高育兵, 梁紅, 杜金香

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欠采樣率下多頻率成分的估計(jì)方法

高育兵1, 梁紅2, 杜金香2

(1. 海軍裝備部駐西安地區(qū)軍事代表局, 陜西 西安, 710054; 2. 西北工業(yè)大學(xué) 航海學(xué)院, 陜西 西安, 710072)

針對(duì)信號(hào)處理和雷達(dá)系統(tǒng)中常見(jiàn)的信號(hào)頻率大于采樣頻率或采樣頻率小于奈奎斯特情況下信號(hào)多個(gè)頻率成分的估計(jì)問(wèn)題, 提出了采用一組非互質(zhì)的模數(shù)和相應(yīng)的一組有誤差的余數(shù), 同時(shí)重構(gòu)多個(gè)任意正實(shí)數(shù)的廣義穩(wěn)健中國(guó)剩余定理(GRCRT), 定理1首先給出了余數(shù)與重構(gòu)的正實(shí)數(shù)的一一對(duì)應(yīng)所需要滿足的條件, 定理2 給出了重構(gòu)正實(shí)數(shù)有唯一解的條件。將該定理用于欠采樣下多個(gè)信號(hào)頻率估計(jì), 仿真實(shí)例驗(yàn)證了余數(shù)估計(jì)有誤差時(shí)同時(shí)估計(jì)多個(gè)正實(shí)數(shù)算法的穩(wěn)健性和實(shí)際工程應(yīng)用前景。

廣義穩(wěn)健中國(guó)剩余定理; 欠采樣; 多頻率估計(jì)

0 引言

信號(hào)的頻率估計(jì)在許多工程領(lǐng)域得到廣泛的應(yīng)用。眾所周知, 當(dāng)信號(hào)的采樣頻率大于奈奎斯特采樣頻率的時(shí)候, 可以唯一地確定信號(hào)頻率, 最簡(jiǎn)單的就是采用快速傅立葉變換(fast Fourier transform, FFT)方法。但是當(dāng)信號(hào)的采樣頻率小于奈奎斯特采樣頻率時(shí), 就不可能從采樣樣本中唯一地獲得信號(hào)頻率的估計(jì)。采樣中國(guó)剩余定理(Chinese remainder theorem, CRT)可以解決這一問(wèn)題。近幾十年來(lái)中國(guó)剩余定理在數(shù)字信號(hào)處理、編碼、密碼學(xué)及計(jì)算機(jī)等領(lǐng)域獲得了廣泛的應(yīng)用[1-3]。傳統(tǒng)的CRT利用一組兩兩互質(zhì)的模和余數(shù)來(lái)估計(jì)一個(gè)整數(shù), 但是只有重構(gòu)的整數(shù)小于這組模的最小公倍數(shù)時(shí), 才能唯一確定該整數(shù)。文獻(xiàn)[4]從相位解卷積的角度出發(fā), 提出了穩(wěn)健的CRT(robust Chinese remainder theorem, RCRT)重構(gòu)算法, 但是采用該算法重構(gòu)的整數(shù)局限于與模相關(guān)的離散點(diǎn), 限制了在工程中的應(yīng)用。文獻(xiàn)[5]提出的廣義RCRT(generalized RCRT, GRCRT)重構(gòu)算法, 給出了余數(shù)有誤差時(shí)重構(gòu)一個(gè)任意正整數(shù)的方法。文獻(xiàn)[6]在文獻(xiàn)[5]的基礎(chǔ)上給出了余數(shù)有誤差時(shí)重構(gòu)一個(gè)任意實(shí)數(shù)的方法, 有效地解決了欠采樣情況下單個(gè)單頻信號(hào)的頻率估計(jì)及陣元間距大于半波長(zhǎng)時(shí)相位解模糊的問(wèn)題, 但是都沒(méi)有涉及多個(gè)數(shù)同時(shí)估計(jì)的問(wèn)題。文獻(xiàn)[7]雖然給出了多個(gè)數(shù)同時(shí)估計(jì)的算法, 但是該方法不能解決所有被估計(jì)數(shù)的余數(shù)都存在誤差時(shí)如何重構(gòu)多個(gè)數(shù)的問(wèn)題, 這一點(diǎn)限制了它在許多領(lǐng)域的應(yīng)用。例如, 在數(shù)字信號(hào)處理中信號(hào)中可能包含多個(gè)單頻成分, 可以采用幾次欠采樣(欠采樣頻率對(duì)應(yīng)CRT中模數(shù))獲得的樣本估計(jì)頻率(CRT中的一組余數(shù)), 如果每次欠采樣下頻率估計(jì)有很小的誤差(在低信噪比下), 則如何利用CRT估計(jì)多個(gè)信號(hào)真實(shí)頻率就成為急需解決的問(wèn)題。

