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 pulse-to-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, 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 and 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 . The elevation angle from the radar phase center to the target is

.(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. 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 . In most cases 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, 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 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 to . 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 ( in Figure 2-1). Chapter 12 of Skolnik’s Radar Handbook contains further discussions of . For our purposes, we will use . That is, we use a backscatter coefficient that doesn’t depend upon . Our justification for this is that, in general is relatively constant (and small). If we were considering an airborne radar, we would need to revert to a model wherein varies with . 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 : , and . 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 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

(2-2)

where the various parameters are shown on Figure 2-1. An assumption in this equation is that is small so that can be taken as a straight line that is perpendicular to .

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

(2-3)

and the other is

.(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 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

(2-5)

where is the elevation pointing angle of the main beam and . The reason for the plus sign on in the above equation is that is negative (see Figure 2-1) but we defined it as a positive angle via the equation . In most applications we assume that the main beam is pointed at the target so that .

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

(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 term to account for the fact that the clutter is in the sidelobes of the transmit and receive antenna.

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

.(2-7)

In this equation, the terms d and are related to range, R, and range resolution, , by and .

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

(2-8)

where is the radar horizon and is defined as

(2-9)

with 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

.(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 . The sidelobe level is assumed to be -30 dB. The radar pulse width is 1 µs so that the range resolution is . The phase center of the antenna is at . The three curves of Figure 2-2 correspond to beam pointing angles () of 0, ½ and 1 beamwidth above horizontal. The assumed value of backscatter coefficient is .

Figure 2-2 – Sample Clutter RCS Plots –

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.

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

(2-11)

where is the Doppler spread of the clutter, 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 is computed from where is the velocity spread of the objects that make up the ground clutter. Sample values for 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 . This will be the approach we will take. Some guesses at k for different conditions would be for hard surfaces like sand, concrete, asphalt, ice, and smooth water; for fields and woods in light winds or for medium rough sea; for rough seas or fields and woods in high winds.

The first term in the clutter spectrum is somewhat justified in Papoulis[footnoteRef:1]. 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. [1: Papoulis, Athanasios “Probability Random Variables and Stochastic Processes” Second Ed, McGraw Hill, Example 10-4 on page 267]

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

(2-12)

where

, , , and . To get the total spectrum due to the internal motion of the spectrum and the scanning, one would convolve with . 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

.(2-13)

This means that the clutter spectrum is normalized to unity power. To get the actual clutter spectrum one would multiply or 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.

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 whereas on the receive mixer we used . 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-of-thumb 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

(2-14)

is a complex signal voltage then the corresponding real, IF voltage is

.(2-15)

Also, if

(2-16)

then

.(2-17)

Let us return to the problem at hand and define the signals of the previous figure. 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 base-band signals, even though their energy is not truly concentrated around 0 Hz.

The in the Equations 2-14 through 2-17 are base-band signals. On the other hand, the are termed IF signals, or more generally, band-pass signals. They are termed band-pass signals because their energy is centered around and thus looks like the response of a band-pass filter.

The typical of interest to us is an infinite sequence of rectangular pulses with a width of and a PRI (pulse repetition interval) of T. A graphical representation of (actually ) is shown in the Figure 2-4.

Figure 2-4 – Depiction of

In equation form, is given by

(2-18)

where

(2-19)

is the Dirac delta and denotes convolution. The notation denotes a 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 would be

(2-20)

where is a complex signal notation of a complicated waveform such as a phase-coded pulse or an LFM pulse.

The STALO signal, , is of the form

.(2-21)

In the above, is the carrier frequency. 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, is a random process. It is such that is wide-sense 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, also includes an amplitude noise component such that . However, is usually made very small by the radar designer and is normally considered to have a much smaller influence on signal processor performance than . 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.

is the transmitted signal and is simply

.(2-22)

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, . If the antenna is scanning, the standard form of is

(2-23)

where

.(2-24)

is the scan period (in sec) and is the azimuth beamwidth (in radians).

In practice, is a periodic function with a period of . However, since the time period of interest, in terms of the signal processor, is small relative to , 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.

is the “clutter signal” and is our means of capturing the power spectrum properties of the clutter, as we discussed earlier. is a random process that is usually assumed to be WSS[footnoteRef:2]. The power spectrum of the clutter is [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.]

