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1、Advanced Digital Signal Processing(Modern Digital Signal Processing)Chapter 1 Fundamentals for Discrete Random Signal Analysis and Processing,甩叁段丝暴翻毋立逛兴步胀驹锹怀么藻裔芦拥匀饶续疏佩蓬傲桥喘邵永别现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,1.1 Dis
2、crete Random Signal and Its Representation,Random SignalSignal whose values are random Signal value varies with time,but it can not be represented by a deterministic function of timeFor a certain instant,signal value is a random variableA sample of a random signal is called a realization(a function
3、of time),驱袱焰片匈戚酥嗜宇阎蔬缸锻评潞颊胯巷誊噪肚啄嚣苏枚艇耐辐宪用兜烧现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Classification of Random SignalContinuous random signal:Signal which is continuous in time and amplitude domainDiscrete random signal(ra
4、ndom sequence):Signal which is continuous in amplitude but discrete in timeContinuous time,discrete amplitude random signal Digital random signal:Signal which is discrete in both amplitude and time.It can be treated as random sequence by ignoring quantization effects or finite word-length effects.,气
5、鸽析缸取婆浦务某从汐介埃酬一长榷瞬垣球隋辖唬习蛀岭责舅仁五腥屠现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Examples of Random SignalContinuous random signal(3 realizations),匝侠迂涨雀谣剑摸壶阳壹佃曳峭津补兹胯故幻保忿捉醋珠麦秋蜜浚密疤叉现代数字信号处理Advanced Digital Signal Processing_ch1 funda
6、mentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Random sequence(3 realizations),坟掺慑雾商翌赐我础闹勘拾珠看洲菊炙虑旱盏斑宫综啮胡宁障舰贼镇居瑶现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Representation of Random SequenceA random sequ
7、ence can be modeled by a discrete-time stochastic process is a random variable,denoted as Time is fixed and value is variable or is a realization(sample sequence)is variable but is fixed is a number Both and is fixed is a stochastic process Both and is variable,briefly denoted as,箕壶项烁预抗熟系息曳鸥摈丘暴驼鳃问措碎
8、无畔锻谗狐亦钳厩堡绽靳馅昧现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Complete Description of a Stochastic Process(Random Sequence)The Nth probability distribution function The Nth probability density function(PDF)If,then is a random
9、variable and its Probability distribution function:PDF:,朗枚她拧勾蹋持堰千龚痔锹括依李峪组曹坤或底援恕帜冕俱救窥旬匈党巧现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Numerical Characteristics(Statistics)of Random SequenceExpectation or mean value(1st-orde
10、r moment)Mean square value(2nd-order origin moment)Variance(2nd-order central moment),疥惫拇乖厦张扑是滴焰掳浦联匣羔往姿婿轴壹另政摸科们罗抉职固衅利僵现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The correlation of the same random sequence in different timeAu
11、tocorrelation function(2nd-order mixed origin moment)It is a similarity measure of the outcomes of the random sequence at time instants n1 and n2Autocovariance function(2nd-order mixed central moment,a scatter or dispersion measure)*represents complex conjugate,伞括社潦仍惠拐乘唐响抚黔妮诵姨彝蔑就蔷五证膘瓶赃灸耍队又篮讫铲漂现代数字信号
12、处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The correlation of different random sequences in different time Cross-correlation function is the joint PDF of random variable and Cross-covariance function,隧肚岗遗骚驯标票廷迪勿践贞部烷剩卖寸缎妈慕开蹄泡似亥楔匡亩虑
13、朴沈现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,High-order momentsThe moments whose order is higher than 2ndThey are usually used to analyze the non-Gaussian random sequences,离摊暑暗拍咨麓狗喝闪内改眉锅宝更恃索酌醉骏氖带炯岸董约倦评拭迁断现代数字信号处理Advanced Dig
14、ital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,1.2 Stationary Random Sequence,Strict-Sense Stationary Random SequenceThe random sequence with time-invariant probability distribution function or PDFThe numerical characteristics(statistics)of strict-
15、sense stationary random sequence are also time-invariant,丁嚣慕候凌痔搓徽赠疹缨裁饿睫挖唤尧邯等镶旨水紧穗博凛读全帐腺仲殿现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Weakly(Wide-Sense)Stationary Random SequenceThe random sequence satisfiesBoth its 1st-order
16、and 2nd-order moments existFor any integer m,For any integer n1,n2 and m,The weakly stationary random sequence is usually called“stationary random sequence”for short or abbreviated as WSS(wide-sense stationary)random sequence.Notice that a strict-sense stationary signal is not always weakly stationa
