Estimation of Unknown Line Spectrum under Colored Noise via Sparse Reconstruction
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摘要: 针对色噪声背景下的未知线谱信号估计问题，该文提出一种基于分子频带处理的稀疏重构类线谱估计方法。首先，利用多速率余弦调制滤波器组对观测信号进行子带分解，得到功率谱相对平坦的子带信号。之后，在每个子带信号上，利用基于迭代最小化的稀疏学习方法进行线谱估计，并将各子带上的线谱估计结果进行频域综合滤波以及门限判决等处理。最终得到色噪声背景下的线谱估计结果。理论推导及仿真实验表明所提方法在色噪声背景下具有较好的线谱估计性能。其能够有效地去除色噪声背景，同时保留稀疏重构类线谱估计方法所具有的高频率分辨力等优点。Abstract: To solve the problem of the line spectrum estimation under colored noise background, a subband line spectrum estimation method using sparse reconstruction is proposed. Firstly, the input signal is divided into several subbands by a multi-rate cosine modulated filter bank. The subband signal has the flatter power spectrum. The sparse learning via iterative minimization method is utilized on each subband to estimate the line spectrum signal. Then, the results of line spectrum estimation on each subband are processed by frequency domain synthesis filtering and threshold decision. Finally, the line spectrum signal under colored noise background is identified. Theoretical derivation and simulation experiments show that the proposed method has better line spectrum estimation performance under colored noise background. The colored noise background can be removed, and the advantage of high frequency resolution of sparse reconstruction method is retained.
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表 1 SLIM算法计算步骤
初始化 $\hat s_k^{(0)}{\rm{ = }}\displaystyle\frac{{{{{a}}^{\rm{H}}}({\omega _k}){{X}}}}{{\left\| {{{a}}({\omega _k})} \right\|_2^2}}, (k = 0,1, ·\!·\!· ,K - 1), \ {\hat \eta ^{(0)}} = \frac{1}{{\gamma N}}\left\| {{{X}} - {{A}}{{{\hat{ S}}}^{(0)}}} \right\|_2^2, \ \gamma {为常数}$ (1)计算信号功率 ${{P}}$ ${{{P}}^{(i)}}{\rm{ = [}}p_0^{(i)}\;p_1^{(i)}\; ·\!·\!· \;p_{N - 1}^{(i)}{\rm{]}}, \ p_k^{(i)} = {\left| {s_k^{(i)}} \right|^{2 - q}}{\rm{,}}\;(k = 0{\rm{,}}\;1{\rm{,}}\; ·\!·\!·\! {\rm{,}}\ K - 1)$ (2)计算信号幅度 ${\hat{ S}}$ ${{\hat{ S}}^{(i + 1)}} = {\rm{diag}}\left({{{P}}^{(i)}}\right){{{A}}^{\rm{H}}}{\left( {{{A}}{\rm{diag}}\left({{{P}}^{(i)}}\right){{{A}}^{\rm{H}}} + {{\hat \eta }^{(i)}}{{I}}} \right)^{ - 1}}{{X}}$ (3)计算噪声功率 $\hat \eta $ $\;{\hat \eta ^{(i + 1)}} = \frac{1}{N}\left\| {{{X}} - {{A}}{{{\hat{ S}}}^{(i + 1)}}} \right\|_2^2$ 注：重复迭代步骤(1)—步骤(3)直至 ${\hat{ S}}$收敛或达到预定迭代次数(i表示迭代次数) 
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