Joint DOA and Frequency Estimation Method Based on Direct Data Domain

Abstract

A method for joint Direction of Arrival (DOA) and frequency estimation based on the Direct Data Domain (DDD-JDFE) is proposed. This algorithm, designed to work within an array antenna reception system, utilizes space-time adaptive processing techniques to acquire short-term samples through both spatial and temporal smoothing. It employs a broadband array direction-finding system to simultaneously determine the frequency and arrival direction of targets. Compared to traditional spatial-domain array signal processing methods, this approach provides improved direction-finding performance, particularly at low signal-to-noise ratios. Meanwhile, it shows great potential for long-distance signal detection, and simulation results have verified the effectiveness of the method.

1. Introduction

As the electronic environment on the battlefield becomes more complex, the same time the demand for passive electronic reconnaissance, especially in terms of signal directionality, is also increasing. Broadband array direction-finding has become a primary method for electronic reconnaissance, offering higher sensitivity compared to traditional techniques such as interferometers. This method is widely used in passive reconnaissance and early warning systems. Currently, array direction-finding equipment operates by forming a high gain antenna beam within the array’s airspace. The target signal is then detected through back-end digital channelization, which helps gather information about the target’s intra-pulse, inter-pulse, direction, etc. However, due to cost and antenna aperture limitations, the number of antennas in the array cannot be excessively large. Additionally, the gain in the air domain is closely tied to the digital channelization process. For example, in electronic reconnaissance systems using uniform line arrays, the overall system sensitivity is determined mainly by the combined effect of antenna gain and back-end receiver sensitivity. The antenna gain primarily depends on the antenna aperture and the receiver sensitivity is influenced by the back-end channel processing bandwidth. Particularly in smaller reconnaissance equipment, there is an urgent need for receivers that can enhance the system’s reconnaissance sensitivity.

Direction of Arrival (DOA) and frequency estimation in low signal-to-noise ratio (SNR) environments are critical issues in array signal processing. Scholars have been working extensively to improve array processing gains and enhance parameter estimation accuracy. For angle estimation, A. N. Lemma first proposed an ESPRIT algorithm for joint angle and frequency estimation [1], along with a performance analysis [2]. L. L. Xu et al. proposed an angle estimation algorithm based on the propagator method (PM) [3]. Although both the ESPRIT and PM algorithms improve computational efficiency, their estimation performance is still limited. The Multiple Signal Classification (MUSIC) algorithm [4] and the Capon algorithm [5] can achieve higher precision but still suffer a significant drop in low SNR conditions. For frequency estimation, the FFT-based approach mainly consists of two stages: coarse estimation and fine estimation. Coarse estimation is achieved by searching for the maximum point of the peak in the FFT spectrum, while fine estimation uses interpolation strategies to estimate the error between the true signal frequency and the coarse estimate. The Rife algorithm uses the two largest FFT spectral lines for interpolation to determine the true frequency, but it is prone to errors when the signal frequency is close to a quantized frequency [6]. The Quinn algorithm uses the ratio of the real parts of the two main lobe spectral lines, but its estimation accuracy depends heavily on the signal frequency, and its root mean square error fluctuates significantly at low SNR [7]. The Jacobsen algorithm proposed in [8] uses the three largest FFT spectral lines to correct the frequency estimate, achieving relatively good results at low SNR with fewer FFT points, though its accuracy is still not ideal. The QIIN algorithm proposed in [9], based on Quinn, introduces a Fourier coefficient interpolation iterative method that achieves estimation variance close to the Cramér-Rao Lower Bound (CRLB) across the entire frequency band after two iterations. However, it requires calculating the DFT coefficients at two points in each iteration, resulting in high computational complexity, and performs poorly at low SNR. From the perspective of array antennas, the study in reference [10] proposes leveraging the orthogonal characteristics of orbital angular momentum (OAM) vortex modes to design continuous aperture distributions and concentric ring arrays with circular support structures. By optimizing the selection of OAM modes and the array layout, it achieves precise control over the field distribution in the target direction while enhancing DOA estimation resolution. However, discretizing continuous aperture sources necessitates frequent optimizations of the element layout, resulting in a substantial increase in hardware implementation costs.

Space-time adaptive processing (STAP) simultaneously performs adaptive processing on signals in both the temporal and spatial domains, thereby increasing the degrees of freedom in processing. For a typical N-channel, M-channel array digital reception system, STAP can enhance signal gain by $\sqrt{NM}$ dB. This makes STAP an ideal solution for improving detection sensitivity in ground-based electronic reconnaissance systems. Jiahua Zhu et al. improved the output signal-to-noise ratio (SNR) and angular resolution by enhancing the Eckart filter and designing an adaptive background normalization beamforming-based weighting matrix, outperforming traditional representative beamforming methods. However, the reliance on adaptive beamforming introduces higher system complexity [11,12,13].

