We assume that the pulse interference is extracted from a wideband continuous signal with rectangular pulse. The time domain waveform is described as j t = c t cos 2 π f 0 t p t = j c t p t (1) where c(t) is the baseband signal, f ₀ is the carrier frequency which is the same with the interest signal, p(t) is a square wave with the expression as p t = A , − τ 2 + n T s ≤ t ≤ τ 2 + n T s   n = 1,2 ⋯ 0 , else. (2) where A represents the amplitude of the rectangular pulse, here we set A as 1. τ is the pulse width, and T _s is the pulse period.

The spectrum of pulse interference is J f = ∑ n = − ∞ + ∞ a n J c f − n f s (3) where J _c (f) is the spectrum of the continuous signal j _c (t), a n = τ f s s a π n f s τ , where f s = 1 T s and sa(x) = sin(x)/x.

Since the low-pass filter exist before analog-to-digital conversion, the bandwidth of the interference entering the space-time filter is limited.

Taking the array satellite navigation receiver as an example, in the signal process flow, the antenna array receivers based on digital signal processing add the array anti-jamming processing segment compared with the single antenna receivers as shown in Figure 1. The signal process of the satellite navigation antenna array receivers mainly includes RF front-end, AD conversion, anti-jamming filter, channel process and other segments. Pulse interferences will have a certain impact on all the above segments. In the paper [11], the impact of pulse interferences on the baseband processing was analyzed. This paper mainly focuses on the impact of pulse interference on anti-jamming processing.

The filter based on SMI use different optimization criteria to solve the filter weight vector. For example, minimum variance distortionless response (MVDR) filter solves the weight vector by minimizing the output power and constraining the satellite signal gain to 1, and the optimal weigth vector is w opt = R y − 1 a s a s H R y − 1 a s (4) where a _s is the steer vector of the interest signal, which is calculated from the incident angle of the signal θ ₀ and the layout of the array. R _y is the covariance matrix of noise and interference, which can not be obtained accurately. Because the satellite signal power is much small than noise power, it is assumed that the sampled signal is the noise and interference signal, which means that R y ≐ R j + R n = R j + σ 2 I (5) where σ ² is the noise power, R _j represents the covariance matrix of the interference signal. Under the assumption that the input signals are stationary, the covariance matrix of the samples is used to replace the real covariance matrix in the processing.

The performance is analyzed by taking the ratio of output SINR of space-time filter under continuous interference and pulse interference as the metric. R _c and R _p are the theoretical covariance matrices under continuous interference and pulse interference respectively. When the signal power is 1, after anti-jamming processing based on MVDR method, the optimal SNR of filter output in two cases are SINR o c = a s H R c − 1 a s and SINR o p = a s H R p − 1 a s respectively. Set v _i as the eigenvector of the covariance matrix and λ _i is the corresponding eigenvalue. Under single interference, for the space-time filter with N elements and M time-taps, the covariance matrix of interference has M larger eigenvalues, and the other eigenvalues are equal to noise power σ n 2 . And under the narrowband assumption, the space-time covariance matrix of the interference can be expressed as [24]. R j = R θ ⊗ R f = v θ v θ H ⊗ R f (6) where v _θ is the steer vector of interference. ⊗ denotes the Kronecker product [25]. R _f is time domain covariance matrix of interference, whose eigenvalues and corresponding eigenvectors are λ _i , v _fi (i = 1, ⋯M). So the larger eigenvalues of R _j are λ _i (i = 1, ⋯M) and the corresponding eigenvectors are v _i = v _θ ⊗v _fi . To simplify the analysis, we assume that the time-domain covariance matrix of the interest signal has only one non-zero eigenvalue and the corresponding eigenvector is a _f , the space-time steer vector of the navigation signal is a _s = a _θ ⊗a _f , a _θ is the spatial steer vector of the interest signal. Then the optimal output SINR of the filter is SINR o = a s H R − 1 a s = a s H ∑ i = 1 N M v i v i H λ i + σ n 2 a s = a s H 1 σ n 2 I − ∑ i = 1 M λ i v i v i H λ i + σ n 2 a s = a s H a s σ n 2 − 1 σ n 2 ∑ i = 1 M λ i λ i + σ n 2 a θ H v θ v θ H a θ ⊗ a f H v f i v f i H a f ≈ a s H a s σ n 2 1 − 1 a s H a s ∑ i = 1 M a θ H v θ v θ H a θ ⊗ a f H v f i v f i H a f (7)

The above expression shows that the angle between steer vectors of interest signal and interference affects the theoretical optimal SINR. When the incident direction of interference and interest signal are not in the same main lobe, the differences of frequency domain characteristics of interference can be ignored. Therefore, under pulse interference and continuous interference, the ratio of theoretical optimal output SINR of the SMI filter is close to 1, and the loss of theoretical optimal SINR caused by pulse interference is 0 dB.

