ON THE IDENTIFICATION OF CERTAIN NOISY FIR CONVOLUTIVE MIXTURES.
W.Hachem,F.Desbouvries,Ph.Loubaton
:INT/SIM,9Rue Charles Fourier,91011Evry Cedex,FRANCE,email:desbou@int-evry.fr
:Laboratoire Syst`e me de Communication-UMLV
Champs sur Marne-5,bvd Descartes,77454Marne la Vallee Cedex2
FRANCE,email:loubaton@univ-mlv.fr
ABSTRACT
In this paper,we address the identi?cation problem of–inputs –outputs MA models corrupted by a white noise with unknown covariance matrix in the case where.Under certain addi-tional conditions,we show that the generating function of the MA model is identi?able up to a constant orthogonal matrix by using the autocovariance function of the observation.
1.INTRODUCTION
Let be a-variate time-series given by
(1)
where is a?nite impulse response transfer function,is a-dimensional(non-observable)white noise sequence for which,and is an additive -dimensional white noise(i.e.if). It is assumed that,i.e.,the dimension of is strictly greater than the dimension of the input and that the transfer function is irreducible,i.e.that
Rank for each(2) If the observations are noiseless(i.e.if),it is well es-tablished that the transfer function can be consistently es-timated from the second order statistics of up to a constant orthogonal matrix(e.g.[13],[3]).A number of ef?cient algorithms based on the so-called linear prediction approach were recently derived in this aim(see e.g.[9]).Therefore,under con-dition(2),the convolutive mixture problem considered here can be reduced to a separation of an instantaneous mixture by using the second order statistics of the observation.The purpose of this paper is to study the behaviour of this approach in the noisy case: in other words,if is non zero,is it still possible to estimate consistently up to a constant orthogonal matrix by using the second order statistics of?The answer is known to be positive if the covariance matrix can be written as where is an unknown scalar parameter.This cor-responds to the case where the components of the additive noise are decorrelated,and have the same variance.In effect,it is possible to identify from the covariance matrix of the vector if the parameter is chosen greater than.To explain this,we denote by the so-called Sylvester matrix associated to
given by
..
.
..
.(3)
Then,it is clear that where and are de?ned as.The covariance ma-trix is thus equal to
As and,is tall,and is
a singular matrix.is thus the smallest eigenvalue of,and can be consistently estimated from an empirical estimate of. Therefore,one can estimate consistently the second order statis-
tics of the noiseless signal,and use the above mentionned linear prediction approach to estimate up to an orthogonal
matrix.
However,the assumption that may be restrictive in certain contexts.Very often,the components of represent the signals sampled behind a sensor array.If the noise is a thermal noise due to the acquisition devices of the sensors,its components
are likely to be decorrelated.If the sensors are not identical,the variances of the components of do not necessarily coincide. The noise may even be spatially correlated if it is due to a super-position of a large number of weak independent sources.Such a situation arises e.g.in underwater acoustics.In this paper,we are going to present certain results showing that,under certain condi-tions,it is possible to identify up to an orthogonal ma-trix from the exact second order statistics of in the case where
no a priori information on is available.
Let us denote by and by the autocovari-ance coef?cients of the useful signal and of respec-tively.As is assumed to be white and is FIR,it is clear that,for,and
for.In particular,does not bring any information on.We thus reformulate our problem as follows: Let be a irreducible
FIR?lter,and let be the autocovariance func-
tion associated to the”spectral density”.
How to identify(up to a orthogonal ma-
trix)from the knowledge of the truncated sequence
?
This problem was?rst introduced in[7]if.It is shown that the unknown transfer function is not necessarily identi?able if.In case of identi?ability,an identi?cation procedure based on the stochastic realization theory is proposed. However,it is based on a dif?cult non convex optimization prob-lem,for which no satisfying solution has been https://www.doczj.com/doc/ac10391216.html,ter, still in the case,[2]showed that the SIMO FIR subspace identi?cation method introduced in[12]if could be gen-eralized if.The case was?rst considered in[5]in the FIR case,and some results in the IIR case are to published([6]).
We now precise the content of the paper.In section2,we?rst present the results provided by an approach aiming at identifying the unknown coef?cient.Subsection2-1outlines the approach, subsection2-2reformulates some results of[5],and subsection2-3 shows how they can be improved.Finally,we introduce in section 3a new identi?ability result based on the so-called Wiener-Hopf factorization theory.
