A Study of Adaptive Canceling for Microwave Radiometry and Spectrometry

Transcription

1 A Study of Adaptive Canceling for Microwave Radiometry and Spectrometry Steven W. Ellingson September 16, 22 Contents List of Figures iii 1 Introduction 6 2 Theory Primary Signal Model Reference Signals Performance Limits, or The Story of Three Little Algorithms A Canceler That Needs Only to Estimate A A Canceler That Needs to Estimate A and H dz A Canceler That Needs to Estimate H dz Without a priori Knowledge of z(t) Conclusions, and Some Caveats Canceling by Adaptive Filtering The MMSE Adaptive Filter Alternatives to MMSE: LMS and RLS MMSE in Simulation, and Some Variations on a Theme CW interferer GPS C/A Realistic Bandpass Effects GPS C/A+P Performance When a Desired Signal is Present Performance When Additional Interference Appears in the Reference Channel A Robust Alternative to MMSE Hard-Limiting to Improve Performance for Constant Modulus Interferers 64 The Ohio State University, ElectroScience Laboratory, 132 Kinnear Road, Columbus, OH 4321, USA. ellingson.1@osu.edu. 1

2 4 MMSE vs. GPS in Field Conditions 66 5 Parametric Estimation/Subtraction Simulation Field Results A The Least Mean Squares (LMS) Algorithm 86 B The Recursive Least Squares (RLS) Algorithm 87 Bibliography 89 2

3 List of Figures 1 The ideal canceler Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Assuming perfect a priori knowledge of H dz (ω) Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p = Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p = +3 db Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p = db Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p =, without assuming that z(t) is a sinusoid Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p = db, without assuming that z(t) is a sinusoid A canceler based on adaptive filtering CW: INR p = db, INR r = db Responses of the computed MMSE filters for 1 iterations of the same run as Figure CW: INR p = db, INR r = +6 db Responses of the computed MMSE filters for 1 iterations of the same run as Figure CW: INR p = db, INR r = db, now unloading the diagonal of R by 99% of the smallest eigenvalue Responses of the computed MMSE filters for 1 iterations of the same run as Figure CW: INR p = db, INR r = +6 db GPS C/A: INR p = INR r = +3 db. M = GPS C/A: INR p = +3 db, INR r = +6 db. M = Same as Figure 17, with simulation extended to 1 iterations Filter response for 1 iteration of the same run as Figure GPS: Same as Figure 16, M = GPS: Same as Figure 16, M = GPS: Same as Figure 16, M = GPS: Same as Figure 16, M = Response of the fixed bandpass filter used in Sections 3.3 onward x(t) and d(t) are both noise (only) passed through a bandpass filter. M = 8. The estimated baseline used for baseline correction is shown as a solid line in the top 2 panels GPS C/A: INR p = +3 db, INR r = +3 db, bandpass filtering. M = GPS C/A: INR p = +3 db, INR r = +1 db, bandpass filtering. M = GPS P: INR p = db, INR r = db, bandpass filtering. M =

4 29 GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db, bandpass filtering. M = Estimation filter response for one iteration of simulation presented in Figure GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =. M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =.1. M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =.5. M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds = 1. M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Also, Reference signal interference: INR = db. M = Same as Figure 35, with 99% diagonal unloading GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = Filter response for w = R 1 r with M = Filter response for w = R 1 r with M = Filter response for w = R 1 r with M = GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. M = 8. Hardlimiting The RPA GPS from RPA. Primary: Antenna 1, Polarization 1; Reference: Antenna 1, Polarization 2. M = Same as Figure 46, but now M = GPS from RPA. Primary: Antenna 5, Polarization 1; Reference: Antenna 1, Polarization 1. M = Same as Figure 48, except primary and reference inputs are swapped GPS from RPA. Primary: Antenna 5, Polarization 1; Reference: Antenna 5, Polarization 2. M = GPS C/A: INR p =INR r =+3 db. MMSE with M = GPS C/A: INR p =+3 db. PES(C/A) GPS C/A: INR p =INR r =+3 db. MMSE with M = GPS C/A+P: INR p =+4.8 db. PES(C/A) GPS C/A+P: INR p =INR r =+4.8 db. MMSE with M = GPS C/A+P: INR p =+4.8 db. PES(C/A,P) GPS C/A+P, BPF: INR p =+4.8 db. PES(C/A,P) GPS C/A+P, BPF: INR p =INR r =+4.8 db. MMSE with M =

5 59 GPS from RPA: MMSE with M = GPS from RPA: PES(C/A) GPS from RPA: PES(C/A,P)

6 1 Introduction Microwave radiometry and spectrometry is the measurement of power and power spectral density, respectively, at microwave frequencies. Radiometry and spectrometry are commonly used in astronomy and earth remote sensing to explore physical properties such as spatial distribution, chemical abundance, temperature, and relative motion. A common limitation is that measurements are typically made at extremely low signal-to-noise ratio (SNR), and are also increasingly interference-limited due to the proliferation of radars and wireless communication systems in the frequency bands of interest. There is growing interest in the scientific community for methods to mitigate this interference in order to achieve better sensitivity. This report addresses adaptive canceling, which is one class of interference mitigation techniques that are potentially useful for this application. In adaptive canceling, the objective is to create an estimate of the interference which can be subtracted from the contaminated data. Canceling is attractive because it nominally allows the user to remove the interference without affecting underlying signals of interest; offering, in effect, a look-through capability. The term adaptive refers to the notion that the canceler has limited or perhaps no a priori knowledge of the interference waveform; instead, the canceler is provided a reference signal which contains a highly-correlated version of the inteference waveform. The canceler must then create an estimate of the interference by comparing the reference signal to the original, contaminated signal. The first compelling presentation of this technique as it applies to radio astronomy appears in a 1998 paper by Barnbaum and Bradley [1]. They describe the use of the Least Mean Squares (LMS) algorithm as the basis for an adaptive canceler for removal of FM radio signals from the output of a radio telescope. Bower (21) performed a similar study using field data captured from various combinations of elements from the Rapid Prototype Array (RPA) [2]. The study reported here covers much of the same ground, using different but related algorithms, filling in some theoretical details, and identifying some more of the abilities and limitations of adaptive canceling. 6