.(2-25)

This is the form we discussed earlier (see Equation 2-11).

To complete our definitions, is the received signal after it goes through the antenna. is the output of the receiver’s mixer and is the matched filter output. is the sampled version of 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 ) of the original pulse train, .

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

.(2-26)

If the antenna is scanning, its pattern modulates the amplitude of . We model this as a multiplication of by . Thus, the signal that leaves the antenna is

.(2-27)

Recall that we set if we consider the tracking problem.

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 , into . We should also include an attenuation of . 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

.(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 by , 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

.(2-30)

and the signal back at the receive antenna is

.(2-31)

This signal next picks up the scan modulation and is then heterodyned by the receiver mixer to produce the matched filter input, . In equation form,

.(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, , we can assume that doesn’t change much over . This means that . With this we get

.(2-33)

For our next step we want to look at the last two terms. We write the product as

(2-34)

where . represents the total (transmit and receive) phase noise on the radar. A standard assumption is that is small relative to unity so that . With this becomes

.(2-35)

Note that we dropped the phase term, . We were able to do this because it is a phase term that we can normalize away in future calculations.

We further simplify the equation by shifting the time origin by . This yields

.(2-36)

We argued earlier that changes slowly relative to so that . Also, and are WSS random processes. This means that their means and autocorrelations do not depend on time origin. Thus, we can replace with and with and not change their means and autocorrelations (the autocorrelation is what we eventually use to find the power spectrum of ). With this we get

.(2-37)

We dropped the prime and reverted to the notation for convenience.

The next steps in our derivation is to process 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 more closely. If we substitute for from Equation 2-20 into Equation 2-37 we get

.(2-38)

We recall that and are WSS random processes. Because of this, the product is also a random process. However, because of the fact that is periodic is not WSS. As is shown in Appendix B, is cyclostationary. As a result, we can use the averaged statistics of and treat it as a WSS process in the following development. With this we write

(2-39)

where we treat as if it was a WSS random process.

We note that, because of the term, 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 , we can write the output of the matched filter as

.(2-40)

We normally derive by saying that the matched filter is matched to some signal . Recalling matched filter theory, this means that we can write

.(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, so that

.(2-42)

However, in some instances, notably when is an LFM pulse, includes an amplitude taper to reduce range sidelobes. In this case will not exactly equal . 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

(2-43)

or

(2-44)

where we have replaced the convolution notation () by the integral it represents.

Figure 2-5 contains depictions of , and for the case where is an unmodulated pulse and is matched to (i.e. ). As expected from matched filter theory, is a series of triangle shaped pulses whose amplitudes depend upon .

Since is a non-stationary random process so is . This makes difficult to deal with since we do not have very sophisticated mathematical tools and 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 . We will only work with samples of . 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 (top plot), (center plot) and (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 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 radar-range equation (see EE619 notes). However, we still assume the radar samples the matched filter output at its peaks.

With the above, the signal that is sent to the signal processor is

(2-45)

where we assume (see Figure 2-5) that the matched filter response peaks occur at .

Since is a continuous-time random process, is a discrete-time random process. As shown in Appendix A, while is not stationary, is a WSS random process. This means that we can write the autocorrelation of as

(2-46)

where . This carries the further implication that we can find the power spectrum of 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

(2-47)

where

(2-48)

and , and are discussed below.

is the matched-range, Doppler cut of the cross ambiguity function of and , the signal to which the matched filter, , is matched. Specifically,

.(2-49)

For uncoded pulses, phase coded pulses and LFM pulses that don’t incorporate range sidelobe reduction, is of the form (see EE619 notes and homework)

(2-50)

where is the uncompressed pulse width.

The scanning function, , is of the form shown earlier (see Equation 2-12). That is

(2-51)

where . If the radar antenna is not scanning, reduces to

.(2-52)

The clutter spectrum, was given earlier (see Equations 2-11 and 2-25) as

.(2-53)

As indicated earlier, represents the phase noise spectrum of the radar. A reasonable expression for is

(2-54)

where is termed the phase noise sideband level. 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 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, reduces to

.(2-55)

is the center spectral line or carrier and represents a pure sinusoid

The final term in the equation for is the constant K. This constant is chosen such that

.(2-56)

Where 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 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 and , we should get that

.(2-57)

A few notes on the forms of the terms of :

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

and 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: .

