground-to-air
communications channels, HF radars, ship-to-shore radio
channels and several very active radio amateur bands which
can add as much as 60 dB more background interference. Therefore,
the sounder signal must be strong enough to be detectable
in the presence of these large interfering signals. To
make matters worse, a pulse sounder signal must have a broad
bandwidth to provide the capability to accurately measure
the reflection height, therefore the receiver must have
a wide bandwidth, which means more unwanted noise is received
along with the signal. The noise is distributed quite evenly
over bandwidth (i.e., white), while interfering signals
occur almost randomly (except for predictably larger probabilities
in the broadcast bands and amateur radio bands) over the
bandwidth. Thus a wider-bandwidth receiver receives proportionally
more uniformly distributed noise and the probability of
receiving a strong interfering signal also goes up proportionally
with increased bandwidth. The
DPS transmits only 300 W of pulsed RF power but compensates
for this low power by digital pulse compression and coherent
spectral (Doppler) integration. The two techniques together
provide about 30 dB of signal processing gain (up to 42
dB for the bi-static oblique waveforms) thus for vertical
incidence measurements the system performs equivalently
with a simple pulse sounder of 1000 times greater power
(i.e., 300 kW). Additional
detail on this topic is contained in Chapter 2 in this section.
Current Applications
of Ionospheric Sounding back to top
Current
applications of ionospheric sounders fall into two categories:
a.
Support of operational systems, including shortwave radio
communications and OTH radar systems. This support can
be in the form of predictions of propagating frequencies
at given times and locations in the future (e.g., over
the ensuing month) or the provision of real-time updates
(updated as frequently as every 15 minutes) to detect
current conditions such that system operating parameters
can be optimized. b.
Scientific research to enable better prediction of ionospheric
conditions and to understand the plasma physics of the
solar-terrestrial interaction of the Earth’s atmosphere
and magnetic field with the solar wind.
There
has been considerable effort in producing global models
of ionospheric densities, temperature, chemical constitution,
etc, such that a few sounder measurements could calibrate
the models and improve the reliability of global predictions.
It has been shown that if measurements are made within a
few hundred kilometers of each other, the correlation of
the measured parameters is very high [Rush, 1978]. Therefore
a network of sounders spaced by less than 500 km can provide
reliable estimates of the ionosphere over a 250 km radius
around them. The
areas of research pursued by users of the more sophisticated
features of the DigisondeTM sounders include
polar cap plasma drift, auroral phenomena, equatorial spread-F
and plasma irregularity phenomena, and sporadic E-layer
composition [Buchau et al., 1985; Reinisch 1987; and Buchau
and Reinisch 1991]. There may be some driving technological
needs (e.g., commercial or military uses) in some of these
efforts, but many are simply basic research efforts aimed
at better understanding the manifestations of plasma physics
provided by nature. Requirements
for a Small Flexible Sounding System back to
top The detailed
design and synthesis of a RF measurement system (or any
electronic system) must be based on several criteria:
a. The
performance requirements necessary to provide the needed
functions, in this case scientific measurements of electron
densities and motions in the ionosphere.
b. The
availability of technology to implement such a capability.
c. The
cost of purchasing or developing such technology.
d. The
risk involved in depending on certain technologies, especially
if some of the technology needs to be developed.
e. The
capabilities of the intended user of the system, and its
expected willingness to learn to use and maintain it;
i.e., how complicated can the operation be before the
user will give up and not try to learn it. The question
of what technology can be brought to bear on the realization
of a new ionospheric sounder was answered in a survey of
existing technology in 1989, when the portable sounder development
started in earnest. This survey showed the following available
components, which showed promise in creating a smaller,
less costly, more powerful instrument. Many of these components
were not available when the last generation of DigisondesTM
(circa 1980) was being developed:
Solid-state
300 W MOSFET RF power transistors High-speed
high precision (12, 14 and 16 bit) analog to digital
(A–D) converters High-speed
high precision (12 and 16 bit) digital to analog (D–A)
converters
Single
chip Direct Digital Synthesizers (DDS) Wideband
(up to 200 MHz) solid state op amps for linear feedback
amplifiers Wideband
(4 octaves, 2–32 MHz) 90°
phase shifters
Proven
DigisondeTM 256 measurement techniques Very
fast programmable DSP (RISC) IC’s Fast,
single board, microcomputer systems and supporting programming
languages
Many
of these components are inexpensive and well developed because
they feed a mass market industry. The MOSFET transistors are
used in Nuclear Magnetic Resonance medical imaging systems to
provide the RF power to excite the resonances. The high speed
D–A converters are used in high resolution graphic video
display systems such as those used for high performance workstations.
