2.0 SIGNAL PROCESSOR ANALYSES
2.1 INTRODUCTION
We now want to turn our attention to the analysis of signal processors. We will
be specifically concerned with analyzing the ability of signal processors to reject clutter
and improve signal-to-noise ratio (SNR). This is an extension of the waveform and
matched filter work of EE 619.
We do not want to discuss clutter and signal processor analyses in general terms.
Instead we want to discuss how one would perform specific analyses. To this end, we
will select a specific radar, target and clutter scenario, and specific signal processors. We
assume that the radar is ground based and has the job of detecting and tracking airborne
targets such as aircraft, helicopters and cruise missiles. We assume that the targets are
flying at low altitudes so that the radar is receiving returns from the target and ground. In
our case, the ground is the clutter source (termed ground clutter). We assume that the
radar is transmitting a pulsed (as opposed to CW or continuous wave) signal. We further
assume that the radar is transmitting an infinite or semi-infinite series of pulses with a
given pulse repetition interval (PRI). In some cases we will let the PRI vary from pulseto-pulse. For now we assume that the target is an ideal point target (SW0) although we
may allow a SW1 or SW3 target later on.
We will consider two types of signal processors: a moving target indicator or MTI
and a pulsed-Doppler signal processor. These are the two most common types of signal
processors in the type of radar indicated above. As an aside, the type of radar we are
considering would be an air defense or air traffic control radar that performs search
and/or track. These types of radars must contend with ground clutter (or sea clutter for
naval radars) while trying to perform these functions. The purpose of the signal
processor is to help the radar mitigate the ground clutter.
We will begin our studies by defining a ground clutter model. After this we will
develop equations that characterize the clutter, target and noise signals at the input to the
signal processor. Finally, we will discuss the characteristics MTI and pulsed-Doppler
signal processors and how they react to the target, clutter and noise signals.
2.2 CLUTTER MODEL
2.2.1 Radar Cross-Section (RCS) Model
A drawing that we will use to develop the ground clutter model is shown in
Figure 2-1. The top drawing represents a side view and the bottom drawing represents a
top view. For the initial development of the ground clutter model we assume that the
earth is flat. Later we will add a correction factor to account for the fact that the earth is
not flat.
The triangle and semicircle on the left represents the radar which is located a
height of h above the ground. When we discuss radar height in the context of clutter or,
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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more specifically, the signal the radar emits, we refer to the “height to the phase center of
the radar”. The phase center is usually taken to be the location of the feed for a reflector
antenna or the center of the phased array for a phased array antenna.
The dashed lines on the side and top views denote the 3-dB boundaries of the
antenna main beam. The angles  E and  A denote the elevation and azimuth 3-dB
beamwidths, respectively.
Figure 2-1 – Geometry for Ground Clutter Model
The horizontal line through the antenna phase center is simply a horizontal
reference line, it is not the elevation angle to which the main beam is steered. The target
is located at a range of R from the radar and at an altitude of hT . The elevation angle
from the radar phase center to the target is
 e  sin 1   hT  h  R  .
(2-1)
In the geometry of Figure 2-1 the clutter patch of interest is also located at a range
of R from the radar. In most applications, this is the region of clutter that is of interest
because we are interested in the clutter that competes with the target. However, for some
cases, most notably pulsed Doppler radars, the ground clutter that competes with the
target will not be at the target range, but at a much shorter range than the target range.
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This will not pose a problem for the clutter model. It is developed so as to handle this
case.
The width of the clutter patch along the R direction is R . In most cases R is
taken as the range resolution of the radar. The reason for this is that almost all signal
processors quantize the incoming signal into range cells that have a width of one range
resolution cell. (Recall our discussions of detection and range cells from EE 619.) In
some cases a range resolution cell is large enough to cause problems in the accuracy of
the ground clutter model. In this case, R is taken to be smaller than a range resolution
cell. If this is necessary, the signal processor calculations must account for this by
integrating across multiple clutter range cells. A discussion of this is beyond the scope of
this course.
With a little thought, it is easy to see that the ground region that extends over R
at a range of R is an annulus centered on the radar. This is depicted in the top view where
a portion of the annulus is shown. For purposes of calculating the radar cross section
(RCS) of the ground in this annulus, the annulus is divided into two regions as indicated
in Figure 2-1. One of these is termed the main beam clutter region and represents the
ground area illuminated by the main beam of the radar. The other is termed the sidelobe
clutter region and represents the ground area illuminated by the sidelobes of the radar.
The standard assumption is that the sidelobe clutter region extends from  2 to  2 .
In other words, it is assumed that there are no clutter returns from the back of the radar.
As implied by the statements above, the ground clutter model incorporates the transmit
and receive antenna beam characteristics. In this development, we are assuming a
monostatic radar that uses the same antenna for transmit and receive.
The size of the clutter RCS will depend upon the size of the ground area
illuminated by the radar (the region discussed in the previous paragraph) and the
reflectivity of the ground. This reflectivity is denoted by the variable  . Consistent with
the previous discussions of target RCS, one can think of clutter reflectivity as the ability
of the ground to absorb and re-radiate energy. In general, clutter reflectivity depends
upon the type of ground (soil, water, asphalt, gravel, sand, grass, trees, etc.) and its
roughness. It also depends upon moisture content and other related phenomena. Finally,
it also depends upon the angle to the clutter patch (  R in Figure 2-1). Chapter 12 of
Skolnik’s Radar Handbook contains further discussions of  . For our purposes, we will
use    0 . That is, we use a backscatter coefficient that doesn’t depend upon  R . Our
justification for this is that, in general  R is relatively constant (and small). If we were
considering an airborne radar, we would need to revert to a model wherein  varies with
 R . Some of these models are discussed in Skolnik’s Radar Handbook. We would also
need to revert to the variable  model for clutter that is very close to the radar.
However, this is not common in pulsed radars. (It is common in CW radars.)
We will use three values for  :   20 dB ,   30 dB and   40 dB .
These are fairly standard values currently used for radars that operate in the 5 to 10 GHz
range. The first case corresponds to moderate clutter and would be representative of
trees, fields and choppy water. The second value is light clutter and would be
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representative of sand, asphalt and concrete. The third value is very light clutter and
would be representative of smooth ice and smooth water.
Table 2-1 – Ground Clutter Backscatter Coefficients
Backscatter Coefficient, 
(dB)
Comment
-20
Moderate Clutter – Trees, fields,
choppy water
-30
Light Clutter – Sand, asphalt,
concrete
-40
Very Light Clutter – Smooth ice,
smooth water
With the above, we can write the RCS of the main beam ground area as
 CMB   ACM    Ad  d
(2-2)
where the various parameters are shown on Figure 2-1. An assumption in this equation is
that  A is small so that  Ad can be taken as a straight line that is perpendicular to d .
Examination of the top part of Figure 2-1 shows that the clutter area is not located
at beam center. This means that the clutter patch is not being fully illuminated, in
elevation, by the main beam. To account for this we include a loss term that depends
upon the antenna pattern. Rather than have to use specific antenna patterns, we will
define a generic gain that is consistent with reasonable antennas. There are two of these.
One is

sinc 2  2.78  E     2.78.
E
G    
0
elsewhere

(2-3)
and the other is
G    e
2.77  E 
2
.
(2-4)
The first is a sinx/x pattern and the second is a Gaussian pattern. In both cases,  is the
angle off of beam center and  E is the elevation beamwidth. It will be left as an exercise
for you to show that both of these models are reasonably good within the mainbeam
region of the antenna pattern. Of the two, the second is easier to use because it is not a
piecewise function.
With this, we can modify the equation for the main beam clutter as
 CMB    Ad  dG2  B   R  m2
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(2-5)
4
where  B is the elevation pointing angle of the main beam and  R  sin 1  h R  . The
reason for the plus sign on  R in the above equation is that  R is negative (see Figure 21) but we defined it as a positive angle via the equation  R  sin 1  h R  . In most
applications we assume that the main beam is pointed at the target so that  B  e .
We now need to examine the side lobe clutter. The basic approach is the same as
for the main beam clutter but in this case we need to account for the fact that the side lobe
clutter represents ground areas that are illuminated through the transmit antenna side
lobes and whose returns enter through the receive antenna sidelobes. The ground area of
concern is the semicircular annulus excluding the mainbeam region. Relatively speaking,
the ground area illuminated by the main beam is small compared to the ground area
illuminated by the side lobes. Because of this, it is reasonable to include the main beam
area in with the side lobe area. With this, the RCS of the clutter in the side lobe region is
 CSL    SL   d d m 2
2
(2-6)
where SL is the average antenna sidelobe level relative to the main beam peak. A typical
value for SL is -30 dB or 0.001 (see your antenna homework problems). This value could
be as high as -20 dB for “cheap” antennas and as low as -40 to -45 dB for “low side lobe
antennas”. The equation above includes a  SL  term to account for the fact that the
clutter is in the sidelobes of the transmit and receive antenna.
2
To get the total clutter RCS from both the main lobe and the side lobes we make
the assumption that the clutter signals are random processes and that they are
uncorrelated from angle to angle. (We also assume that the clutter signals are
uncorrelated from range cell to range cell.) Since the clutter signals are uncorrelated
random processes, and since RCS is indicative of power, we can get the total clutter RCS
by adding the main beam and side lobe RCSs. Thus


 C   CMB   CSL   G 2  B   R   A    SL  d d m2 .
2
(2-7)
In this equation, the terms d and d are related to range, R, and range resolution, R , by
d  R cos  R and d  R cos  R .
For the final step we need a term to account for the fact that the earth is round and
not flat. We do this by including a pattern propagation factor. This pattern propagation
factor allows the clutter RCS to gracefully decrease as clutter cells move beyond the
radar horizon. David Barton has developed some sophisticated models to account for this
that, I believe, are in his latest book. He also provided a simple approximation that works
very well. Specifically, the defines a loss factor as
L  1   R Rh 
4
(2-8)
where Rh is the radar horizon and is defined as
Rh  2  4 RE 3 h
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(2-9)
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with RE  6,371,000 m being the mean radius of the earth. The 4/3 factor in the above
equation invokes the so-called 4/3 earth model. This model states that, to properly
account for diffraction we need to increase the earth radius to effectively reduce its
curvature. This is discussed in Skolnik’s text book as well as other places. With the
pattern propagation factor, the equation for the clutter RCS becomes
C 


 G 2  B   R   A    SL  RR
2
1   R Rh 
4
m2 .
(2-10)
Figure 2-2 contains a plot of clutter RCS for a typical scenario. In particular, the
radar uses a circular beam with a beamwidth of 1.5 degrees. Thus  A  E  1.5  180 .
The sidelobe level is assumed to be -30 dB. The radar pulse width is 1 µs so that the
range resolution is R  150 m . The phase center of the antenna is at h  5 m . The
three curves of Figure 2-2 correspond to beam pointing angles (  B ) of 0, ½ and 1
beamwidth above horizontal. The assumed value of backscatter coefficient is
  20 dB .
Figure 2-2 – Sample Clutter RCS Plots –   20 dB
The first observation from Figure 2-2 is that the ground clutter RCS is quite large
for low beam elevation angles. This means that for low altitude targets at short ranges
(less than about 30 Km) the clutter will be larger than typical (6 to 10 dBsm) targets.
Thus, unless the radar includes signal processing to reduce the clutter returns, they will
dominate the target returns. At larger elevation angles the problem is less severe because
the ground is no longer being illuminated by the main beam.
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The shape of the curves of Figure 2-2 requires some discussion. Examination of
the equation for clutter RCS indicates that the numerator term increases with increasing
range to the clutter because d depends directly upon R. However, for ranges past the
radar horizon, which is at 9.2 Km for this radar, the pattern propagation factor starts to
predominate and reduces the clutter RCS. This is what causes the curves of Figure 2-2 to
first increase and then decrease.
2.2.2 Clutter Spectral Model
To reduce the clutter return in the radar, it is necessary to have a clutter
characteristic that is different from the target so that that characteristic can be used as the
basis for designing a signal processor. The characteristic that will be used is Doppler
frequency. (Range and angle can’t be used because the target and clutter are at the same
range and angle.) Specifically, the signal processor will exploit the fact that the ground
clutter is at zero Doppler while the target is at some non-zero Doppler (usually).
In practice the clutter signal in the receiver is not concentrated at zero Doppler. In
fact, it has some spread about zero Doppler because of motion of objects (leaves, waves,
grass, etc.) that make up the clutter. For scanning radars (i.e. search radars) there will be
a Doppler spread of the return clutter signal caused by the fact that the radar beam is
moving across the clutter. A standard model for the spectrum of ground clutter is
C f  
k
 fC 2
e
f2
 2 
2
fC
 1  k    f 
(2-11)
where  fC is the Doppler spread of the clutter,   f  is the Dirac delta function and k is
a constant that apportions the clutter power between the spectral line at zero and the
portion that is spread. The quantity  fC is computed from  fC  2 v  where  v is the
velocity spread of the objects that make up the ground clutter. Sample values for  v can
be obtained from Chapter 15 (page 15.9) of Skolnik’s Radar Handbook. To the author’s
knowledge, there is so set method of apportioning the spectrum between the impulse and
Gaussian parts. Most analysts completely ignore the impulse by setting k  1. This will
be the approach we will take. Some guesses at k for different conditions would be
k  0.1 for hard surfaces like sand, concrete, asphalt, ice, and smooth water; k  0.5 for
fields and woods in light winds or for medium rough sea; k  0.9 for rough seas or fields
and woods in high winds.
The first term in the clutter spectrum is somewhat justified in Papoulis1. In the
example cited he shows that if the density function for the velocity distribution of the
clutter is Gaussian (which is a reasonable assumption by the central limit theorem) then
the spectrum of the signal returned from the clutter will also be Gaussian.
Papoulis, Athanasios “Probability Random Variables and Stochastic Processes” Second Ed, McGraw Hill,
Example 10-4 on page 267
1
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As the beam scans by the target its amplitude will vary. If we assume a Gaussian
beam shape, the amplitude variation of the returned clutter voltage will have a Gaussian
shape. Since the Fourier transform of a signal with a Gaussian shape is also a Gaussian
shape the spectrum will be described by a Gaussian function. Skolnik’s Radar Handbook
gives the form of this spectrum as
CS  f  
1
 S 2
e
f2
2 
2
S
(2-12)
where
 S  0.265 f r n , f r  PRF , n  f r ATscan  2  ,  A  Azimuth Beam Width and
Tscan  scan period . To get the total spectrum due to the internal motion of the spectrum
and the scanning, one would convolve C  f  with CS  f  . The justification for this
convolution is discussed below and in Appendix B.
A tacit implication of the terminology used above is that the clutter is a stationary
random process. Also, it will be noted that