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: .

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 . As an example, to represent a target with non-zero Doppler frequency, we would choose , where 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 and equivalent frequency response of

.(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 from the sum given earlier, multiply it by 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 over (-1/2T, 1/2T] we use over . We also use over . As before, we multiply these and integrate to find the power; except that this time we integrate over . With this approach we are “unfolding” and and then “refolding” them when we find the power. This approach had the advantage of avoiding the sum and allowing us to work with only . 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

.(2-58)

We substitute for and bring inside of the sum to yield

.(2-59)

Note that we canceled the T’s. We next note that is periodic with a period of . This allows us to replace with since . Doing this and reversing the order of summation and integration results in:

.(2-60)

In each of the integrals of the sum, we make the change of variables to get

.(2-61)

Finally, we recognize the above as an infinite sum of non-overlapping integrals, which we can write as a single integral over . Specifically,

(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 first-order, non-recursive, high-pass, digital filter.

Figure 2-6 – Two-Pulse MTI

A time domain model of the filter is

.(2-63)

Note that if then . Thus, the MTI perfectly cancels DC, or zero-frequency signals. If we take z-transform of both sizes we get

(2-64)

which can be solved to yield the filter transfer function as

(2-65)

where we use the subscript U to denote the fact that the filter transfer function is un-normalized. We will discuss normalization of the MTI shortly. From Equation 2-65 we can find the filter frequency response as

.(2-66)

A plot of 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

,(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.

Earlier we wrote the equation for the MTI time response equation as

.(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

(2-69)

and find the value of such that the noise power out of the MTI,

, (2-70)

is equal to the noise power, , into the MTI.

If we let in the above equation we get

.(2-71)

We note that the noise power into the MTI is

.(2-72)

We can then write the noise power at the output of the MTI as

.(2-73)

In the above, the cross expectations on the third line are zero because of the assumption that is white. The fact that comes from the assumption that is WSS. From the above, it is apparent that for we must have that . If we apply this to our previous derivation, it is easy to see that

(2-74)

rather than the we derived earlier. A plot of the normalized is shown in Figure 2-8.

Figure 2-8 – Normalized Frequency Response of a 2-pulse MTI

Note: we can also find from

.(2-75)

The proof that

(2-76)

for the normalized version of , 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

.(2-77)

In this equation is the input clutter power and is the same we used earlier. Similarly, is the noise power we used earlier. The definitions of the rest of the variables should be obvious.

For the next step, we want to eliminate the explicit dependence on . The clutter power out of the (normalized) MTI is

.(2-78)

But,

.(2-79)

Thus

(2-80)

and

.(2-81)

This means that, to compute clutter attenuation, we only need to compute

.(2-82)

We can assume that for practical radars. With this we get

.(2-83)

For our first computation of clutter attenuation, we will ignore the phase noise and let . Using the forms for and given earlier we can write

(2-84)

where .

Lets further simplify our problem by assuming that . With this we get

.(2-85)

It should be obvious how one would extend this to the case where .

If we substitute this, and the equation for , into the equation for G we get

.(2-86)

We can simplify this further by observing that, over the region of f where is non zero, we can approximate by . Over the rest of f, is very small so that the approximation to is not very important (i.e., is good enough). With this, G becomes

.(2-87)

From random variable theory we recognize the term in parentheses as . This gives

,(2-88)

and

.(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, . With this and the clutter attenuation results from above we get

.(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 in this case since the peak gain through the MTI is 2 w/w.

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 . From this we can derive the frequency spread of the clutter as

.(2-91)

If we assume we are in a tracking environment and ignore scanning for now, we get and

.(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. . With this, the standard deviation on the spectrum due to scanning is

.(2-93)

The combination of the clutter spread and scanning gives a total spectrum spread of

.(2-94)

The resulting clutter attenuation is

.(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

· 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 .

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 (non-scanning 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.

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 we can write as

.(2-96)

We considered the first term above and will now turn our attention to the second term,

.(2-97)

We note that the total power in is

.(2-98)

We also note that, if , the spectrum of is wide relative the sample frequency (PRF) and thus that we can assume that is the spectrum of white noise with a power of . 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 . If we combine this with the clutter power contributed by the term we now find that the total clutter power out of the MTI is

(2-99)

and the resulting is

.(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, . 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.