In this paper,we concentrate on identi?ability results based on the knowledge of the true autocovariance coef?cients. However,concrete estimation algorithms can be derived immedi-ately from the results of section2.The practical use of the material presented in section3is more involved,and is out of the scope of this paper.
2.IDENTIFICATION OF.
2.1.Outline of the results.
The results of this section are based on the following observation. As,it exists certain FIR?lters
(where is to be determined)for which
for each(4) or equivalently for which for each.Let us assume that a family of degree FIR?lters
satisfying(4)is available,and let us set
.Put
,where is supposed to be known.As,it is clear that
(5) Equating the coef?cients of both sides,this equation allows to compute the matrix where.If is full column rank,can be retrieved from(5),and can be identi?ed from the whole sequence.
This approach is based on the FIR?lters satisfying(5).There-fore,it is useful to to present some of their properties.for this, we have to recall some well known results related to rational sub-spaces.
A review of rational subspaces Let us?rst recall that the set
of all rational transfer functions is a-dimensional sub-space over the?eld of all scalar rational transfer functions.Let be a-dimensional()subspace of.It therefore ad-mits bases characterized by the fact that Rank for almost all:the rational matrix valued function is said to have a normal rank equal to.admits polynomial bases.A polynomial basis is said to be minimal if deg is minimum(see[10]for more details).All minimal bases share the same degrees,and are characterized by the well known criterion(see[8],[10]): Proposition1The polynomial basis is mini-mal if and only if the matrix polynomial
is irreducible and column reduced.
Usually,the minimal degrees are called the Kronecker indices associated to.The”orthogonal”of is the-dimensional subspace of all rational transfer functions satisfying for each.admits Kronecker indices denoted,which satisfy the important equal-ity:
(6)
Finally,we recall that if is a mini-mal polynomial basis of,then the rank of the Sylvester matrix is given by[11]:
Rank(7)
As satis?es condition(2),the rational space generated by its columns is–dimensional.From now on,this subspace is denoted by and its–dimensional dual space by.By the very de?nition of,it is clear that a degree FIR?lter satis?es(4)if and only if is a degree polynomial of .If we denote by the dual Kronecker indices of ,it turns out that if,then holds if and only if for each,while if,then it exists exactly linearly independant degree FIR?lters for which.In particular,if, then it exists a polynomial basis of degree FIR?lters in.
2.2.The use of the block-Hankel matrix associated to the se-quence.
In this paragraph,we propose an approach which allows to com-pute degree FIR?lters of from.In order the approach to be effective,we assume in this subsection that satis?es the following extra-assumptions:
The columns of share the same degree
Rank(
Let be the block Hankel matrix given by
..
.(8) can be factored as
..
.
..
.
..
.
(9) As and are full rank column,it is clear that and have are also full column rank.Therefore,the rank of
is also equal to.Let be the block exchange matrix:
,where denotes the Kronecker product,and
.It is easy to check that a row –dimensional vector(where each is –dimensional)satis?es
and(10) if and only if.Therefore,Ker
Ker Ker(Ker stands for the left Kernel).Let
be the degree FIR?lter associated to. Then,it is easily seen that if and only if
for each.Therefore,there is a one to one correspondance between the space Ker Ker and the set of all de-gree less than or equal to FIR?lters of.We have?rst to check if this subset of is not reduced to.For this,we have to compare with the dual Kronecker indices of.As is irreducible(condition(2))and column reduced(because Rank(),is a minimal polynomial basis.The Kronecker indices of all coincide therefore with.Hence,by relation(6),
If,we get that,i.e.
,or equivalently. Consequently,if,it exists at least a non zero degree FIR?lter satisfying
for each.This condition,which is assumed to hold from now on in this paragraph,implies in particular that.It re-mains to investigate under which conditions it exist degree
FIR?lters such that the matrix associated to
is full rank column.For this,we ?rst note that full rank is equivalent to the condition
for each(11) In order to guarantee the validity of this condition,the number of rows of have of course to be as large as possible.How-ever,what is really important is the number of linearly indepen-dent(over the?eld)rows of:if say belongs to the rational space generated by the?rst rows of,is full rank column if and only if the matrix associated to
is itself full column rank.The number of linearly independent degree FIR?lters of is de?ned by the fact that
The maximum value of is of course equal to(the dimension of,and is reached if. More precisely,we have the following result:
Theorem1Let us assume that(in particular )and let be lin-early independent degree FIR?lters satisfying
for each.Then,the matrix associated to is full rank column,so that can be identi?ed.from re-lation(5).