To address the application of space-time adaptive processing (STAP) in electronic reconnaissance, a new method based on Direct Data Domain for joint Direction of Arrival (DOA) and frequency estimation (DDD-JDFE) is proposed. Under the array reception system, this receiver acquires short-term samples through spatial and temporal smoothing while simultaneously estimating both the DOA and frequency of targets using adaptive processing. By processing signals in both the spatial and frequency domains, the method achieves higher signal gain compared to spatial beamforming. Additionally, it can determine DOA estimates without requiring prior knowledge of the signal’s precise frequency and demonstrates superior direction-finding performance at lower signal-to-noise ratios (SNR) than spatial domain array signal processing alone. Simulation results validate the effectiveness of the proposed method, showing it outperforms traditional spatial DOA and conventional frequency estimation methods in terms of noise resistance and estimation accuracy.

2. Signal Model

The array signal model is illustrated using a uniform linear array (ULA), with the arrangement structure of each array element shown in Figure 1:

In Figure 1, the array elements are equally spaced and arranged in a straight line, forming a uniform linear array. Taking the uniform linear array as the receiver antenna, the signals sampled simultaneously by all array elements are represented by a vector x(t). Using the first array element as the reference point, the spacing between adjacent elements is d. Assuming a plane wave propagates toward the array at an angle $\theta$ relative to the array normal (broadside direction), under narrowband conditions, if the signal is s(t), the received array signal can be expressed as:

$$x(t)=s(t)·{\left[1,e^{j{\displaystyle \frac{2\pi d\sin \theta }{\lambda }}},\cdots ,e^{j(N-1){\displaystyle \frac{2\pi d\sin \theta }{\lambda }}}\right]}^T=s(t)·a(\theta )$$

(1)

where $\lambda$ is the wavelength, $a\left(\theta \right)$ is the array steering vector. If there are P signals arriving at an N-element array from the directions of angles ${\theta }_1,{\theta }_2,\dots ,{\theta }_P$, respectively, N > P. The signal received by the array can be represented as:

$$X(t)=A(\theta )S(t)+N(t)={\displaystyle \sum _{p=1}^P{s}_p(t)e^{j\omega t}a({\theta }_p)}+N(t)$$

(2)

where N(t) is the additive white Gaussian noise. It is easy to understand from the above signal model that the signal vector X(t) received by the N-array elements at any moment t is composed of the instantaneous amplitude values of the P signals. The signal vector X(t) received by N-array elements at any time t is a linear combination of the instantaneous amplitude values $s_1(t)e^{j\omega t}$, $s_2(t)e^{j\omega t},\cdots ,s_P(t)e^{j\omega t}$ of P signals and the steering vectors $a({\theta }_1),a({\theta }_2),\cdots ,a({\theta }_P)$ of these P signals.

If the frequency of the signal is $f_0$ and the sampling frequency is $f_s$, then the signal at a certain element over time can be represented as:

$$x_i(n)=a_t\left(n\right)\exp \left(j2n\pi f_0/f_s\right)+N(n),n=1,2,3,\dots ,M$$

(3)

where M represents the number of time sampling points, i denotes the index of the array element, i = 1, 2, …, N, $a_t\left(·\right)$ represents amplitude. Generally, due to the relatively low attenuation of electromagnetic waves in the air, the received signal can be approximately regarded as a stationary random process over time. Therefore, the samples obtained through time-domain smoothing can be approximately considered to satisfy the characteristics of being independent and identically distributed.

3. The Proposed Method

3.1. Design of Digital Receiver

Due to this study on the joint DOA and frequency estimation method based on a uniform linear array, its engineering implementation does not require large-scale active-phased array circuits. The input signals from the array antennas can be directly sampled via analog-to-digital converters (ADCs) after passing through filtering and amplification circuits, thereby forming spatio-temporal two-dimensional snapshot data. The proposed method primarily focuses on the joint spatio-temporal processing of array snapshot data and the estimation of DOA and frequency. The designed receiver processing architecture is illustrated in Figure 2.

In the diagram, N antennas are arranged in a uniform linear array. The data received by each antenna undergoes radio frequency transformation followed by digital sampling, forming a two-dimensional data matrix. This matrix is then segmented, with each segment consisting of M pulses. For each data segment, space-time joint adaptive processing is applied to perform direction finding and frequency measurement simultaneously. After processing each segment, the detection results are temporally fused to form the signal detection results for all data over the entire period. Due to the limited data volume in each time segment, the processing gain is constrained. The fundamental idea behind space-time joint processing, due to the small amount of data in each time segment, is to forward and backward smooth the data received by each antenna array channel in both the spatial and temporal domains to obtain several samples. The samples within the window are sub-aperture received data in the spatio-temporal domain as illustrated in Figure 3. This approach is used to estimate the signal’s covariance matrix for the design of an optimal two-dimensional space-time filter.