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 , the transfer function of an n-pulse MTI is

(2-101)

where the constant is included to normalize so that it provides unity noise gain.

The specific transfer functions for 2-, 3-, 4-, and 5-pulse MTIs are

(2-102)

It will be noted that the coefficients of the powers of z are binomial coefficients (see the CRC Handbook) with alternating signs.

Following the method we used for the 2-pulse MTI, we can compute the normalizing coefficient as

(2-103)

where the are the MTI coefficients (binomial coefficients) given above. Specific values of for the 2-, 3-,4- and 5-pulse MTI are 1/2, 1/6, 1/20 and 1/70, respectively. for an n-pulse MTI with binomial coefficients is given by

(2-104)

or

.(2-105)

where and .

The above values are summarized in Table 3.

Table 2-3 - for Various Size MTIs

Number of pulses in MTI – n

2

1/2

3

1/6

4

1/20

5

1/70

n

If we extend the results of our 2-pulse analysis, we can write the normalized frequency response of an n-pulse MTI as

.(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.[footnoteRef:3] We again use the approximation that . 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.]

(2-107)

where G now becomes

.(2-108)

Evaluation of this integral yields

(2-109)

where . We can write the clutter attenuation as

.(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

.(2-111)

Specific values of and for a 3- and 4-pulse MTI are

(2-112)

and

.(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 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.

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 and . 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:

.(2-114)

If we adjust the time origin so that it is centered on the middle pulse we get

.(2-115)

This will make some of the math to come a little easier.

To find for this filter we find the Fourier transform of and take its magnitude squared. That is,

(2-116)

or

.(2-117)

as for the regular, 3-pulse MTI. Figure 2-15 contains a plot of 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 2-position 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.

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 . Thus, if we had a 4-position stagger with PRIs of , , and we would find: using three pulses with PRIs of and , using three pulses with PRIs of and , using three pulses with PRIs of and and using three pulses with PRIs of and . We would then form the averaged response as

.(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 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. Since the target is at a range of it will have a range attenuation of . The clutter in the target range cell is at a range of and will undergo a range attenuation of (recall that clutter attenuation varies as ). Now, since 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.

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 single-pulse 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 pulsed-Doppler curves were obtained by folding the clutter power calculated for a single pulse. That is, if is the clutter power due to a single pulse, the clutter power due to a string of high PRF pulses is

.(2-119)

In the above equation, the sum can usually be limited to a small number of terms since drops off rapidly past the radar horizon. Also, in practice for . When generating , one only considers ranges of

. (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.

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 pulsed-Doppler 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.

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 , 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 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, , by the ’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 so that

(2-121)

where is the signal power computed from the radar-range equation. With this, the power spectrum at the output of the matched filter is

.(2-122)

From Equation 2-62, with

(2-123)

the signal power at the output of the signal processor is

.(2-124)

where

.(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 . Also, the target Doppler is 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 and . Combining these results in the observation that . 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 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.

2.4.3.4.2 Noise

For noise we have

(2-126)

and

.(2-127)

In these equations is the receiver noise power spectral density from the radar-range equation. The definition of 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

.(2-128)

From earlier, we know that the noise power into the signal processor (out of the ADC) is

(2-129)

so that we can write as

(2-130)

which results in

.(2-131)

2.4.3.4.3 Clutter

For the clutter we use the clutter model of Equation 2-53 with . 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

,(2-132)

and

.(2-133)

The clutter power at the output of the signal processor is thus

.(2-134)

We can write Equation 2-134 as

(2-135)

with

(2-136)

and as defined in Equation 2-131.

In these equations, is the folded clutter power, given by Equation 2-119.

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 function is in the central lobe, it is usually sufficient to integrate over limits of about or .