Proof.In order to establish that is full rank column,we consider a vector such that for each.As the rows of are linearly independent,is a basis of the rational space.Therefore,the condition for each holds if and only if the constant FIR?lter belongs to the dual space of,i.e.to.But,as,the space does not con-tain non zero constant vectors.This in turn implies that.
This in particular implies that for each.There-
fore,using the derivative of allows to extract certain FIR de-
gree?lters of,whatever is.However,these elements do
not span in general the whole set because they generate the dual
space of which has dimension.We now
check if one can build from these elements of a matrix polyno-
mial for which is full column rank.
Theorem2Assume that the rational space does not contain
non zero constant vectors,i.e.that its smallest Kronecker index
is non zero.Then,if deg deg,
it exists degree linearly independent FIR?lters
for which the associated matrix is
full rank column.
Proof.Denote by the dimensional dual space of,and
by and its Kronecker and dual Kro-
necker indices.It is clear that
deg deg
Therefore,deg deg.If is
chosen as in the statement of the theorem,is greater than.
It thus exists a polynomial basis
for which deg
for each.Let be the corresponding matrix.If is
not full column rank,it exists a non zero–dimensional constant
vector for which for each.As is a ba-
sis of,this holds if and only if the constant polynomial vector
belongs to the dual space of,i.e.to.As,
must be reduced to0,i.e.is full rank column.
This identi?ability result can be used in order to derive a con-
crete estimation algorithm of(up to a constant factor).This
algorithm of course requires,at least implicitely the calculation of
a left Wiener-Hopf factor of.This non obvious problem is
out of the scope of the present paper,and is under investigation.
4.CONCLUSION
In this paper,we have proposed two second order based statis-tics approaches to blindly identify up to a constant matrix a noisy MA model.The problem has been formulated as the identi?ca-tion of the?lter from the truncated autocovariance sequence associated to the spectral density.
The?rst approach consists in identifying directly by calculat-ing certain FIR?lters?lters satisfying.The extraction of these?lters use either the block Hankel matrix associ-ated to the sequence,either the derivative of
w.r.t.the variable.The use of require less restrictive assumptions on.Finally,we have introduced an alternative approach based on the Wiener-Hopf factorization theory.In this context,an identi?ability result has been shown.The practical use of the Wiener-Hopf factorization approach is under study.
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M A: Has the case been closed yet? B: No, the magistrate still needs to decide the outcome. magistrate n.地方行政官，地方法官，治安官 A: I am unable to read the small print in the book. B: It seems you need to magnify it. magnify vt.１．放大，扩大；２．夸大，夸张 A: That was a terrible storm. B: Indeed, but it is too early to determine the magnitude of the damage. magnitude n.１．重要性，重大；２．巨大，广大 A: A young fair maiden like you shouldn’t be single. B: That is because I am a young fair independent maiden. maiden n.少女，年轻姑娘，未婚女子 a.首次的，初次的 A: You look majestic sitting on that high chair. B: Yes, I am pretending to be the king! majestic a.雄伟的，壮丽的，庄严的，高贵的 A: Please cook me dinner now. B: Yes, your majesty, I’m at your service. majesty n.１．［Ｍ－］陛下（对帝王，王后的尊称）；２．雄伟，壮丽，庄严 A: Doctor, I traveled to Africa and I think I caught malaria. B: Did you take any medicine as a precaution? malaria n.疟疾 A: I hate you! B: Why are you so full of malice? malice n.恶意，怨恨 A: I’m afraid that the test results have come back and your lump is malignant. B: That means it’s serious, doesn’t it, doctor? malignant a.１．恶性的，致命的；２．恶意的，恶毒的 A: I’m going shopping in the mall this afternoon, want to join me? B: No, thanks, I have plans already. mall n.（由许多商店组成的）购物中心 A: That child looks very unhealthy. B: Yes, he does not have enough to eat. He is suffering from malnutrition.