7 Section 2 ( Theory ) presents a introduction to the concepts and defines some nomenclature. A very general (and already well-known) class of algorithms based on the minimum mean-square error (MMSE) principle is derived, and it s relationship to the similarly well-known LMS and Recursive Least Squares (RLS) algorithms is briefly discussed. The issue of the ultimate performance of a canceler is also considered. Section 3 ( MMSE in Simulation, and Some Variations on a Theme ) presents a series of simulations intended to demonstrate the performance and behaviors of MMSE-based canceling schemes. Interferers considered include sinusoids as well as the narrow- and wideband components of the transmission from the satellites of the U.S. Global Positioning System (GPS) [3], which is a major source of interference to radio astronomy. Section 4 ( MMSE vs. GPS in Field Conditions ) describes the results of a study using real-world data collected from the RPA, which is contaminated by GPS. Section 5 ( Parametric Estimation/Subtraction ) discusses an alternative to MMSE for implementing an adaptive canceler when there is some advance information about the interfering signals. This approach is applied to GPS in simulation and also to the RPA field data. The specific technique described here is a simplied version of a technique previously applied to the narrowband component of GLONASS [6]. However, a new, simple method for dealing with the wideband component of GLONASS and GPS signals is described and demonstrated here. 7

8 Figure 1: The ideal canceler. 2 Theory Numerous treatments of the theory of adaptive signal processing already exist in the literature; see for example [4] for general background and [1] for an applicationspecific treatment. The purpose of this section is not to repeat material presented in those works, but rather to offer a minimal treatment that concisely explains some nomenclature, assumptions, and the justification for considering certain algorithms and not others. The material presented in Section 2.3 ( Theoretical Limits ) may not have appeared in previous treatments. 2.1 Primary Signal Model Figure 1 describes the ideal canceler. In this figure, x(t) is the contaminated signal which is to be processed, referred to as the primary signal, or channel. x(t) is assumed to consist of the following components: x(t) = s(t) + z(t) + n x (t) (1) where s(t) is the desired signal, z(t) is the interference, and n x (t) is the noise associated with the measurement. s(t) and z(t) can in turn be the sum of multiple components. For example, s(t) may be multiple spectral lines appearing in the passband, and z(t) may be multiple interferers. It will be assumed that n x (t) has the following properties: That it has white gaussian power spectrum, a mean of zero in the time domain, and that it is stationary in the sense that the variance of its power spectrum goes to zero as the integration time increases to infinity. It is important to note that bandpass filtering results in a violation of the white noise spectrum assumption. Thus, the cancelers we will consider in this study will 8

9 be tempted to interpret the shape of the bandpass as an interference contribution that needs to be corrected. This is a real problem, but usually the effects are subtle, as will be demonstrated later in this report. There are in fact rigorous methods for dealing with the colored noise problem; See [5] for an excellent introduction. However, since there are other, more serious problems lurking, this report will not address these more sophisticated approaches. 2.2 Reference Signals The ideal canceler produces as output: y(t) = x(t) z(t) = s(t) + n x (t). (2) A key feature that distinguishes canceling from other interference mitigation techniques like nulling or blanking is that nominally neither s(t) or n x (t) are affected by the cancellation process. This is in contrast to other approaches that do alter components of x(t) other than z(t). Nulling, for example, intentionally removes part of the original spectrum of x(t) to eliminate z(t). For the canceler to be able to cancel z(t), it must have a method to discriminate between z(t) and the other components of x(t). Therefore, a reference signal d(t) is provided. The ideal reference signal is simply d(t) = z(t), for then the operation of the canceler is trivial. In practice, the situation is not so simple. Typically, one has a reference signal which is merely correlated with z(t) in some sense. Also, z(t) may be contaminated with a contribution from the desired signal s(t), and usually some noise is also present. Consider a radio astronomy example. In this example, x(t) is the output of a dish antenna. Let us assume the source of the interference is localized; e.g., a satellite. One can use a second dish, pointed at the satellite, to provide the reference signal. However, what the second dish measures is not exactly z(t), because the second dish receives the interference with a different gain and phase. There may also be multipath present in the first or second dishes, further distorting z(t) in the reference signal. Also, the second dish will receive s(t) through it s sidelobes, just as the first dish 9

10 probably received z(t). Finally, the second dish contributes it s own, uncorrelated noise n d (t). So, a very general and realistic model for d(t) is d(t) = h dz (t) z(t) + h ds (t) s(t) + n d (t) (3) where h dz (t) is the impulse response that relates z(t) to what actually appears in the reference signal, h ds (t) is the impulse response that relates s(t) to what appears in the reference signal, and denotes convolution. The same assumptions are imposed on n d (t) that were imposed on n x (t), and we further assume that n d (t) and n z (t) are perfectly uncorrelated. The latter is a reasonable approximation, although probably not exact since x(t) and d(t) are usually bandlimited. So, under what conditions is a reference signal of the form shown in (3) useful? Clearly, it is desireable for the time-average power associated with the quantities h ds (t) s(t) h dz (t) z(t) and h ds (t) s(t) n d (t) (4) to be as small as possible, for otherwise the canceler may interpret s(t) as something to be suppressed. Also, it is important for the reference channel interference-to-noise ratio (INR r ), which is the time-average power associated with the term h dz (t) z(t) n d (t) (5) to be as large as possible, for two reasons. First, all canceling algorithms are first and foremost estimation algorithms. That is, a canceler must estimate z(t) given x(t) and d(t). The estimation accuracy is always a function of SNR (in this case, INR r ); thus, canceling performance will be degraded as INR r decreases. The second issue is that for many canceling algorithms, a portion of the noise input to a canceler via the reference signal is added to the output. This additional noise can potentially degrade the sensitivity of a radiometer or spectrometer. The third criterion for a useful reference signal is that h dz (t) be well behaved. What specifically do we mean? As explained in a later section, the canceler will probably be implemented by generating an estimate of the interference by filtering d(t) using an adaptively-determined impulse response g(t), and then subtracting that 1

11 estimate from x(t). For this to work, we will need the quantity x(t) g(t) [h dz (t) z(t)] to be close to zero most of the time. This in turn requires that G(ω) H 1 dz (ω) (upper case variables refer to the Fourier transforms of the associated lower case variables). If H dz (ω) is simple; e.g., equal to some constant, then there is no problem. If on the other hand H dz (ω) has one or more zeros, then d(t) is missing spectral information which cannot be recovered, and a naively-implemented canceler is likely to exhibit unstable operation. Spectral zeros are not uncommon: a typical culprit is a multipath that cancels the direct path at a certain frequency. Thus, some caution is required when designing practical cancelers to ensure that nothing bad happens when this situation occurs. It should be noted that not all cancelers use reference signals of the kind described by Equation 3. Instead, one might choose generate a reference signal based on something other than an additional measurement. For example, one might make an informed guess as to the value of z(t) and use this directly as d(t). This is not entirely unreasonable: for example, if you know z(t) is a sinusoid, then there is not much else that the canceler needs to be able to figure out (specifically, gain and phase). This is also the philosophy of the GLONASS canceler of Ellingson, Bunton, and Bell [6], and why it is effective even at very low INR r. On the other hand, if the assumed form of z(t) turns out to have some unanticipated error, then the canceller may not be able to construct an adaptive filter G(ω) that is good enough. There is one remaining consideration that has been glossed over so far: It may be that the reference signal is significantly delayed (or advanced) in time with respect to the primary signal. in which the elements may be very far apart. This happens, for example, in large radio telescope arrays However, note that the frequencydomain interpretation of a delay is simply a phase shift which varies as a linear function of frequency. So, a delay simply means that H zd (ω) has a non-flat linear phase contribution. compensation can be included in G(ω). This delay can be compensated prior to canceling, or delay 11