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

(2-137)

where 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

(2-138)

where

(2-139)

with SNR the single-pulse SNR from the radar-range equation and CNR is the “single-pulse” 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, is the SCR improvement offered by the signal processor. From Equation 2-138 it is given by

(2-140)

where is as define above and represents the ability of the signal processor to reject the central line clutter (not the clutter that enters through the phase noise). is given as

(2-141)

where is computed from Equation 2-136. As we will see in the following example, 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

· 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

.(2-142)

We will let . We will use a 5th-order, Butterworth HPF. Thus we have

(2-143)

with

(2-144)

and

.(2-145)

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 , 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

(2-146)

where

(2-147)

and

.(2-148)

In these equations,

(2-149)

is the frequency to which the BPF is tuned and

(2-150)

is the bandwidth of the BPF. As with , is an approximation of the frequency response of a 6th-order Butterworth, band pass filter in that it has flat pass and stop bands.[footnoteRef:4] [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.]

We will use the same clutter spectrum model we used in the MTI analyses. Specifically, we let

.(2-151)

with

.(2-152)

Note that this assumes that . We will treat the phase noise level, , as a parameter for now.

Since our pulse is a 1-s, unmodulated pulse we get

.(2-153)

With the above, we get that the signal power out of the signal processor is

,(2-154)

which implies that the signal gain through the signal processor, , (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,

.(2-155)

Combining this with yields a SNR gain of

.(2-156)

The clutter gain through the signal processor can be found from Equation 2-136 as

.(2-157)

This results in a central-line SCR improvement of

(2-158)

which, as indicated earlier, is very large. From Equation 2-140 we can compute the SCR improvement as

(2-159)

If we assume a phase-noise level of -120 dBc/Hz we get

.(2-160)

Clearly, the predominant term in is the phase noise term. This is typical of pulsed-Doppler 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 by decreasing the phase noise. If we were able to decrease the phase noise to -135 dBc/Hz, 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.

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

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 . 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 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

(2-161)

where 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 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 that includes the effects , phase noise and the ADC is

.(2-162)

where several of the terms are familiar from Section 2.4.3.4. 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 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 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 is

(2-163)

where 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 ADC, q=1; if it toggles the lower two bits, q=3.) If the ADC specifications give an ADC SNR then , in dB, is the negative of that SNR. If the negative of the SNR is not equal to or greater that the of Equation 2-163 with , then dither noise of ½ to 1 quanta () should be added to the ADC input. However, adding too much dither noise to will degrade .

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, . If the radar uses IF sampling with digital down conversion, can be much larger than the modulation bandwidth.

The equation above is written in term used for the pulsed-Doppler signal processor. It is also applicable to the MTI processor with and .

A derivation of Equation 2-163 is given in Appendix C.

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

(A-1)

where represents one of the transmit pulses and is the pulse to which the single-pulse matched filter is matched. We assume that is a WSS random process with an autocorrelation of and a power spectral density of . It will be recalled that 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, 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,, once per PRI, , at the peak of the matched filter response. We also, without loss of generality, assume that the matched filter peaks occur at . With this we can write the output of the ADC as

.(A-2)

If we assume that and are of the form

(A-3)

where

(A-4)

and that , then all of the terms of the summation of Equation A-2 are zero except for the case where . With this Equation A-2 reduces to

.(A-5)

To find the power spectrum of we must first show that is WSS. To that end we form

.(A-6)

From previous discussions we note that

(A-7)

and

.(A-8)

By making use of

(A-9)

we can write

.(A-10)

We now make the change of variables, to yield

.(A-11)

We next make the change of variables and get

.(A-12)

Rearranging yields

.(A-13)

For the next change of variable we let to yield

.(A-14)

The first thing to note about Equation A-14 is that the right side is a function of , and constitutes the proof that 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

(A-15)

where is the matched-range, Doppler cut of the cross ambiguity function of and . In the remainder we will use the notation . With all of the statements in this paragraph we can write

.(A-16)

We next want to find the power spectrum of . We could do this by taking the discrete-time Fourier transform of . However, the math associated with this will probably be quite involved. We will take a more indirect approach.

Let be a WSS random process with an autocorrelation of and a power spectrum of . Further assume that we can sample to get . That is

.(A-17)

is the same as the random process defined by Equation A-2.

From random processes theory we can write

.(A-18)

Further, from the theory of discrete-time signals and their associated Fourier transforms, if is the power spectrum of we can write the power spectrum of as

.(A-19)

From this same theory we can write

(A-20)

If we substitute Equation A-19 into A-20 we get

(A-21)

or