The DDS chips are used in cellular telephone technology, in
which the chip manufacturer, Qualcomm, is an industry leader.
The DSP chips are widely used in speech processing, voice recognition,
image processing (including medical instrumentation). And of
course, fast microcomputer boards are used by many small systems
integrators which end up in a huge array of end user applications
ranging from cash registers to scientific computing to industrial
process controllers. The
performance parameters were well known at the beginning of the
DPS development, since several models of ionospheric pulse sounders
had preceded it. The frequency range of 1 to 20 MHz for vertical
sounding was an accepted standard, and 2 to 30 MHz was accepted
as a reasonable range for oblique incidence measurements. It
was well known that radio waves of greater than 30 MHz often
do propagate via skywave paths, however, most systems relying
on skywave propagation don’t support these frequencies,
so interest in this frequency band would only be limited to
scientific investigations. A required power level in the 5 to
10 kW range for pulse transmitters had provided good results
in the past. The measurement objectives were to simultaneously
measure all seven observable parameters outlined at Paragraph
107 above in order to characterize the following physical features:
The height
profile of electron density vs. altitude Position
and spatial extent of irregularity structures, gradients and
waves Motion vectors
of structures and waves
As
mentioned in the section above dealing with Current Applications
of Ionospheric Sounding (Paragraph 127 et seq. above),
the accurate measurement of all of the parameters, except frequency
(it being precisely set by the system and need not be measured)
depends heavily on the signal to noise ratio of the received
signal. Therefore vertical incidence ionospheric sounders capable
of acquiring high quality scientific data have historically
utilized powerful pulse transmitters in the 2 to 30 kW range.
The necessity for an extremely good signal to noise ratio is
demanded by the sensitivity of the phase measurements to the
random noise component added to the signal level. For instance,
to measure phase to 1 degree accuracy requires a signal to noise
ratio better than 40 dB (assuming a Gaussian noise distribution
which is actually a best case), and measurement of amplitude
to 10% accuracy requires over 20 dB signal to noise ratio. Of
course, is it desirable that these measurements be immune to
degradation from noise and interference and maintain their high
quality over a large frequency band. This requires that at the
lower end of the HF band the system’s design has to overcome
absorption, noise and interference, and poor antenna performance
and still provide at least a 20 to 40 dB signal to noise ratio.
METHODOLOGY,
THEORETICAL BASIS AND IMPLEMENTATION back to
top General The
VIS/DPS borrows several of the well proven measurement techniques
used by the DigisondeTM 256 sounder described in
[Bibl, et al, 1981; Reinisch et al., 1989] and [Reinisch, 1987],
which has been produced for the past 12 years by the UMLCAR.
The addition of digital pulse compression in the DPS makes the
use of low power feasible, the implementation in software of
processes that were previously implemented in hardware results
in a much smaller physical package, and the high level language
control software and standard PC-DOS (i.e., IBM/PC) data file
formats provide a new level of flexibility in system operation
and data processing. A technical
description of the DPS (sounder unit and receive antennas sub-systems)
are contained in Section 2 of this manual. Coherent
Phase Modulation and Pulse Compression back to
top
The
DPS is able to be miniaturized by lengthening the transmitted
pulse beyond the pulse width required to achieve the desired
range resolution where the radar range resolution is defined
as,
DR=c/2b where b is the system bandwidth, or (1–4)
DR=cT/2 for a simple rectangular pulse
waveform, with T being the width
of the rectangular pulse
The
longer pulse allows a small low voltage solid state amplifier
to transmit an amount of energy equal to that transmitted by
a high power pulse transmitter (energy = power x time, and power
= V2/R) without having to provide components to handle
the high voltages required for tens of kilowatt power levels.
The time resolution of the short pulse is provided by intrapulse
phase modulation using programmable phase codes (user selectable
and firmware expandable), the Complementary Codes, and M-codes
are standard. The use of a Complementary Code pulse compression
technique is described in this chapter, which shows that at
300 W of transmitter power the expected measurement quality
is the same as that of a conventional sounder of about 500 kW
peak pulse power. The
transmitted spread spectrum signal s(t) is a biphase (180° phase
reversal) modulated pulse. As illustrated in Figure 1–3,
bi-phase modulation is a linear multiplication of the binary
spreading code p(t) (a.k.a. a chipping sequence, where each
code bit is a "chip") with a carrier signal sin(2pf0t) or in complex form,
exp[j2pf0t],
to create a transmitted signal,
s(t)=p(t)exp[j2pf0t] (1–5)
Figure 1-3 Generation of a Bi-phase Modulated Spread Spectrum Waveform
NOTE
Notation throughout this chapter
will use s(t) as the transmitted signal, r(t) the received
signal and p(t) as the chip sequence. Functions r1(t)
and r2(t) will be developed to describe
the signal after various stages of processing in the receiver.