 C  f  df 


 C  f  df
S
 1.
(2-13)

This means that the clutter spectrum is normalized to unity power. To get the actual
clutter spectrum one would multiply C  f  or CS  f  by the clutter power. The clutter
power would be computed from the clutter RCS and the radar range equation.
Now that we have a clutter model, we want to develop the equations that
characterize the clutter, noise and target spectra at the input to the signal processor. This
is the topic of the next section.
2.3 SIGNAL ANALYSIS
2.3.1 Background and Definitions
We want to develop equations that allow us to analyze what a radar signal
processor does to signals returned from clutter (or targets). We will eventually work in
the frequency domain, but we start in the time domain.
A simplified block diagram of a radar transmitter and receiver is shown in Figure
2-3. The block diagram contains only the elements essential to our development.
Specifically, it does not contain any of the intermediate frequency (IF) amplifiers and
filters, nor the mixers needed to up- and down-convert the various signals. We have not
lost any generality with this technique because we will use normalized, complex signal
notation. This allows us to ignore all IF processes. Recall the detection discussions from
EE 619.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Figure 2-3 – Transmitter, Receiver and Signal Processor
Complex signal notation has an advantage of being easy to manipulate since the
signals are represented by complex exponentials rather than sines and cosines.
Operations such as filtering, sampling, transforms, etc. are treated the same with complex
signals and real signals. The place where one must take care when using complex signals
is in non-linear operations such as mixing. For example, in the transmit mixer of the

previous figure we used vLO  t  whereas on the receive mixer we used vLO
t  . We knew
we needed to do this based on real signal analyses. Specifically, we performed real
signal analyses and used the results to determine what we needed to do with complex
signals.
As a caution, be very careful when using complex signal analysis with other
nonlinearities such as limiters, saturating amplifiers, squarers, diodes, etc. The rule-ofthumb I use is to perform careful, real, analysis and look for ways to extend it to complex
signals. I find that I must revert to the real, IF signal, go through the non-linearity, and
then reconstruct the complex signal. A key point to remember is that the magnitude of
the complex signal is the magnitude of the IF signal and the phase of the complex signal
is the phase of the IF signal. In equation form, if
vc  t   A  t  e j t 
(2-14)
is a complex signal voltage then the corresponding real, IF voltage is
vIF  t   A  t  cos IF t    t   .
(2-15)
vc  t   I  t   jQ  t   A t  cos  t   jA t  sin  t 
(2-16)
Also, if
then
vIF  t   I  t  cos IF t  Q  t  sin IF t
 A  t  cos   t  cos IF t  A  t  sin   t  sin IF t
© M. C. Budge, Jr., 2012 – merv@thebudges.com
.
(2-17)
9
Let us return to the problem at hand and define the signals of the previous figure.
v p  t  is the pulse train and is generally a complex, base-band signal. This means that its
energy, or power, is generally concentrated around 0 Hz, as opposed to some IF. It
should be noted that signals that have a Doppler frequency are usually considered baseband signals, even though their energy is not truly concentrated around 0 Hz.
The vc  t  in the Equations 2-14 through 2-17 are base-band signals. On the other
hand, the vIF  t  are termed IF signals, or more generally, band-pass signals. They are
termed band-pass signals because their energy is centered around  IF and thus looks like
the response of a band-pass filter.
The typical v p  t  of interest to us is an infinite sequence of rectangular pulses
with a width of  p and a PRI (pulse repetition interval) of T. A graphical representation
of v p  t  (actually v p  t  ) is shown in the Figure 2-4.
Figure 2-4 – Depiction of v p  t 
In equation form, v p  t  is given by
 t  kT 
t 
t 
v p  t    rect 
  rect       t  kT   rect    i  t 
k
  p 
 p  k
 p 
(2-18)
1
rect  x   
0
(2-19)
where
x  12
x  12
  t  is the Dirac delta and  denotes convolution. The notation

denotes a
k
summation over all integers. This implies an infinite number of pulses. In practice,
radars use only a finite number of pulses. The techniques we develop for the case of an
infinite number of pulses will apply to a finite number of pulses provided the number of
pulses in a burst (i.e. a burst of N pulses) satisfies some constraints. We will discuss this
when we consider specific signal processor.
A more general form of v p  t  would be
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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v p  t   p  t   i  t    p  t  kT 
(2-20)
k
where p  t  is a complex signal notation of a complicated waveform such as a phasecoded pulse or an LFM pulse.
The STALO signal, vLO  t  , is of the form
vLO  t   e jct e j t  .
(2-21)
In the above, fc  c 2 is the carrier frequency.   t  is termed the phase noise on the
STALO signal and represents the instability of the oscillator that generates the STALO
signal. As implied by its name,   t  is a random process. It is such that e j t  is widesense stationary (WSS), or at least this is the standard assumption. We will address the
phase noise later. Phase noise is included because it is often the limiting factor on signal
processor performance.
In most radars, vLO  t  also includes an amplitude noise component such that
vLO  t   1  A  t   e jct e j t  . However, A  t  is usually made very small by the radar
designer and is normally considered to have a much smaller influence on signal processor
performance than   t  . For this reason it is almost always ignored in signal processor
analyses. Having said this, it should be noted that modern STALOs are becoming so
stable that the amplitude noise will soon overtake phase noise as the limiting factor in
signal processor performance.
vT  t  is the transmitted signal and is simply
vT  t   v p  t  vLO  t  .
(2-22)
vS  t  is a term we include to account for the fact that the antenna may be
scanning (which is generally taken to mean that the beam is rotating horizontally, as in a
search radar). If we are considering a tracking radar, vS  t   1 . If the antenna is
scanning, the standard form of vS  t  is
vS  t   e  t
2
2
2 TS
(2-23)
where
1   ATSCAN 


 .
2.77   
2

2
TS
(2-24)
TSCAN is the scan period (in sec) and  A is the azimuth beamwidth (in radians).
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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In practice, vs  t  is a periodic function with a period of TSCAN . However, since the time
period of interest, in terms of the signal processor, is small relative to TSCAN , it is assumed
that the radar beam scans by the target only one time. Incidentally, the “time period of
interest, in terms of the signal processor” is termed a CPI, or coherent processing interval.
C  t  is the “clutter signal” and is our means of capturing the power spectrum
properties of the clutter, as we discussed earlier. C  t  is a random process that is usually
assumed to be WSS2. The power spectrum of the clutter is
C  f     Rc       E C  t    C   t 

k
 fC 2
e
 f 2 2 2fC
 1  k    f 
.
(2-25)
This is the form we discussed earlier (see Equation 2-11).
To complete our definitions, vR  t  is the received signal after it goes through the
antenna. vm  t  is the output of the receiver’s mixer and vMF  t  is the matched filter
output. vo  k  is the sampled version of vMF  t  and is the signal that goes to the signal
processor. The matched filter is matched to a single pulse (i.e. it is matched to p  t  ) of
the original pulse train, v p  t  .
2.3.2 Signal Analysis
We start our analysis by noting that the mixer is a multiplication process. Thus,
the signal sent to the antenna is
vT  t   v p  t  vLO  t  .
(2-26)
If the antenna is scanning, its pattern modulates the amplitude of vT  t  . We model this
as a multiplication of vT  t  by vS  t  . Thus, the signal that leaves the antenna is
vantT  t   v p  t  vLO  t  vS  t  .
(2-27)
Recall that we set vS  t   1 if we consider the tracking problem.
2
A note about stationarity: Realistically, none of the random processes we are dealing with are truly WSS.
However, over the CPI we can reasonably assume they are stationary. From random processes theory, we
know that if a process is stationary, in the wide sense, over a CPI then we can reasonably assume that it is
WSS. This stems from the fact that we are only interested in the random process over the CPI.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
12
After the signal leaves the antenna, it propagates a distance of R to the clutter (or
target). We represent this propagation by incorporating a delay, which we denote as
 d 2 , into vantT  t  . We should also include an attenuation of 1 R . However, we are
dealing with normalized signals for now so we can ignore it. We will consider the actual
power in the signal at a later time.
With the above, the signal that arrives at the clutter (or target) is
vCT  t   vantT  t   d 2 
 v p  t   d 2  vLO  t   d 2  vS  t   d 2  .
(2-28)
The clutter “reflects” the signal back to the radar and imposes its temporal, or spectral,
characteristics on the reflected signal. We represent this operation by multiplying vCT  t 
by C  t  , the function that we use to represent the temporal (and spectral) properties of
the clutter. The fact that we represent the operation by multiplication is due to the fact
that the interaction of the signal with the target (clutter) is essentially a modulation
process. We derived this in EE619 when we found that the motion of the target caused a
shift in the frequency of the signal (Doppler shift) and that the amplitude of the return
signal was a function of the target RCS.
The signal reflected by the clutter is
vCR  t   vCT  t  C  t 
 v p  t   d 2  vLO  t   d 2  vS  t   d 2  C  t 
.
(2-30)
.
(2-31)
and the signal back at the receive antenna is
vantR  t   vCR  t   d 2 
 v p  t   d  vLO  t   d  vS  t   d  C  t   d 2 
This signal next picks up the scan modulation and is then heterodyned by the receiver
mixer to produce the matched filter input, vm  t  . In equation form,

vm  t   vantR  t  vS  t  vLO
t 

 v p  t   d  vLO  t   d  vS  t   d  C  t   d 2  vS  t  vLO
t 
.
(2-32)
We now want to study and manipulate this equation. We start by simplifying the
equation and making some approximations. Since the antenna will not move far over the
round trip delay,  d , we can assume that vS  t  doesn’t change much over  d . This
means that vS  t   d   vS  t  . With this we get

vm  t   v p  t   d  vS2 t  C t   d 2 vLO t   d  vLO
t  .
(2-33)
For our next step we want to look at the last two terms. We write the product as
© M. C. Budge, Jr., 2012 – merv@thebudges.com
13