12 2.3 Performance Limits, or The Story of Three Little Algorithms Before moving on to a discussion of particular canceling techniques, it is worthwhile to consider the ultimate limits that apply to cancelers in general. Specifically, one would like to know the maximum possible attenuation that can be achieved for a given interferer. As stated, this is a poorly-posed and difficult question to answer. However, there is a useful special case that provides insight into many other practical cases. This is described next. Let z(t) be the complex-valued sinusoid Ae jωt, where A is a complex constant. Let n x (t) and n d (t) be ideal noise, as described previously. Finally, assume s(t) = and H dz (ω) = 1. Then, the only factor limiting attenuation is the reference signal noise, which is completely described by the parameter INR r. Let us further assume that we know we are dealing with a sinusoid, and have perfect a priori knowledge of ω. Then, z(t) is completely described by a single parameter, A A Canceler That Needs Only to Estimate A Under the conditions stated above, the optimum canceling algorithm is as follows: 1. Estimate A given d(t) and the a priori knowledge of ω. Let this estimate be known as Â. 2. Synthesize ẑ(t) = Âejωt. 3. Subtract ẑ(t) from x(t). Due to our carefully chosen conditions, the optimum estimate of A is very easy to compute. Given L samples of d(t) at sample rate F S = 1/T S, the answer is [5]: Â = 1 L L k=1 d(kt S ) e jωkt S (6) In other words, just downconvert d(t) to a frequency of zero and calculate the mean of the resulting DC signal. 12

13 4 Interference Power Relative to Input (db) L=1 L=1 L= Reference Signal INR (db) Figure 2: Attenuation acheived for a sinusoid in noise as a function of INR r number of samples, L. Assuming perfect a priori knowledge of H dz (ω). and Figure 2 shows the attenuation achieved as a function of INR r using this approach. The data are generated by Monte Carlo simulation, and each plotted point is the mean of 1 trials. From this plot, one obtains emperically the amazingly simple result that the attenuation of the interferer is equal to INR r L A Canceler That Needs to Estimate A and H dz To this point, it has been assumed that the canceler has perfect a priori knowledge of H dz (ω) (specifically, that H dz (ω) = 1). Closer to reality is the problem where H dz (ω) is also unknown. Let us assume that we only know that H dz (ω) = H dz, some unknown complex constant. Then, we must estimate H dz as well as A to make the canceler work. A revised algorithm is: 1. Estimate H dz A given d(t) and the a priori knowledge of ω. 2. Estimate H dz, using a method to be described shortly. Let this estimate be known as Ĥdz. 3. Â is the result of the first step divided by the result of the second step. 13

14 4. Synthesize ẑ(t) = Âejωt. 5. Subtract ẑ(t) from x(t). In Step 2, H dz is estimated as Ĥ dz = < d(t)x (t) > < x(t)x (t) > (7) where < > denotes the time-average of the argument and the superscript denotes conjugation. To see why this is so, note that: < d(t)x (t) >=< (H dz z(t) + n d (t))(z(t) + n x (t)) >= H dz z(t) 2 (8) and z(t) 2 =< x(t)x (t) > under the assumptions made so far. Finally, note that we should expect that the canceling performance will now depend on INR p as well as INR r, since the primary signal noise n x (t) degrades the estimate of H dz when a finite number of samples is used to compute (8). The performance of this algorithm is shown in Figures 3, 4, and 5 for INR p =, +3 db, and db respectively. Note that the act of estimating H dz has resulted in an immediate 6 db penalty in canceling performance, even when INR p =. This is because Ĥdz depends on d(t), which is affected by INR r. Also, note that INR p limits the performance that can be acheived with increasing INR r. In summary, we find that the ultimate attenuation that can be acheived by this canceler is the lesser of INR r L/4 and INR p L A Canceler That Needs to Estimate H dz Without a priori Knowledge of z(t) In many practical situations, it may not be convenient or even possible to determine in advance what kind of interferer z(t) actually is. Consider the following algorithm, which uses no a priori knowledge about z(t) and assumes only that H dz is a complex constant. 1. Estimate H dz using Equation Synthesize ẑ(t) = Ĥdzd(t). 14

15 4 2 Interference Power Relative to Input (db) 4 6 L=1 L=1 L= Reference Signal INR (db) Figure 3: Attenuation acheived for a sinusoid in noise as a function of INR r number of samples, L. Estimating H dz with INR p =. and 4 2 Interference Power Relative to Input (db) 4 6 L=1 L=1 L= Reference Signal INR (db) Figure 4: Attenuation acheived for a sinusoid in noise as a function of INR r number of samples, L. Estimating H dz with INR p = +3 db. and 15

16 4 2 Interference Power Relative to Input (db) 4 6 L=1 L=1 L= Reference Signal INR (db) Figure 5: Attenuation acheived for a sinusoid in noise as a function of INR r number of samples, L. Estimating H dz with INR p = db. and 3. Subtract ẑ(t) from x(t). Note that for signals with non-zero bandwidth (as opposed to sinusoids), this is not necessarily the optimal estimator: We could possibly do better by taking into account the complete cross-spectrum of x(t) with d(t), as opposed to considering only the zero lag. For sinusoids, however, there is no additional information in the non-zero lags. The performance of this algorithm is shown in Figures 6 and 7 for INR p = and db respectively. It is immediately clear that the penalty for not making use of a priori knowledge of z(t) is that the canceler stops working as the INR r approaches and then drops below db. On a happier note, we see that we have recovered the 6 db for large INR r that we lost going from the optimum canceler to the second canceler considered above. Also, note that the INR p L limit is still in effect Conclusions, and Some Caveats Taking into account the findings of the previous three sections, we can conclude that the high-inr r limit of attenuation is probably on the order of min(inr r L,INR p L) for a broad class of algorithms, but that the low-inr r performance can be potentially much worse than this. Also, we conclude that there is really no substitute for model 16

