IRO0140 Advanced space time-frequency signal processing
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1 IRO4 Advanced space tme-frequency sgnal processng Lecture Toomas Ruuben Takng nto account propertes of the sgnals, we can group these as followng: Regular and random sgnals (are all sgnal parameters determned or not Contnous tme t ( and dscreet tme sgnals (s the sgnal tme varalble contnous or sampled ( n ( nt T / F c s s Analog and quantzsed sgnals (s the sgnal values contnousvalued or dscrete-valued Dgtal sgnals, perhaps quantzsed dcreet tme sgnals where quantzsed levels are presented wth bnary code Real and comple sgnals Sgnals wth nfnte or fnte tme duraton Perodcal sgnals may descrbed by the followng formulas: ( t ( t T T ( k, t ( t kt k ( n ( n ( k, n ( n k k Here T s the sgnal tme perod and s the perood n samples Takng nto accout symmetcty of the sgnals n tme doman t, we can dvde them as the sgnals wth even or odd symmetrcty To obtan real or comple valued dscreet tme sgnals, samplng of the correspondng contnous tme sgnals wth frequency F s must be done (usually, the samplng frequency s constant Man task of the sgnal analyss s developng of algortms to estmate parameters, propertes, smlartes and varatons of the sgnal Random sgnals may be analyzed by usng the followng methods: Statstcal analyss of the sgnal ampltudes (sample values Analyss and modellbg of correlaton propertes of the sgnal dscreet values. Smultaneous estmatng of multple sgnals, ther mutual nfluence and relatonshps 3 4 Analyss of random sgnals The sgnal parameters whch are only depend on values of samples but not wth the order of samples are: Arthmetcal and statstcal (specfc n-th sample mean value Analyss of the sngle sgnal sample values Mean value, probablty densty functon, peak value Analyss of the mutual nfluence of the sngle sgnal sample values Autocorrelaton, Power Spectrum Densty, parametrcal models, hgh order statstcs Smultaneous estmatng of multple sgnals Crosscorrelaton, mutual power, coherency, hgh order statstcs ( n ( n / n ( n k(, n W (, n R - amount of samples R - amount of measurements (relzatons - Values of random measure n level k(, n - amount of n th dscreet measurement values on level W(, n - probablty densty functon 5 6
2 4 3 Representaton of a random dscrete-tme sgnal n case of = *** * ****** *** **** * ***** **** ** * n= n= =3 The dstrbuton of dscrete values of sgnals are represented by probablty densty functons. They show the representaton probablty (amount, percentage of the dscrete values of sgnals n a certan range. The result s represented as a hstogram. k(, n W (, n R ormal dstrbuton (amount Unform dstrbuton (amount 7 8 Sgnal power Sgnal RMS (Root Mean Square Sgnal AC (floatng component Sgnal energy ( n ( n n n n ef ( ( n ( n ( n ( n E n n ( Dsperson s equal wth the centralzed moment of the second order of a random measure and characterze the wdth of the probablty densty functon. Its formula: ( n M ( n ( n W (, n ( n ( n Hgh order dstrbuton moments are obtaned by averagng stages of random values. k k ( n W (, n k M ( n ( n W (, n k 9 If the sgnal probablty densty functon s ndependent from tme, then the sgnal s statonary. W(, n W(, n y W( y...,,,,... In case of statonary sgnals, dstrbuton moments are fed. k ( n k ( n y k Random sgnal s ergodc f ts statstcal mean value s equvalent wth arthmetcal mean value calculated by usng one measurement r ( n of the random process ( n lm ( n In case of ergodc sgnal, fndng of dstrbuton moments s much more spler by usng one measurement averagng durng enough long tme perood. n ( n
3 In case of ergodc sgnals: Frst order of dstrbuton moment s equal wth DC component of the sgnal ( n lm ( n ( n Second order of dstrbuton moment s equal wth the power of the sgnal ( n lm ( n ( n n n Dsperson (centralzed moment of the second order of a random measure s equal wth the power of AC component of the sgnal ( n ( n Sgnal RMS (Root Mean Square value s equal wth standard devaton M The devaton of a random measure equals wth the dstncton of a random measure and ts mean value. ( n ( n ormalzed devaton appears as a dvson of the devaton of a random measure wth the standard devaton: ( n 3 4 Analyss of the mutual nfluence of the sngle sgnal sample values and multple sgnals. Man defntons Autocorrelaton functon (ACF n power dmenson