vLO  t   d  vLO
t   e
jc  t  d 
e
j  t  d   jc  t   j  t 
e
e
 e jc d e j t 
(2-34)
where   t     t   d     t  .   t  represents the total (transmit and receive) phase
noise on the radar. A standard assumption is that   t  is small relative to unity so that
e j t   1  j  t     t  . With this vm  t  becomes
vm  t   v p  t   d  vS2 t  C t   d 2  t  .
(2-35)
Note that we dropped the phase term, e  jc d . We were able to do this because it is a
phase term that we can normalize away in future calculations.
We further simplify the vm  t  equation by shifting the time origin by  d . This
yields
v 'm  t   vm  t   d   v p  t  vS2  t   d  C t   d 2  t   d  .
(2-36)
We argued earlier that vS  t  changes slowly relative to  d so that vS2  t   d   vS2  t  .
Also, C  t  and   t  are WSS random processes. This means that their means and
autocorrelations do not depend on time origin. Thus, we can replace C  t   d 2 with
C  t  and   t   d  with   t  and not change their means and autocorrelations (the
autocorrelation is what we eventually use to find the power spectrum of vm  t  ). With
this we get
vm  t   v p  t  vS2  t  C t   t  .
(2-37)
We dropped the prime and reverted to the notation vm  t  for convenience.
The next steps in our derivation is to process vm  t  through the matched filter and
to then sample the matched filter output via the analog-to-digital converter (ADC) (see
Figure 2-3). Before we do this we need to examine vm  t  more closely. If we substitute
for v p  t  from Equation 2-20 into Equation 2-37 we get
vm  t   vS2  t  C  t    t   p  t  kT 
k
  p  t  kT  v  t  C  t    t 
2
S
.
(2-38)
k
We recall that C  t  and   t  are WSS random processes. Because of this, the
product r  t   vS2  t  C  t    t  is also a random process. However, because of the fact
that vS  t  is periodic r  t  is not WSS. As is shown in Appendix B, r  t  is
cyclostationary. As a result, we can use the averaged statistics of r  t  and treat it as a
WSS process in the following development. With this we write
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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vm  t    p  t  kT  r  t 
(2-39)
k
where we treat r  t  as if it was a WSS random process.
We note that, because of the p  t  kT  term, vm  t  is not stationary. This is
something we will need to deal with in the following discussions.
If we represent the impulse response of the matched filter as m  t  , we can write
the output of the matched filter as
vMF  t   m  t   vm  t    m  t    p  t  kT  r  t   .
(2-40)
k
We normally derive m  t  by saying that the matched filter is matched to some
signal q  t  . Recalling matched filter theory, this means that we can write
m  t   q  t  .
(2-41)
In this application, the matched filter is termed a single-pulse matched filter. The
matched filter is often matched to the transmit pulse, p  t  so that
q t   p t  .
(2-42)
However, in some instances, notably when p  t  is an LFM pulse, m  t  includes an
amplitude taper to reduce range sidelobes. In this case q  t  will not exactly equal p  t  .
In the remainder of this derivation we will use the more general form of Equation 2-41.
Substituting Equation 2-41 into Equation 2-40 yields
vMF  t    q  t    p  t  kT  r  t  
(2-43)
k
or

vMF  t     q   t  p   kT  r   d
(2-44)
k 
where we have replaced the convolution notation (  ) by the integral it represents.
Figure 2-5 contains depictions of vm   , q   t  and vMF  t  for the case
where p  t  is an unmodulated pulse and m  t  is matched to p  t  (i.e. q  t   p  t  ).
As expected from matched filter theory, vMF  t  is a series of triangle shaped pulses
whose amplitudes depend upon r  t  .
Since vm  t  is a non-stationary random process so is vMF  t  . This makes vMF  t 
difficult to deal with since we do not have very sophisticated mathematical tools and
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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procedures that would allow us to efficiently analyze non-stationary random processes.
Fortunately, because of the ADC we don’t need to deal directly with vMF  t  . We will
only work with samples of vMF  t  . As stated below, and proved in Appendix A, the
presence of the ADC greatly simplifies our ability to work with its output.
In Figure 2-3, and ensuing discussions, we assume that the ADC produces digital
outputs and that the signal processor is a digital signal processor (DSP). This is an
accurate model for modern radars since almost all of them employ DSPs. However, the
analyses to be discussed also apply, with minor modifications, to older radars that use
analog signal processors.
Figure 2-5 – Depictions of vm   (top plot), q   t  (center plot) and vMF  t 
(bottom plot)
We note that, consistent with most DSP analyses, we are ignoring the amplitude
quantization performed by the ADC. We will only be concerned with its time
quantization, or time sampling. In this sense, we are assuming that the ADC is an
infinite-bit ADC, which is simply a sampler.
We will assume that we are sampling the matched filter once per PRI (every T
seconds) on the peak of the matched filter response. The fact that we assume we are
sampling the matched filter output on its peaks allows us to use the radar-range equation
to compute the SNR and the CNR at the signal processor input. Recall that the
assumption associated with the radar-range equation is that SNR is computed at the peak
of the matched filter output. That is, SNR is the peak signal power at the matched filter
output divided by the average noise power at the matched filter output.
If we find that the radar timing is such that it does not sample the matched filter
output on its peaks, we incorporate a range-gate, or range straddling, loss in the radarrange equation (see EE619 notes). However, we still assume the radar samples the
matched filter output at its peaks.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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With the above, the signal that is sent to the signal processor is
vo  k   vMF  kT 
(2-45)
where we assume (see Figure 2-5) that the matched filter response peaks occur at t  kT .
Since vMF  t  is a continuous-time random process, vo  k  is a discrete-time
random process. As shown in Appendix A, while vMF  t  is not stationary, vo  k  is a
WSS random process. This means that we can write the autocorrelation of vo  k  as
Ro  k1 , k2   E vo  k1  vo  k2   Ro  k1  k2   Ro  k 
(2-46)
where k  k1  k2 . This carries the further implication that we can find the power
spectrum of vo  k  and use this to perform our signal processor analyses in the frequency
domain.
In Appendix A we show that the power spectrum of the ADC output is given by
So  f   K  MF  f  l T  S r  f  l T 
2
(2-47)
l
where
Sr  f   Cs  f   C  f    f 
(2-48)
and Cs  f  , C  f  and   f  are discussed below.
MF  f  is the matched-range, Doppler cut of the cross ambiguity function of
p  t  and q  t  , the signal to which the matched filter, M  t  , is matched. Specifically,
MF  f  

 p t  q t  e

j 2 ft
dt .
(2-49)

For uncoded pulses, phase coded pulses and LFM pulses that don’t incorporate range
sidelobe reduction, MF  f  is of the form (see EE619 notes and homework)
MF  f   sinc  f  p 
(2-50)
where  p is the uncompressed pulse width.
The scanning function, CS  f  , is of the form shown earlier (see Equation 2-12).
That is
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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CS  f  
1
 S 2
e
f2
2 
2
S
(2-51)
 2 
where  s  0.265 
 . If the radar antenna is not scanning, CS  f  reduces to
  ATscan 
CS  f     f  .
(2-52)
The clutter spectrum, C  f  was given earlier (see Equations 2-11 and 2-25) as
C f  
k
 fC 2
e
 f 2 2 2fC
 1  k    f  .
(2-53)
As indicated earlier,   f  represents the phase noise spectrum of the radar. A
reasonable expression for   f  is
  f     f   0
(2-54)
where  0 is termed the phase noise sideband level.  0 is caused by noise in the stable
local oscillator (STALO) circuitry. It has the units of dBc/Hz which means dB relative to
the power in the carrier of the radar, measured in a 1 Hz bandwidth. Typical values for
 0 are -125 to -140 dBc/Hz for radars that use STALOs that employ very narrow band
filters or phase locked loops (such as Klystron-based STALOs); around -110 dBc/Hz for
radars that use frequency multiplied or digitally synthesized STALOs; around -90
dBc/Hz for radars that use Magnetron transmitters. I should caution you that these
numbers are based on my experiences with specific hardware. Other people have
reported better (meaning lower) and worse (meaning higher) values of phase noise for
radars similar to those I have worked with. There is still a fairly large controversy over
phase noise levels. However, the general consensus seems to be that modern, well
designed radars that use good STALOs have phase noise values in the vicinity of -125 to
-135 dBc/Hz. Some advanced radar designs appear to be pushing phase noise to -150 to 160 dBc/Hz.
If we ignore phase noise,   f  reduces to
 f     f .
(2-55)
  f  is the center spectral line or carrier and represents a pure sinusoid
The final term in the equation for So  f  is the constant K. This constant is
chosen such that

Pc  K  MF  f  Sr  f  df .
2
(2-56)

Where Pc is the clutter power calculated from the radar-range equation and the clutter
model that we developed earlier. The integral simply states that the total power in the
© M. C. Budge, Jr., 2012 – merv@thebudges.com
18
matched filter output, for a single received pulse, is the integral of its power spectrum
over all frequencies. If we have correctly specified the terms MF  f  and Sr  f  , we
2
should get that
K  Pc .
(2-57)
A few notes on the forms of the terms of Sr  f  :
1. All of the terms contain either impulses, Gaussian functions or constants. We
chose them that way because of the next three notes and the fact that they
represent the “real world” fairly well.
2. The convolution of two Gaussian functions is a Gaussian function with a variance
equal to the sum of the variances of the two Gaussian functions. That is, if
G1  f  
2
1
e f
 1 2
2 12
and G2  f  
1
G1  f   G2  f  
 12   22 2
e

2
1
e f
 2 2
 f 2 2  12  22

2 22
then
.
3. The convolution of an impulse with any other function is the other function
shifted so that it is centered on the impulse. That is:
G  f    f  f 0   G  f  f 0  .
4. The convolution of a constant with any other function is a constant with a value
equal to the product of the original constant and the area under the function. That

is: G  f   K  K  G  f  df .

As a final note, the spectrum equations we have derived here are general and can
be used to represent any clutter, or target, by changing C f  . As an example, to
represent a target with non-zero Doppler frequency, we would choose C  f     f  f d 
, where f d is the Doppler frequency of the target. When we use the model to represent
targets, we usually ignore phase noise and the spectrum spreading caused by scanning.
2.4 SIGNAL PROCESSOR ANALYSES
2.4.1 Introduction and Background
Now that we have an equation for the clutter (and target) spectrum at the ADC
output, we want to turn our attention to considering how to use it to perform signal
processor analyses. As indicated earlier, we will consider digital signal processors. For
now, we assume we have a signal processor with a z-transfer function of H  z  and
equivalent frequency response of
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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H  f   H  z  z e j 2 fT .
2
(2-57)
We note that we are using the form of frequency response usually used in analyzing
random processes because, by assumption, our clutter (and target) signal at the input to
the signal processor is a random process.
The standard way of performing digital signal processor analyses, in the
frequency domain, is to find So  f  from the sum given earlier, multiply it by H  f  and
integrate the result over (-1/2T, 1/2T] to find the power at the output of the signal
processor. Recall that we use this approach because, for digital signals, the only valid
frequency region is (-1/2T, 1/2T].
We propose a different approach here. Rather than use So  f  over (-1/2T, 1/2T]
we use SMF  f   MF  f  Sr  f  over  ,   . We also use H  f  over  ,   . As
2
before, we multiply these and integrate to find the power; except that this time we
integrate over  ,   . With this approach we are “unfolding” So  f  and H  f  and
then “refolding” them when we find the power. This approach had the advantage of
avoiding the So  f  sum and allowing us to work with only SMF  f  . It has the added
advantage allowing us to analyze staggered waveforms without having to derive a new
set of equations. Staggered waveforms are waveforms whose pulse repetition interval
(PRI) changes from pulse to pulse.
We want to digress to show that the approach we propose is valid, in terms of
computing the power out of the signal processor. We start by noting that the power at the
output of the digital signal processor is
1 2T

Pout  T
H  f  So  f  df .
(2-58)
1 2 T
We substitute for So  f  and bring H  f  inside of the sum to yield
1 2T
1 2T
1
Pout  T  H  f   S MF  f  l T df    H  f  S MF  f  l T df . (2-59)
T l
1 2T
1 2 T l
Note that we canceled the T’s. We next note that H  f  is periodic with a period of 1 T .
This allows us to replace H  f  with H  f  l T  since H  f   H  f  l T  . Doing
this and reversing the order of summation and integration results in:
Pout  
l
1 2T

H  f  l T  S MF  f  l T  df .
(2-60)
1 2T
In each of the integrals of the sum, we make the change of variables   f  l T to get
Pout  
l
 lT 1 2 T

H   S MF   d .
(2-61)
 lT 1 2T
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Finally, we recognize the above as an infinite sum of non-overlapping integrals, which
we can write as a single integral over  ,   . Specifically,