本文在文獻(xiàn)[5]的基礎(chǔ)上, 提出了采用一組非互質(zhì)的模數(shù)和相應(yīng)的有誤差的余數(shù)同時(shí)估計(jì)多個(gè)任意正實(shí)數(shù)的廣義穩(wěn)健中國(guó)剩余定理, 很好地解決了欠采樣頻率下多個(gè)頻率成分信號(hào)的估計(jì)問(wèn)題, 仿真算例驗(yàn)證了算法的穩(wěn)健性及工程實(shí)用性。

1 問(wèn)題的提出

2 同時(shí)估計(jì)多個(gè)正實(shí)數(shù)的廣義穩(wěn)健中國(guó)剩余定理

定義

根據(jù)上述定義及文獻(xiàn)[5]~[7]的啟發(fā), 可以得到以下定理。

定理2的證明可參見(jiàn)文獻(xiàn)[6]。文獻(xiàn)[8]和[9]給出了定理2中重構(gòu)實(shí)數(shù)的快速算法。

估計(jì)的誤差上限為

3 欠采樣下利用GRCRT估計(jì)多個(gè)單頻信號(hào)頻率

在實(shí)際應(yīng)用中, 有些情況下獲得的樣本是不滿足Nyquist采樣定理要求的, 即欠采樣樣本。例如信號(hào)頻率非常高時(shí), 由于實(shí)時(shí)處理的需要, 采樣頻率不能太高, 或硬件采樣頻率達(dá)不到要求。此時(shí)如果利用欠采樣的樣本進(jìn)行信號(hào)的參數(shù)估計(jì)就不能獲得準(zhǔn)確的估值。如果需要估計(jì)的頻率成分只有1個(gè), 文獻(xiàn)[6]給出了重構(gòu)單個(gè)實(shí)數(shù)的方法。而信號(hào)處理中很多情況下信號(hào)包含的頻率成分較多, 此時(shí)可以采用本文提出的同時(shí)估計(jì)多個(gè)正數(shù)的GRCRT 算法, 對(duì)欠采樣下多個(gè)頻率成分進(jìn)行估計(jì)。文中對(duì)含有2個(gè)單頻成分的信號(hào)頻率進(jìn)行估計(jì), 并給出仿真結(jié)果。

仿真中復(fù)信號(hào)為

圖1 M變化信號(hào)頻率為f1時(shí)信噪比與檢測(cè)概率之間的關(guān)系曲線

圖2 M變化信號(hào)頻率為f2時(shí)信噪比與檢測(cè)概率之間的關(guān)系曲線

圖3 M變化信號(hào)頻率為f1時(shí)信噪比與估計(jì)的均方誤差之間的關(guān)系曲線

4 結(jié)束語(yǔ)

本文提出了解余數(shù)有誤差時(shí)同時(shí)估計(jì)多個(gè)正實(shí)數(shù)的廣義穩(wěn)健中國(guó)剩余定理, 得出算法估計(jì)的誤差僅與余數(shù)估計(jì)的誤差有關(guān), 克服了傳統(tǒng)CRT算法極小的余數(shù)誤差帶來(lái)相當(dāng)大整數(shù)估計(jì)誤差的局限。用信號(hào)處理中的仿真實(shí)驗(yàn)驗(yàn)證了所提GRCRT算法的有效性和穩(wěn)健性。算法適用于估計(jì)的所有余數(shù)有誤差時(shí)重構(gòu)多個(gè)正實(shí)數(shù)的場(chǎng)合, 可以應(yīng)用于數(shù)字信號(hào)處理中的雷達(dá)、聲納及生物醫(yī)學(xué)等領(lǐng)域。

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Multiple-Frequency Estimation Method with Undersampled Waveform

GAO Yu-bing1, LIANG Hong2, DU Jin-xiang2

(1. XI′an Representative Bureau, Naval Armament Department, Xi′an 710054, China; 2. College of Marine Engineering, Northwestern Polytechnical University, Xi′an 710072, China)

In some applications, such as signal processing and radar systems, it is preferred that the range of the frequencies is as large as possible for a given sampling rate and the sampling rate is below the Nyquist rate. In both cases, frequencies estimation from undersampled waveforms is necessary. In this paper, a generalized robust Chinese remainder theorem (GRCRT) for reconstructing multiple positive real numbers is presented, where modules are not pair-wisely co-prime and the remainders with errors. In theorem 1, the sufficient condition for the multiple real numbers to satisfy is given, where all remainders have errors and we can determine which one in the remainder set is the remainder of any real number. And an approach to determine unique solution of multiple real numbers from the remainder set with errors is proposed in theorem 2. Simulation results show that the present method is efficient for estimating multiple frequencies from multiple undersampled waveforms with sampling rate below the Nyquist rate, and it can be applied to such areas as digital signal processing.

generalized robust Chinese remainder theorem(GRCRT); undersample; multiple-frequency estimation

TJ630.34；TN911.7

A

1673-1948(2012)01-0019-05

2011-04-11;

2011-06-10.

國(guó)家自然科學(xué)基金(60702067).

高育兵(1964-), 男, 高級(jí)工程師, 研究方向?yàn)樗曅盘?hào)處理.

(責(zé)任編輯: 楊力軍)