5.0 SIDELOBE CANCELLATION
5.1 INTRODUCTION
Sidelobe cancellation (SLC) can be considered an extension of STAP. It is a spatial adaptive processing technique that is aimed at removing interference. In STAP, or more accurately SAP, the processor places nulls in the antenna pattern at the angular locations of the interference sources. In SLC, the processor attempts to subtract the interference from the antenna output. In fact, SLC is also likened to adaptive cancellation techniques used in communication systems for removing interference and multipath signals
SLC is designed to operate against active electronic attack (EA) devices (jammers) and not against clutter or passive interference such as chaff. It is usually assumed that the EA signal is noise-like with a bandwidth that exceeds the intermediate frequency (IF) bandwidth of the radar receiver. It is also assumed that the EA signal is entering the radar antenna through one of its sidelobes. The fact that SLC cancels interference entering the radar through the antenna sidelobes is believed to be the origin of the term sidelobe cancellation.
5.2 BACKGROUND
We will derive the configuration of a sidelobe canceller (abbreviated SLC) by first examining the interference cancellation problem. To this end, Figure 1 contains a block diagram of an interference canceller. In this figure, the top antenna represents the main radar antenna and the bottom antenna is an auxiliary antenna used to gather information on the interference signal. The block with is the gain, or weight, (scalar in this case) analogous to the weight in STAP and the arrow through the box denotes that the weight is adjusted based on the error voltage, . The error voltage is formed by subtracting a weighted version of the auxiliary channel signal, , from the main channel signal, . That is
.(1)
The resulting error signal is sent to the rest of the radar receiver and signal processor. (The interference canceller is implemented early in the receiver.)
If the interference canceller is working correctly will contain only the target, or desired, signal. Indeed, suppose we assume that the signal from the main antenna consists of a desired signal, , and an interference signal, . That is
.(2)
Further, assume the auxiliary channel signal consists of only interference[1]. That is
.(3)
Now, suppose we are able to choose the weight, , such that it is
.(4)
If we can choose this we would get
.(5)
Thus, the error signal is, in fact, the desired signal. What we need now is a criterion for computing so that it behaves as indicated.
5.3 A METHOD FOR FINDING
Before we propose a criterion we need to recognize that is a random process. Because of this and are also random processes. This means that, for a particular , , andare random variables. The implication of this is that we must use statistical methods to characterize the random variables and develop the criterion. With the above restrictions the criterion we will use is to choose so as to minimize the mean-squared value of . Mathematically
.(6)
We use magnitudes because we recognize that and must, in general, be represented by complex variables so as to capture their amplitude and phase. We use expectations because is a random variable, which causes and to be random variables[2]. We use the square because it is fairly easy to work with. The criterion given in Equation (6) is termed a least mean-square (LMS) criterion.
From our experience with quadratic, or squared error, minimization (recall least-squared curve fitting), we know that we can select as the at which the partial derivative of the mean-squared error with respect to is zero. That is
(7)
If we perform the indicated operation we get
(8)
which leads to the solution
.(9)
Although this result is interesting we want to see if it leads to an error signal that is reasonable. That is, is , the desired signal. We will assume that is a zero-mean, wide-sense stationary (WSS), random process with a variance of
.(10)
We let and be as defined in Equations (2) and (3). With this we can write
,(11)
which is the result we said we needed to obtain (See Equation (4)). Thus, we conclude that if we use the LMS methodology to determine the weight (gain), , the error signal out of the interference canceller will indeed contain and not – at least in theory.
5.4 PRACTICAL IMPLEMENTATION CONSIDERATIONS
Direct implementation of the LMS canceller as discussed thus far requires a priori knowledge of . In general, this is not a practical assumption in that it requires knowledge of the interference source, which is not generally available. Because of this and must be estimated based on measurements of and .
Strictly speaking, the expected values are ensemble averages and can’t be evaluated from a single set of and measurements. To be valid one must average across many radars, targets, environments and interference sources (all of the same type and in the same location) to obtain a true ensemble average. Clearly this is not possible since we have only one radar, etc. To get around this problem we invoke the concept of ergodicity. This concept states that, if a random process is ergodic, ensemble averages can be replaced by time averages. Proving that a process is ergodic is very difficult, if not impossible.