Pout 
 H  f  S  f  df
MF
(2-62)

which is the desired result. Note that we changed the variable of integration from 
back to f.
2.4.2 Moving Target Indicator (MTI)
We are now ready to consider our first signal processor: a moving target indicator,
or MTI. An MTI is a high-pass digital filter that is designed to reject clutter, but not
targets that are moving. A block diagram of a two-pulse MTI is shown in Figure 2-4. It
is termed a two-pulse MTI because it operates on two pulses at a time. It successively
subtracts the returns from two adjacent pulses. For signal processor buffs, it is a firstorder, non-recursive, high-pass, digital filter.
Figure 2-6 – Two-Pulse MTI
A time domain model of the filter is
vSP  k   vo  k   vo  k  1 .
(2-63)
Note that if vo  k   K then vSP  k   vo  k   vo  k  1  K  K  0 . Thus, the MTI
perfectly cancels DC, or zero-frequency signals. If we take z-transform of both sizes we
get
VSP  z   Vo  z   z 1Vo  z 
(2-64)
which can be solved to yield the filter transfer function as
HU  z  
VSP  z 
 1  z 1
Vo  z 
(2-65)
where we use the subscript U to denote the fact that the filter transfer function is unnormalized. We will discuss normalization of the MTI shortly. From Equation 2-65 we
can find the filter frequency response as
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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HU  f   HU  z  z e j 2 fT  1  z 1
2
2
z  e j 2  fT
 1  e  j 2 fT
2
 e j fT  e j fT  e j fT   e j fT  2 j sin  fT  .
2
2
(2-66)
 4sin 2  fT 
A plot of HU  f  is shown in Figure 2-7 for the case where T = 400 s.
Figure 2-7 – Frequency Response of an un-normalized 2-pulse MTI
2.4.2.1 MTI Response Normalization
Before we turn our attention to computing the clutter rejection capabilities of an
MTI we need to normalize the MTI response to something. Without normalization, it is
difficult to quantify the clutter rejection capabilities of the MTI because we have no
reference. The instinct is to say that the clutter rejection is a measure of the clutter power
out of the MTI relative to the clutter power into the MTI. However, we can make this
anything we want by changing the gain of the MTI. To avoid this problem, we normalize
the gain of the MTI so that it has a noise gain of unity. In this way we can easily
compare the CNR at the output of the MTI to the CNR at the input since we have noise
power as a common reference. In a similar fashion, we will be able to characterize the
SNR improvement, or degradation, through the MTI.
To carry out the computations we consider that the MTI is digital and work in the
digital domain. We assume that the noise into the MTI is white and has a power, and
power spectrum, (the power and power spectrum of white noise in digital systems is
equal) of
PN  kT0 F  p ,
(2-67)
which is the effective noise power computed from the radar range equation. The
assumption of white noise is good because the bandwidth of the noise at the matched
filter output is large compared to the sampling frequency.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Earlier we wrote the equation for the MTI time response equation as
vSP  k   vo  k   vo  k  1 .
(2-68)
We want to add a gain to this equation and adjust the gain so that the noise power out of
the MTI is the same as the noise power into the MTI. Thus, we rewrite the MTI equation
as
vSP  k   K MTI vo  k   vo  k  1 
(2-69)
and find the value of K MTI such that the noise power out of the MTI,

PNout  E vSPn  k 
2
,
(2-70)
is equal to the noise power, PN , into the MTI.
If we let vo  k   n  k  in the above equation we get
vSPn  k   K MTI  n  k   n  k  1  .
(2-71)
We note that the noise power into the MTI is

PN  E n  k 
2
.
(2-72)
We can then write the noise power at the output of the MTI as

 E K
  E K n  k   n  k 1 
n  k    E K
n  k  1 
PNout  E vSPn  k 
2
MTI
2
2
2
MTI
2
2
2
MTI
2
2
 E  K MTI
n  k  n  k  1  E K MTI
n  k  n  k  1

2
 2 E K MTI
n k 
2
  2K
2
MTI
.
(2-73)
PN
In the above, the cross expectations on the third line are zero because of the assumption

2
n k 
that n  k  is white. The fact that E K MTI
2
  E K
2
MTI
n  k  1
2
 comes from the
assumption that n  k  is WSS. From the above, it is apparent that for PNout  PN we must
have that KMTI  1
2 . If we apply this to our previous derivation, it is easy to see that
2
H  f   KMTI
HU  f   2sin 2  fT 
(2-74)
rather than the 4sin 2  fT  we derived earlier. A plot of the normalized H  f  is
shown in Figure 2-8.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Figure 2-8 – Normalized Frequency Response of a 2-pulse MTI
Note: we can also find PNout from
 1 2T
 T  PN H  f  df PN  T  H  f  df
 1 2T
1 2 T

1 2T
PNout

  PN .


(2-75)
The proof that
1 2T
T

H  f  df  1
(2-76)
1 2T
for the normalized version of H  f  , is left as an exercise.
2.4.2.2 MTI Clutter Performance
Now that we have normalized our MTI we want to compute its clutter attenuation
and SCR improvement. (Skolnik and other authors also call SCR improvement,
Improvement Factor, a term that we will also use.) We start with clutter attenuation.
Clutter attenuation is defined as the ratio of the CNR at the input to the MTI to the CNR
at the output of the MTI. The CNR at the input to the MTI is the CNR given by the radar
range equation . The CNR at the output of the MTI is the clutter power out of the
(normalized) MTI divided by the noise power at the output of the MTI. However, the
noise power at the output of the MTI is equal to the noise power at the input. Thus, the
clutter attenuation is the ratio of the clutter power at the input to the MTI divided by the
clutter power at the output of the MTI. In equation form
CA 
CNRin
P P
P
 c N  c .
CNRout Pcout PN Pcout
(2-77)
In this equation Pc is the input clutter power and is the same Pc we used earlier.
Similarly, PN is the noise power we used earlier. The definitions of the rest of the
variables should be obvious.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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For the next step, we want to eliminate the explicit dependence on Pc . The clutter
power out of the (normalized) MTI is

Pcout 
 H  f  S  f  df .
(2-78)
MF

But,
SMF  f   Pc MF  f  Sr  f  .
2
(2-79)
Thus

Pcout  Pc
 H  f  MF  f 
2
Sr  f  df  PcG
(2-80)

and
CA 
Pc

Pc
 H  f  MF  f 
2
S r  f  df

Pc
1
 .
PcG G
(2-81)

This means that, to compute clutter attenuation, we only need to compute

G
 H  f  MF  f 
2
Sr  f  df .
(2-82)

We can assume that MF  f   1 for practical radars. With this we get

G
 H  f  S  f  df .
(2-83)
r

For our first computation of clutter attenuation, we will ignore the phase noise and
let   f     f  . Using the forms for CS  f  and C  f  given earlier we can write
Sr  f  

1
 S 2
k
 T 2
e f
2
e f
2
 k

 f 2 2 2fC
 
e
 1  k    f      f 
  C 2

2
2
1
2 T2
 1  k 
e  f 2 S
 s 2
2 S2
(2-84)
where  T2   2fC   S2 .
Lets further simplify our problem by assuming that k  1. With this we get
Sr  f  
1
 T 2
e f
2
2 T2
.
(2-85)
It should be obvious how one would extend this to the case where k  1.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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If we substitute this, and the equation for H  f  , into the equation for G we get

G
2
 2sin  fT 

1
 T 2
e f
2
2 T2
df .
(2-86)
We can simplify this further by observing that, over the region of f where Sr  f  is non
zero, we can approximate sin  fT  by  fT . Over the rest of f, Sr  f  is very small so
that the approximation to sin  fT  is not very important (i.e.,  fT is good enough).
With this, G becomes

G


2  fT 
 T 2
2
e f
2
2 T2
1
2
df  2  T  
  T 2



f 2e f
2
2 T2

df  .

(2-87)
From random variable theory we recognize the term in parentheses as  T2 . This gives
G  2 2T 2 T2 ,
(2-88)
and
2
 PRF 
1
CA 
 2
 .
2 2 2
2 T  T
 2 T 
(2-89)
We next want to look at improvement factor, or SCR improvement. Improvement
in defined as the SCR out of the MTI divided by the SCR into the MTI, averaged over all
Doppler frequencies of interest. The need for averaging comes from the fact that the
signal power out of the MTI will depend upon the target Doppler frequency. Indeed, if
we look at the frequency response plot in Figure 2-8 we note that the gain of the MTI
varies from 0 to 2 w/w. In order to remove this frequency dependency from the final
answer, we average across frequency. It should be noted that some people quote SCR
improvement as that measured at the peak response of the MTI. This is called peak SCR
improvement.
From the frequency response of Equation 2-74, the signal gain through the MTI,
averaged over one cycle, is unity. Also, recall that we normalized the MTI so that its
noise gain was also unity. With this, the SNR gain through the MTI is unity. That is,
SNRout  SNRin . With this and the clutter attenuation results from above we get
I SCR 
SCRout SNRout CNRout CNRin


 CA .
SCRin
SNRin CNRin
CNRout
(2-90)
Thus, because of the normalization we have performed, the SCR improvement is equal to
the clutter attenuation. It should be noted that the peak SCR improvement is 2CA in this
case since the peak gain through the MTI is 2 w/w.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
26
2.4.2.3 An Example
Let’s compute the clutter attenuation for an example radar. This radar has a
carrier frequency of 8 GHz and uses a PRI of 400 µs. We assume that the clutter is
wooded hills in a 20 knot wind. From Table 15.1 on page 15.9 of Skolnik’s Radar
Handbook, the appropriate standard deviation on clutter velocity is  v  0.22 m/s . From
this we can derive the frequency spread of the clutter as
 fC 
2 v


2  0.22
 11.7 Hz .
0.0375
(2-91)
If we assume we are in a tracking environment and ignore scanning for now, we get
 T   fC  11.7 Hz and
2
CA  I scr
 PRF 
 2
  2313 w/w or 33.6 dB .
 2 T 
(2-92)
As an extension, lets assume the same radar parameters but use a scanning radar.
We assume that the radar has a 2 second scan period. We use the beam width associated
with the example of Figure 2-2, i.e.  A  1.5 deg . With this, the standard deviation on
the spectrum due to scanning is
 2 
  31.8 Hz .
  ATscan 
 S  0.265 
(2-93)
The combination of the clutter spread and scanning gives a total spectrum spread of
 T   2fc   s2  33.9 Hz .
(2-94)
The resulting clutter attenuation is
2
CA  I scr
 PRF 
 2
  276 w/w or 24.4 dB .
 2 T 
(2-95)
Let us carry this example further and examine SNR, CNR, and SIR. In addition to
the aforementioned parameters we assume the following for the radar

Peak Power is 100 Kw

Noise Figure is 6 dB

Total Losses for the target and clutter are 13 dB

The height of the antenna phase center is 5 m

The rms antenna side lobes are 30 dB below the peak gain

The clutter backscatter coefficient is -20 dB

The target RCS is 6 dBsm
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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
The ranges of interest are 2 Km to 50 Km
From the beam width and the equations from EE619 we compute the antenna gain as
G  25,000  A E  11,111 w/w  40.5 dB .
Using these parameters, the SNR, CNR, and SIR vs. R at the matched filter output
is as shown in Figure 2-9. It will be noted that the SNR is reasonable but the SIR is much
too low to support detection and track.
Figure 2-9 – SNR, CNR, SIR at Matched Filter Output
Figure 2-10 contains plots similar to those of Figure 2-9 for the two cases (nonscanning and scanning) where and MTI is used. As can be seen, the MTI significantly
reduced the CNR and allowed the SIR to approach the SNR. Thus, in these cases the
radar should be able to do a reasonable job of detecting and tracking the target.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Figure 2-10 – SNR, CNR, SIR at MTI Output
2.4.2.4 Phase Noise
We next want to examine how to handle phase noise in the MTI. Referring to
Equations 2-48 and 2-79, we can see that if we use   f     f   0 we can write
SMF  f  as
S MF  f   Pc MF  f  CS  f   C  f    f  
2
 Pc MF  f  CS  f   C  f    Pc MF  f   0 .
 Sco  f   Sc  f 
2
© M. C. Budge, Jr., 2012 – merv@thebudges.com
2
(2-96)
29
We considered the first term above and will now turn our attention to the second
term,
Sc  f   Pc MF  f   0 .
2
(2-97)
We note that the total power in Sc  f  is

P 

Sc  f  df 

We also note that, if  p


Pc MF  f   0df  Pc 0  p .
2
(2-98)

T , the spectrum of Sc  f  is wide relative the sample
frequency (PRF) and thus that we can assume that Sc  f  is the spectrum of white noise
with a power of P . Since we have normalized the MTI so that it has unity gain for white
noise, the phase noise component of the clutter at the output of the MTI will be P . If we
combine this with the clutter power contributed by the Sco  f  term we now find that the
total clutter power out of the MTI is
Pcout  PcG  Pc  0  p
(2-99)
and the resulting CA is
CA 
Pc
1
.

