AM-FM Interference Excision in Spread Spectrum Communications via Projection Filtering

Recently,Amin et al. introduced a projection ﬁltering method for excising constant amplitude FM jammers from DSSS communications, with minimal distortion to the PN sequence. In this paper, we show that this approach can be applied to AM-FM jammers as well, with a simple modiﬁcation. Theoretical performance measures (correlator SNR) of the AM-FM projection method are derived, and demonstrate that near ideal performance is achieved for unbiased estimates of the jammer parameters. Results showing the effects of estimation errors in the AM and FM of the jammer on SNR are also provided. In general, FM errors cause greater performance degradation than the same level of error in estimating the AM.


INTRODUCTION
Spread spectrum communications employ a modulation technique in which the bandwidth of the transmitted signal is much larger than, and effectively independent of, the bandwidth of the baseband message. The advantage of spread spectrum communications is that it makes the system less sensitive to interfering signals other than thermal noise. These interfering signals might arise from intentional jamming, multipath, and multi-users coexisting on the same bandwidth. In the presence of strong interferences, the performance of the spread spectrum communication system can be further improved by employing various signal processing techniques at the receiver prior to despreading [1,2].
Several approaches have been proposed for removing wideband FM interference from direct sequence spread spectrum (DSSS) communications [3,4,5,6,7,8]. Amin introduced a time-frequency based technique for excising FM jammers, wherein estimates of the instantaneous frequency of the jammer are used to adaptively notch-filter the received signal, prior to despreading [3]. Extensions of this technique include the use of the Hough transform to more accurately estimate the instantaneous frequency of FM jammers [6], and the use of instantaneous bandwidth information as well to excise AM-FM jammers [9,10].
While the adaptive notch-filter approach to jammer excision is an effective means for removing wideband FM and AM-FM jammers from DSSS communications, the notch filter has the undesirable effect of distorting the desired information encoded in the PN sequence. Accordingly, even if estimates of the jammer parameters (i.e., instantaneous frequency and instantaneous amplitude/bandwidth) are ideal, the performance (e.g., bit-error rate (BER)) of such excision systems is less than ideal [3,5,9,10].
To overcome this undesired distortion of the PN sequence, Amin, Ramineni, and Lindsey recently introduced a projection filtering technique for excising constant amplitude FM jammers [4]. For unbiased estimates of the instantaneous frequency of the jammer, performance of this system approaches ideal levels. In this paper, we extend this method for excising AM-FM jammers, by a simple modification, and we provide theoretical performance analyses, including the effects of estimation errors in the jammer parameters (i.e., its AM and FM). For unbiased estimates of the jammer parameters, the performance of the AM-FM projection filter is nearly ideal.
be expressed in vector form as where p is the desired PN sequence modulated by the transmitted data bit b (±1 for BPSK modulation), w is white noise, and j is the jammer; all vectors are of dimension L × 1.
With the projection technique [4], jammer excision is achieved by projecting the input data vector over one bit period onto a subspace orthogonal to the jammer subspace. The projection matrix is defined as where u is a basis vector that spans the subspace of the jammer. For FM jammers of the form j(n) = Ae jφ(n) , the subspace is estimated from the instantaneous frequency of the jammer, which is then integrated (summed in discrete time) to obtain the instantaneous phase of the jammer. The orthonormal basis vector u is then given by Applying the projection filter to the received signal yields for ideal estimates of the jammer instantaneous frequency (note that we have taken b = 1, without loss of generality, in the preceding equations). For large L, the distortion to the PN sequence is negligible, This method readily extends to handle the case of FM jammers that also have AM, as we show next. We then derive performance measures for the AM-FM projection filter.

AM-FM JAMMER EXCISION BY ORTHOGONAL PROJECTIONS
We extend Amin's work to the AM-FM jammer case by incorporating the amplitude information into the basis vector. For AM-FM jammers of the form j = A a(1)e jφ(1) a(2)e jφ (2) · · · a(L)e jφ(L) T , where L k=1 a 2 (k) = L (so that the jammer power is A 2 ), the orthonormal basis vector becomes a(1)e jφ(1) a(2)e jφ (2) · · · a(L)e jφ(L) T , where a(k) and φ(k) are the instantaneous amplitude and instantaneous phase of the jammer, respectively. It is straightforward to show that the orthogonality condition is maintained, that is, Vj = 0. For the special case of constant AM, a(k) = 1, 1 ≤ k ≤ L, the basis vector u defined in (9) reduces to that of (3).

