MUSIC算法的數學模型
Generally, we assume there are M far-field narrow-band
signals which are irrelevant to each other and come in
different angles m θ ,received by N ULA element, with array
space is d. The data received by N-array is
x(ti) , i = 1,2,...snap _ nu , snap _ num is the sampling points,
the signal receivied by the antenna can expressed as:
壹般來說,我們假定有相互獨立的遠場窄帶信號,以不同的角度被N ULA接收為陣列式內的元素,陣列式記為D.被N-陣列式記錄的數據記為x(ti) , i = 1,2,...以單元序號為例,被天線接收的信號可以表示為:
Where X (t) is the received data matrix, A(θ) is vector
matrix, S(t) is the signal data matrix , N(t) is the additive omplex Gaussian white noise matrix.
X (t)是已接受教育信號矩陣, A(θ)是向量矩陣,S(t)是信號數據矩陣,N(t)是復合AGWN矩陣
is the
carrier wavelength, λ =c/ f ,where c is the speed of light,
f is the carrier frequency.
Covariance matrix of the receiving data is shown in the
following:
λ =c/ f,λ為載波長,c為光速,f為載波頻.已接收信息的協方差矩陣表示如下:
The first step of MUSIC algorithm is the eigendecomposition
of the array covariance matrix:
MUSIC算法的第壹步是用二階修正算法對協方差矩陣行列式進行矩陣特征分解
Then sort the eigenvalue, the vector corresponds to M
(source number) large eigenvalues generate
subspa 1 2 1 { , } { ( ),..., ( )} M M S=spanvvv =spanθ θ , known
as the signal subspace. The vector corresponds to
N?M small eigenvalues generate subspace
1 { ,..., } M N span + N= v v , called the noise subspace. Clearly,
the signal subspace and noise subspace is orthogonal, that is
S⊥N.The projection matrix of noise subspace is as follows
然後對特征值分類,對應M(來源編號)的向量,大特征值生成subspa 1 2 1 { , } { ( ),..., ( )} M M S=spanvvv =spanθ .作為信號的子空間.對應N-M的向量生成子空間1 { ,..., } M N span + N= v v ,稱為噪音子空間.顯然地,二者成直角.投影矩陣如下
Scanning vector to search the target DOA, as the signal
subspace and noise sub-space orthogonal, so when θ is the
DOA of the signal, the vector project to the noise subspace
equal to zero, the theoretical P(θ) towards infinity, but in
practice subspace and noise subspace adopted by a limited
sample of the estimated signal are not entirely orthogonal,
then there will be a peak in the spectrum function.
從信號子空間和正交噪音子空間中找出DOA向量。若θ為DOA信號，映射到噪音子空間裏為0，理論P(θ)趨向無窮大，但實際上用壹個有限樣本估計可以發現子空間和噪音子空間並非嚴格正交，在頻譜函數中會形成壹個峰值。