PcG  Pc 0  p G   0  p
(2-100)
To get a feeling for the impact of phase noise on MTI signal processors, let’s
revisit the previous example and plot clutter attenuation vs. the phase noise level,  0 .
This plot is shown in Figure 2-11. It will be noted that, for the clutter only case, the
phase noise doesn’t start degrading the clutter attenuation until the phase noise level is
above about -105 dBc/Hz. For the case where the scanning effects are included, the
phase noise doesn’t degrade clutter attenuation until the phase noise level exceeds about 95 dBc/Hz. As we increase the order of the MTI processor, we will see that phase noise
starts to contribute more to the overall degradation in clutter attenuation.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Figure 2-11 – Phase Noise Effects on Clutter Attenuation
2.4.2.5 Higher Order MTI Processors
If we want to make the radar of the previous example operate in a noise limited
environment we would need more than the 33.6 dB of clutter attenuation offered by the
2-pulse MTI. This leads us to ask the question of how much the clutter attenuation could
we obtain if we used a 3 or 4 pulse MTI. We address this issue now.
To obtain an n-pulse MTI we cascade n-1, 2-pulse MTIs. Specifically, if the
transfer function of a 2-pulse MTI is H  z  , the transfer function of an n-pulse MTI is
H n  z   K MTI  H  z  
n 1
(2-101)
where the constant K MTI is included to normalize H n  z  so that it provides unity noise
gain.
The specific transfer functions for 2-, 3-, 4-, and 5-pulse MTIs are
H 2  z   K MTI 1  z 1 
H 3  z   K MTI 1  z 1   K MTI 1  2 z 1  z 2 
2
H 4  z   K MTI 1  z 1   K MTI 1  3 z 1  3 z 2  z 3 
3
(2-102)
H 5  z   K MTI 1  z 1   K MTI 1  4 z 1  6 z 2  4 z 3  z 4 
4
It will be noted that the coefficients of the powers of z are binomial coefficients (see the
CRC Handbook) with alternating signs.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Following the method we used for the 2-pulse MTI, we can compute the
normalizing coefficient as
2
K MTI

1
(2-103)
n
a
m 1
2
m
where the am are the MTI coefficients (binomial coefficients) given above. Specific
2
values of K MTI
for the 2-, 3-,4- and 5-pulse MTI are 1/2, 1/6, 1/20 and 1/70, respectively.
2
K MTI
for an n-pulse MTI with binomial coefficients is given by
n
a
m 1
2
m

2
 2  n  1  1!!
 2  n  1 !!
(2-104)
 2  n 1 !! .
 2  n 1 1!!
(2-105)
2 n 1
or
2
K MTI

2
2 n 1
where  2m  1!!  1  3  5 
  2m  1 and  2m!!  2  4 
 2m,  0!!  1 .
The above values are summarized in Table 3.
2
Table 2-3 - K MTI
for Various Size MTIs
Number of pulses in MTI – n
2
K MTI
2
1/2
3
1/6
4
1/20
5
1/70
n
2
© M. C. Budge, Jr., 2012 – merv@thebudges.com
 2  n  1 !!

 2  n  1  1!!
2 n 1
32
If we extend the results of our 2-pulse analysis, we can write the normalized
frequency response of an n-pulse MTI as
2
H n  f   KMTI
 2sin  fT 
2 n 1
.
(2-106)
Figure 2-12 contains plots of the normalized frequency responses of 3- and 4-pulse
MTIs. It will be noted that peaks of the response become narrower and the valleys
become wider as the order of the MTI increases. This means that we should expect
higher CA and SCR improvement as the order of the MTI increases.
Figure 2-12 – Normalized Frequency Response of a 3- and 4- pulse MTI
We can compute the clutter attenuation for the general n-pulse MTI by extending
the work we did for the two pulse MTI.3 We again use the approximation that
sin  fT    fT . With this we get that
3
In this derivation we are using the clutter spectrum with the assumption that k=1 and no phase noise. See
the derivation for the 2-pulse MTI.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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CA 
1
G
(2-107)
where G now becomes

GK
2
MTI


 2 fT  
2 n 1
e f
 T 2
2
2 T2
df .
(2-108)
Evaluation of this integral yields
2
G  K MTI
 2  n  1  1!! 2 T T 
where  2m  1!!  1  3  5 
CA 
2 n 1
(2-109)
  2m  1 . We can write the clutter attenuation as


1
1
1


 2
G K MTI  2  n  1  1!!   2 T  T 2 n 1 

.
 PRF 
1
 2


K MTI  2  n  1  1!!  2 T 
2 n 1
(2-110)
As with the 2-pulse MTI case, we can show that the signal power averaged across
all expected target velocities is equal to one so that the average SNR gain through the
MTI is unity. With this, the SCR improvement, as before, is
I SCR  CA .
(2-111)
Specific values of CA and I SCR for a 3- and 4-pulse MTI are
 PRF 
I SCR 3  CA3  2 

 2 T 
4
(2-112)
and
6
I SCR 4
4  PRF 
 CA4  
 .
3  2 T 
(2-113)
If we revisit the previous example, we find that the CA for the non scanning case
is 64.3 dB for the 3-pulse MTI and 93.1 dB for the 4-pulse MTI. The CA for the
scanning case are 45.8 dB for the 3-pulse MTI and 65.4 dB for the 4-pulse MTI. Since
the clutter attenuations are so high for the non-scanning case, it is likely that phase noise
© M. C. Budge, Jr., 2012 – merv@thebudges.com
34
will become the limiting factor on CA for the 3- and 4-pulse MTIs. This is left as a
homework assignment.
As a closing note, the most common order MTIs in use are 2- and 3- pulse MTIs.
On rare occasions one will encounter a radar that uses a 4-pulse MTI and one almost
never encounters a radar that uses a 5- or higher-pulse MTI. The reason for this is that
higher order MTIs can’t achieve their theoretical potential because of phase noise, timing
jitter, instrumentation errors, round-off errors and the like. Therefore, there is usually no
reason to use higher than a 3-pulse MTI.
2.4.2.6 Staggered PRIs
Examination of the MTI frequency response plots of presented earlier indicate
that the SNR gain through the MTI can vary considerably with target Doppler frequency.
This is quantified in Figure 2-13 below which is a plot of the percent of time that the
MTI gain will be above some level. For example, the MTI gain will be above -5 dB 73%
of the time for the 2-pulse MTI, and 60% and 52% of the time for the 3- and 4- pulse
MTIs. If we say, arbitrarily, that the MTI is blind when the gain drops below -5 dB, we
can say that the 2-pulse MTI is blind 27% of the time and the 3- and 4-pulse MTIs are
blind 40% and 48% of the time. We would like to improve this situation. A method of
doing this is to use staggered PRIs. That is, we use waveforms in which the spacing
between pulses changes on a pulse-to-pulse basis. Through this approach we “break up”
the orderly structure of the MTI frequency response and “fill in” the nulls. We also
reduce the peaks in the frequency response. The net effect is to provide an MTI
frequency response that doesn’t have deep nulls and large peaks but, rather, a somewhat
constant level. The response still has the null at zero frequency and thus still provides
clutter rejection.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
35
Figure 2-13 – Percent of Time MTI Gain Above Specified Levels
To illustrate how to work with staggered waveforms we consider a 3-pulse MTI
and a waveform that alternates between two PRIs of T1 and T2 . A sketch of the
waveform is shown in Figure 2-14. This type of waveform is termed a two-position
stagger because it uses two PRIs. An n-position stagger would use n PRIs, some of
which could be the same.
Figure 2-14 – Two-Position Stagger Waveform
To determine the frequency response of an MTI with a staggered waveform we
work in the continuous time domain and imply sampling by using impulse functions.
Thus, the impulse response of a 3-pulse MTI with binomial coefficients and the
waveform above is:
h  t   KMTI   t   2  t  T1    t  T1  T2   .
(2-114)
If we adjust the time origin so that it is centered on the middle pulse we get
h  t   KMTI   t  T1   2  t     t  T2   .
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(2-115)
36
This will make some of the math to come a little easier.
To find H  f  for this filter we find the Fourier transform of h  t  and take its
magnitude squared. That is,
H  f     h  t       h  t       h  t   
2

(2-116)
or
2
e j 2 fT1  2  e  j 2 fT2  e  j 2 fT1  2  e j 2 fT2 
H  f   K MTI
. (2-117)
2
6  4 cos  2 fT1   4 cos  2 fT2   2 cos  2 f T1  T2   
 K MTI
2
K MTI
 1 6 as for the regular, 3-pulse MTI. Figure 2-15 contains a plot of H  f  for a 3-
pulse and PRIs of 385 and 415 s. The operating frequency of the radar is 8 GHz, which
was used to convert frequency (f in the above equation) to range-rate as shown on the
plot. The plot also contains the response of the MTI for the unstaggered waveform. It
will be noted that the use of the stagger fills-in the nulls that are present in the
unstaggered case.
Figure 2-15 – 3-pulse MTI Response With and Without Stagger
The response with the staggered waveform still has a considerable variation in
MTI gain as a function of range-rate. This is due to the fact that we only used a 2position stagger. According to your Skolnik (Page 15.36 in the Radar Handbook) the use
of a 4-position stagger would provide a better response. However, the responses he
shows are not much better than the one in Figure 2-15.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
37
The use of a 2-position stagger with a 3-pulse MTI is a sort of “matched”
condition. In other words, the complete characteristics and effects of the stagger are
captured by looking at three pulses. Had we used a 4-position stagger we would use only
three pulses at a time and would be able to capture only two PRIs at a time. This means
that the frequency response of the MTI actually varies with time as different sets of three
pulses are processed through the MTI. You will look at this phenomenon in a future
homework. To get around this time variation of MTI response we usually determine the
response to the different sets of PRIs and then average the results. The average is done
on H  f  . Thus, if we had a 4-position stagger with PRIs of T1 , T2 , T3 and T4 we
would find: H1  f  using three pulses with PRIs of T1 and T2 , H 2  f  using three
pulses with PRIs of T2 and T3 , H3  f  using three pulses with PRIs of T3 and T4 and
H 4  f  using three pulses with PRIs of T4 and T1 . We would then form the averaged
response as
H  f    H1  f   H 2  f   H 3  f   H 4  f   4 .
(2-118)
To determine the clutter attenuation of an MTI with a staggered waveform we use
the same formulas as for the unstaggered case. To find the SNR gain through the MTI
we find the average signal gain from the MTI response and use this as the SNR gain. We
can do this because we have still normalized the MTI so that it provides unity noise gain.
We often find the average MTI gain via the “eyeball” method; we estimate it from the
plot. A better method would be to numerically average the gain (in w/w) across the
range-rates of interest. The MTI gain indicated via the “eyeball” method for the response
below is about 0 dB. The calculated gain is -0.08 dB.
2.4.3 Pulsed-Doppler Processors
2.4.3.1 Introduction
We now want to turn our attention to pulsed-Doppler signal processors. The
exact origin of the phrase “pulsed-Doppler” is not clear. It probably derives from the fact
that early pulsed-Doppler radars did CW processing using pulsed waveforms.
Specifically, classical CW radars work primarily in the frequency (and angle) domain
whereas pulsed radars work primarily in the time (and angle) domain. It is assumed that
the phrase pulsed-Doppler was coined when designers started using pulsed radars that
worked primarily in the frequency, or Doppler, domain. Early pulsed-Doppler radars
used a 50% duty cycle pulsed waveform and had virtually no range resolution capability,
only Doppler resolution. The use of a pulsed waveform was motivated by the desire to
use only one antenna and to avoid isolation problems caused by CW operation. Modern
pulsed-Doppler radars are actually high PRF pulsed radars with duty cycles in the 10%
range. They are used for both range and Doppler measurement. Most pulsed-Doppler
radars are ambiguous in range and unambiguous in Doppler. However, because of
waveform constraints, some are ambiguous in both range and Doppler. For purposes of
© M. C. Budge, Jr., 2012 – merv@thebudges.com
38
our signal processor analyses we will consider pulsed-Doppler waveforms that are
unambiguous in Doppler.
The reason for using pulsed-Doppler radars is not clear. One of the claims is that
they provide better clutter rejection capabilities than pulsed radars. However, they need
to do so since they must contend with more clutter than do pulsed radars. It is believed
that designers may be forced to pulsed-Doppler radars when the radar must operate at
long ranges. However, this is not clear. An argument is that long-range radars must use
low PRFs for range unambiguous operation. Since clutter attenuation in MTI processors
decreases as the PRF decreases such radars might not be able to achieve suitable
performance against short range targets in clutter. However, with current computer
capabilities and transmitter flexibility, one could use higher PRF waveforms for short
ranges and lower PRFs at longer ranges. This way it would be possible to obtain good
MTI performance against short range targets. The MTI performance against long range
targets would not be good. However, the clutter will be past the radar horizon at long
ranges and good MTI performance may not be necessary.
Another possible reason for using pulsed-Doppler radars is that they provide the
capability of measuring target Doppler frequency, and thus have the ability to
discriminate on the basis of Doppler frequency. The former could be helpful in an ECM
environment because it provides a cross-check on the range-rate measured by the target
tracker. This, in turn, could help in attempting to counter range-gate deception jamming.
Pulsed-Doppler radars could be helpful in rejecting weather clutter and chaff
because of their ability to provide good Doppler measurement capability. Also, because
of the potential of using narrow bandwidth Doppler filters, the radar could be less
susceptible to noise jamming.
2.4.3.2 Pulsed-Doppler Clutter
The ground clutter environment in pulsed-Doppler radars is generally more severe
than in pulsed radars that are unambiguous in range. This has to do with the fact that, in
pulsed-Doppler radars, the signal returned from long-range targets must compete with
clutter at short ranges. Figure 2-16 is an attempt to illustrate this. In this figure, the solid
triangle is a target return from the first (leftmost) pulse in the string of pulses. The
dashed triangles are returns from the same target but different pulses. The solid, curved
line through the solid triangle represents the clutter from the pulse immediately preceding
the triangle. (The other, dashed, curved lines are clutter returns from other pulses.) The
significance of what signal comes from which pulse has to do with range attenuation.
4
Since the target is at a range of Rtgt it will have a range attenuation of Rtgt
. The clutter in
3
the target range cell is at a range of Rclut and will undergo a range attenuation of Rclut
(recall that clutter attenuation varies as R 3 ). Now, since Rtgt Rclut the target will
undergo much more attenuation than the clutter. The result of this is that the SCR at the
input to the signal processor in pulsed-Doppler radars is much lower than for the same
scenario in pulsed radars.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
39
Figure 2-16 – Target and Clutter Returns in a Pulsed Doppler Radar
As an illustration of the difference in SCR ratios in pulsed and pulsed-Doppler
radars, Figure 2-17 contains plots of CNR and SCR for a radar similar to the one we
considered in the previous MTI example. These CNRs, and SCRs are termed singlepulse CNRs and SCRs.
The bottom two curves correspond to the case where the radar uses a 50 KHz PRF
pulsed-Doppler waveform (with 1-s pulses) and the top two curves correspond to the
case where the radar uses the 2.5 KHz PRF of the previous examples. The pulsedDoppler curves were obtained by folding the clutter power calculated for a single pulse.
That is, if Pc1  R  is the clutter power due to a single pulse, the clutter power due to a
string of high PRF pulses is
Pcpd  R  