In Figure 3, the horizontal axis represents temporal snapshot data, and the vertical axis denotes spatial array channels. The red solid-line box indicates the scanning sub-aperture, with spatial and temporal ranges of Nm and Km, respectively. The process begins by acquiring N × K snapshot data through N-array element channels receiving K-point temporal signals. The scanning sub-aperture is first horizontally smoothed to the rightmost edge. During each smoothing iteration, an Nm × Km data matrix is extracted. Subsequently, for every downward shift of one point, horizontal smoothing is repeated, as illustrated by the blue and green dashed-line boxes. After L-smoothing iterations, a spatio-temporal data matrix of size Nm × Km × L is obtained for space-time adaptive processing. Following STAP, joint estimation of DOA and frequency is performed on the processed data.

3.2. DDD-JDFE Method

The essence of the Direct Data Domain (DDD) [14] approach lies in spatially and temporally filtering out the target from the array-received data and then performing subspace smoothing to obtain sufficient samples for adaptive processing. However, the DDD algorithm, in order to garner an adequate number of training samples from the current range cell, must incur a significant aperture loss in both the spatial and temporal domains. Conversely, to ensure minimal aperture loss in these domains, it must accept the trade-off of insufficient training samples. Both of these scenarios contribute to a decrease in the detection performance of space-time adaptive processing. In practice, a balanced consideration of these two aspects is typically employed to make an informed selection.

If the spatial domain phase corresponding to the direction of the target signal is ${\phi }_s({\psi }_{so})$, the time domain phase is ${\phi }_T(f_o)$, and the phase difference in the space-time domain direction is ${\phi }_s({\psi }_{so})$ + ${\phi }_T(f_o)$. According to the phase differences in the spatial domain, the temporal domain, and the space-time domain of the signal matrix, after performing two-element (two-pulse) cancellation and signal filtering in the spatial domain, the temporal domain, and the space-time domain, respectively, on the matrix X, the following three matrices can be obtained:

$$X_s=X(1:N-1,:)-\exp (-j{\phi }_s({\psi }_{so}))X(2:N,:)$$

(4)

$$X_T=X(:,1:K-1)-\exp (-j{\phi }_T(f_o))X(:,2:K)$$

(5)

$$X_{ST}=X(1:N-1,1:K-1)-\exp (-j{\phi }_T(f_o)-j{\phi }_s({\psi }_{so}))X(2:N,2:K)$$

(6)

where $X_S$, $X_T$, and $X_{ST}$ are data matrices of (N − 1) × K dimensions, (K − 1) × N dimensions, and (N − 1) × (K − 1) dimensions, respectively. They correspond to the data matrices of the spatial, temporal, and space-time domains, respectively. And the K is the number of snapshots. Assume that the sliding sub-apertures in the spatial domain and the time domain are Nm and Km, respectively. The total number of samples that can be obtained after the forward and backward sliding of matrices $X_S$, $X_T$ and $X_{ST}$ is:

$$L=2((N-Nm)(K-Km+1)+(N-Nm+1)(K-Km)+(N-Nm)(K-Km))$$

(7)

Take these L samples as training samples, denoted as $X_m^l,l=1,2,\dots ,L$. The clutter covariance matrix estimated from these training samples is:

$$R={\displaystyle \frac{1}{L}}{\displaystyle \sum _{l=1}^{L}Vec\left(X_m^l\right)Ve{c}^H\left(X_m^l\right)}$$

(8)

In the equation, Vec represents the following operation on a matrix: Place the second column of the matrix below the first column, place the third column below the second column, and so on, transforming the matrix into a column vector. The adaptive weight is proposed according to the following optimization problem:

$$\left\{\begin{array}{l}\underset{W}{\min } W^{H}RW\\ s.t W^H{S}_m=1\end{array}\right.$$

(9)

In the equation, $S_m$ is the joint space-time steering vector, pointing to the target signal, is composed of the first Nm and Km pulses. The space-time joint steering vector is obtained by computing the Kronecker product of the temporal domain steering vector $V_t$ and the spatial domain steering vector $V_a$.

$$S_m=V_t\otimes V_a$$

(10)

$$V_t=\exp \left(j2n\pi f_0/f_s\right),n=1,2,3,\dots ,M$$

(11)

$$V_a=a\left(\theta \right)=\exp \left(j2\pi \left(n-1\right)d\sin \theta /\lambda \right),n=1,2,3,\dots ,N$$

(12)

After determining the target and conditions, the optimal weight vector can be calculated by constructing the Lagrange function:

$$J\left(W\right)=W^{H}RW+\eta \left(1-W^{H}a\left(\theta \right)\right)$$

(13)