The
term chip is used rather than bit because for
spread spectrum communications many chips are required to transmit
one bit of message information, so a distinct term had to be
developed. Figure 1-4 on the following page depicts the modulation
of a sinusoidal RF carrier signal by a binary code (notice that
the code is a zero mean signal, i.e., centred around 0 volts
amplitude). Since the mixer in Figure 1-3 can be thought of
as a mathematical multiplier, the code creates a 180o
(p radians)
phase shift in the sinusoidal carrier whenever p(t) is negative,
since –sin(wt) = sin(wt+p). The
binary spreading code is identical to a stream of data bits
except that it is designed such that it forms a pattern with
uniquely desirable autocorrelation function characteristics
as described later in this chapter. The 16-bit Complementary
Code pair used in the DPS is 1-1-0-1-1-1-1-0-1-0-0-0-1-0-1-1
modulated onto the odd-numbered pulses and 1-1-0-1-1-1-1-0-0-1-1-1-0-1-0-0
modulated onto the even-numbered pulses. This pattern of phase
modulation chips is such that the frequency spectrum of such
a signal (as shown in Figure 1-4) is uniformly spread over the
signal bandwidth, thus the term "spread spectrum".
In fact, it is interesting to note that the frequency spectrum
content of the spread spectrum signal used by the DPS is identical
to that of the higher peak power, simple short pulse used by
the DigisondeTM 256, even though the physical pulse
is 8 times longer. Since they have the same bandwidth, Equation
1–4 would suggest that they have the same range resolution.
It will be shown later in this chapter, that the ability of
the DigisondeTM 256 and the DPS to determine range
(i.e., time delay), phase, Doppler shift and angle of arrival
is also identical between the two systems, even though the transmitted
waveforms appear to be vastly different.
 Figure
1–4 Spectral Content of a Spread-Spectrum Waveform Since
the transmitted signal would obscure the detection of the much
weaker echo in a monostatic system the transmitted pulse must
be turned off before the first E-region echoes arrive at the
receiver which, as shown in Figure 1-5, is about TE
= 600 m
sec after the beginning of the pulse. Also, since the receiver
is saturated when the transmitter pulse comes on again, the
pulse repetition frequency is limited by the longest time delay
(listening interval) of interest, which is at least 5 msec,
corresponding to reflections from 750 km altitude. To meet these
constraints, a 533 m
sec pulse made
up of eight 66.67 m sec phase
code chips (15 000 chips/sec) is selected which allows detection
of ionospheric echoes starting at 80 km altitude. To avoid excessive
range ambiguity, a highest pulse repetition frequency of 200
pps is chosen, which allows reception of the entire pulse from
a virtual height of 670 km (the pulse itself is 80 km long)
altitude before the next pulse is transmitted. This timing captures
all but the highest multihop F-region echoes which are of little
interest. Under conditions where higher unambiguous ranges,
and therefore longer receiver listening intervals, are desired
100 pps or 50 pps can be selected under software control. 
Figure
1-5 Natural Timing Limitations for Monostatic Vertical Incidence
Sounding
The
key to the pulse compression technique lies in the selection
of a spreading function, p(t), which possesses an autocorrelation
function appropriate for the application. The ideal autocorrelation
function for any remote sensing application is a Dirac delta
function (or instantaneous impulse, d
(t) since this would provide perfect range accuracy and infinite
resolution. However, since the Dirac delta function has infinite
instantaneous power and infinite bandwidth, the engineering
tradeoffs in the design of any remote sensing system mainly
involve how far one can afford to deviate from this ideal (or
how much one can afford to spend in more closely approximating
this ideal) and still achieve the accuracy and resolution required.
More to the point, for a discussion of a discrete time digital
system such as the DPS, the ideal signal is a complex unit impulse
function, with the phase of the impulse conveying the RF phase
of the received signal. The many different pulse compression
codes all represent some compromise in achieving this ideal,
although each code has its own advantages, limitations, and
trade-offs. The autocorrelation function as applied to code
compression in the VIS/DPS is defined as: k
R(k)=S p(n) p(n+k) (1–6)
n
Therefore
the ideal as described above is R(k) = d(k).