One of the key characteristics of ergodic processes is that their mean and variance are constant. Thus designers usually try to rationalize an assumption of constant mean and variance and invoke, without justification, ergodicity. Although not strictly legitimate from a random processes perspective, this approach works well.
Thus, if we assume and are WSS over the measurement interval, [3], we can write
(12)
and
(13)
where means evaluate the integral over some interval . In the above must be large relative to the reciprocal of the bandwidth of and . This is necessary to obtain good averages.
If we assume digital processing we get
(14)
and
.(15)
The spacing between samples, , should be greater than or equal to the reciprocal of the bandwidth of and and should be such that is large relative to the reciprocal of the bandwidth of and .
Given the above, one can compute from
(16)
and implement the canceller as
.(17)
In the above, the notation was used in place of to indicate that the weight is on approximately optimum (because of having to estimate the expected values).
In a typical application, one would periodically compute , and , and use if for a while. The time between updates would be determined by how often one could assume and remain constant. A good rule of thumb is that they should be updated once per PRI in a low PRF radar or once per coherent dwell in a pulsed-Doppler radar.
The above implementation is feasible if the radar has a digital computer fast enough to compute , and , and apply it via Equation (17). With modern radars that use DSP this may be a feasible approach. However, in older analog radars it was not. Therefore, the algorithm had to be modified to work in such radars. The result was the SLC.
5.4 SIDELOBE CANCELLER
The weight calculation technique upon which the SLC is based is termed a gradient search technique. The gradient technique iteratively computes weights so as to eventually minimize the mean-squared error
.(18)
In the implementation of the technique we can’t really evaluate the expected value so we approximate it by the simple squared error, or
.(19)
The gradient algorithm is given by the equation
.(20)
In Equation (20), is the gradient of the error evaluated at and is given by
.(21)
Basically, the gradient updates the latest estimate by adding a correction that is proportional to the negative of the slope, or gradient, of the error evaluated at the latest estimate. This is illustrated in Figure 2. In this figure and we note that the slope is positive. We also note that we want to be less than if we are to move toward . Thus, we see that we want to move in a direction that is opposite to the sign of the slope. With some thought, we also note that if is far away from we would like to change by a large amount; whereas if is close to we want to change by a small amount. Thus the amount of change is related to the magnitude of the slope. This is what the algorithm of Equation (20) does.
The parameter controls the rate at which the estimate approaches . If is small will approach in small steps and if is large will approach in large steps. If is too small, convergence will be very slow. On the other hand, if is too large the solution could diverge. Thus, choosing is one of the important parts of implementing a SLC.
We recognize that Equation (20) is a difference equation. However, SLCs are usually implemented in the continuous time domain. From our experience with differential and difference equations, we can convert Equation (20) to a differential equation of the form
(22)
where we have made use of Equation (21). Note that the parameter has been changed to . We did this to note that the scaling constant will be different for discrete-time and continuous-time implementations.
Equation (22) also contains another, subtle, change over Equation (20). Specifically, in Equation (22) we allow the error voltage and auxiliary channel voltage to change with time as the weight is being updated. In Equation (20), we used one sample of the error voltage and auxiliary channel voltage to iterate on the weight. Allowing the voltages to change will incorporate averaging into the SLC to help stabilize the performance.
If we represent Equation (22) as a block diagram we have the classical Howells-Applebaum SLC. This is shown in Figure 3.
5.6 PRACTICAL CONSIDERATIONS
The Howells-Applebaum loop is usually implemented at some intermediate frequency (IF). As such, the lower multiplication of Figure 3 is generally performed by a mixer whereas the upper multiply is a variable gain amplifier. The , and voltages are IF signals while is a baseband signal. The conjugation at the bottom is implemented as a 90º phase shift. The block diagram of Figure 3 uses complex signal notation. In an actual implementation, one needs to use quadrature signals to produce an IF implementation that captures the operations implied by the complex signal notation.
An example block diagram for an IF implementation is contained in Figure 4. In this figure, the circles with crosses are mixers and the squares with crosses are variable gain amplifiers. The gain is bipolar. That is, the weight can vary the amplifier gain and the sign of the product depending upon the signs of the inputs.