意见应以事实为根据. 3 来自辞典例句 192. The bombers swooped ( down ) onthe air base. 轰炸机 突袭 空军基地. 来自辞典例句 193. He mounted their engines on a rubber base. 他把他们的发动机装在一个橡胶垫座上. 14 来自辞典例句 194. The column stands on a narrow base. 柱子竖立在狭窄的地基上. 14 来自辞典例句 195. When one stretched it, it looked like grey flakes on the carvas base. 你要是把它摊直, 看上去就象好一些灰色的粉片落在帆布底子上. 18 来自辞典例句 196. Economic growth and human well - being depend on the natural resource base that supports all living systems. 经济增长和人类的福利依赖于支持所有生命系统的自然资源. 12 1 来自辞典例句 197. The base was just a smudge onthe untouched hundred - mile coast of Manila Bay. 那基地只是马尼拉湾一百英里长安然无恙的海岸线上一个硝烟滚滚的污点. 6 来自辞典例句 198. You can't base an operation on the presumption that miracles are going to happen. 你不能把行动计划建筑在可能出现奇迹的假想基础上.
英语造句大全English sentence 在句子中，更好的记忆单词！ 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character. （2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n.
主语+谓语 1. 理解主谓结构 1) The students arrived. The students arrived at the park. 2) They are listening. They are listening to the music. 3) The disaster happened. 2．体会状语的位置 1) Tom always works hard. 2) Sometimes I go to the park at weekends.. 3) The girl cries very often. 4) We seldom come here. The disaster happened to the poor family. 3. 多个状语的排列次序 1) He works. 2) He works hard. 3) He always works hard. 4) He always works hard in the company. 5) He always works hard in the company recently. 6) He always works hard in the company recently because he wants to get promoted. 4. 写作常用不及物动词 1. ache My head aches. I’m aching all over. 2. agree agree with sb. about sth. agree to do sth. 3. apologize to sb. for sth. 4. appear (at the meeting, on the screen) 5. arrive at / in 6. belong to 7. chat with sb. about sth. 8. come (to …) 9. cry 10. dance 11. depend on /upon 12. die 13. fall 14. go to … 15. graduate from 16. … happen 17. laugh 18. listen to... 19. live 20. rise 21. sit 22. smile 23. swim 24. stay (at home / in a hotel) 25. work 26. wait for 汉译英： 1.昨天我去了电影院。 2.我能用英语跟外国人自由交谈。 3.晚上7点我们到达了机场。 4.暑假就要到了。 5.现在很多老人独自居住。 6.老师同意了。 7.刚才发生了一场车祸。 8.课上我们应该认真听讲。9. 我们的态度很重要。 10. 能否成功取决于你的态度。 11. 能取得多大进步取决于你付出多少努力。 12. 这个木桶能盛多少水取决于最短的一块板子的长度。
【it's time to和it's time for】 ——————这其实是一个句型,只不过后面要跟不同的东西. ——————It's time to跟的是不定式（to do）.也就是说,要跟一个动词,意思是“到做某事的时候了”.如： It's time to go home. It's time to tell him the truth. ——————It's time for 跟的是名词.也就是说,不能跟动词.如： It's time for lunch.(没必要说It's time to have lunch) It's time for class.(没必要说It's time to begin the class.) They can't wait to see you Please ask liming to study tonight. Please ask liming not to play computer games tonight. Don’t make/let me to smoke I can hear/see you dance at the stage You had better go to bed early. You had better not watch tv It’s better to go to bed early It’s best to run in the morning I am enjoy running with music. With 表伴随听音乐 I already finish studying You should keep working. You should keep on studying English Keep calm and carry on 保持冷静继续前行二战开始前英国皇家政府制造的海报名字 I have to go on studying I feel like I am flying I have to stop playing computer games and stop to go home now I forget/remember to finish my homework. I forget/remember cleaning the classroom We keep/percent/stop him from eating more chips I prefer orange to apple I prefer to walk rather than run I used to sing when I was young What’s wrong with you There have nothing to do with you I am so busy studying You are too young to na?ve I am so tired that I have to go to bed early
《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译：我依然能记得哈桑坐在树上的样子，阳光穿过叶子，照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃，鼻子大而扁平，双眼眯斜如同竹叶，在不同光线下会显现出金色、绿色，甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译：从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high
一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中，一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情（心情）。 4，make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床)，( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天，我懂得了读书不是我认为的简单的娱乐消遣，也不只是躺在吊床上，一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话，他将能改变一个人以后的人生道路。 二、词组造句 1. on purpose 特意，故意 This is especially true here, and it was ~. (这一点在这里尤其准确，并且他是故意的) 2.think up 虚构，编造，想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时 助记：in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢？ In the meantime, we may not know how it works, but we know that it works. 在此期间，我们不知道它是如何工作的，但我们知道，它的确在发挥作用。 4.as though 好像，仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关，房