 P  R  kcT 2  .
k 
c1
(2-119)
In the above equation, the sum can usually be limited to a small number of terms since
Pc1  R  drops off rapidly past the radar horizon. Also, in practice Pc1  R   0 for
R  2  c p 2  . When generating Pcpd  R  , one only considers ranges of
c
c

R    kT   p  ,   k  1 T   p   , k  0 .
2
2

(2-120)
This accounts for the fact that the receiver is shut off during the transmit pulse and for
one pulse width before the transmit pulse. This is what gives rise to the blank regions in
the CNR and SCR plots of Figure 2-17.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
40
Figure 2-17 – CNR and SCR for a Pulsed and Pulsed-Doppler Radar
In Figure 2-17 it will be noted that the CNR steadily decreases with target range
for the low PRF case. However, the CNR stays high for the pulsed-Doppler case.
Similarly, the SCR for the pulsed case rises and eventually goes above 0 dB at ranges
greater than about 23 Km. On the other hand, the SCR for the pulsed Doppler case
steadily decreases with increasing target range. The result of all of this is that the pulsedDoppler signal processor must provide considerably more clutter rejection capability than
the MTI processors we previously considered.
2.4.3.3 Signal Processor Configuration
The actual implementation of a particular, pulsed-Doppler signal processor will
depend upon the specific radar design and whether the signal processor is in the search
receiver or the track receiver. We will discuss some sample configurations later. For
purposes of analyzing what the signal processor does to the signal, noise and clutter we
can use the somewhat generic signal processor chain shown in Figure 2-18.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Figure 2-18 – Pulsed Doppler Signal Processor
For our purposes, the signal processor starts with the single-pulse matched filter
followed by the ADC. In practical systems, the ADC includes an anti-aliasing filter to
limit the bandwidth of the signals that are sampled by the ADC. Since we did not include
the anti-aliasing filter in our original development of Section 2.3 we will not include it
here. It turns out that this will not affect our analysis results. As a reminder, the ADC
samples the matched filter output once per PRI, every T , on the peak of the matched
filter response.
The high-pass filter following the ADC is used to reduce the clutter power that is
located near zero Doppler. In addition to reducing clutter power it also serves to reduce
the dynamic range requirements on the band-pass filter following the high-pass filter.
The high-pass filter is almost always included in ground based radars because of these
dynamic range considerations. It is almost never included in airborne radars because one
cannot guarantee that the clutter will be a zero Doppler.
Sometimes the high-pass filter is implemented before the ADC to limit the
dynamic range of the signal into the ADC. In the past it was thought that the dynamic
range of the ADC needed to be greater than the SCR at the ADC input. However, recent
analyses indicated that this is not the case. In any event, it turns out that the analyses
presented herein do not depend upon whether the high-pass filter is before or after the
ADC since we will not consider ADC quantization or dynamic range in these analyses.
(ADC quantization and dynamic range will be discussed later.)
The final device in the signal processing chain is the band-pass filter. This filter
usually has a small bandwidth (200 to 1000 Hz) and is centered on the target Doppler
frequency.
As a note, the implementation of pulsed-Doppler signal processors has evolved
over the years from all analog to all, or almost all, digital. The evolution has generally
been driven by the speed, availability and cost of ADCs and digital signal processing
components. Older radars (pre 1980’s or so) used all analog signal processors. Radars
designed between about 1980 and a few years ago used a mix of digital and analog
components. Pulsed-Doppler radars being designed today are almost exclusively digital.
Some go to the digital domain at the matched filter output, as in Figure 2-18. Others go
to the digital domain at the IF amplifier output and implement the matched filter in the
digital domain. Future plans (hopes) are to use phased array radars with solid-state
transmit-receive (TR) modules, digitize the signals at the output of each TR module and
do the beam forming and signal processing in the digital signal processor. This concept
goes by the name of space-time signal processing and has been touted for many years as
having the potential of offering huge (but somewhat vaguely stated) potential in all
© M. C. Budge, Jr., 2012 – merv@thebudges.com
42
aspects of interference (clutter, jamming, etc.) rejection. Hardware technology is not
quite at the state where it can support space-time signal processing.
2.4.3.4 Analysis Techniques
The analysis of digital, pulsed-Doppler signal processors is performed using
techniques very similar to those used in analyzing MTI processors. Specifically, one
multiplies the spectrum of the ADC output, So  f  , by the G  f  ’s of the signal
processor and integrates the resulting spectrum, using Equation 2-62, to find the signal,
noise and clutter power at the output of the signal processor. These operations are
usually performed on the computer because of the difficulty of analytically evaluating the
integral of the (rather complicated) output spectrum.
2.4.3.4.1 Signal
For the case of the target, we assume that the signal is a sinusoid at some Doppler
frequency of f d so that
Srs  f   Ps  f  f d 
(2-121)
where Ps is the signal power computed from the radar-range equation. With this, the
power spectrum at the output of the matched filter is
SMFs  f   Ps MF  f d    f  f d  .
2
(2-122)
From Equation 2-62, with
H  f   GH  f  GB  f 
(2-123)
the signal power at the output of the signal processor is
Psout  GS Ps .
(2-124)
where
GS  MF  f d  GH  f d  GB  f d  .
2
(2-125)
In most applications, the main lobe of the matched-range Doppler cut of the ambiguity is
wide relative to target Doppler so that MF  f d   1 . Also, the target Doppler is
2
frequency is normally in the pass band of the HPF and the BPF is centered very close to
the target Doppler frequency. This means that GH  f d   1 and GB  f d   1 . Combining
these results in the observation that GS  1 . In practice we account for the fact that the
various terms of Equation 2-125 are not exactly unity by including a loss term in the
radar range equation. The most likely term to be less than unity is GB  f d  since it is not
easy to perfectly match the center frequency of the BPF to the target Doppler. The
resulting loss term for this is the Doppler straddling loss we discussed in EE619.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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2.4.3.4.2 Noise
For noise we have
Srn  f   kT0 F
(2-126)
and
SMFn  f   kT0 F MF  f  .
2
(2-127)
In these equations kT0 F is the receiver noise power spectral density from the radar-range
equation. The definition of Srn  f  implies that the receiver noise, at the input to the
matched filter, is white. The noise power out of the signal processor is given by

PNout  kT0 F  MF  f  GH  f  GB  f  df .
2
(2-128)

From earlier, we know that the noise power into the signal processor (out of the ADC) is
PN  kT0 F  p
(2-129)
so that we can write PNout as
PNout  PN p


MF  f  GH  f  GB  f  df  GN PN
2
(2-130)

which results in
GN   p


MF  f  GH  f  GB  f  df .
2
(2-131)

2.4.3.4.3 Clutter
For the clutter we use the clutter model of Equation 2-53 with k  1 . We will
assume that the radar is not scanning so that we do not need to include a scanning
spectrum. We use the phase noise spectrum of Equation 2-54. With this, we get
Src  f   PcC  f     f    0   PcC  f   Pc  0 ,
(2-132)
SMFc  f   PcC  f  MF  f   Pc 0 MF  f  .
(2-133)
and
2
2
The clutter power at the output of the signal processor is thus
© M. C. Budge, Jr., 2012 – merv@thebudges.com
44

Pcout  Pc

MF  f  C  f  GH  f  GB  f  df
2

.

(2-134)
 Pc  0  MF  f  GH  f  GB  f  df
2

We can write Equation 2-134 as
Pcout  Pc  GC   0GN  p 
(2-135)
with

GC 

MF  f  C  f  GH  f  GB  f  df
2
(2-136)

and GN as defined in Equation 2-131.
In these equations, Pc is the folded clutter power, Pcpd  R  given by Equation 2119.
All of the integrals above are over the limits of -∞ to ∞. Clearly, it is not possible
to numerically integrate over these limits. However, given that most of the area of the
MF  f  function is in the central lobe, it is usually sufficient to integrate over limits of
2
about  3  p ,3  p  or  5  p ,5  p  .
2.4.3.4.4 SNR and SCR Improvement.
If we combine Equations 2-124 and 2-130 we see that the SNR at the signal
processor output is
SNRout 
Psout
G P
G
 S S  S SNR  GSNR SNR
PNout GN PN GN
(2-137)
where GSNR is the SNR gain of the signal processor and SNR is the single-pulse SNR
computed from the radar-range equation.
If we combine Equations 2-124 and 2-135 we see that the SCR at the signal
processor output is
SCRout 
Psout
GS Ps

 I SCR SCR
Pcout Pc  GC   0GN  p 
(2-138)
where
SCR 
SNR
CNR
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(2-139)
45
with SNR the single-pulse SNR from the radar-range equation and CNR is the “singlepulse” CNR at the output of the matched filter, also from the radar-range equation (see
Section 3.4.3.2). Alternately, one could directly use the “single-pulse” SCR (again, see
Section 3.4.3.2).
In Equation 2-138, I SCR is the SCR improvement offered by the signal processor.
From Equation 2-138 it is given by
I SCR 
GS
1

GC   0GN  p 1 GSCR  1 GSNR    0  p 
(2-140)
where GSNR is as define above and GSCR represents the ability of the signal processor to
reject the central line clutter (not the clutter that enters through the phase noise). GSCR is
given as
GSCR 
GS
GC
(2-141)
where GC is computed from Equation 2-136. As we will see in the following example,
GSCR is usually very large.
2.4.3.5 Example
As an example of how to perform a pulsed-Doppler signal processor analysis we
will consider the example discussed in connection with Figure 2-17. The parameters of
the radar are similar to those we used in the MTI example with the exception that the
peak power is 10 KW instead of 100 KW and the PRF is 50 KHz rather than 2.5 KHz.
We also reduced the antenna height to 3 m and parked the beam at 0.75º in elevation (1/2
beamwidth). The pertinent parameters are:

Peak Power= 10 Kw

Operating Frequency = 8 GHz

Noise Figure = 4 dB

PRF = 50 KHz (PRI = 20 µs)

Pulse Width = 1 µs

Total Losses for the target and clutter = 6 dB

Height of the antenna phase center = 3 m

rms antenna side lobes are 30 dB below the peak gain

Clutter backscatter coefficient = -20 dB

Target RCS = 6 dBsm
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
Ranges of interest are 2 Km to 50 Km
Figure 2-19 contains plots of SNR, CNR, SCR and SIR at the matched filter output for this
example. It will be noted that, at long ranges, the SNR is lower than desired. Also the,
the SCR is very low. As result of this, the overall SIR is also very low. This means that
the signal processor needs to offer a fairly reasonable SNR increase and a considerable
increase in SCR.
Figure 2-19 – SNR, CNR, SCR and SIR at Matched Filter Output
We want our radar to be able to track targets with range-rates down to about 20
m/s. This means that we must choose the cutoff frequency of the high-pass filter (HPF)
to be
© M. C. Budge, Jr., 2012 – merv@thebudges.com
47
f ch 
2 Rmin


2  20
 1067 Hz .
0.0375
(2-142)
We will let f ch  1000 Hz . We will use a 5th-order, Butterworth HPF. Thus we have
  ch 
GH  f   1 

10
10
1     ch 
1     ch 
(2-143)
  tan  fT 
(2-144)
ch  tan  fchT  .
(2-145)
10
1
with
and
We note that this is an approximation to the frequency response of a 5th-order, digital
Butterworth filter. It has the flat pass and stop band shapes associated with Butterworth
filters. Its response is a periodic function of 1 T , which is expected since it is a digital
filter.
We typically want to choose the bandwidth of the band-pass filter (BPF) to be as
small as possible since this sets the ultimate SNR and SCR improvement of the radar.
The lower limit is generally set by track requirements, target decorrelation time, number
of BPFs required to cover a PRF, desired Doppler resolution and dwell time (in phased
array radars). The first two set ultimate lower limits of about 10 Hz. The second two set
more practical limits of 200 to 1000 Hz. We will choose a bandwidth of 1000 Hz in this
example. This choice gives a Doppler resolution of about 20 m/s and means we will need
50 filters to cover our PRF of 50 KHz. If we are using a phased array radar, it would
impose a dwell time minimum of about 4 ms (4/(Doppler filter bandwidth)). We assume
that the BPF is a 3rd order Butterworth filter. We further assume that it is centered on the
target Doppler frequency of 8000 Hz. (which corresponds to a target range-rate of 150
m/s). With this, the frequency response of the BPF is
GB  f  
1
  
