平成25年度電気関係学会四国支部連合大会 講演論文集 （2013 徳島大学）
2013 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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IMPLEMENTATION OF SPECTRAL MUSIC FORUNIFORM LINEAR ARRAY
Xiang Cao, Jingmin Xin

Yoshifumi Nishio

Xi’an Jiaotong University

The University of Tokushima
The above function is a one-to-one mapping that takes the unit
upper semicircle to the interval [−1, 1] of real line. Substituting (3) into (4), we get the relationship between ω ∈ [−π, π]and u ∈ [−1, 1] as follows:ω .
(5)u = tan4

1. INTRODUCTION
The multiple signal characterization (MUSIC) algorithm fordirection finding has been advanced for many years. To a certain degree, the more search points one use, the more accurate results one will obtain. The main computational burden
locate the process of spectral search. In this paper, a polynomial form with real coefficients of spectral music is presented.
Our method treats the whole range of interest in uniform linear array. The complexity of spectral music can be reduced
by replacing a large number of search points by real roots belonging to [−1, 1].

This is a monotonic function in [−π, π] and its range is[−1, 1]. Thus, once the variable u is estimated, the uniquespatial frequency ω can be obtained by ω = 4 arctan u.
For this purpose, the array vector response a(ω) must beconnected to u.
jω= −z 2 ( the two equations
Using z = −j u−j
u+j and e
can be easily obtained from (3) and (4) ), we get

2. DATA MODEL AND SPECTRAL MUSICa(ω) = (u + j)

Assume p far-field narrowband signals {sk } impinging on a
Uniform linear array (ULA) with M (M > p) sensors. The
output vector y(t) of the array at time t can be written asy(t) = A(θ)s(t) + n(t),(1)(2)

This is an univariate polynomial of degree 4(M − 1) with realcoefficients.
4. CONCLUSIONS

3. REAL POLYNOMIAL FORM OF SPECTRALMUSIC

An univariate polynomial with real coefficients is presented
for reducing the search points in spectral MUSIC. The realroots belonging to [−1, 1] only need to be solved and the process of solution can avoid all complex arithmetic.

By defining a translationωπ(3)the real point ω belonging to the domain [−π, π] can bemapped to the unique complex point z belonging to the unit
upper semicircle. Furthermore, let us introduce a conformaltransformationu = −jz−j.
z+j(6)

where Re{} is a real operator and diagonal matrix W =diag[0, j, · · · , j(M − 1)]. Applying (6),(7), v H (u) = v T (u)and dropping the scalar givenov T (u)Re C H W U s U H(8)s C v(u) = 0.

The total number of complex flops in (2) is Jp(2M + 1),
where J (J ≫ M > p) is the number of spectral searchpoints.

z = ej( 2 + 2 ) ,Cv(u).

Meanwhile,a function’s differential in its local extremum
is zero. We can obtain [1]no∂f (ω)= 2Re aH (ω)W U s U H(7)
s a(ω) = 0,∂ωIt is clear that the columns of A(θ) have the maximum projection on the signal subspace U s . With this observation, A
natural estimation criterion is to find the local maxima, viz.
ω̂ = arg max f (ω) = arg max aH (ω)U s U Hs a(ω)ωω

2(1−M )

5. REFERENCES[1] J. Selva, “Computation of spectral and root music
through real polynomial rooting,” IEEE Trans. Signal
Process., vol. 53, pp. 1923–1927, May 2005.

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