The effective sentences:(improve the sentences!) 1.She hopes to spend this holiday either in Shanghai or in Suzhou. 2.Showing/to show sincerity and to keep/keeping promises are the basic requirements of a real friend. 3.I want to know the space of this house and when it was built. I want to know how big this house is and when it was built. I want to know the space of this house and the building time of the house. 4.In the past ten years,Mr.Smith has been a waiter,a tour guide,and taught English. In the past ten years,Mr.Smith has been a waiter,a tour guide,and an English teacher. 5.They are sweeping the floor wearing masks. They are sweeping the floor by wearing masks. wearing masks,They are sweeping the floor. 6.the drivers are told to drive carefully on the radio. the drivers are told on the radio to drive carefully 7.I almost spent two hours on this exercises. I spent almost two hours on this exercises. 8.Checking carefully,a serious mistake was found in the design. Checking carefully,I found a serious mistake in the design.
M1 U1 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： add up (to) be concerned about go through set down a series of on purpose in order to according to get along with fall in love (with) join in have got to hide away face to face 1 We’ve chatted online for some time but we have never met ___________. 2 It is nearly 11 o’clock yet he is not back. His mother ____________ him. 3 The Lius ___________ hard times before liberation. 4 ____________ get a good mark I worked very hard before the exam. 5 I think the window was broken ___________ by someone. 6 You should ___________ the language points on the blackboard. They are useful. 7 They met at Tom’s party and later on ____________ with each other. 8 You can find ____________ English reading materials in the school library. 9 I am easy to be with and _____________my classmates pretty well. 10 They __________ in a small village so that they might not be found. 11 Which of the following statements is not right ____________ the above passage? 12 It’s getting dark. I ___________ be off now. 13 More than 1,000 workers ___________ the general strike last week. 14 All her earnings _____________ about 3,000 yuan per month. 二.用以下短语造句： 1.go through 2. no longer/ not… any longer 3. on purpose 4. calm… down 5. happen to 6. set down 7. wonder if 三. 翻译： 1.曾经有段时间，我对学习丧失了兴趣。（there was a time when…） 2. 这是我第一次和她交流。（It is/was the first time that …注意时态） 3.他昨天公园里遇到的是他的一个老朋友。（强调句） 4. 他是在知道真相之后才意识到错怪女儿了。（强调句） M 1 U 2 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： play a …role (in) because of come up such as even if play a …part (in) 1 Dujiangyan(都江堰) is still ___________in irrigation(灌溉) today. 2 That question ___________ at yesterday’s meeting. 3 Karl Marx could speak a few foreign languages, _________Russian and English. 4 You must ask for leave first __________ you have something very important. 5 The media _________ major ________ in influencing people’s opinion s. 6 _________ years of hard work she looked like a woman in her fifties. 二.用以下短语造句： 1.make (good/full) use of 2. play a(n) important role in 3. even if 4. believe it or not 5. such as 6. because of
English sentence 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能 句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的 句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character.（2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n. 句子：He is in America. 10、appear 出现v. He appears in this place. 11. artist 艺术家n. He is an artist. 12. attract 吸引 He attracts the dog. 13. Australia 澳大利亚 He is in Australia. 14.base 基地 She is in the base now. 15.basket 篮子 His basket is nice. 16.beautiful 美丽的 She is very beautiful. 17.begin 开始 He begins writing. 18.black 黑色的 He is black. 19.bright 明亮的 His eyes are bright. 20.good 好的 He is good at basketball. 21.British 英国人 He is British. 22.building 建造物 The building is highest in this city 23.busy 忙的 He is busy now. 24.calculate 计算 He calculates this test well. 25.Canada 加拿大 He borns in Canada. 26.care 照顾 He cared she yesterday. 27.certain 无疑的 They are certain to succeed. 28.change 改变 He changes the system. 29.chemical 化学药品