17 4 2 Interference Power Relative to Input (db) 4 6 L=1 L=1 L= Reference Signal INR (db) Figure 6: Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p =, without assuming that z(t) is a sinusoid. 4 2 Interference Power Relative to Input (db) 4 6 L=1 L=1 L= Reference Signal INR (db) Figure 7: Attenuation acheived for a sinusoid in noise as a function of INR r and number of samples, L. Estimating H dz with INR p = db, without assuming that z(t) is a sinusoid. 17

18 knowledge (that is, some a priori information that lets us parameterize z(t)) if good performance is desired when INR r is small. Before continuing, it is worth noting a few practical issues that are most likely to degrade performance with respect to the ideal presented above. The noise contained in the L samples must be statistically independent. Bandpass filtering can lead to a significant violation of this assumption. As the passband becomes narrower, the number of statistically-independent samples becomes a proportionally smaller fraction of the total number of samples. In the limit as the filter bandwidth approaches zero, the noise becomes indistinguishable from a sinusoid, so that the effective value of L approaches 1 regardless of the total number of samples. The passband of a practical microwave radiometer or spectrometer is typically between 1% and 9% of the Nyquist bandwidth, depending on the amount of oversampling used. Thus, the effective number of samples is usually less than the number of samples available by approximately this fraction. The stationarity of physical signals is usually limited. If the signal is varying over time, the effective number of samples is limited to those that are collected while the signal is approximately stationary. So, there is usually a limit to the improvement in canceling performance that can be achieved simply by using more samples. In practice, a canceler may need to be able to deal with arbitrary H dz (ω). If there is a significant and complex difference in the response of the receivers or propagation channels associated with x(t) and d(t), then compensating for H dz (ω) may become a challenge. 2.4 Canceling by Adaptive Filtering Most practical cancelers do some form of estimation. Estimation is the process of finding a reasonable approximation of z(t) by comparing x(t) and d(t). The key distinction between the various classes of cancelers is how they do estimation. In 18

19 this report, we will focus on estimation by adaptive filtering. The basic strategy is indicated in Figure 8. In this architecture, d(t) is applied to a filter, which produces ẑ(t), an estimate of z(t). The filter should perform the following three functions: Apply H 1 dz (ω), or a similar response, that achieves the goal of making h dz(t) z(t) become something as close as possible to z(t). Suppress n d (t) to whatever extent possible. In certain high-sensitivity applications, it may also be necessary to limit the extent to which the filter colors the power spectrum of n d (t). Suppress the h ds (t) s(t) component of d(t), to prevent inadvertent canceling of the desired signal. These days, this filter will almost certainly be digital. Specifically, let us assume that it is a finite impulse response (FIR) filter consisting of M taps. We can define the filter as follows: ẑ(kt S ) = w H d(kt s ), where (9) d(kt S ) = [ d(kt S ) d([k 1]T S ) d([k M + 1]T S ) ] T (1) where w is the M 1 vector of coefficients which define the filter, the superscript H denotes the conjugate transpose, and the superscript T denotes the transpose. w is determined by an algorithm which examines x(t) and d(t). The method for doing this is what distinguishes the various classes of adaptive filters. 2.5 The MMSE Adaptive Filter In this report, the focus will be on the minimum mean square error (MMSE) approach. The relationship between MMSE and a few other popular algorithms will be addressed below. The MMSE algorithm is the answer to the mathematical The filter to be described is also sometimes known as the Wiener or Wiener-Hopf filter. In fact, many mathematical problems and derivations lead to the Wiener solution. We will continue to refer to this filter as the MMSE filter here, as a reminder of the mathematical problem that it solves in our case. 19

20 Figure 8: A canceler based on adaptive filtering. question: arg min w < x(t) wh d(t) 2 > (11) In English: Find the filter w that minimizes the difference between x(t) and the filter output in a mean-square time-average sense. Let ɛ(t) = x(t) w H d(t) 2. (12) Applying a little algebra: ɛ(t) = (x(t) w H d(t))(x(t) w H d(t)) (13) = (x(t) w H d(t))(x (t) d H (t)w) (14) = x(t)x (t) x(t)d H (t)w x (t)w H d(t) + w H d(t)d H (t)w (15) Now, taking the time-average: < ɛ(t) >=< x(t)x (t) > < x(t)d H (t)w > < x (t)w H d(t) > + < w H d(t)d H (t)w > (16) = x(t) 2 < x(t)d H (t)w > < x (t)w H d(t) > +w H Rw (17) Note: At this point, the terms primary signal and reference signal might become a bit confusing. Reason is that with respect to conventional MMSE interpretation, we are using the reference signal as a primary signal and vise-versa. This is because the vast majority of adaptive filters are not cancelers per se, and the filter is allowed to operate directly on the primary signal. In contrast, the filter in a canceler operates on the reference signal. The convention in this report will be to always use the term primary to refer to x(t) and reference to refer to d(t). 2

21 where R is defined as the quantity < d(t)d H (t) >. R is sometimes referred to as the covariance matrix. Note that it s elements are the correlations between the taps of the filter, so it is in some sense a measurement of the power spectrum of d(t). A classical and useful method for minimizing < ɛ(t) > is to take the derivative with respect to w and find the solution that equals zero. In other words, we want to solve w < ɛ(t) >=. So, we need to evaluate: w < ɛ(t) >= w x(t) 2 w < x(t)d H (t)w > w < x (t)w H d(t) > + w w H Rw (18) Going term by term: w x(t) 2 = (19) w < x(t)d H (t)w >= (2) w < x (t)w H d(t) >= 2 < x (t)d > (21) w w H Rw = 2Rw (22) So we have: 2r + 2Rw = (23) where r is defined as < x (t)d > (Look familiar? See Equation 8). r is sometimes known as the reference correlation vector; it is literally a measurement of the crossspectrum of x(t) with d(t). Continuing, we find: Rw = r (24) This is the solution to the problem posed in (11), and constitutes the MMSE algorithm. The more common form of the solution is: w = R 1 r (25) Although the difference seems trivial, one should take a moment to consider the implications of the second form. First, R is not necessarily invertible, and sometimes even when it is, it is poorly conditioned leading to strange results. This possibility See [4] (Section 3.4) for an explanation of the vector differentiation. 21