Further, the gradient formula with respect to W is obtained as:

$$\nabla J\left(W\right)=2RW+2\eta a\left(\theta \right)=0$$

(14)

The expression of the optimal weight vector W* obtained by solving is:

$$W^{\ast }={\displaystyle \frac{R^{-1}{S}_m}{S_m^H{R}^{-1}{S}_m}}$$

(15)

After applying adaptive filtering to the space-time two-dimensional data $X_m^l$ within the sub-aperture smoothing window using the optimal weight vector W*, the resulting output is:

$${X_m^l}^{\prime }=W^{\ast H}{X}_m^l$$

(16)

Next, compute the covariance matrix $R^{\prime }$ from the output results and derive the two-dimensional spectrum in the spatial and frequency domains using the space-time joint steering vector $S_m$. Finally, obtain the target frequency and angle information via a joint search over both Direction of Arrival (DOA) and frequency.

$$\left(f_i,{\theta }_i\right)=\arg \underset{(\mathrm{f},\theta )}{\max }\left|{\displaystyle \frac{1}{S_m{R^{\prime }}^{-1}{S}_m^H}}\right|,i=1,2,\dots ,K$$

(17)

It is important to note that in passive electronic reconnaissance, since the frequency of the signal is unknown, the wavelength is also unknown. Conventional pure spatial domain array signal processing requires obtaining the signal frequency in advance, followed by the calculation of the signal’s DOA. For space-time adaptive processing, due to the degrees of freedom in the time dimension, a pre-set reference wavelength (${\lambda }_0$) can be used as the wavelength corresponding to the spatial domain steering vector in advance. After performing the two-dimensional space-time adaptive processing, after obtaining the signal frequency, the true DOA angle of the signal can be obtained through calculation. If the signal DOA angle obtained after the space-time adaptive processing is ${\theta }_0$, then the relationship between the true signal angle ${\theta }_i$ and ${\theta }_0$ can be given by:

$${\displaystyle \frac{2\pi d\cos {\theta }_0}{{\lambda }_0}}={\displaystyle \frac{2\pi d\cos {\theta }_i}{c}}{f}_i$$

(18)

where $f_i$ is signal frequency, c is the propagation speed of electromagnetic waves, and the true value of ${\theta }_i$ can be calculated by using Equation (14).

The implementation steps of the joint space-time frequency measurement and direction-finding algorithm are as follows (Algorithm 1):

Algorithm 1. The Proposed Algorithm for Joint DOA and Frequency Estimation
Input:	$X(1:N,1:K)$; N is the number of antenna elements, K is the number of snapshots;
Initialize:	$\mathrm{Search} \mathrm{range} \mathrm{of} \mathrm{angles}: {\psi }_{s0}=[1:0.01:90]$ degree; $\mathrm{Search} \mathrm{range} \mathrm{of} \mathrm{frequency}: f_0=[0.5e9:0.01e6:1e9]$ Hz; Aperture: Nm = 12; Km = 24;
1:	$\mathrm{for} {\psi }_{s0}=[1:0.01:90]$    $\mathrm{for} f_0=[0.5e9:0.01e6:1e9]$
2:	Two-array phase-elimination filtering:    $X_S=X(1:N-1,:)-\exp (-j{\phi }_S({\psi }_{S0}))X(2:N,:)$;    $X_T=X(:,1:K-1)-\exp (-j{\phi }_T(f_0))X(:,2:K)$; $X_{ST}=X(1:N-1,1:K-1)-\exp (-j{\phi }_T(f_0)-j{\phi }_S({\psi }_{S0}))X(2:N,2:K)$;
3:	Forward-backward windowing processing:    $X_m^l=windowing\left(X_s,X_T,X_{ST}\right)$;
4:	$R={\displaystyle \frac{1}{L}}{\displaystyle \sum _{l=1}^{L}Vec\left(X_m^l\right)Ve{c}^H\left(X_m^l\right)}$;
5:	$W^{\ast }={\displaystyle \frac{R^{-1}{S}_m}{S_m^H{R}^{-1}{S}_m}}$;
6:	$R^{\prime }={\displaystyle \frac{1}{L}}{\displaystyle \sum _{l=1}^L\left(W^{\ast H}{X}_m^l\right){\left(W^{\ast H}{X}_m^l\right)}^H}$;
7:	$P\left({\psi }_{s0},f_0\right)=\left|{\displaystyle \frac{1}{S_m{R^{\prime }}^{-1}{S_m}^H}}\right|$;
8:	end end
9:	Peak search in two-dimensional matrices: $[ro{w}_i,co{l}_i]=peak\left\{P\left({\psi }_{s0},f_0\right)\right\},i=1,2,\dots ,sN$, sN is the number of signals.
10:	${\widehat{\theta }}_i={\psi }_{s0}\left(ro{w}_i\right), {\widehat{f}}_{0i}=f_0\left(co{l}_i\right), i=1,2,\dots ,sN$
11:	${\displaystyle \frac{2\pi d\cos {\widehat{\theta }}_{oi}}{{\lambda }_0}}={\displaystyle \frac{2\pi d\cos {\widehat{\theta }}_i}{c}}{\widehat{f}}_{oi}$
Return:	$\widehat{\theta }=\left[{\widehat{\theta }}_{oi}\right]$$, {\widehat{f}}_0=\left[{\widehat{f}}_{0i}\right]$$, i=1,2,\dots ,sN$