With the analysis of last section, it is obvious that if the covariance matrix of the interference is accurately estimated, performances of SMI filter under pulse interference and continuous interference are the same. Next, the convergence characteristics of SMI filter under pulse interference are derived. Assuming that one or more pulse cycles are in training samples, and total interest signal power of the SMI filter output is P s = 1 K t s ∑ k = 1 K v s H R ̂ ptl − 1 s t l + k t s v s H R ̂ ptl − 1 s t l + k t s H (8) where, K is the length of the training samples, t _s is the sampling interval, v _s = a _s is the steer vector of the interest signal, and R ̂ ptl signal covariance matrix used in the lth data cell, which is generally calculated from the samples of the (l − 1)th data cell and t _l represents the starting time of the lth data cell. The interference and noise signal output power of the SMI filter is P i + n = ∑ k = 1 K β v s H R ̂ ptl − 1 j t l + k t s v s H R ̂ ptl − 1 j t l + k t s H + ∑ k = 1 K v s H R ̂ ptl − 1 n t l + k t s v s H R ̂ ptl − 1 n t l + k t s H (9)

Assuming that the pulse duty is β, the average output SINR of all the L data cells is SINR a = E P s E P i + n = E ∑ l = 1 L ∑ K = 1 K v s H R ̂ ptl − 1 s t l + k t s v s H R ̂ ptl − 1 s t l + k t s H ∑ l = 1 L ∑ K = 1 βK v s H R ̂ ptl − 1 j t l + k t s v s H R ̂ ptl − 1 j t l + k t s H + ∑ K = 1 K v s H R ̂ ptl − 1 n t l + k t s v s H R ̂ ptl − 1 n t l + k t s H ≐ E v s H R ̂ p − 1 R s v s H R ̂ p − 1 H v s H R ̂ p − 1 β R c + σ n 2 I v s H R ̂ p − 1 H (10)

The covariance matrix in a sampling period is approximate as R ̂ p ≈ β R ̂ c + σ 2 I = β R ̂ c + σ 2 β I (11) where R ̂ c is the covariance matrix of interference estimated from βK pulse samples. Without affecting the conclusion, we take the noise power as the signal power unit, set σ ² as 1, and set ε = 1 β , record that R ̂ = R ̂ c + ε I , R ₂ = R _c + I, R ₁ = R _p ≐β R _c + I, Then the loss of SINR caused by the error of estimated covariance is ρ = SINR a SINR o p = v s H R ̂ − 1 v s 2 v s H R ̂ − 1 R 1 v s H R ̂ − 1 H v s H R 1 − 1 v s (12)

Next, the theoretical distribution of ρ under different training samples number and pulse interference parameters is analyzed. Let y k = R 2 − 1 / 2 j k , R ̂ 2 = 1 ε R 2 − 1 / 2 R ̂ R 2 − 1 / 2 , then R ̂ 2 = 1 K ∑ i = 1 β K y i y i H + R 2 − 1 (13) and Eq. 12 can be written as ρ = v s H R 2 − 1 2 R ̂ 2 − 1 R 2 − 1 2 v s 2 v s H R 2 − 1 2 R ̂ 2 − 1 R 2 − 1 2 R 1 R 2 − 1 2 R ̂ 2 − 1 R 2 − 1 2 v s v s H R 1 − 1 v s (14) Decomposing R ₁ and R ₂ into R 1 = U c β Σ c + I r c U c H + U n U n H (15) R 2 = U c Σ c + I r c U c H + U n U n H (16)

Assuming that the pulse interference power is much greater than the noise power and β is not too small, which means that β Σ c ≫ I r c . Then we have the following approximate expression R 2 − 1 2 R 1 − 1 R 2 − 1 2 ≈ β U c U c H + U n U n H , R 2 − 1 2 ≈ U n U n H , R 2 − 1 ≈ U n U n H (17) Let C = 1 K ∑ i = 1 Kβ y i y i H + U n U n H − 1 , then ρ = v s H U n U n H C U n U n H v s 2 v s H U n U n H C β U c U c H + U n U n H C U n U n H v s v s H R 1 − 1 v s (18)

According to the derivation of Ref. [23], the distribution of ρ is as follows f ρ ρ = ε r c ρ K β − r c 1 − ρ r c − 1 B r c , K β − r c + 1 ε + 1 − ε ρ K β + 1 I ρ ≥ 0 (19) where B r c , K β − r c + 1 = r c − 1 ! K β − r c ! / ( ( K β ) ! ) is Beta function, and I () is the indicator function, i.e., I x ≥ 0 = 1 , x ≥ 0 0 , x < 0 When Kβ ≥ r _c , the expression of average SINR loss is E ρ ≈ ε K β − r c ε K β − ε − 1 r c × 1 + K β r c ε K β − ε − 1 r c 2 K β − r c − 1 = K − ε r c K − ε − 1 r c 1 + K r c K − ε − 1 r c 2 K − ε r c − ε (20)

In the above derivation, the first assumption, i.e., β Σ c ≫ I r c , is also a necessary condition for the effectiveness of pulse interference. If β Σ c → I r c , which means the average power of interference is less than that of the noise, the effective jamming cannot be achieved. Then the approximation that R _p ≈ β R _c + I is completely valid in spatial anti-jamming processing. In space-time anti-jamming processing, if the baseband of interference is wideband, according to 3) and Wiener-Khinchin theorem, it can be inferred that the approximation is also valid.

The above derivation shows that, the convergence rate is consistent with that of LSMI filter under the amplitude heterogeneous clutter environment when taking the length of interference samples as the reference variable. Combined with the implementation process of the SMI filter, the conclusion can be explained as follow. If the sampling length is long enough, the estimation of the covariance matrix of the noise signal can be approximately accurate. At this time, it can be considered that the unit matrix with amplitude σ n 2 is loaded on the estimated interference covariance matrix, which is same as the LSMI. From the perspective of power, the average power of the pulse interference is proportional to the power of interference samples, which reflects the amplitude heterogeneous of interference power.