22 should not be taken lightly it frequently occurs in common practice. One way it can happen is when multipath results in spectral zeros in H dz (ω) and INR r is large. Another way it can happen is that a bandpass filter in the reference channel proceeding the canceler can make D(ω) effectively go to zero in the stop band. Implementation issues usually strongly favor (24), from which w can frequently be more efficiently and safely computed using, for example, LU decomposition. Nevertheless, a literal implementation of the second form is usually possible but, you have been warned! One final note before moving on: Note well that the MMSE criterion does not require that the filter to suppress z(t) and do nothing else. It is merely the best thing that we can ask for given the limited information available. MMSE is like a story book genie, granting wishes but interpreting the wisher s statements literally. Since MMSE is not specifically required to cancel z(t), it frequently does not, and in many situations MMSE does things that the user may not have intended to happen. Some examples will be presented later. 2.6 Alternatives to MMSE: LMS and RLS There are at least two other adaptive filtering algorithms that are well-known in fact, probably better known than MMSE. These are Least Mean Squares (LMS) and Recursive Least Squares (RLS) [4]. LMS and RLS are intimately related to MMSE. In fact, it is easy to show that both are approximate solutions to the MMSE criterion (11). A legitimate reason for considering LMS or RLS is reduced computational effort, but certainly not performance. In fact, LMS and RLS are notorious for troublesome behaviors like erratic convergence that are simply not an issue with MMSE. By focussing on MMSE in this report, we can avoid such issues, and will nevertheless be confident that the performance we observe will also be the best possible performance that can be expected from LMS, RLS, or their many variants. However, because LMS and RLS are so frequently proposed for this application, some additional information is provided in the appendix. 22

23 3 MMSE in Simulation, and Some Variations on a Theme In this section, we explore the behavior of the MMSE canceler under the carefullycontrolled conditions offered by computer simulation. To begin, we will assume that both x(t) and d(t) are digital signals sampled at F S = 1/T S = 3 million samples per second (MSPS). The details of the MMSE canceller shall be as follows: The input sample streams will be partitioned into blocks of L = 124 samples, representing about 34 µs of data each. The samples will be used to compute R and r, and the filter will be computed using w = R 1 r. Initially, this filter will be of length M = 8 taps (about 267 ns), although this parameter will be modified to better understand it s effect later. The filter is applied to d(t) to get ẑ(t), which is subtracted from x(t). The resulting L samples are then processed through an unwindowed Fast Fourier Transform (FFT) of length L. The power spectrum is computed and added to the previous result. This process of filtering and integration continues for 1 iterations (about 3.4 ms). 3.1 CW interferer First, let us examine the performance when z(t) is a continuous wave (CW) signal; in other words, a sinusoid. We ll make this a relatively weak interferer, with INR in the reference channel (INR p ) of db. To keep things very simple, let H dz (ω) = 1 for now. INR r will be varied. Let x(t) and d(t) have perfect noise satisfying the assumptions stated in Section 2.1. radiometry. Also, let s(t) =, as is normally the case for Although this is perhaps a little bit boring, this is in fact a pretty good approximation to the problem of high-sensitivity radio astronomy, in which the signals of interests would not be detectible in the relatively short time scales over which the filter weights would normally be updated. Further, setting s(t) = eliminates one potential source of problems that are better addressed later in this report (Section 3.5). 23

24 1 Variance Time (s) Figure 9: CW: INR p = db, INR r = db. Figure 9 shows the result for INR r = db. The top and middle panels show the integrated spectra of x(t) and y(t), respectively. The bottom panel shows the integrated noise power as a function of time. The top, middle, and bottom curves in this panel correspond to x(t), y(t), and n x (t) respectively, each normalized to have the same value after the first block of L samples is processed. The middle panel is of primary interest for spectrometry, whereas the bottom panel is of primary interest for radiometry. Figure 9 shows that the canceler is marginally effective in these conditions. The interferer is suppressed by about 2 db, but at the same time an undesireable bulge appears in the integrated noise baseline. Since the interference is not fully suppressed, it is not suprising to see that the integrated noise power begins to level off. For radiometry, such as continuum measurements in radio astronomy, this leveling-off 24

25 represents the limit of sensitivity. From this perspective, the canceler can maintain the theoretical sensitivity of the measurement only for integrations less than 8 µs, and is fundamentally limited to the equivalent of 4 µs of interference-free integration. For spectroscopy too, the performance is not really acceptable. Not only is the interferer still visible, but the bulge in the noise baseline could potentially mask weak spectral lines. Furthermore, the distortion of the baseline complicates the already-onerous task of baseline correction. In Section 2.3, the ultimate limit of attenuation was found to be upper-bounded by min( INR r L, INR p L ); in this case, about 3 db. About 2 db was achieved here. The reason for the shortcoming is quite simply that satisfying the MMSE criterion does ensure the best possible canceling performance. On the bright side, this performance is considerably better than that achieved by the canceler in Section when INR r = db (Figure 7). To understand some of the other undesirable features of the canceler in more detail, consider the frequency response of the estimation filter. This response is shown for 1 iterations in Figure 1. Not suprisingly, MMSE strives to pass the CW component of d(t) and attempts to suppress noise elsewhere. However, note that the response of the filter is slightly different each time, and never exactly passes through db at the frequency of the CW signal. The reason for this is simply that the MMSE criterion (Equation 11) does not require it. With INR r = db, the criterion is actually better met by diverting some attention from the CW signal and paying attention instead to spurious noise features on a block-by-block basis. Another unwelcome behavior is that the selective response of this filter preferentially passes reference signal noise in the 1 MHz or so surrounding the interferer, resulting in the bulge in the baseline in the middle panel of Figure 9. How can one do better? Clearly, increasing INR r would be a good idea, and this will be considered next. Increasing L would also help. First, though, let us consider some alternatives, just to point out some additional issues. One possibility is to increase the number of taps M. This would result in a narrower filter, and thus the width of the baseline bulge would decrease. However, 25

26 response (db) freq (MHz) Figure 1: Responses of the computed MMSE filters for 1 iterations of the same run as Figure 9. 26

