Two beams of coherent signals satisfy the condition is referred to as coherent signals,
coherence condition (Coherent Condition):
two beams meet in the signal area: ① the same vibration direction;
② the same oscillation frequency;
same phase or a phase difference remains constant ③
then the two beam signals Interference will occur in the area where they meet.
Can issue
Mathematical Model of Coherent Signal Source
When looking at multiple signals, these signals can be uncorrelated, correlated, or coherent. For two
A 
sum of stationary signals 
, define their correlation coefficient as
From Schwartz's inequality 
, the correlation between signals is defined as follows:
From the above definition, it can be seen that when the signal sources are coherent, the mathematical performance is: there is only one difference between coherent signal sources.
A complex constant. Coherent signal source model:
Where 
is a dimensional vector composed of a series of complex constants 
. n is the number of coherent signal sources incident on the array.
From the above mathematical model of the coherent signal source, it can be seen that when the signal source is completely coherent, the rank of the data covariance matrix received by the array is reduced to 1, which obviously causes the dimension of the signal subspace to be smaller than the number of signal sources. That is to say, the signal subspace "spreads" into the noise subspace, which will cause the steering vectors of some coherent sources to be not completely orthogonal to the noise subspace, so that the signal source direction cannot be estimated correctly.
So how to correctly estimate the signal direction in the case of a coherent signal source? The answer is decoherence or decorrelation . It is through a series of processing or transformation that the rank of the signal covariance matrix is effectively restored, so as to correctly estimate the signal source direction. At present, there are basically two types of decoherence processing: one is dimensionality reduction processing; the other is non-dimensionality reduction processing.
The dimensionality reduction processing algorithm is a commonly used decoherence processing algorithm, which can be divided into two types of algorithms based on spatial smoothing and matrix reconstruction. Among them, the spatial smoothing algorithm is mainly composed of forward spatial smoothing algorithm, two-way spatial smoothing algorithm, modified spatial smoothing algorithm and spatial filtering method; the algorithm based on matrix reconstruction mainly refers to matrix decomposition algorithm and vector singular value method. The difference between these two types of algorithms is that the covariance matrix modified by the matrix reconstruction algorithm is a rectangular matrix (singular value decomposition is required to estimate the signal subspace and the noise subspace), while the matrix modified by the spatial smoothing algorithm is a square matrix ( The estimated signal subspace and noise subspace can be decomposed by eigenvalues).
Non-dimensionality reduction processing algorithms are also an important class of decoherent processing methods, such as frequency-domain smoothing algorithms, Toeplitz methods, virtual array transformation methods, and so on. Compared with the dimensionality reduction algorithm, the biggest advantage of this type of algorithm is that there is no loss of the array aperture, but this type of algorithm is often aimed at specific environments, such as broadband signals, non-equidistant arrays, and mobile arrays. Here is an algorithm: MUSIC algorithm based on spatial smoothing.
Spatial smoothing algorithm
It is only suitable for uniform array (ULA) in general. The following briefly introduces the principle of spatial smoothing MUSIC using sub-array smoothing to restore the data covariance matrix.
Suppose the base matrix is an M-element equally spaced linear array with the spacing d. Divide the M-element linear array into q sub-arrays, and the number of elements in each sub-array is p, as shown in Figure 1. Where the relationship between p and q satisfies
Figure 1 Divide the array element into multiple sub-arrays
The signal matrix of each sub-array is expressed as
Find the covariance matrix for each sub-array separately, and then take the average to get the forward covariance matrix estimate
In the estimation of the most covariance matrix, the forward and backward spatial smoothing sub-matrix divisions are often used for joint coordination.
Variance matrix estimation:
Among them, J is the exchange matrix of order p, except the element on the subdiagonal line is 1, and the others are zero.
Now that decoherence is achieved through spatial smoothing, then we will summarize the calculation process of the MUSIC algorithm based on spatial smoothing.
1. Obtain the data covariance matrix R from the received data of the array. (For details, see the 3-MUSIC algorithm of the positioning algorithm series. The array model and algorithm process are described in detail, so I won’t repeat it.)
2. Use the spatial smoothing algorithm introduced in this article to revise R.
3. Use the corrected covariance matrix to estimate the MUSIC spectrum and find the signal direction corresponding to the maximum value. (Decompose the modified covariance matrix into the new subspace and substitute it in. If 
you want to know the source of the specific formula, please go back to the 3-MUSIC algorithm of the positioning algorithm series).
For the MUSIC algorithm based on spatial smoothing, there is one thing to note! The essence of the spatial smoothing algorithm is the process of restoring the rank of the data covariance matrix, but this process is usually only suitable for equidistant uniform linear arrays , and the dimension of the modified matrix is smaller than that of the original matrix, which means that the coherence is resolved. Performance is traded by reducing degrees of freedom.
The two signal sources of signals that interfere with each other are called coherent signal sources.