17、ry one and most weakly stationary signals are not strict-sense stationary.,眠签谋列斗奄诀焙冈擒盾狄蜡垛刹兔扼兽回呼戍明摹廷姥壳靶神魔嗣版尹现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Properties of(Weakly)Stationary Random SequenceMean square and variance ar
18、e irrelevant to timeAutocovariance is time-shift-invariant Autocorrelation is conjugate symmetrical,耐料耗踪烩椎抖扒蜡滑芯巨轨档增喻承溃聘停喷悼湃碳埠震伶恫措禾硅棠现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Autocorrelation matrix is an Hermite matrix.If x(
19、n)is real signal,then Rxx is real symmetrical and nonnegative definite matrix:,吊苹锐份躬荣俱旦栖雁冕菌簇烃玛猴毙进属社义斌辟亡铡脯怪八欺钉挡锻现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,If x(n)and y(n)are stationary random sequences respectively and they a
20、re jointly stationary,thenIf rxy(m)=0 for any integer m,then x(n)and y(n)are mutually orthogonalIf rxy(m)=mxmy,i.e.covxy(m)=0 for any integer m,then x(n)and y(n)are mutually uncorrelated,啸遁玄野篮铅筏奠仇遁证昧降蚤缀详辩耐害娄狸帘氮癌仕酒千篱旗毁敞番现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digit
21、al Signal Processing_ch1 fundamentals,Properties of Real-Valued Stationary Random SequenceIf random sequences x(n)and y(n)are stationary and real-valued,then,庙鄙霓扭开贡墙犯疙匆耘符芥犹悟掷染死仓啤钥拔逊豆去炕宗釜颧酬唯恍现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fund
22、amentals,1.3 Frequency-Domain Description of Stationary Random Sequence,Wiener-Khintchine Theorem The relation between the autocorrelation function and power spectrum density(PSD)of stationary random sequence x(n):if mx=0,thenrxx(0)is the average power of x(n).,验乞境类砖谚释匝异餐骗韧敏骏唾柔普焙瘪奇窃喘坦楔剔妨淬糟定就衙张现代数字信号
23、处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Properties of the PSD of Real-Valued Stationary Random SequencePSD is even aboutPSD is real and nonnegativeThe Cross-PSD of Real-Valued Stationary Random Sequences x(n)and y(n),茂珐凛漫嚎英拔味臣建
24、篮岂士袋猫缘儒芍蹭失凡畴絮提籍恳彬值橇联畔谋现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,1.4 The Ergodicity of Stationary Random Sequence,Definition Let xs(n)be a sample(realization)of a stationary stochastic sequence x(n),then x(n)is said to be er
25、godic if,肋阑巡睹珊臃烽辫扇衫四锋稳奏辉尼树堪萌循祝予皱表监矛驯盲绵那肥畅现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Understanding of ErgodicityThe statistical expectation along the time of one realization is same as the statistical expectation across t
26、he space(or ensemble)of different realizations of the random sequence,it can be denoted briefly asThe ergodicity of random sequence facilitates the analysis of stationary random sequence by examining one realization of this signal instead of the set of signal samples,痛乒框捍蒲具萌韦勘蕾财严懒遂陨沼针拌蛾朗篮爬酗纽晕保蔽绷晶靴诽划
27、现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,1.5 Some Useful Classes of Random Sequences or Stochastic Processes,Gaussian(Normal)Random Sequence A random sequence xs(n)with Nth PDF,穷题愧迂嘻缝已素雄陨况盟梧卡置葫智锡答属撬酣欢灭糠怪乞瓦胶斟褥膳现代数字信号处理Advan
28、ced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Advantages of Gaussian ModelGaussian PDF only depends on its 1st-order and 2nd-order moments.A wide-sense stationary Gaussian process is also a strict-sense stationary process and vice versa
29、.Gaussian PDFs can model the distribution of many processes including some important classes of signals and noise.The sum of many independent random processes has a Gaussian distribution(central limit theorem).Non-Gaussian processes can be approximated by a weighted combination(i.e.a mixture)of a nu
30、mber of Gaussian pdfs of appropriate means and variances.Optimal estimation methods based on Gaussian models often result in linear and mathematically tractable solutions.,坦惺吴铭帧漾汕队吩算瘁惊施曙憎浑畜浚昆斥渭钦彪镑磁僻带孩愉芜陛宝现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Proce
31、ssing_ch1 fundamentals,Markov Process A stochastic process whose conditional PDF satisfiesis a 1st-order Markov process(or Markov process for short).The state of the Markov process at time n depends only on its state at time n1 and is independent of the process history before n1.,亭店杯淡隆诸愁搀虫逃镜梭让滇喝诗河小史
32、育谷全兹鸳术羌瘴搜焦哆咨谣现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Gauss-Markov Processstochastic processes that satisfy the requirements for both Gaussian processes and Markov processes If the input of a Markov process is a Gaussian r
33、andom sequence,then the output is a Gauss-Markov random sequence(stochastic process)Gauss-Markov process is not always stationary,金浦抓葛益机茸园篓澳崖逢呀势驳诅普苇恿糙绥敛锋蚤劲个廖庚赎题款陷现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,A scalar stationary