THEORETICAL PERFORMANCE ANALYSIS
To quantify the performance of the excision filter, we use the correlator SNR which is defined as the square of the output mean divided by the output variance [1] where y is the output of the excision filter. From the SNR we can obtain the bit-error-rate (BER) as which is an approximation that holds under the assumption that y is Gaussian distributed with mean E{y} and variance σ 2 For the case of no-jammer, the SNR is where σ 2 n is the variance of the channel noise (AWGN). This value of the SNR represents the upper bound on the receiver performance for any excision filter. For a jammer of power A 2 , the SNR degrades as which represents the worst-case performance (i.e., no excision filter is used, or the excision filter is completely ineffective).

Performance analysis with perfect jammer parameter estimation
We derive the correlator SNR for the AM-FM projection filter with perfect knowledge of the jammer parameters, following the approach in [4] for the FM case. The result is (see Appendix A.1) Note that for the constant amplitude (a(k) = 1) FM jammer, this reduces to in agreement with the results in [4]. Observe that for large L the correlator output SNR expressions for AM-FM (equation (14)) and the FM (equation (15)) are almost identical since Note also that both expressions approach the no-jammer SNR (equation (12)) as L increases. For ideal estimates of the jammer parameters, the technique completely eliminates the jammer (independent of the AM depth-see Figure 1; note the plot was generated via theoretical calculations using equations (11) and (14)). For AM-FM jammer, the distortion to the PN sequence after AM-FM projection filtering depends on the instantaneous amplitude a(k), but for large L, the distortion to the PN sequence is negligiblē (17) Figure 2 shows the correlator SNR (theoretical) of the AM-FM projection filter compared to both FM projection filter and 5-tap FIR notch filter [3]. From the figure we can observe that the two correlator SNR expressions are almost identical and approach the ideal upper bound.

Effects of jammer estimation errors
Estimation error in the jammer parameters will degrade performance. In this section, we derive the correlator SNR that  includes errors in the estimation of AM and FM of the jammer. We define the estimated basis vector that has errors as whereã(k) = a(k) + ∆a(k) is the estimate of the jammer amplitude with estimation error ∆a(k) andφ(k) = φ(k) + ∆φ(k) is the estimate of the jammer phase with estimation error ∆φ(k). The estimated vectorũ is assumed to have AM and FM errors over the length of the PN sequence, and the errors at different chips are assumed to be i.i.d. random variables with a zero mean Gaussian distribution and variance σ 2 ∆a and σ 2 ∆φ , respectively. We also assume that the two random variables are independent. We consider single block processing and assume the error is spread over the entire PN length, that is, k = 1, m = L case in [4].

A. FM error case
For the case of FM error only with variance σ 2 ∆φ , the correlator SNR for the AM-FM projection filter is given by (see where A 2 is the jammer power. Note that as the error increases, that is, σ 2 ∆φ → ∞, the SNR approaches the lower performance limit given by (13), for large L.  For the constant amplitude (a(k) = 1) FM jammer case, the correlator SNR is which we note is a special case of the general expression given in (19), and agrees with the results in [4]. Figure 3 shows BER, calculated via (11), versus FM error for the AM-FM projection filter with ideal AM at several JSRs. The RMS error (%) for the FM parameter is defined as where P FM is the average power of the ideal FM parameter. At small to moderate JSRs (<10 dB), the performance is quite good, but degrades significantly for higher JSRs. However, we point out that in practice as the JSR increases, it is quite likely that errors in estimation of the jammer parameters will decrease.

B. AM error case
For the case of AM error only with variance of σ 2 ∆a , the correlator SNR for the AM-FM projection filter is given by (see Appendix A.2)  Here too we see that as the error increases, that is, σ 2 ∆a → ∞, the SNR approaches the worst-case performance for large L. Also, note that since σ 2 /(1 + σ 2 ) grows slower than 1 − e −σ 2 as the error variance increases, we see that a given error in the estimation of the jammer FM (equation (19)) degrades SNR more than the same error in the estimation of the AM (equation (22)), although for very small-and very largeerrors, the two SNRs are about the same. Figure 4 shows BER versus AM error for the AM-FM projection filter with ideal FM at several JSRs. As in the case of FM error, we have defined the RMS error (%) for AM parameter as where P AM is the average power of the ideal AM parameter. At small to moderate JSRs (< 20 dB), the performance is quite good.