3.3. Array Channel Uncertainty Calibration

In practical engineering applications, after the design of an array antenna is completed, element channel errors inevitably exist. These errors are typically categorized into three types: array position errors, amplitude-phase inconsistency errors across array channels, and antenna mutual coupling errors [15]. Array position errors primarily arise from manufacturing tolerances and mechanical vibrations. Amplitude-phase inconsistency errors across channels mainly result from environmental factors (e.g., atmospheric pressure and temperature variations) and component parameter mismatches, forming unpredictable random errors. These errors ultimately lead to distortions in the antenna radiation pattern, beam pointing deviations from the intended direction, and amplitude-phase distortions in received signals. Consequently, they degrade antenna gain and DOA (Direction of Arrival) estimation accuracy. Therefore, amplitude-phase error calibration must be performed prior to DOA estimation. This paper adopts an auxiliary source calibration algorithm [16] to address these errors.

According to Equation (1), the received snapshot data of a uniform linear array in the presence of amplitude-phase errors can be expressed as:

$$x^{\prime }\left(t\right)=\left[{\rho }_1{e}^{j{\phi }_1},{\rho }_2{e}^{j{\phi }_2},\dots ,{\rho }_N{e}^{j{\phi }_N}\right]\times a(\theta )\times s(t)+N\left(t\right)=\Gamma \times a(\theta )\times s(t)+\mathrm{N}\left(t\right)$$

(19)

where, $\Gamma =diag\left[{\rho }_1{e}^{j{\phi }_1},{\rho }_2{e}^{j{\phi }_2},\dots ,{\rho }_N{e}^{j{\phi }_N}\right]$, let ${\rho }_i$ and ${\phi }_i$ denote the gain and phase of the i-th array element, respectively, where i = 1, 2, …, N. N is the total number of elements, and N(·) represents the internal white noise of the receiver.

An auxiliary source is deployed at a known spatial direction ${\theta }_s$ with power ${\sigma }_s$. The array covariance matrix can be expressed as:

$$R_s={\sigma }_s^2\Gamma a\left({\theta }_s\right)a^H\left({\theta }_s\right)\Gamma +{\sigma }^{2}I$$

(20)

Perform eigenvalue decomposition on the array covariance matrix $R_s$, obtaining:

$$\Gamma a\left({\theta }_s\right)=k\epsilon$$

(21)

where $\epsilon$ is the eigenvector corresponding to the largest eigenvalue of $R_s$, and k is an unknown complex constant. Let

$$a\left({\theta }_s\right)=\left[1,a_2\left({\theta }_s\right),\dots ,a_N\left({\theta }_s\right)\right]$$

(22)

$$\epsilon ={\left[{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_N\right]}^T$$

(23)

From Equation (21), we obtain:

$$\left[\begin{array}{l}1\\ {\Gamma }_2{a}_2\left({\theta }_s\right)\\ ⋮\\ {\Gamma }_N{a}_N\left({\theta }_s\right)\end{array}\right]=\left[\begin{array}{l}k{\epsilon }_1\\ k{\epsilon }_2\\ ⋮\\ k{\epsilon }_N\end{array}\right]$$

(24)

And:

$$k={\displaystyle \frac{1}{{\epsilon }_1}}$$

(25)

Therefore, the amplitude-phase calibration values for the array channels are:

$${\Gamma }_i={\displaystyle \frac{k{\epsilon }_1}{a_i\left({\theta }_s\right)}}={\displaystyle \frac{k{\epsilon }_1}{e^{\left(j2\pi \left(i-1\right)d\sin \theta /\lambda \right)}}},i=1,2,\dots ,N$$

(26)

3.4. Complexity Analysis

In practical operation, radar reconnaissance equipment typically requires real-time processing capability. If the computational complexity of the processing algorithm is excessively high, it not only poses challenges for system implementation but may also hinder real-time online processing. The computational load of the proposed method in this paper primarily arises from two components: STAP and the two-dimensional spatial-frequency spectrum calculation. Let N denote the spatial degrees of freedom (DOFs) in STAP, K the temporal DOFs, resulting in total DOFs of NK. Assuming L is the number of samples used to estimate the covariance matrix, the algorithmic steps and computational complexity are summarized as follows:

Select L samples of N × K degrees of freedom data to compute the covariance matrix, requiring $O\left(L\times {\left(NK\right)}^2\right)$ complex multiply-accumulate (MAC) operations. Calculate the spatio-temporal two-dimensional adaptive optimal weight vector based on the covariance matrix, with a computational load of $O\left(3{\left(NK\right)}^3+{\left(NK\right)}^{2}K\right)$ complex MAC operations. Apply the optimal weight vector for spatio-temporal filtering, requiring $O\left(L\times {\left(NK\right)}^2\right)$ complex MAC operations. Recompute the covariance matrix using the filtered spatio-temporal data, involving $O\left(L\times {\left(NK\right)}^2\right)$ complex MAC operations. Compute the spatial-frequency two-dimensional spectrum using the covariance matrix and spatio-temporal steering vector, consuming $O\left({\left(NK\right)}^3\right)$ complex MAC operations. Perform a joint search over S grid points (total search length S), requiring $O\left(S\times N^2\right)$ operations. The overall computational complexity is $O\left(\left(3L+K\right){\left(NK\right)}^2+4{\left(NK\right)}^3+S{N}^2\right)$ operations. For engineering implementation, it is recommended to consider employing FPGA to accomplish spatio-temporal two-dimensional smoothed data acquisition and covariance matrix computation, leveraging the FPGA’s pipelined architecture to achieve accelerated parallel computing.

4. Simulation Results

4.1. Simulation Setup

In the simulation experiment, the number of array antennas is selected as 16. The frequency band of the measured signal ranges from 0.6 GHz to 0.9 GHz, with an antenna spacing of 0.17 m and a preset reference wavelength of 0.37 m. The design ensures that the antenna reconnaissance can cover 90° instantaneously. Four targets are simulated within the 90° range, and all of them have pulse overlapping. The target parameters are shown in the following

Table 1. List of signal parameters.

No.	Signal Type	Signal Parameters
1	Single-frequency signal	Frequency: 0.74 GHz, Direction: 30°
2	Chirp Signal	Frequency: 0.62 GH, Band: 20 MHz, Direction: 50°
3	Single-frequency signal	Frequency: 0.83 GH, Direction: 50°
4	Single-frequency signal	Frequency: 0.74 GH, Direction: 60°

:

The digital sampling frequency is set to 2.4 GHz, and the pulse width of the four signals is 5 microseconds (us), resulting in 11,500 sampled data points per pulse. Assuming the number of snapshots K selected for each processing segment is 256, within these snapshots, the targets can be approximated as single-frequency signals. For the second signal, the bandwidth variation within 256 snapshots is approximately 0.25 MHz.

To evaluate the performance of the proposed algorithm, the number of Monte Carlo experiments is set to 1000, with P representing the number of sources. The root mean square error (RMSE) for angle and frequency estimation is defined as:

$$RMS{E}_{\theta }={\displaystyle \frac{1}{P}}{\displaystyle \sum _{p=1}^P\sqrt{\frac{1}{1000}{\displaystyle \sum _{i=1}^{1000}{\left({\theta }_{k,i}-{\theta }_k\right)}^2}}}$$

(27)

$$RMS{E}_f={\displaystyle \frac{1}{P}}{\displaystyle \sum _{p=1}^P\sqrt{\frac{1}{1000}{\displaystyle \sum _{i=1}^{1000}{\left(f_{k,i}-f_k\right)}^2}}}$$

(28)

During the experiments, the added noise is Gaussian white noise, and the estimation errors for angle and frequency follow a normal distribution. By constructing the likelihood function and deriving the Fisher information matrix, the Cramér-Rao Lower Bound (CRLB) [6,17] for the estimation of angle and frequency can be calculated after inversion:

$$CRL{B}_{\theta }={\displaystyle \frac{6}{{\left(2\pi \right)}^{2}M\frac{M+1}{M-1}SNR{\left(\frac{L}{\lambda }\right)}^2{\sin }^2\theta }}$$

(29)

$$CRL{B}_f={\displaystyle \frac{3{f_s}^2}{2{\pi }^{2}SNR\times N\left(N^2-1\right)}}$$

(30)

In Equation (17), M represents the number of antenna array elements, and L represents the total length of the linear array.

4.2. Comparative Simulation

When the sub-aperture is selected as Nm = 10 and Km = 24, the total number of samples is 8828. The dimensionality of the joint space-time sub-aperture processing is 240, which means the number of samples is significantly greater than twice the system processing dimensionality.