27 the computational burden of the filter would increase. Another possibility is to do just the opposite and decrease M to a smaller value. In fact, in this scenario note that the filter needs only to adjust the gain and phase of the reference signal, so MMSE can be pretty effective with M = 1. In fact, there is a strong argument for doing this in that there is no possibility of coloring the noise spectrum if M = 1. On the other hand, an M = 1 filter is incapable of dealing with wideband signals with multipath, delay differences between x(t) and d(t), or spurious signals that appear in d(t) but not in x(t). Because we may want to be able to deal with these things, we will continue for the time being with M = 8. Another possibility for improving performance is a technique known as diagonal loading. In this technique, one adds the quantity βi to the computed covariance matrix R, where I is the M M identity matrix and β is a real, positive loading coefficient. If β is chosen to be sufficiently large, the effect is to make R appear as if n d (t) had been closer to being perfectly spectrally white, even within the very short time period over which R is computed in this problem. This reduces the tendency of MMSE to pay attention to spurious noise features, which in turn results in a more stable frequency response in which the response does not rumble as shown in Figure 1. Unfortunately, diagonal loading also tends to desensitize MMSE to the non-noise signal components, so using this technique when the INR r is already low can further reduce the amount of cancelation that can be achieved. With this in mind, yet another approach that can be considered is diagonal unloading [7]. In this approach, one subtracts the quantity βi from R. This has the effect making MMSE more sensitive to the non-noise signal components, but with the side effect that the response rumble becomes worse. Nevertheless, it is sometimes useful and will be considered shortly. First, let us verify that increasing INR r is a useful thing to do. Figure 11 shows the result when INR r is increased to 6 db. The improvement is indeed dramatic, but a new problem emerges: the canceler has now managed to place a null in the I am the only person I know who calls it this, but no other term appears in common use and my term certainly seems appropriate! 27

28 1 Variance Time (s) Figure 11: CW: INR p = db, INR r = +6 db. output spectrum. If the purpose of using the canceler was to have a look through capability, then this is clearly bad news. To understand how this happened, consider the following. In the process of computing w, MMSE obtains an accurate estimate of interferer-plus-noise spectrum at the interferer s frequency. Since the reference signal is virtually noise-free in this case, MMSE is able to compute a filter whose output exactly matches that spectrum of x(t) not just that of z(t) at that one frequency. Thus, this suprising behavior is a direct consequence of the MMSE criterion in (11). Figure 12 shows the response of the estimation filter computed for 1 iterations. Although it is not easy to see in this plot, the response does in fact go through db at the interference frequency for each realization of the filter. Elsewhere, however, the filter response is quite erratic. The reason for this is simply that R is becoming badly conditioned; i.e., vanishingly-small eigenvalues of R are converted 28

29 5 4 3 response (db) freq (MHz) Figure 12: Responses of the computed MMSE filters for 1 iterations of the same run as Figure 11. into extraordinarily-large-valued eigenvalues by matrix inversion. This behavior is not necessarily a problem, since there is virtually no power in d(t) at the affected frequencies. In terms of implementation, however, the numerical dynamic range requirements implied by this example may be very difficult to support. Finally, let us consider diagonal unloading. For R to be invertible, all of it s eigenvalues should be greater than zero. Since the sum of the eigenvalues equals the sum of the diagonal elements, β must be less than the smallest eigenvalue. Let us arbitrarily set β to be 99% of the smallest eigenvalue. The results are shown in Figure 13, and some filter responses are shown in Figure 14. Now, the cancelation of the interferer is nearly perfect very close to the theoretical baseline and no further. 29

30 1 Variance Time (s) Figure 13: CW: INR p = db, INR r = db, now unloading the diagonal of R by 99% of the smallest eigenvalue. On the other hand, the effect everywhere else is disasterous, with a great deal of noise injection. To avoid giving the impression that MMSE canceling of weak CW interferers is hopeless, let us consider one final case. In Figure 15, INR r is increased to +6 db (and no diagonal unloading). Note the results are now quite good, although a slight amount of baseline bulging is still evident. It turns out that the range of useful INR r is just a few db around +6 db. As we have seen, going just a few db lower leads to ineffective canceling and noise injection, whereas going higher leads to pathological canceling behavior, specifically, driving nulls in the output spectrum. 3

31 3 2 1 response (db) freq (MHz) Figure 14: Responses of the computed MMSE filters for 1 iterations of the same run as Figure

32 1 Variance Time (s) Figure 15: CW: INR p = db, INR r = +6 db. 32

33 1 Variance Time (s) 3.2 GPS C/A Figure 16: GPS C/A: INR p = INR r = +3 db. M = 8. Next, let us consider an example where the interferer is the C/A (narrowband) component of the GPS signal. This signal is simply a sinusoid whose phase is advanced or 18 degrees once every.977 µs according to a repeating sequence of 123 bits [3]. This results in a relatively wideband signal, as shown in the top panel of Figure 16. H dz (ω) = 1 and s(t) = as before. Figure 16 shows the results for M = 8 and INR p = INR r = +3 db. About 15 db of canceling is evident, and we see that the effective sensitivity is limited to that of approximately 3 µs of interference-free integration. The INR r L bound on performance in this case is 33 db, so the MMSE canceler is performing significantly worse for this interferer than for the CW case considered in the previous section. 33

34 1 Variance Time (s) Figure 17: GPS C/A: INR p = +3 db, INR r = +6 db. M = 8. A pleasant surpise occurs when we increase INR r to +6 db, as shown in Figure 17. Unlike the CW case, we see that the canceler performance is nearly ideal up to the end of the simulation at about 3 ms. Figure 18 shows the results when the simulation is extended to 1 iterations (about 34 ms). Here, we see that the actual performance is just beginning degrade at the very end of the simulation. The upper-bound for performance in this scenario should be limited by primary channel noise at about 33 db. This appears to be consistant with the result. Figure 19 shows the filter response. Note that the response is similar in form to that encountered in the CW example when INR r = +6 db (Figures 11 and 12), except that the extent of the variation is much less just a few db around the db response near the center frequency of the interferer. Again, this is a symptom of the eigenvalue spread associated with R. 34

35 4 4 1 Variance Time (s) Figure 18: Same as Figure 17, with simulation extended to 1 iterations. 35

36 3 2 1 response (db) freq (MHz) Figure 19: Filter response for 1 iteration of the same run as Figure

37 1 Variance Time (s) Figure 2: GPS: Same as Figure 16, M = 1. Next, consider the effect of varying M. Figures 2, 21, 22, and 23 show the results for M = 1, 2, 4, and 32 respectively. The results suggest best performance for M is in the range of 4 to 8. Some additional study is required to understand exactly why this is, although it is comforting to know that the M = 4 canceler is as good as or better than cancelers with many more taps. 37