C. AM and FM errors case
For the case of both AM and FM errors, the correlator SNR for the AM-FM projection filter is given by (see Appendix A.2)   Figure 5 shows the BER versus JSR for the AM-FM projection filter with some fixed combination of AM and FM errors. In all cases, the performance is near ideal at lower JSR ranges, but the performance gets worse with increasing JSR. Figure 6 compares the theoretical performance and numerical calculation of the BER via simulation for FM error case, with trials averaged per JSR data point to obtain the BER. We report the comparison result in terms of percent error between theory and simulation, defined as % error = simulation − theory theory × 100. (25) From Figure 6 we can see that the simulation and theoretical results match closely for sufficiently large number of trials.
In Table 1, we have summarized the correlator SNR of the AM-FM projection filter for AM-FM jammer in terms of a lower bound (the bounds follow from (16)).

CONCLUSION
We have derived correlator SNR measures for an AM-FM projection filter for excising jammers from DSSS communications. For unbiased estimates of the jammer instantaneous amplitude and instantaneous frequency, the AM-FM projection filter is able to effectively remove the interference with negligible distortion to the desired PN sequence. For large PN sequence length, the performance approaches the ideal upper bound. Estimation errors on the AM or FM of the jammer degrade performance, and in general FM errors cause more degradation than the same level of error in the estimation of the jammer AM.

A.1. Performance analysis of AM-FM projection filter with perfect jammer parameter estimation
We derive the SNR as defined by (10) where y 1 is the contribution of the PN sequence and y 2 is the contribution of the white noise sequence to the decision variable. The mean value of y is The mean value of y 1 is given as where Tr(·) is the matrix trace. Since the PN sequence and white noise are uncorrelated, the mean-square value of y is since E{y 1 y * 2 } and E{y * 1 y 2 } are zero. The mean-square value of y 1 is where · 2 F is the square of the Frobenius norm of the matrix. In deriving (A.6), we have used the constraint defined in (8) to maintain the power of the AM-FM jammer to A 2 . From (A.4) and (A.6), the variance of y 1 can be given as (A.7) The mean-square value of y 2 is Since the PN and white noise sequences are uncorrelated, and from the whiteness property of both sequences, (A.8) can be simplified to Since E{y 2 } = 0, the variance of y 2 is therefore From (A.7) and (A.10), the variance of y is (A.11) Therefore, the correlator SNR for the AM-FM projection filter is given by which is (14).

A.2.1 FM error case
The decision variable y is obtained by where y 1 is the contribution of the PN sequence, y 2 is the contribution of the white noise sequence, and y 3 is the contribution of the jammer to the decision variable (ideally zero). The mean value of y is since E{y 2 } = 0 and E{y 3 } = 0 because the PN sequence is uncorrelated with the white noise and jammer. The mean value of y 1 is the same as given in (A.4) (A.16) The mean-square value of y is and E{y * 3 y 1 } are all zero. The mean-square value of y 1 is the same as in (A.7) The variance of y 2 is the same as in (A.10) The mean-square value of y 3 is  The last term in (A.21) is given as E u HũũH pp HũũH u (A.24) The second and third terms in (A.21) has the same results as that of the last term shown in (A.24). Therefore, Since E{y 3 } = 0, the variance of y 3 is Therefore, the correlator SNR for the AM-FM projection filter with FM error is SNR=(L−1) (A.28)

A.2.2 AM error case
The decision variable y can be obtained in the same manner as in the previous section. The mean value of y 1 is the same as in (A.4). The mean-square value of y 1 is (a(l) + ∆a(l)) 2 . (A.30) In deriving (A.29), we have used the following approximation which is valid for σ 2 (A.32) The variance of y 2 is the same as in (A.10). Therefore, Since E{y 3 } = 0, the variance of y 3 is (A.38) Therefore, the correlator SNR for the AM-FM projection filter with AM error is (A.39)

A.2.3 AM and FM errors case
The decision variable y can be obtained in the same manner as in Section A.2.1. The variance of y 1 is the same as in (A.32), and the variance of y 2 is the same as in (A.10). The mean-square value of y 3 can be calculated the same way as in (A.21),