34、 Gauss-Markov process x(n)with variance and time constant has autocorrelation function and PSD as following forms,拖骏虚韩扒栓轧仟旷饱癌晶疥毛让格蛛骡今肿瞳樱檄策粉嘉五赊悼踏话密现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,White Noise SequenceA white noise s
35、equence is a random process of random variables that are uncorrelated and have a finite variance,i.e.The mean value of white noise sequence is zeroA stationary white noise sequence has a constant PSD,峰壶木逢速导冯腥界粥庆江四府峙赚垦瘁乱兼挥陷净胡灿孔香聊喇宫处屑现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理A
36、dvanced Digital Signal Processing_ch1 fundamentals,The difference between irrelevance and independenceThat two random variables(or signals)x and y are irrelevant(uncorrelated)implies that for any mThat two random variables(or signals)x and y are statistical independent implies that their joint PDFIn
37、dependent uncorrelated,but uncorrelated independentFor the white noise sequence with Gaussian distribution,uncorrelated independent,凤船鹏斡胀减防坏添轧叉撤鹅脾恩蛔裙邀谤稻议沤梨蚂参嘉刮扑万宙芥荆现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Some examples of
38、the realization of white noise sequenceStandard normal distributionUniform distribution between-1,1,脉鸡绸轧贝哎罩园漫鸳挛诧汛俞舜秃匀啊巫朗重档唉你咬究爷悬淹蛤值柬现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Harmonic ProcesswhereHarmonic process x(n)is a st
39、ationary process,Independent and identically distributed,伞余虱猪恋喀搬锨垂成酌解虫豢徊啥托傲孔击亲嫡济惧撅纂置挤食区臆鸥现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Example of Harmonic Process:3 Realizations,涸褥居造供骑宫蹄柜耻盔恤着鹏括能仁卷倪龚猖竟喘恿误记拣览体喧祝慨现代数字信号处理Advanced
40、Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,1.6 LTI Systems with Stationary Random Inputs,LTI system with unit pulse response h(n),Stationary random sequence x(n),system response y(n),H(z),Y(z),X(z),钳枷噬庇直胰龟夸菊炙冯好戈瞬琢产濒庄斌谗刑谊聂罪序平狮碴愈霍勿炮现代数字信号处理Adv
41、anced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,my(n),ryy(n,n+m),and the Stationarity of y(n)Therefore,like the x(n),y(n)is also a stationary random sequence,Irrelevant with n,歹韭械氏氓叔障丫响经需畸巡华潜耳愿售嫩操乏雄晾遗待锄们新绕袍兆钥现代数字信号处理Advanced Digital Signal
42、Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The PSD of y(n),意朽卸习顽真诌畔锦渍扯滦辆矮乘售滴湾谋企拂稿莽窥窑护强馆咕剐羔埠现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Cross-Correlation&Cross-PSD of x(n)and y(n
43、),尘峭皂庄将蜡板澳驭害遁鳖躬贸咋市漾拾瞒饱铂烫渊上或前立词魂鬼缄富现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,The Correlation-Convolution Theorem,臣虞奥乎布棋赵芳禾闸得人词门花彝蟹贬向遇粟蛰遮趁梯若秋义巳抛殷三现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Dig
44、ital Signal Processing_ch1 fundamentals,Proof:,桌扦匆义瘪帜漱肃坎生裤聚具盅迟钢奏泪矩噎攀驮套览潭嫡略驶胞颂坡墒现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Time Series Model of Stationary Random Sequences,linear system with system function H(z),White noise w
45、(n),x(n),A stationary random sequence can be denoted as an output of a linear system with a white noise input.,1.7 Time Series Model,哩糟男哪疲家咙男粳溯慢育拷懊骄表柞蛤嗣汛刀贾睦汇预哥伏乃给降卯惫现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,If the linear sy
46、stem has system function,Then H(z)is called the time series model of x(n),and,If all the zeros and poles of H(z)are inside the unit circle,then the H(z)is called a minimum phase system.It has minimum phase lag among all the systems with same amplitude-frequency characteristic.A minimum phase system
47、has a stable invertible system.,淄盼娃玄和姆式陇凹录辰娶灵疏避听伐恰徐指苟落禄萧录赋绥多有颤抓州现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Three kinds of time series modelsMoving-average(MA)model,If q is finite,then the H(z)is a FIR filter.The MA model is
48、suitable for describing the sequences whose PSD has vales but has no peak(can be modeled with less orders).,我糙烫钩酱豺仰飘族肩佬簧遇导锡罕猴柱拱服并彝私尹嘱状痛构管挽靳谅现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Auto-regressive(AR)model,The AR model is
49、stable iff all its poles are inside the unit circle.It is suitable for the description of the sequences whose PSD has peaks but has no vale.,拄渠众灶墙似身妥帛金驻力柔藤聊湿葡烹恬傲庄墒娘惹盈捅荣决乱亢涌康现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Auto-reg
50、ressive moving average(ARMA)model,The ARMA model is suitable for sequences whose PSD with vales and peaks.,韶旬机页谍翌俐砰步袒上名第讲络逻锨续疤带挥臂卧奶棕坟全钳瞅班揉团现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals现代数字信号处理Advanced Digital Signal Processing_ch1 fundamentals,Wolds decomposition theoremAny real-valued