1   fT 
  cb 2 
6
(2-146)
where
 fT  tan   f  fT  T 
(2-147)
cb  tan  fcbT  .
(2-148)
and
In these equations,
fT  8000 Hz
(2-149)
is the frequency to which the BPF is tuned and
© M. C. Budge, Jr., 2012 – merv@thebudges.com
48
fcb  1000 Hz
(2-150)
is the bandwidth of the BPF. As with GH  f  , GB  f  is an approximation of the
frequency response of a 6th-order Butterworth, band pass filter in that it has flat pass and
stop bands.4
We will use the same clutter spectrum model we used in the MTI analyses.
Specifically, we let
C f  
1
 fc 2
e
 f 2 2 2fc
.
(2-151)
with
 fc  11.7 Hz .
(2-152)
Note that this assumes that k  1 . We will treat the phase noise level,  0 , as a parameter
for now.
Since our pulse is a 1-s, unmodulated pulse we get
MF  f   sinc2  f  p  .
2
(2-153)
With the above, we get that the signal power out of the signal processor is
Psout  Ps MF  f d  GH  f d  GB  f d   Ps ,
2
(2-154)
which implies that the signal gain through the signal processor, GS , (see Equation 2-125)
is unity. Some thought will confirm the veracity of this statement in that the signal
processor is tuned to the signal.
The noise gain of the signal processor is can be found from Equation 2-131 as,
GN   p


MF  f  GH  f  GB  f  df  0.021 .
2
(2-155)

Combining this with GS yields a SNR gain of
GSNR 
GS
1

 47.6 or 16.8 dB .
GN 0.021
(2-156)
4
The BPF of Equation 2-146 is a 3rd order BPF with complex coefficients. It is centered at +8000 Hz.
Obviously, this is not realizable with standard hardware. It’s equivalent filter, with real coefficients, is a 6th
order BPF. This filter has responses centered at +8000 Hz and -8000 Hz. Thus the doubling of the filter
order.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
49
The clutter gain through the signal processor can be found from Equation 2-136
as

GC 

MF  f  C  f  GH  f  GB  f  df  1.63 1024 .
2
(2-157)

This results in a central-line SCR improvement of
GSCR 
GS
1

 6.14 1024 or 238 dB
GC 1.63 1024
(2-158)
which, as indicated earlier, is very large. From Equation 2-140 we can compute the SCR
improvement as
I SCR 
1
1

24
1 GSCR  1 GSNR    0  p  1.63 10  0.021  0  p 
(2-159)
If we assume a phase-noise level of -120 dBc/Hz we get
I SCR 
1
12
10
106
 4.76 107 or 76.8 dB
1.63 1024  0.021

1
1.63 10  2.1108
24
.
(2-160)
Clearly, the predominant term in I SCR is the phase noise term. This is typical of pulsedDoppler radars and signal processors and is why designers strive to make the STALOs in
these radars very stable and quiet.
Figure 2-20 contains plots similar to those of Figure 2-19 at the output of the
signal processor. It will be noted that the SNR improvement provides for good values of
SNR at ranges out to about 50 Km. SCR and SIR at these ranges is not as good as hoped.
There are ranges where SIR drops below 13 to 15 dB (the “standard” rule-of-thumb that
we use for “reasonable” detection and track performance) for certain ranges. Because of
this, we would like to either obtain more SCR improvement from the radar or reduce the
clutter input to the radar.
From Equations 2-159 and 2-160 we see that we could increase I SCR by
decreasing the phase noise. If we were able to decrease the phase noise to -135 dBc/Hz,
I SCR would increase to about 92 dB and the performance of our radar, at long ranges,
would become good.
Another means of improving the SCR at the signal processor output would be to
decrease the clutter power at the input to the radar. There are several ways that the clutter
power into the radar could be reduced. One might be to use a smaller beamwidth.
Another might be to place the radar on a tower and reduce the antenna sidelobes.
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50
3.4.3.6 Variations
The analysis presented above assumed that the signal processor was digital. In
some applications the signal processor is analog or incorporates analog components. A
block diagram of an all-analog signal processor is shown in Figure 2-21. It will be noted
that this block diagram is similar to the block diagram of Figure 2-18 except that the
ADC is replaced by a sampler and a band limit filter. The combination of the sampler
and the band limit filter is often referred to as a sample-and-hold device or as a range
gate.
We can approach the analysis of an analog signal processor via several paths.
One would be to fold the spectrum at the matched filter output using Equation 2-47. One
would then use this folded spectrum with the frequency responses of the analog filters
(band limit, high pass, band pass) filters of Figure 2-21 to find the appropriate spectra at
the output of the signal processor. To find the powers out of the signal processor one
would simply integrate these spectra.
Figure 2-20 – SNR, CNR, SCR and SIR at Signal Processor Output
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51
An alternate approach, would be to use the theory associated with discrete-time
systems and samplers to “fold” the frequency responses of the analog filters. One would
then use the techniques of Section 3.4.3.4 to perform the analyses.
Figure 2-21 – Analog Signal Processor
In still another variation, some pulsed-Doppler signal processors include a
combination of analog and digital signal processing. An example might be where the
band limit and high pass filters are analog filters and the band pass filter is digital.
Again, the analysis could be approached by folding the spectrum at the output of the
matched filter or using discrete time system theory to fold the frequency responses of the
analog filters (the digital filter response would already be folded). If the spectrum is
folded, it would need to be refolded after passing through the analog filter. The refolded
spectrum would then be multiplied by the (folded) spectrum of the band pass filter.
However, in this case, the power out of the signal processor would need to be computed
using the Equation 2-58 since both the signal and filter would be represented in the
discrete-time domain.
In some cases the band pass filter is implemented with a FFT. In fact, if one uses
an N-point FFT one has a bank of N band pass filters. To perform the analyses using the
techniques of this section, one would compute the frequency response of the appropriate
FFT tap and use this in the analysis. This frequency response can be found by
recognizing that the FFT provides samples of the Fourier transform of the weights
applied to the input taps of the FFT. Thus, one could compute the Fourier transform of
the input weights, shift it to the frequency of the FFT tap of interest and square the result
to get its frequency response, as we have defined frequency response in this section.
Keep in mind that the frequency response of the FFT tap is periodic with a period of 1 T .
When computing the Fourier transform one would need to be sure that the resulting
response was periodic.
An assumption of the analyses presented in this section is that the waveform
consists of an infinite number of pulses. In phased array radars this will not be the case;
the waveforms will be finite duration. This introduces complications in the design of the
signal processor, and possibly its analysis. The reason for this has to do with transients in
the filters of the signal processor. Generally, the designer is careful to set the filter
bandwidths, and use time gating, such that the transients have settled fairly well and one
© M. C. Budge, Jr., 2012 – merv@thebudges.com
52
can treat the analysis as if the waveform consisted of an infinite number of pulses. That
is, one can use continuous time analysis techniques.
2.4.5 ADC Effects
In both the MTI and pulsed-Doppler signal processors we assumed that the ADC
had an infinite number of bits and an infinite dynamic range. In essence, we analyzed the
processors as if the ADC was simply a sampler. We now want to address the impact that
a realistic ADC will have on performance. In particular, we want to account for the
number of bits in the ADC, the quantization and internal ADC noise and ADC dynamic
range.
Until very recently, the rule of thumb used to characterize the impact of the ADC
on SCR improvement was to say that the ADC imposed an absolute limit on performance
of
I SCR  6  Nbit 1 dB
(2-161)
where Nbit is the number of bits in the ADC. In the past, this has resulted in design
constraints that were unnecessary.
Engineers at Dynetics, Inc. have shown that, while I SCR is influenced by the
number of bits in the ADC, the hard limit on SCR improvement given by Equation 2-161
is not valid. A more representative equation for I SCR that includes the effects GSCR ,
phase noise and the ADC is
I SCR 
GSNR PADC
GSCR GSNR PADC
.
 GSCR PADC  0  p  GSCR PNADC  p Fs 
(2-162)
where several of the terms are familiar from Section 2.4.3.4. PADC is the level of the
clutter at the ADC input relative to the ADC saturation level. It is normally taken to be 6 dB to assure that the Swerling nature of clutter doesn’t occasionally cause ADC
saturation. The presence of PADC implies that there is some type of gain control that
monitors the clutter level into the ADC and adjusts the gain to keep the clutter power 6
dB below ADC saturation.
The term PNADC accounts for the quantization noise of the ADC, the ADC internal
noise and any additional dither noise that is added to the ADC input to assure linear
operation of the ADC. This sentence raises an important issue concerning the ADC. In
order for the ADC to preserve the relative sized of signal, clutter and noise after
quantization, there must always be sufficient noise at the ADC input. Generally, only
quantization noise is not sufficient since it is too small. In modern ADCs with a large
number of bits (>10) the internal noise is usually sufficient. If it is not, dither noise must
be added to the input to the ADC. A reasonable value of PNADC is
6 Nbit 1log2 q  10
PNADC  10
(2-163)
where Nbit is the number of bits in the ADC q is the number of quantization levels of the
ADC noise. Typical values of q are 1 to 3 (note: if we say noise toggles the lsb of the
© M. C. Budge, Jr., 2012 – merv@thebudges.com
53
ADC, q=1; if it toggles the lower two bits, q=3.) If the ADC specifications give an
ADC SNR then PNADC , in dB, is the negative of that SNR. If the negative of the SNR is
not equal to or greater that the PNADC of Equation 2-163 with q  0 , then dither noise of
½ to 1 quanta ( q  1 2 to 1 ) should be added to the ADC input. However, adding too
much dither noise to will degrade I SCR .
Fs is the ADC sample rate. It is normally taken to be the modulation bandwidth
of the waveform if range gating is performed after the ADC, as in a full digital processor.
For an unmodulated pulse, Fs  1  p . If the radar uses IF sampling with digital down
conversion, Fs can be much larger than the modulation bandwidth.
The I SCR equation above is written in term used for the pulsed-Doppler signal
processor. It is also applicable to the MTI processor with GSNR  1 and GSCR  CA .
A derivation of Equation 2-163 is given in Appendix C.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
54
APPENDIX A – DERIVATION OF EQUATION 2-47
In this appendix we want to go through the derivation of Equation 2-47 of the
notes. We start with the output of the matched filter as

vMF  t     q   t  p   lT  r   d
(A-1)

l
where p  t  represents one of the transmit pulses and q  t  is the pulse to which the
single-pulse matched filter is matched. We assume that r  t  is a WSS random process
with an autocorrelation of Rr   and a power spectral density of Sr  f  . It will be
recalled that r  t  is not WSS for the case of clutter. However, it is shown in Appendix
B that it is wide-sense cyclostationary. Because of this, it can be represented by its
averaged autocorrelation, which is then used in this development. In this instance, r  t 
represents some hypothetical WSS random process whose autocorrelation is equal to the
averaged autocorrelation of the actual random process.
As indicated in the notes, we will assume that the ADC samples the output of the
matched filter, vMF  t  , once per PRI, T , at the peak of the matched filter response. We
also, without loss of generality, assume that the matched filter peaks occur at t  kT .
With this we can write the output of the ADC as

vo  k   vMF  t  t kT    q   kT  p   lT  r   d .
l
(A-2)

If we assume that p  t  and q  t  are of the form
t 
p  t   p '  t  rect  
 p 
(A-3)
1
rect  x   
0
(A-4)
where
x 1 2
x 1 2
and that  p  T 2 , then all of the terms of the summation of Equation A-2 are zero except
for the case where l  k . With this Equation A-2 reduces to
vo  k  

 q   kT  p   kT  r   d .

(A-5)

To find the power spectrum of vo  k  we must first show that vo  k  is WSS. To
that end we form
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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Ro  k1 , k2   E vo  k1  vo  k2 


 
 E    q   k1T  p   k1T  r   d 
  

 

  q  t  k2T  p  t  k2T  r  t  dt 
 

 




.
(A-6)
  q   k T  p   k T  q  t  k T  p  t  k T 



1
 
1
2
2
E r   r   t  d dt
From previous discussions we note that
E r   r   t   Rr   t 
(A-7)
and
Ro  k1 , k2  
 
  q   k T  p   k T  q t  k T  p t  k T R   t  d dt .