(Several examples of autocorrelation functions of the codes
described in this Section can be seen in Figures 1-9 through
1-13.) For
ionospheric applications, the received spread-spectrum coded
signal, r(t), may be a superposition of several multipath echoes
(i.e., echoes which have traveled over various propagation paths
between the transmitter and receiver) reflected at various ranges
from various irregular features in the ionosphere. The algorithm
used to perform the code compression operates on this received
multipath signal, r(t), which is an attenuated and time delayed
(possibly multiple time delays) replica of the transmitted signal
s(t) (from Equation 1–5), which can be represented as: P
r(t)=S ai s(t-ti) or (1–7)
i=1
P
r(t)=S ai p(t–ti)exp[j2pf0t – fi]
i=1
where
S
shows that the P multipath signals sum linearly at the receive
antenna, ai is the amplitude of the ith
multipath component of the signal, and ti
is the propagation delay associated with multipath i. The carrier
phase fi
of each multipath could be expressed in terms of the carrier
frequency and the time delay t
i
; however, since the multiple carriers (from the various multipath
components) cannot be resolved, while the delays in the complex
code modulation envelope can be, a separate term, f
i,
is used. Next, when the carrier is stripped off of the signal,
this RF phase term will be represented by a complex amplitude
coefficient ai
rather than ai. 
Figure
1-6 Conversion to Baseband by Undersampling
By
down-converting to a baseband signal (a digital technique is
shown in Figure 1-6), the carrier signal can be stripped away,
leaving only the superposed code envelopes delayed by P multiple
propagation paths. Figure 1-6 presents one way to strip the
carrier off a phase modulated signal. This is the screen display
on a digital storage oscilloscope looking at the RF output from
the DPS system operating at 3.5 MHz. Notice that the horizontal
scan spans 2 msec, which if the oscilloscope was capable of
presenting more than 14 000 resolvable points, would display
7 000 cycles of RF. The sample clock in the digital storage
scope is not synchronized to the DPS, however, the digital sampling
remains coherent with the RF for periods of several milliseconds.
The analog signal is digitized at a rate such that each sample
is made an integer number of cycles apart (i.e., at the same
phase point) and therefore looks like a DC level until the phase
modulation creates a sudden shift in the sampled phase point.
Therefore the 180º phase reversals made on the RF carrier show
up as DC level shifts, replicating the original modulating code
exactly. The more hardware intensive method of quadrature demodulation
with hardware components (mixers, power splitters and phase
shifters) can be found in any communications systems textbook,
such as [Peebles, 1979]. After removing the carrier, the modified
r(t), now represented by r1(t) becomes:
P
r1(t)=S ai p(t–ti) (1–8)
i=1
where
the carrier phase of each of the multipath components is now
represented by a complex amplitude a
i which carries along the RF phase term, originally defined
by f
i in Equation 1–7, for each multipath. Since
the pulse compression is a linear process and contributes no
phase shift, the real and imaginary (i.e., in-phase and quadrature)
components of this signal can be pulse compressed independently
by cross-correlating them with the known spreading code p(t).
The complex components can be processed separately because the
pulse compression (Equation 1–9B) is linear and the code
function, p(n), is all real. Therefore the phase of the cross-correlation
function will be the same as the phase of r1(t). The
classical derivation of matched filter theory [e.g., Thomas,
1964] creates a matched filter by first reversing the time
axis of the function p(t) to create a matched filter impulse
response h(t) = p(–t). Implementing the pulse compression
as a linear system block (i.e., a "black box" with
impulse response h(t)) will again reverse the time axis of
the impulse response function by convolving h(t) with the
input signal. If neither reversal is performed (they effectively
cancel each other) the process may be considered to be a cross-correlation
of the received signal, r(t) with the known code function, p(t).
Either way, the received signal, r2(n) after matched
filter processing becomes:
r2(n)=r1(n)*h(n)=r1(n)*p(–n) (1–9A)
or by substituting
Equation 1–8 and writing out the discrete convolution,
we obtain the cross-correlation approach,
P M P
r2(n)=S ai S p(k-ti)p(k-n)=S Mai d(n-ti) (1–9B)
i=1 k=1 i=1
where
n is the time domain index (as in the sample number, n, which
occurs at time t = nT where T is the sampling interval), P is
the number of multipaths, k is the auxiliary index used to perform
the convolution, and M is the number of phase code chips. The
last expression in Equation 1–9B, the d(n), is only true if the autocorrelation
function of the selected code, p(t), is an ideal unit impulse
or "thumbtack" function (i.e., it has a value of M
at correlation lag zero, while it has a value of zero for all
other correlation lags). So, if the selected code has this property,
then the function r2(n), in Equation 1–9 is
the impulse response of the propagation path, which has a value
ai,
(the complex amplitude of multipath signal i) at each time n
= t
i (the propagation delay attributable to multipath
I). 