Figure 4a shows the detection results of four incoming signals without space-time filtering, while Figure 4b shows the detection results after space-time filtering. It can be observed that after two-dimensional space-time joint processing, the overall signal-to-noise ratio (SNR) improvement is approximately 12 dB. Additionally, the method effectively resolves and detects two signals with the same direction but different frequencies (Signal 1 and Signal 4) as well as two signals with the same frequency but different directions (Signal 2 and Signal 3).

Figure 5 shows the detection results of traditional pure spatial domain DOA estimation methods: Capon, MUSIC, and PM. It can be observed that direct spatial domain processing cannot directly distinguish signals of the same frequency. Moreover, since only the spatial degrees of freedom are utilized, the gain extraction for targets is significantly lower compared to joint space-time processing. In comparison with existing array reception systems, the SNR improvement of the method proposed in this paper is shown in the

Table 2. Comparison with the signal gain of existing receivers.

Comparison Content	The Proposed Method	Existing Receivers
signal gain	Nm × Km = 240	$\sqrt{\mathrm{N}\mathrm{M}}=\sqrt{16\times 64}$ = 32

below:

It can be observed that by employing space-time joint processing, an improvement of nearly 10 dB in signal detection gain can be achieved under the condition of an unchanged antenna array configuration.

4.3. Array Error Robustness Simulation

Usually, after the design of an array antenna is completed, there will inevitably be array element channel error due to engineering precision issues. To analyze the adaptability of the proposed method to array element channel errors, a sub-aperture of Nm = 10 and Km = 24 was established, and simulations were conducted under conditions of array element channel errors of 4%, 6%, and 8%, respectively. The simulation results are as follows.

As can be seen from the simulation results in Figure 6, it is evident that channel errors significantly impair the direction-finding performance in both spatial and temporal dimensions. When the channel error is at 6%, although there is a degradation in performance, the fundamental direction-finding capability can still be maintained. However, when the channel error reaches 8%, the accuracy and resolution of direction finding are no longer sufficient to distinguish multiple targets. Therefore, in practice, the channel errors of the array should be calibrated and controlled to within 6% as much as possible.

Building upon the aforementioned experiments, a further analysis was conducted to examine the response of sub-aperture sizes to array element channel errors. A comparative simulation experiment was performed to evaluate the DOA estimation performance across different sub-aperture sizes. The sub-aperture sizes were set to (Nm, Km) = (6, 24), (8, 24), and (10, 24), with channel errors of 2%, 4%, and 6%, respectively. The simulation results are as follows.

Figure 7 reveal that channel errors can significantly impair the direction-finding performance in both spatial and temporal dimensions. However, by increasing the processing dimensions, a certain degree of resistance to channel errors can be achieved. For the same level of channel error, the performance with smaller processing dimensions is markedly inferior to that with higher processing dimensions. Nevertheless, in practical engineering applications, indiscriminately increasing the processing dimensions can lead to a substantial increase in computational load. Therefore, it is advisable to select an appropriate processing dimension that balances the engineering requirements with computational efficiency.

4.4. Estimation Performance Comparison Simulation

To verify that the space-time joint direction-finding processing can both enhance the signal detection SNR and parameter estimation accuracy, Signal 4 from Table 1 was selected for Monte Carlo comparative experiments under different SNR conditions, different snapshot numbers, and different numbers of array elements. The DOA estimation performance was compared with traditional methods such as CAPON, ESPRIT, MUSIC, and PM, while the frequency estimation performance was compared with typical frequency estimation methods, including Jacobsen, Quinn, Rife, and QIIN. In the three sets of experiments, the variables were set as $SNR\in \left[-20 \mathrm{dB},10 \mathrm{dB}\right]$, $K\in \left[64,512\right]$, $M\in \left[8,18\right]$, and the invariant default values were set at SNR = 0 dB, K = 256, and M = 16. The simulation results are shown in the figure below.

It is evident that the parameter estimation performance of various algorithms degrades as the SNR decreases. The proposed space-time joint processing method in this paper outperforms other traditional pure spatial domain DOA estimation algorithms and conventional frequency estimation methods in both DOA estimation performance and frequency estimation performance, delivering superior performance even in low SNR conditions. Particularly in the frequency estimation performance comparison, when the SNR is below −10 dB, the performance of conventional frequency estimation algorithms deteriorates significantly, whereas the proposed algorithm still performs well, maintaining a consistent trend with the Cramér-Rao Lower Bound (CRLB). The simulation results of Figure 8 demonstrate that the two-dimensional joint frequency and direction estimation is also related to the signal-to-noise ratio, but the coherent integration gain of this two-dimensional approach far surpasses traditional pure spatial direction-finding methods and conventional frequency estimation methods. Additionally, comparative experiments under different snapshot numbers and different array element numbers confirm that the performance of the proposed method is consistently better than the others.