1
1
2
2
(A-8)
r
 
By making use of

Rr   
 S  f e
j 2 f 
r
df
(A-9)

we can write
Ro  k1 , k2  
 

 


j 2 f  t 


df d dt . (A-10)
 q   k1T  p   k1T  q t  k2T  p t  k2T   Sr  f  e
We now make the change of variables,     t , d  d to yield
Ro  k1 , k2  


 q t  k T  p t  k T   S  f 

2
2

r


.
 q   t  k T  p   t  k T  e

1
1
j 2 f 
(A-11)
d dfdt

We next make the change of variables     t  k1T d   d and get
Ro  k1 , k2  



 q t  k T  p t  k T   S  f   q    p    e

2

2
r


j 2 f   t  k1T 
d  dfdt .
(A-12)

Rearranging yields
© M. C. Budge, Jr., 2012 – merv@thebudges.com
56
Ro  k1 , k2  

 S  f e
j 2 fk1T
r


  


 j 2 ft
dt   q    p    e j 2 f  d   df
  q  t  k2T  p  t  k2T  e
 
 

.
(A-13)
For the next change of variable we let   t  k2T d   dt to yield
Ro  k1 , k2  

 S  f e
j 2 f  k1  k2 T
r


  


 j 2 f 
d    q    p    e j 2 f  d   df .(A-14)
  q   p   e
 
 

The first thing to note about Equation A-14 is that the right side is a function of
k1  k2 , and constitutes the proof that vo  k  is WSS. The next thing to note is that the
two integrals in the brackets are conjugates of each other. Finally, from ambiguity
function theory, we recognize that

 p q  e

j 2 f 
d    pq  0, f 
(A-15)

where  pq  0, f  is the matched-range, Doppler cut of the cross ambiguity function of
p  t  and q  t  . In the remainder we will use the notation  pq  0, f   MF  f  . With all
of the statements in this paragraph we can write
Ro  k  


MF  f  Sr  f  e j 2 fkT df .
2
(A-16)

We next want to find the power spectrum of vo  k  . We could do this by taking
the discrete-time Fourier transform of Ro  k  . However, the math associated with this
will probably be quite involved. We will take a more indirect approach.
Let v  t  be a WSS random process with an autocorrelation of R   and a power
spectrum of S  f  . Further assume that we can sample v  t  to get vo  k  . That is
vo  k   v  t  t  kT .
(A-17)
vo  k  is the same as the random process defined by Equation A-2.
From random processes theory we can write
Ro  k   R     kT .
(A-18)
Further, from the theory of discrete-time signals and their associated Fourier transforms,
if S  f  is the power spectrum of v  t  we can write the power spectrum of vo  k  as
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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So  f  
1
S  f  m T .
T m
(A-19)
From this same theory we can write
1 2T
Ro  k   T

So  f  e j 2 kfT df
(A-20)
1 2T
If we substitute Equation A-19 into A-20 we get
Ro  k   T
1 2T
1
S  f  m T e j 2 kfT df

T
m
1 2T

(A-21)
or
Ro  k   
1 2T
 S  f  m T e
j 2 kfT
df .
(A-22)
m 1 2T
We now make the change of variables x  f  m T to get
Ro  k   
1 2T  m T
S  x  e j 2 k  x  m T T dx

m 1 2T  m T
  e j 2 km
m

1 2T  m T

S  x  e j 2 kxT dx .
(A-23)
1 2T  m T
1 2T  m T

S  x  e j 2 kxT dx
m 1 2T  m T
Where we made use of the fact that e j 2 km  1 .
We recognize that the last term is an infinite summation of integrals over nonoverlapping intervals, and that the total of the non-overlapping intervals cover the range
of x   ,  . With this we can write
Ro  k  

 S  f e
j 2 kfT
df
(A-24)

where we have let x  f .
If we compare Equation A-24 to Equation A-16 we have
S  f   MF  f  Sr  f  .
2
(A-25)
With this and Equation A-19 we arrive at the desired result that the power spectrum of
the signal at the ADC output is
So  f  
2
1
MF  f  m T  S r  f  m T  .

T m
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(A-26)
58
APPENDIX B – PROOF THAT r  t  IS WS CYCLOSTATIONARY
In this appendix we show that the process
r  t   cs2  t  c  t    t 
(B-1)
is wide-sense cyclostationary (WSCS). We first note that since c  t  is zero-mean so is
r  t  . For ease of notation, we replace the symbol cs2  t  by the symbol cs 2  t  . To show
that r  t  is WSCS we must show that
E r  t  kTs    r   t  kTs   E r  t    r   t 
(B-2)
for some Ts . That is, we must show that the autocorrelation of r  t  is a periodic
function of t .
We recall that c  t  and   t  are WSS random processes. Thus the product
c  t    t  is also WSS. The function cs 2  t  is a deterministic function and is periodic
with a period of Ts where Ts is the scan period of the antenna. If we form
Rr  t ,   E r  t    r   t 
(B-3)
we get
Rr  t ,   cs 2  t    cs 2  t  E c  t    c  t  E   t       t 
(B-4)
 cs 2  t    cs 2  t  Rc   R  
where we have made use of the fact that c  t  and   t  are independent and WSS. In a
similar fashion we can write
Rr  t  kTs ,   cs 2  t  kTs    cs 2 t  kTs  Rc   R   .
(B-5)
But, since cs 2  t  is periodic with a period of Ts we have
cs 2  t  kTs   cs 2  t 
(B-6)
and thus that
Rr  t  kTs ,   Rr  t , 
(B-7)
which says that r  t  is WSCS.
From the theory of WSCS random processes we can use the averaged
autocorrelation of r  t  to characterize the average behavior of r  t  . Specifically, in
place of Rr  t ,  we use
© M. C. Budge, Jr., 2012 – merv@thebudges.com
59
Rr   
1
Ts
 R  t ,  dt
(B-8)
r
Ts
where the integral notation means to perform the integration over one period of Rr  t ,  .
As a note, what Equation B-8 is saying is that, on average, a system will respond to r  t 
in the same manner that it will respond to a WSS process that has the autocorrelation
Rr   . In the notes we have dispensed with the overbar and used the notation Rr   .
Also, in Appendix A and the notes we assumed that r  t  was a WSS random process.
What we really mean is that the r  t  used in those places is a WSS random process
whose autocorrelation, Rr   is equal to the averaged autocorrelation, Rr   of the
actual, WSCS random process.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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APPENDIX C – DERIVATION OF EQUATION 2-163
Figure C-1 contains a block diagram of the receiver/ADC/signal processor
configuration we will use in this analysis. In this configuration it is assumed that the
ADC is sampling the IF signal and that the signal processor contains the matched filter,
digital down conversion to baseband and any other signal processor components such as
the clutter rejection filters and coherent integrators. This model will also apply to the
case where the ADC is sampling a baseband signal and the matched filter and/or down
conversion is performed in the block labeled IF section.
Figure C-1 – Block Diagram Used in the Derivation
In Figure C-1, PS and PC are the (normalized) signal and clutter powers at the
output of the IF section and are the powers computed from the radar range equation (see
EE619 notes and the clutter model discussions of Section 2.2 and the examples). PS and
PC are single-pulse powers.
PkT is the (normalized) receiver noise power at the output of the IF section and is
the noise power term that is also encountered in the radar range equation (see EE619
notes). It is given by
PkT  kT0 Fn B
(C-1)
where k is Boltzman’s constant, T0  290 K is the reference temperature, Fn is the
system noise figure and B  1  p is the effective noise bandwidth of the radar.  p is the
uncompressed pulsewidth.
P is the component of clutter that is manifest through the phase noise sidebands
of the transmitter and local oscillator (LO) (see Section 2.4). If we assume a flat phase
noise spectrum with a power spectral density of 0 w/Hz relative to the transmit power
then P is given by
P  PC0 B
(C-1)
where PC and B are as defined earlier.
The ADC normalizer is a gain, Gnorm , that is used to adjust the total signal-plusclutter-plus-noise power so as to avoid ADC saturation. Generally, Gnorm is selected so
© M. C. Budge, Jr., 2012 – merv@thebudges.com
61
that the power level, PADC , at the ADC input is 3 to 6 dB below ADC saturation. This
should provide enough of a margin to avoid saturation in the presence of fluctuations in
the signal, clutter and noise. In this analysis, it is assumed, without loss of generality,
that the ADC saturates at a normalized power of 1 watt. With this, the normalizer gain is
given by
Gnorm 
PADC
PS  PC  PkT  P
(C-3)
where PADC is between 106 10 and 103 10 .
The standard performance metrics used in radar and signal processor analyses are
signal-to-noise ratio (SNR), signal-to-clutter ratio (SCR) and signal-to-interference ratio
(SIR). These will be the performance metrics used here.
The SCR at the ADC input is defined as
SCRADC 
Gnorm PS PS

 SCR
Gnorm PC PC
(C-4)
where SCR it the single-pulse SCR derived from the radar-range-equation (see Section
2.4). It has been shown that if sufficient noise is present in the ADC (and the ADC is not
allowed to saturate), SCR is preserved through the ADC. This means that the SCR at the
signal processor input is also the SCR of Equation C-4. That is,
SCRSPin  SCR 
PS
.
PC
(C-5)
It will be assumed that the signal processor provides an SCR gain of GSCR as
discussed in Section 2.4.3.4. To repeat the statements of Section 2.4.3.4, GSCR is not to
be confused with signal-to-clutter improvement ( I SCR ). The latter includes factors such
as phase noise and, as will be shown, ADC noise. GSCR is a measure of how well the
signal processor can reject clutter when the transmit and LO signals are perfect sinusoids
(no phase noise) and the ADC has an infinite number of bits and infinite dynamic range.
The noise into the signal processor consists of the receiver noise ( PkT ), the phase
noise ( P ) and the ADC noise, which is represented by PNADC in Figure C-1. The phase
noise term is included as part of the noise, and not the clutter, because the signal
processor acts on it the same as it does on receiver noise.
The ADC noise consists of noise that is generated by the ADC circuitry,
quantization noise and, in some instances, intentionally injected dither noise. The
presence of ADC noise is critical to the preservation of SCR through the ADC. Without
it, the SCR at the output of the ADC would drop to zero (w/w, not dB) once the SCR at
the input to the ADC dropped below the dynamic range of the ADC.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
62
A standard way of representing ADC noise is in terms of ADC quanta, q. One
can think of q as the count of the ADC. Thus, if the least significant bit (lsb) of the ADC
is 1 and the rest are 0, q  1 . If the two lsb’s are 1 and the rest are zero, q  3 (binary 11
= decimal 3). For an N-bit ADC with q quanta of ADC noise and an ADC saturation
level of 1 watt,
 6 N 1 log 2 q  10
PNADC  10 
(C-6)
where N 1 is used to reflect the fact that the most significant bit (msb) of an N-bit ADC
is normally taken to be a sign bit.
With the above, the total noise power spectral density at the input to the signal
processor can be written as
NT  Gnorm kT0 Fn  Gnorm PC0  PNADC Fs .
(C-7)
The division by Fs in the last term of Equation C-7 reflects the assumption that the ADC
noise power is uniformly distributed over a bandwidth equal to the ADC sample rate of
Fs . For the reasonable assumption that Fs is greater than the effective noise bandwidth,
B, the effective total noise power into the signal processor is
PT  Gnorm kT0 Fn B  Gnorm PC0 B  PNADC B Fs .
(C-8)
With Equation C-8, we can write the SNR at the input to the signal processor as
SNRSPin 
Gnorm PS
.
Gnorm kT0 Fn B  Gnorm PC0 B  PNADC B Fs
(C-9)
Given that the SNR gain of the signal processor is GSNR , the SNR at the signal
processor output is
SNRout  GSNR SNRSPin 
GSNR Gnorm PS
.
Gnorm kT0 Fn B  Gnorm PC0 B  PNADC B Fs
(C-10)
The SIR at the signal processor output can be determined from
1
1
1


SIRout SNRout SCRout
(C-11)
with Equation C-10 and SCRout  GSCR PS PC .
In this form, Equation C-11 is not very easy to use. However, if we make the
reasonable assumption that the signal into the normalizer is clutter limited, we can write
Gnorm 
PADC
PC
(C-12)
and
SNRout 
GSNR PADC PS PC
PADC kT0 Fn B PC  PADC0 B  PNADC B Fs
© M. C. Budge, Jr., 2012 – merv@thebudges.com
(C-13)
63
which can be substituted into Equation C-11 to yield
P kT BF P  PADC0 B  PNADC B Fs
PC
1
 ADC 0 n C

.
SIRout
GSNR PADC PS PC
GSNR PS
(C-14)
After some manipulation, Equation C-14 can be written as
1
1
1


SIRout GSNR SNR I SCR SCR
(C-15)
where the SCR improvement, I SCR is given by
I SCR 
GSNR PADC
GSCR GSNR PADC
 GSCR PADC0 B  GSCR PNADC B Fs
(C-16)
which is Equation 2-163.
© M. C. Budge, Jr., 2012 – merv@thebudges.com
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