Figure
1-7 Illustration of Complementary Code Pulse Compression
Figure
1-7 illustrates the unique implementation of Equation 1–9
employed for compression of Complementary Sequence waveforms.
A 4-bit code is used in this figure for ease of illustration
but arbitrarily long sequences can be synthesized (the DPS’s
Complementary Code is 8-chips long). It is necessary to transmit
two encoded pulses sequentially, since the Complementary Codes
exist in pairs, and only the pairs together have the desired
autocorrelation properties. Equation 1–8 (the received
signal without its sinusoidal carrier) is represented by the
input signal shown in the upper left of Figure 1-7. The time
delay shifts (indexed by n in Equation 1–9 are illustrated
by shifting the input signal by one sample period at a time
into the matched filter. The convolution shifts (indexed by
k in Equation 1–9 sequence through a multiply-and-accumulate
operation with the four ±
1 tap coefficients. The accumulated value becomes the output
function r2(n) for the current value of n. The two
resulting expressions for Equation 1–9 (an r2(n)
expression for each of the two Complementary Codes) are shown
on the right with the amplitude M=4 clearly expressed. The non-ideal
approximation of a delta function, d(n–ti), is apparent from the spurious a
and –a amplitudes. However, by summing the two r2(n)
expressions resulting from the two Complementary Codes, the
spurious terms are cancelled, leaving a perfect delta function
of amplitude 2M. The
amplitude coefficient M in Equation 1–9 is tremendously
significant! It is what makes spread-spectrum techniques practical
and useful. The M means that a signal received at a level of
1 mv would
result in a compressed pulse of amplitude M mv, a gain
of 20 log10(M) dB. Unfortunately, the benefits of
all of that gain are not actually realized because the RMS amplitude
of the random noise (which is incoherently summed by Equation
1–9B) which is received with the signal goes up by a factor
of \/M. However, this still represents a power gain (since
power = amplitude2) equal to M, or 10log10(M)
dB. The \/M coefficient for the incoherent summation of multiple
independent noise samples is developed more thoroughly in the
following section on Coherent Spectral Integration, but the
factor of M-increase for the coherent summation of the signal
is clearly illustrated in Figure 1-7. The
next concern is that the pulse compression process is still
valid when multiple signals are superimposed on each other as
occurs when multipath echoes are received. It seems likely that
multiple overlapping signals would be resolved since Equation
1–9 and the free space propagation phenomenon are linear
processes, so the output of the process for multiple inputs
should be the same as the sum of the outputs for each input
signal treated independently. This linearity property is illustrated
in Figure 1-8. Two 4-chip input signals, one three times the
amplitude of the other, are overlapped by two chips at the upper
left of the illustration. After pulse compression, as seen in
the lower right, the two resolved components, still display
a 3:1 amplitude ratio and are separated by two chip periods.
Figure
1-8 Resolution of Overlapping Complementary Coded Pulses
The
phase of the received signal is detected by quadrature sampling;
but, how is the complex quantity, a
i, or ai exp[fi],
related to the RF phase (fi)
of each individual multipath component? It can be shown that
this phase represents the phase of the original RF signal components
exactly. As shown in Equations 1–10 and 1–11, the
down-converting (frequency translation) of r(t) by an oscillator,
exp[j2pf0t]
results in: P P
r1(t)=Saip(t-ti)exp[j2pf0t–jfi]exp[j2pf0t]=Saip(t–ti)exp[jfi]
i=0 i=0
(1–10)
or P
r1(t)=Saip(t–ti) where ai=aiexp[jfi] is a complex amplitude (1–11)
i=0
This signal
maintains the parameter fi
which is the original phase of each RF multipath component.
Note that the oscillator is defined as having zero phase (exp[j2pf0t]). Alternative
Pulse Compression Codes back to top Due
to many possible mechanisms the pulse compression process will
have imperfections, which may cause energy reflected from any
given height to leak or spill into other heights to some degree.
This leakage is the result of channel induced Doppler, mathematical
imperfection of the phase code (except in the Complementary
Codes which are mathematically perfect) and/or imperfection
in the phase and amplitude response of the transmitter or receiver.