The blocks with integral signs in them are typically implemented using low-pass filters where the bandwidth of the low-pass filter is set somewhat lower than the reciprocal of the integration time of the SLC.
Earlier, it was indicated that SLCs are used to cancel interference entering through the sidelobes of the main antenna. In theory, a SLC can also cancel signals entering through the mainlobe of the main antenna. The restriction to sidelobes is an implementation constraint. Generally, the relative gains of signals entering through the main lobe and sidelobes differ by 30 to 50 dB. Further, the gain of the auxiliary antenna is usually only 3 to 10 dB higher than the gain through the sidelobes of the main antenna. If the SLC were to cancel interference entering through the mainlobe, the gain of the variable gain amplifiers would need to be variable between 20 and 47 dB (30-10 and 50-3). Further, if the SLC was to also be able to accommodate interference entering through the sidelobes of the main antenna, the gain of the variable gain amplifiers would need to be between roughly -3 and -40 or -50 dB. Thus, the variable gain amplifiers would need a capability of providing variable gain between about -50 and +50 dB, and they would need to remain fairly linear. This is quite a stringent set of requirements to place on the amplifiers. Thus, the designer must usually choose between canceling mainbeam or sidelobe interference. Canceling sidelobe interference is the most common approach.
When we were deriving the SLC algorithm we made the assumption that was WSS. This is generally not a good assumption because (see Equation (2)) is not stationary. One method of avoiding this problem is to have the SLC compute during a time when only the interference is present and hold the weight during the rest of the PRI. One approach might be to activate the adaptive part of the SLC a few 10’s of µs before the transmit pulse and then hold the weight through the PRI. That way, during the adaptation time, will (hopefully) contain only the interference signal, which can reasonably be assumed to be WSS over the time the SLC is adapting the weight.
For the case of a pulsed-Doppler signal, the SLC would adapt before the coherent dwell, during a dead time, and maintain the SLC weight during the coherent dwell.
In phased array radars, the SLC weight should be computed after the beam has been steered to the desired angles and the beam should not be re-steered during the time the SLC is attempting to cancel the interference.
In scanning antennas, the SLC designer must account for antenna motion in designing and specifying the performance of the SLC since beam motion during weight computation, and the cancellation interval, will degrade performance. This is due to the fact that the characteristics of the interference entering through the sidelobes will change if the antenna moves.
When we developed the cancellation algorithm we assumed that the auxiliary channel signal did not contain the target signal. If the weight calculation is performed as indicated in the previous discussions, this is a good assumption since the weight is calculated when no target signal is expected in either the main or auxiliary channel. However, during the cancellation interval, both target and interference signal will be in both the main and auxiliary channel signals. Let us suppose that the signal-to-interference ratio is SIR. Let us further assume, for simplicity, that the main antenna offers a gain of G to the target signal and unity gain to the interference signal. We assume that the auxiliary antenna offers a (power) gain of to both the target and interference signal. If we relate this to Equations (2) and (3) we would write
(23)
and
.(24)
We can show that the optimum weight is
.(25)
With this, the signal out of the SLC is
.(26)
If the auxiliary channel did not contain a target signal component, the output of the canceller would be
.(27)
It will be noted that the presence of the target in the auxiliary channel degrades the output of the canceller, but only slightly – much less than 1 dB.
The SLCs of Figures 3 and 4 are termed single-loop SLCs in that they are capable of canceling only one interference source. If two or more interference sources are present, the SLC will attempt to partially cancel all of the sources. However, it will generally not cancel any of them very well. If one were to add another auxiliary antenna and build a duplicate loop, but keep the summers in Figures 3 and 4, one would have a two-loop canceller that would be capable of canceling two interference sources. Adding more auxiliary antennas and loops increases the potential of canceling multiple sources. The rule of thumb is that N SLC loops will cancel N interference sources. There is some practical limit to the number of loops and interference sources that can be canceled but I am not sure what the limit is. Also, there is some concern as to whether the various loops will “fight” each other. I’ve seen this discussed in the literature, but I’ve not heard of it being encountered in actual radars.
© 2014 M. C. Budge - 1
[1] This may not be a good assumption in some cases but may be valid in others. This will be discussed later
[2]We assume is a deterministic (complex) variable.
[3]If and are WSS over , by definition their mean and variances are constant over .