38 1 Variance Time (s) Figure 21: GPS: Same as Figure 16, M = 2. 38

39 1 Variance Time (s) Figure 22: GPS: Same as Figure 16, M = 4. 39

40 1 Variance Time (s) Figure 23: GPS: Same as Figure 16, M = 32. 4

41 2 mag. response (db) freq (MHz) Figure 24: Response of the fixed bandpass filter used in Sections 3.3 onward. 3.3 Realistic Bandpass Effects In a practical receiver, the bandpass is not likely to be perfectly flat to Nyquist. To prevent aliasing, the bandpass is usually filtered (non-adaptively) to a bandwidth significantly less than Nyquist before sampling. This may affect the behavior of the adaptive filter and the interpretation of the results. In this section, we consider this issue in simulation. The bandpass filter used in this simulation is a 33-tap FIR filter based on the Hamming window, designed to have a 3-dB bandwidth of about 7 MHz at F S = 3 MSPS. It s magnitude response is shown in Figure 24. The ripple in the central 5 MHz is less than.1 db and is very slowly varying. Figure 25 shows the results when noise only is passed through the MMSE canceler; i.e., s(t) = z(t) =. The bandpass filter changes the shape of the integrated noise 41

42 1 Variance Time (s) Figure 25: x(t) and d(t) are both noise (only) passed through a bandpass filter. M = 8. The estimated baseline used for baseline correction is shown as a solid line in the top 2 panels. baseline, so we account for this before computing the integrated noise powers in the bottom panel. The technique used here is to fit a 17th-order polynomial to the integrated noise baseline of x(t) (i.e., before canceling) after 1 iterations, and then subtract this polynomial from both the pre- and post-canceling results. Although the baseline correction is effective, the bottom panel in Figure 25 shows that it is not perfect the small error in the baseline correction eventually comes to dominate the variance as the integration time increases. Nevertheless, we learn two things from this simulation: (1) The canceler is not noticably affected by the bandpass filter in absence of interference, and (2) Any degredation in variance vs. time worse than that indicated in Figure 25 should not be attributed to errors in baseline correction. 42

43 In Figure 26 we repeat the experiment, but now we reintroduce the GPS C/A signal used in the previous section. As before, s(t) = and H dz (ω) = 1. The GPS signal is added before the bandpass filter, and INR r and INR p refer to the values before the bandpass filter. Since the bandpass filter suppresses about three-fourths of the noise power and very little of the GPS signal, INR r and INR p are about 6 db higher at the input of the canceler. At the same time, the effective number of samples is also about 75% less due to the resulting noise correlation. Thus, we should not expect any net change in the canceling performance simply because the noise spectrum is now bandlimited. Baseline correction is done using the polynomial fit computed in the previous (noise-only) case. With respect to the previous non-bandpass-filtered case (Figure 16), we see that there is a significant degradation in the performance of the canceler. Note that this occurs despite the fact that both INR p and INR r are significantly greater than before. Apparently, the coloring of the noise resulting from bandpass filtering is to blame. Recall that this issue was discussed in Section 2.1. Figure 27 shows the results when the experiment is repeated, but increasing INR r from +3 db to +1 db. Now, the performance is closer to that demonstrated in the non-bandpass-filtered case (Figure 16). 3.4 GPS C/A+P The actual GPS signal includes a second component, the P (wideband) signal. This signal is identical to the C/A signal except: (1) different codes are used, (2) it is 3 db weaker in total power, and (3) the symbol rate is 1 times greater, so that the bandwidth is 1 times greater [3]. A consequence of (3) is that not only is the P signal difficult to see in a 7 MHz bandpass, but the power spectral density is 13 db less, so it is very difficult to distinguish from the noise contribution. So, it is worthwhile to investigate the performance of the MMSE canceler in the presence of the complete C/A+P signal with bandpass filtering. Figure 28 shows the results when z(t) is the P signal only, with INR p =INR r = db. Note that the canceler is able to suppress the P signal to about the same level as the 43

44 1 Variance Time (s) Figure 26: GPS C/A: INR p = +3 db, INR r = +3 db, bandpass filtering. M = 8. 44

45 1 Variance Time (s) Figure 27: GPS C/A: INR p = +3 db, INR r = +1 db, bandpass filtering. M = 8. 45

46 1 Variance Time (s) Figure 28: GPS P: INR p = db, INR r = db, bandpass filtering. M = 8. C/A signal was canceled in the previous section. Figure 29 shows the results when z(t) is the complete C/A+P signal. Now, INR p = INR r = +4.8 db. The combined signal is again canceled to about the same level. Thus, the performance of the MMSE canceler is apparently not significantly degraded by an increase in the complexity of the signal in this manner. Figure 3 shows one iteration of the estimation filter response for the previous example. Note that the filter tends to want to create a flat response covering the noise bandwidth, at the expense of a well-behaved response elsewhere. This is not necessarily a problem, since there is very little signal power where the filter response becomes large. Nevertheless, this situation can very easily lead to implementation problems due to the large dynamic range. 46

47 1 Variance Time (s) Figure 29: GPS C/A+P: INR p = +4.8 db, INR r M = 8. = +4.8 db, bandpass filtering. 47

48 response (db) freq (MHz) Figure 3: Estimation filter response for one iteration of simulation presented in Figure

49 1 Variance Time (s) Figure 31: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =. M = Performance When a Desired Signal is Present So far, we have set s(t), which is essentially true for radiometry and not a bad approximation for high-sensitivity spectrometry. Nevertheless, one may encounter situations in which a specific relatively-strong narrowband signal, such as a spectral line, is of interest. This is investigated next. Figure 31 continues the work of the previous section, except now we add a signal s(t) consisting of a single sinusoid at +1 MHz with SNR in the primary channel (SNR p ) equal to db. We also assume for the moment perfect isolation from the reference channel; i.e., H ds (ω) =. Comparison with Figure 29 shows that the canceling performance is not noticably affected and, importantly, neither is the desired signal. 49

50 1 Variance Time (s) Figure 32: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =.1. M = 8. In practice, it is not possible to achieve H ds (ω) = ; some portion of the desired signal is likely to leak into the reference channel. The danger of this is that the MMSE canceler is then tempted to suppress s(t) in the primary channel. Figure 32 shows the effect when H ds (ω) =.1; i.e., 2 db isolation. The basic performance of the canceler is not affected, however s(t) is reduced by about 1 db in the output. Figure 33 shows what happens when H ds (ω) is increased to.5; i.e., 6 db isolation. Again, the canceling is not significantly affected, however the desired signal is now severely reduced. Finally, in Figure 35, H ds (ω) = 1; i.e, the primary and reference channels both receive s(t) equally. It is perhaps not suprising that this results in s(t) being supressed beyond the limit of detection. 5

51 1 Variance Time (s) Figure 33: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds (ω) =.5. M = 8. 51

52 1 Variance Time (s) Figure 34: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Simulated spectral line: SNR p = db, H ds = 1. M = 8. 52