Several codes were simulated and analyzed for leakage from one
height to another and for tolerance to signal distortion caused
by band-limiting filters. All of the pulse compression algorithms
used are cross-correlations of the received signal with a replica
of the unit amplitude code known to have been sent. Therefore,
since Equation 1–9B represents a "cross-correlation"
(the unit amplitude function p(t) is cross-correlated with the
complex amplitude weighted version) of p(k) with itself,
it is the leakage properties of the autocorrelation functions
which are of interest. The autocorrelation
functions of several codes were computed either on a PC or a
VAX computer for several different codes and are shown in the
following figures:
a. Complementary
Series (Figure 1-9) b. Periodic
M-codes (Figure 1-10) c. Non-periodic
M-codes (Figure 1-11) d. Barker
Codes (Figure 1-12) e. Kasami
Sequence Codes (Figure 1-13)
Figure
1-9 Autocorrelation Function of the Complementary Series
Figure
1-10 Autocorrelation Function of a Periodic Maximal Length Sequence
Figure
1-11 Autocorrelation Function of a Non-Periodic Maximal Length
Sequence
Figure
1-12 Autocorrelation Function of the Barker Code
Figure
1-13 Autocorrelation Function of the Kasami Sequence
Since
the Complementary Series pairs do not leak energy into
any other height bin this phase code scheme seemed optimum and
was chosen for the DPS’s vertical incidence measurement
mode in order to provide the maximum possible dynamic range
in the measurement. If there is too much leakage (for instance
at a –20 dB level) then stronger echoes would create a
"leakage noise floor" in which weaker echoes could
not be detectable. The autocorrelation function of the Maximal
Length Sequence (M-code) is particularly good since for M =
127, the leakage level is over 40 dB lower than the correlation
peak and the correlation peak provides over 20 dB of SNR enhancement.
However, since these must be implemented as a continuous transmission
(100% duty cycle) they are not suitable for vertical incidence
monostatic sounding. Therefore the M-Code is the code of choice
for oblique incidence bi-static sounding, where the transmitter
need not be shut off to provide a listening interval. The
M-codes which provide the basic structure of the oblique waveform,
all have a length of M = (2N–1). The attractive
property of the M-codes is their autocorrelation function, shown
in Figure 1-10. This type of function is often referred to as
a "thumbtack". As long as the code is repeated at
least a second time, the value of the cross correlation function
at lag values other than zero is –1 while the value at
zero is M. However, if the M-Code is not repeated a second time,
i.e., if it is a pulsed signal with zero amplitude before and
after the pulse, the correlation function looks more like Figure
1-11. The characteristics of Figure 1-11 also apply if the second
repetition is modulated in phase, frequency, amplitude, code
# or time shift (i.e., starting chip). So to achieve the "clean"
correlation function with M-Codes (depicted in Figure 1-10),
the identical waveform must be cyclically repeated (i.e., periodic). The
problem that occurs using the M-codes is if any of the multipath
signal components starts or ends during the acquisition of one
code record, then there are zero amplitude samples (for that
multipath component) in the matched filter as the code is being
pulse compressed. If this happens then the imperfect cancellation
of code amplitude (which is illustrated by Figure 1-11) at correlation
lag values other than zero will occur. In order to obtain the
thumbtack pulse compression, the matched filter must always
be filled with samples from either the last code repetition,
the current code repetition or the next code repetition (with
no significant change), since these sample values are necessary
to make the code compression work. "Priming" the channel
with 5 msec of signal before acquiring samples at the receiver
ensures that all of the multipath components will have preceding
samples to keep the matched filter loaded. Similarly after the
end of the last code repetition an extra code repetition makes
the synchronization less critical. This
"priming" becomes costly however, for when it is desired
to switch frequencies, antennas, polarizations etc., the propagation
path(s) have to be primed again. The 75% duty cycle waveform
(X = 3) allows these multiplexed operations to occur, but as
a result, only 8.5 msec out of each 20 msec of measurement time
is spent actually sampling received signals. The 100% duty cycle
waveform (X = 4) does not allow multiplexed operation, except
that it will perform an O polarization coherent integration
time (CIT) immediately
after an X polarization CIT has been completed. Since the simultaneity
of the O/X multiplexed measurement is not so critical (the amplitude
of these two modes fade independently anyway), this is essentially
still a simultaneous measurement. Because the 100% mode performs
an entire CIT without changing any parameters, it can continuously
repeat the code sequence and therefore the channel need only
be primed before sampling the very first sample of each CIT.
After this subsequent code repetitions are primed by the previous
repetition. Even
though the Complementary Code pairs are theoretically perfect,
the physical realization of this signal may not be perfect.