4.5. Multi-Target Resolution Simulation

Multi-target resolution, a critical performance metric in DOA estimation research, refers to the minimum angular separation between two closely spaced sources that a system can distinguish. The following section provides the formal definition of resolution and the criteria for determining resolvability.

First, define:

$$P_{peak}=\left[P\left({\theta }_1\right)+P\left({\theta }_2\right)\right]/2$$

(31)

In the equation, $P\left({\theta }_i\right)$ denotes the value of the signal spectrum (excluding the noise spectrum) at the corresponding angle ${\theta }_i$, as derived from the combined result of Equations (10) and (17):

$$P\left({\theta }_i\right)=\left|{\displaystyle \frac{1}{\left[\exp \left(j2n\pi f_0/f_s\right)\otimes a\left({\theta }_i\right)\right]{R^{\prime }}^{-1}{\left[\exp \left(j2n\pi f_0/f_s\right)\otimes a\left({\theta }_i\right)\right]}^H}}\right|$$

(32)

Then, define:

$${\theta }_m=\left({\theta }_1+{\theta }_2\right)/2$$

(33)

$$Q\left(\Delta \right)=P_{peak}-P\left({\theta }_m\right)$$

(34)

In the equation, $\Delta =\left|{\theta }_1-{\theta }_2\right|$. When $Q\left(\Delta \right)>0$, $Q\left(\Delta \right)$ is resolvable with a resolution of $\Delta$. This constitutes the formal definition of resolution [18].

Typically, the angular resolution of an array antenna is primarily determined by its aperture length [19]. The aperture length governs the minimum detectable phase difference between signals, which in turn affects the theoretical angular resolution. However, the practical resolution is limited by the measurement accuracy of these phase differences. Under low SNR conditions, noise disrupts the statistical properties of signals and exacerbates ambiguity effects, leading to a significant degradation in practical resolution.

To analyze the multi-target resolution performance of the proposed method, simulation experiments were conducted under fixed array errors to measure resolution across varying SNR and array aperture lengths. The aperture length is defined as D = (M − 1)d, where M is the number of array elements. Signal 4 from Table 1 was selected as the fixed target signal, with a snapshot count K = 256. The number of array elements M was set to 4, 8, 12, and 16, while the SNR range was configured between −10 and 50 dB. The DOA and frequency parameters of the other target were adjusted to gradually approach those of the fixed target. When the detection results failed to resolve multiple targets, the difference $\Delta$ between the last resolvable parameters and those of the fixed target was recorded as the resolution metric. Monte Carlo experiments were conducted for each SNR and array length condition. The simulated resolution performance for DOA and frequency is shown in the following figure(s).

As observed from the simulation results in Figure 9a, under low SNR conditions, noise significantly distorts the array phase measurements. Consequently, the practical angular resolution deviates from the ideal performance, being influenced not only by the number of array elements (i.e., aperture length) but also by noise. In contrast, at high SNR levels, the angular resolution is predominantly governed by the number of array elements (aperture length), consistent with the conclusions in [19].

From Figure 9b, the frequency resolution exhibits similar trends to the angular resolution analysis. Under low SNR conditions, frequency resolution suffers severe noise interference. However, increasing the number of array elements enhances the array data sample size and signal gain, thereby improving frequency resolution. At high SNR levels, frequency resolution becomes less affected by noise, with performance primarily dependent on the number of array elements. This aligns with the frequency estimation performance discussed in Section 4.4.

In summary, the joint estimation algorithm proposed in this paper achieves superior parameter estimation accuracy for single-target scenarios while maintaining robust multi-target resolution capability. However, in practical applications where low SNR environments are prevalent, the multi-target resolution performance is predominantly limited by noise, given fixed array aperture lengths and calibration errors. To address this, noise resilience can be further enhanced through techniques such as noise subspace weighting, thereby improving resolution reliability under adverse SNR conditions.

5. Conclusions

This paper addresses the application of space-time adaptive processing in electronic reconnaissance and proposes a fast method for frequency and direction finding, based on the Direct Data Domain. In an array reception system, the receiver acquires samples over a short period through spatial and temporal smoothing. It then employs adaptive processing to identify the frequency and direction finding of targets. This approach allows for simultaneous determination of the signal’s Direction of Arrival (DOA) and frequency. By integrating spatial-frequency joint processing, it enhances the signal processing gain beyond what spatial beamforming alone can achieve. Simulations of typical spatially overlapping signals have been carried out, validating that the proposed method improves the resolution in both azimuth and frequency domains, as well as the ability to detect signal-to-noise ratio (SNR). Furthermore, comparative simulations with traditional frequency and direction-finding methods show that the proposed method offers improved detection capabilities at low SNR levels and is more resistant to superior noise resistance.