Convoluton ( n ( n ( ( n Mutual power B( n ( n ( n n n Most mportant propertes: B( ( n ( n ( n P P ( n ( n ( n ( n n Crosscorrelaton functon (CCF n power dmenson B( n B( n B ( n B ( n B ( n ( n ( n n n B ( n B ( n 5 6 Sgnals are orthogonal f the followng equaton s vald n ( n ( n.. Orthogonalty n case of comples sgnals ( n ja ( n. * ( n ( n AC and DC components of the sgnal are orthogonal Autocorrelaton functon of harmoncal sgnal s ndependent from the ntal phase of the sgnal Sgnal wth fnte duraton have fnte ACF and perodcal sgnal have perodcal sgnaalde ACF Random processes may change n tme. One mportant proeperty s the rate of change. Ths property s not represented n dstrbutn moments and dstrbuton functons. Correlaton functons used nstead. Averagng multplcatons of random sgnal values on dferent tme moments, autocorrelaton functon of random process (ACF s obtaned. B ( n, n W (,, n, n j j j and j are random sgnal measurement values on levels and j. Quantszed levels don't change durng the sgnal
4 In case of statonary sgnals ACF s ndependent from concrete n and nand depend only from nterval of the samples. n n n B ( n W (,, n j j j In case of ergodc sgnals, only measurement needed B ( n lm ( n ( n n ( n ( n n n To descrbe relaton between the two sgnals, crosscorrelaton functon s used ( n ja y( n CCR (two dmensonal assorted dstrbuton moment B ( n, n y W (, y, n, n y j j j CCR n case of statonary sgnals B ( n y W (, y, n y j j j CCR n case of ergodc sgnals B ( n lm ( n y( n n ( n y( n n y n 9 In case of aggregated sgnals, ACF and CCF may be represented as correlaton matr. Elements on correlaton matr dagonaal are ACF-s and another matr elements are correspondng CCF-s. Tsentralzed ACF: ormalzed ACF: K ( n B ( n P B ( n ( n ( K ( n R n Autocorrelaton functon of nose sgnals have usually narrow peak contcentrated around delay tme n. n k Correlaton tme s used to descrbe wdth of the autocorrelaton functon man lobe. Autocorrelaton functon propertes of determnstc sgnals are the same as ergodc random sgnals. How to generate sgnal manually for t=:.:., // sgnal samples, length ms f t <.5, //Sgnal value s f t <.5 ms a=; else a=4*t/3-5/3; //Lne equaton end z=(t*/.+; //Indeng of the sgnal vector zz=round(z; (zz=a; //Recordng of sgnal values nto array end t=(:.:.; //Tme vector for sgnal plottng How to plot the sgnal fgure(; plot(t,,ttle('sigaal ',grd,pause Plottng multple sgnals subplot(,plot(t,,ttle('sigaal ', grd, subplot(,plot(t,,ttle('sigaal ',grd, pause 3 4 4
![Several methods for sgnal generaton: How to calculate mean value n MATLAB: =[ 3 4 5 6 3 4] // Manually as vector X=[ 3;4 5 6;7 8 9] // Manually as matr Z=zeros(,4 //*4 matr wth zeros F=ones(3,3 //3*3](/docs-images/94/119825872/images/5-0.jpg "matr wth ones R=rand(M,,P... //Unf. dstrbuton wth sze M,, P.. R=rand('twster',sum(*clock //Wth dferent ntal values //method twster ( state, seed R = a + (b-a.")
5 Several methods for sgnal generaton: How to calculate mean value n MATLAB: =[ ] // Manually as vector X=[ 3;4 5 6;7 8 9] // Manually as matr Z=zeros(,4 //*4 matr wth zeros F=ones(3,3 //3*3 matr wth ones R=rand(M,,P... //Unf. dstrbuton wth sze M,, P.. R=rand('twster',sum(*clock //Wth dferent ntal values //method twster ( state, seed R = a + (b-a.*rand(,; //Unf, dstrbuton between a and b r = +.*randn(,; //ormal dstrbuton mean value //and standard devaton =[ ] //Sgnal vector dsp( Sgnal mean value'; dsp(mean(; How to calculate sgnal power n MATLAB: dsp( Sgnal power'; P=mean(.*; 5 6 How to calculate mutual energy of the sgnals How to calculate autocerrelaton functon dsp( Mutual energy of and y'; dsp(mean(.*y; How to calculate RMS dsp( Sgnal Root Mean Square value'; dsp(sqrt(p; t=(:.:.; //Tme vector dsp( Sgnal ACF'; A=corr(/(t(length(t/.; ta=(-t(length(t:.:t(length(t; subplot(,plot(t,,ttle('sigaal',grd,subplot(,plot(ta,a, ttle('akf',grd,pause 7 8 Sgnal and ts ACF n MATLAB: How to calculate crosscorrelaton t=(:.:.4; //Tme vector dsp( Calculatng of CCF of and y'; A=corr(,y/(t(length(t/.; B=corr(y,/(t(length(t/.; ta=(-t(length(t:.:t(length(t; plot(ta,a,ttle( RKF SS',grd,pause plot(ta,b,ttle( RKF SS',grd,pause 9 3 5
6 Crosscorrelaton plot n MATLAB (S S Crosscorrelaton plot n MATLAB (S S 3 3 How to calculate Hstogram of the sgnal MATLAB =-4:.:4 y = randn(,; hst(y, 33 6
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