The Complementary Code pairs achieve zero leakage by producing
two compressed pulses (one from each of the two codes) which
have the same absolute amplitude spurious correlation peaks
(or leakage) at each height, but all except the main correlation
peak are inverted in phase between the two codes. Therefore,
simply by adding the two pulse compression outputs, the leakage
components disappear. Since the technique relies on the phase
distance of the propagation path remaining constant between
the sequential transmission of the two coded pulses, the phase
change vs. time caused by any movement in the channel geometry
(i.e., Doppler shift imposed on the signal) can cause imperfect
cancellation of the two complex amplitude height profile records.
Therefore, the Complementary Code is particularly sensitive
to Doppler shifts since channel induced phase changes which
occur between pulses will cause the two pulse compressions
to cancel imperfectly, while with most other codes we are only
concerned with channel induced phase changes within the duration
of one pulse. However, if given the parameters of the propagation
environment, we can calculate the maximum probable Doppler shift,
and determine if this yields acceptable results for vertical
incidence sounding. With
200 pps, the time interval between one pulse and the next is
5 msec. If one pulse is phase modulated with the first of the
Complementary Codes, while the next pulse has the second phase
code, the interval over which motions on the channel can cause
phase changes is only 5 msec. The degradation in leakage cancellation
is not significant (i.e., less than –15 dB) until the phase
has changed by about 10 degrees between the two pulses. The
Doppler induced phase shift is:
Df=2pTfD radians (1–12)
where fD
is the Doppler shift in Hz and T is the time between pulses. The Doppler
shift can be calculated as:
fD=(f0vr)/c< (or for a 2-way radar propagation path)
fD=(2f0vr)/c (1–13)
where
f0 is the operating frequency and vr is
the radial velocity of the reflecting surface toward or away
from the sounder transceiver. The radial velocity is defined
as the projection of the velocity of motion (v) on the
unit amplitude radial vector (r) between the radar location
and the moving object or surface, which in the ionosphere is
an isodensity surface. This is the scalar product of the two
vectors:
vr=v.r=|v|cos(q) (1–14)
A
phase change of 10° in 5 msec
would require a Doppler shift of about 5.5 Hz, or 160 m/sec
radial velocity (roughly half the speed of sound), which seldom
occurs in the ionospheric except in the polar cap region. The
8-chip complementary phase code pulse compression and coherent
summation of the two echo profiles provides a 16-fold increase
in signal amplitude, and a 4-fold increase in noise amplitude
for a net signal processing gain of 12 dB. The 127-chip Maximal
Length Sequence provides a 127-fold increase in amplitude and
a net signal processing gain of 21 dB. The Doppler integration,
as described later can provide another 21 dB of SNR enhancement,
for a total signal processing gain of 42 dB, as shown by the
following discussion. Coherent
Doppler (Spectral or Fourier) Integration back
to top The
pulse compression described above occurs with each pulse transmitted,
so the 12 to 21 dB SNR improvement (for 8-bit complementary
phase codes or 127-bit M-codes respectively) is achieved without
even sending another pulse. However, if the measurement can
be repeated phase coherently, the multiple returns can be coherently
integrated to achieve an even more detectable or "cleaner"
signal. This process is essentially the same as averaging, but
since complex signals are used, signals of the same phase are
required if the summation is going to increase the signal amplitude.
If the phase changes by more than 90° during
the coherent integration then continued summation will start
to decrease the integrated amplitude rather than increase it.
However, if transmitted pulses are being reflected from a stationary
object at a fixed distance, and the frequency and phase of the
transmitted pulses remain the same, then the phase and amplitude
of the received echoes will stay the same indefinitely. The
coherent summation of N echo signals causes the signal amplitude,
to increase by N, while the incoherent summation of the noise
amplitude in the signal results in an increase in the noise
amplitude of only \/N. Therefore with each N pulses integrated,
the SNR increases by a factor of \/N in amplitude which is a
factor of N in power. This improvement is called signal processing
gain and can be defined best in decibels (to avoid the confusion
of whether it is an amplitude ratio or a power ratio) as:
Processing Gain = 20 log10 {(Sp/Qp)/ (Si/Qi)} (1–15)
where
Si is the input signal amplitude, Qi the
input noise amplitude, Sp the processed signal amplitude,
and Qp the processed noise amplitude. Q is chosen
for the random variable to represent the noise amplitude, since
N would be confusing in this discussion. This coherent summation
is similar to the pulse compression processing described in
the preceding section, where N, the number of pulses integrated
is replaced by M, the number of code chips integrated. Another
perspective on this process is achieved if the signal is normalized
during integration, as is often done in an FFT algorithm to
avoid numeric overflow. In this case Sp is nearly
equal to Si, but the noise amplitude has been averaged.