53 3.6 Performance When Additional Interference Appears in the Reference Channel To this point, we have assumed that d(t) contains no interference that is not already present in x(t). However, this may not nessessarily be the case in practice. For example, consider two dish antennas separated by some distance. Although one antenna might be used as a reference signal for canceling interference in the other antenna, there may be interference local to the reference signal antenna that is not received at the primary signal antenna. This situation is considered next. Figure 35 continues the work of previous sections, but now a reference signal interferer in the form of a sinusoid at.5 MHz with INR = db is added. The results are catastrophic: the reference signal interference is introduced into the primary signal at nearly full strength, and the performance of the canceler is severely degraded. Ideally, an adaptive filter should be able to suppress reference signal interference, and MMSE does in fact have this ability. The problem in this case is that the INR for the sinusoid is so low that MMSE finds that it can do a better job of meeting it s criterion (Equation (11)) by operating on the noise instead. One possible means to overcome this problem is to use diagonal unloading, discussed in Section 3.1. Figure 36 shows the results with β =.99. Only about 7 db of suppression of the reference signal interferer is achieved. Clearly, reference signal interference has the potential to be a vexing problem if it exists. 3.7 A Robust Alternative to MMSE As observed in previous sections, the problem with MMSE canceling is not always just that the attenuation of interference is less than desired, but also that MMSE canceling exhibits some additional, harmful behavior such as wild filter responses. Often, one would prefer to sacrifice some amount of canceling performance if this would guarantee that the canceler remains well behaved. In other words, a robust filter may be preferred to the basic MMSE filter. 53

54 1 Variance Time (s) Figure 35: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Also, Reference signal interference: INR = db. M = 8. 54

55 1 Variance Time (s) Figure 36: Same as Figure 35, with 99% diagonal unloading. 55

56 A simple, robust variant of the MMSE solution that is useful in noise-dominated applications is simply: w = R 1 r (26) where denotes the matrix norm; specifically, the absolute value of the largest singular value of the argument. This is essentially the MMSE solution with diagonal loading built in. In fact, for small INR r this is about the same as: w = r σ 2 d (27) where σ 2 d is the time average power of n d(t). Note that these weights are potentially very easy to compute, since it is not really required to compute the full covariance matrix. Only it s norm, or, alternatively, σd 2, is required. The reason this solution is robust is because the filter coefficients are simply the scaled elements of the reference correlation vector; thus, the stability and dynamic range problems associated with matrix inversion are avoided. Part of the tradeoff is that this estimation filter is unable to cope with reference signal interference. However, the results shown in Section 3.6 suggest that this is no great loss, since this ability of MMSE turns out not to be very helpful anyway. Nevertheless, since (27) does not satisfy the MMSE criterion, we should be prepared for a penalty in canceling performance. This performance is illustrated in the following examples. Figures 37, 38, 39, and 4 show the performance using w = R 1 r in the same scenario considered in previous sections, with M set to 1, 4, 8, and 32 respectively. Ironically, the performance is actually as good as the best MMSE performance for M = 1, and degrades with increasing M. This can be understood by examining the associated estimation filter responses. Figures 41, 42, and 43 show the estimation filter response for M = 4, 8, and 32 respectively. Note that compared to the analogous MMSE results, this response is very well behaved. However, increasing M only narrows the estimation filter response, so increasing M turns out to be detrimental. 56

57 1 Variance Time (s) Figure 37: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = 1. 57

58 1 Variance Time (s) Figure 38: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = 4. 58

59 1 Variance Time (s) Figure 39: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = 8. 59

60 1 Variance Time (s) Figure 4: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. Using w = R 1 r with M = 32. 6

61 5 response (db) freq (MHz) Figure 41: Filter response for w = R 1 r with M = 4. 61

62 5 response (db) freq (MHz) Figure 42: Filter response for w = R 1 r with M = 8. 62

63 5 response (db) freq (MHz) Figure 43: Filter response for w = R 1 r with M =

64 Before moving on, it should be noted that that when M = 1, w = R 1 r is the MMSE solution. Thus, there is considerable merit to considering an M = 1 design if feasible. 3.8 Hard-Limiting to Improve Performance for Constant Modulus Interferers Finally, consider one other variation of the MMSE strategy that is useful in certain cases. A broad class of signals has a property known as constant modulus, which means that all their information is conveyed using phase variations, and their magnitude is nominally constant. signals using phase shift keying. Signals falling into this class include analog FM and digital The C/A and P components of the GPS signal fall into this catagory, because they are each BPSK. Collectively, the GPS signal is only approximately constant modulus, because the sum of the C/A and P complexvalued waveforms is not exactly on the unit circle. But, because the P waveform is transmitted with half the power of the C/A waveform, it is pretty close. The constant modulus property amounts to a priori knowledge of z(t) that can be exploited to improve the performance of the canceler. One way to realize this improvement is to hard-limit d(t) before giving it to the canceler. Hard limiting means to force each sample of d(t) onto the unit circle; this can be achieved simply by dividing each sample by it s magnitude. This has the effect of suppressing the magnitude component of the noise. Figure 44 shows the result when d(t) is hard-limited before canceling, using the GPS C/A+P example from previous sections. Comparing to Figure 29, we see that some improvement is in fact achieved; in fact, we are now close to the original performance obtained without bandlimiting (Figure 16). 64

65 1 Variance Time (s) Figure 44: GPS C/A+P: INR p = +4.8 db, INR r = +4.8 db. M = 8. Hard-limiting. 65

66 Figure 45: The RPA. 4 MMSE vs. GPS in Field Conditions In this section, we consider the performance of MMSE canceling in field conditions. The data were obtained from the Rapid Prototype Array (RPA), a joint project of the University of California at Berkeley and the SETI Institute. The RPA is an array of 7 1-ft. dishes, as shown in Figure 45. It is located near Lafayette, CA. Each dish has independently-instrumented orthogonal linear feeds, and operates at L-band. The data used in this study were obtained at about 17:58 UT 1 April 21. The GPS satellite NAVSTAR 35 was selected as the the primary target of the observations. The expected elevation of this satellite from the RPA site was Only two of the seven antennas are utilized in this study. Antenna 1 was pointed to put the satellite in the main beam. Antenna 5 was pointed about 8 (about one half-power beamwidth) away. Data were obtained in two polarizations for each antenna using 8-bit A/Ds at 3 MSPS. The RPA s receivers downconvert the nominal GPS center frequency Thanks to Dr. G. Bower of U.C. Berkeley for providing the data. 66