Thus by invoking the central limit theorem [Freund, 1967 or
any basic text on probability], we would expect that as long
as the input noise is a zero mean (i.e., no DC offset) Gaussian
process, the averaged RMS noise amplitude, snp (p for processed) will
approach zero as the integration progresses, such that after
N repetitions:
snp2=sni2/N (the variance represents power) (1–16)
Since
the SNR can be improved by a variable factor of N, one would
think, we could use arbitrarily weak transmitters for almost
any remote sensing task and just continue integrating until
the desired signal to noise ratio (SNR) is achieved. In practical
applications the integration time limit occurs when the signal
undergoes (or may undergo, in a statistical sense) a phase change
of 90°. However,
if the signal is changing phase linearly with time (i.e., has
a frequency shift, Dw
), the integration
time may be extended by Doppler integration (also known as,
spectral integration, Fourier integration, or frequency domain
integration). Since the Fourier transform applies the whole
range of possible phase shifts needed to keep the phase of a
frequency shifted signal constant, a coherent summation of successive
samples is achieved even though the phase of the signal is changing.
The unity amplitude phase shift factor, e–jwt, in the Fourier Integral (shown
as Equation 1–17) varies the phase of the signal r(t) as
a function of time during integration. At the frequency (w) which
stabilizes the phase of the component of r(t) with frequency
w over the
interval of integration (i.e., makes r(t) e–jwt
coherent) the value of the integral increases with time rather
than averaging to zero, thus creating an amplitude peak in the
Doppler spectrum at the Doppler line which corresponds to w:
F[r(t)]=R(w)=òr(t)e-jwtdt (1–17)
Does
this imply that an arbitrarily small transmitter can be used
for any remote sensing application, since we can just integrate
long enough to clearly see the echo signal? To some extent this
is true. There is no violation of conservation of energy in
this concept since the measurement simply takes longer at a
lower power; however, in most real world applications, the medium
or environment will change or the reflecting surface will move
such that a discontinuous phase change will occur. Therefore
a system must be able to detect the received signal before a
significant movement (e.g., a quarter to a half of a wavelength)
has taken place. This limits the practical length of integration
that will be effective. The
discrete time (sampled data) processing looks very similar (as
shown in Equation 1–18). For a signal with a constant frequency
offset (i.e., phase is changing linearly with time) the integration
time can be extended very significantly, by applying unity amplitude
complex coefficients before the coherent summation is performed.
This stabilizes the phase of a signal which would otherwise
drift constantly in phase in one direction or the other (a positive
or negative frequency shift), by adding or subtracting increasingly
larger phase angles from the signal as time progresses. Then
when the phase shifted complex signal vectors are added, they
will be in phase as long as that set of "stabilizing"
coefficients progress negatively in phase at the same rate as
the signal vector is progressing positively. The Fourier transform
coefficients serve this purpose since they are unity amplitude
complex exponentials (or phasors), whose only function is to
shift the phase of the signal, r(n), being analyzed. Since
the DigisondeTM sounders have always done this spectral
integration digitally, the following presentation will cover
only discrete time (sampled data rather than continuous signal
notation) Fourier analysis.
N
F[r(t)]=R[k]=S r[n]exp[–jnk2p/N] (1–18)
n=0
where
r[n] is the sampled data record of the received signal at one
certain range bin, n is the pulse number upon which the sample
r[n] was taken, T is the time period between pulses, N is the
number of pulses integrated (number of samples r[n] taken),
and k is the Doppler bin number or frequency index. Since a
Doppler spectrum is computed for each range sampled, we can
think of the Fourier transforms as F56[w]
or F192[w]
where the subscripts signify with which range bin the resulting
Doppler spectra are associated. By
processing every range bin first by pulse compression (12 to
21 dB of signal processing gain) then by coherent integration,
all echoes from each range have gained 21 to 42 dB of processing
gain (depending on the waveform used and the length of integration)
before any attempt is made to detect them.
NOTE
Further explanation of Equation
1–18 which can be gathered from any good reference on
the Discrete Fourier Transformation, such as [Openheim &
Schaefer, Prentice Hall, 1975], follows. The total integration
time is NT, where T is the sampling period (in the DPS, the
time period between transmitted pulses). The frequency spacing
between Doppler lines, i.e., the Doppler resolution, is 2p/NT rads/sec (or 1/NT Hz) and the entire Doppler
spectrum covers 2p/T rad/sec (with complex input samples
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