Content Chate 8 Samling Lectue D Noaat Wattanamongkhol Samling Theoem Samling of Continuou-Time Signal 3 Poceing Continuou-Time Signal 4 Samling of Dicete-Time Signal 5 Multi-ate Samling Deatment of Electical Engineeing, Engineeing Faculty, Buaha Univeity 8 Intoduction ti The amling oce hould not yield any lo of the infomation In othe wod, the oiginal analog ignal hould be econtucted (etoed) baed on the time-dicete equence Signal econtuction i needed eone to the Nyquit amling theoem It i band-limited dli it d ignal and amling fequency i moe than two time of maximum fequency of continuou-time ignal The oblem i how to chooe the amling inteval T o that the oiginal analog ignal can be econtucted amling xt xn xnt Analog ignal xnt x 0, x T, x T,, xnt Continuou-time ignal Dicete-time equence i called amle o data equence T: amling eiod
8 Samling Continuou-time Signal Let x(t) be non-eiodic continuou-time time ignal and eal function and thei Fouie tanfom i X j Then, ignal x(t) in called bandlimited ignal when 0, m X j fo Shannon Samling Theoem A continuou-time ignal x(t) with fequencie no highe than f max can be econtucted exactly fom it amle x[n] = x(nt), if the amle ae taken at a ate f = /T that i geate than f max, which i called Nyquit amling ate (o fequency) when i the maximum fequency (adian/ec) of ignal x(t) and m when /T T i the amling fequency The minimum i amling fequency m i called Nyquit ate If the amling fequency m will haen henomenon aliaing i which h did not deied d and we then can not econtuct the ignal x(t) m Claude Shannon fathe of infomation theoy Hay Nyquit o With diffeent amling fequency, amling of an analog ignal will diffeent dicete equence o Samling of diffeent analog ignal may yield the ame dicete equence o Samling fequency mut be emloyed in ode to econtuct the oiginal analog ignal o Audio CD ue a amling ate of 44 khz fo toage of digital audio ignal, which i lightly highe than 0 khz, the ue limit fo human heaing and ecetion of muical ound Nyquit ate i 0 = 40 khz that the ytem will eeve bandwidth 4 khz fo taking the guad band o In the telehone ytem, eech ignal ha bandwidth 34 khz and ue amling ate of 8 khz Nyquit ate i 34 = 68 khz that the ytem will eeve bandwidth khz fo the guad band 8 Samling oce Data amle which deive fom amling continuou-time ignal x(t) can be exeed of multilication between ignal x(t) and delta imule tain It will often be called amling function that i when t tnt x t x t t n Fom the oety of delta imule function, xt tt xt tt x t xnt tnt n 0 0 0
That i, x ha imule tain ignal that amlitude of each imule i t equal to xnt Ditance between of each imule ignal i T which i hown in the figue, t t nt P j k T CtFT j n, and the multilication oety of FT i n F x t t X j P j d Theefoe, the eult of FT fo t ignal i X j =F x t X j k T k The FT ai of amling goce fo continuou-time ignal i x t x nt t nt X j X j k T CtFT n T k In the amling oce, if it ue fequency m, then xt xnt the ignal can be econtucted a to the data equence by taking thi data equence into the ideal low-a filte Thi filte ha an imule eone h t with gain equal to T and ha the cut-off fequency equal to ignal i, thu and thei eult of FT i H m c m Tin ct Tc h t incct t T, c j 0, ele Fom the figue, x t i the econtucted ignal when the amling ignal eond to amling theoem Then, it get ectum fequency X j X j and will be ummaized a x t xt 8 Samling with zeo-ode hold Nomally, ignal x t get fom amling imule tain ignal that imule ignal i a naow ignal and moe high amlitude In actical, ceating and tanmiion ignal to the detination i difficult Then, the ignal wa aleady amled will be fomed of zeo-ode hold by ending ignal x t into zeo-ode hold filte with imule eone h0 t when h 0 t and the eult of FT fo h t i 0, 0 t T 0, ele j T / in T / 0 H j e
It mean that amling ignal ytem x t will kee the eult until to the conecutive amling Theefoe, thi eult alo wa called zeo-ode ode hold ignal x t 0 Similaly, the econtuction ignal xt fom the zeo-ode ode hold ignal x0 t can take by tanmiion ignal x0 t though the econtuction filte with imule eone h t By doing thi, the econtucted ignal t will be the ame a oiginal ignal xt when the ummation of imule eone filte of two cicuit i equal to ideal low-a filte That i, o in fequency domain, h t h t h t 0 H j H j H j 0 We will have H j jt / jt / j H j Te e H0 j in T / inc T / fo c Figue how amlitude ectum and hae ectum of the econtuction filte cicuit, h t Geneally, the filte cicuit H j can not be ceated in the actical Howeve, the zeo-ode hold ignal x 0 t in many cae can be eeented the oiginal ignal x t by doe not ue filte cicuit h t If the amling fequency i high value, o uing etimation in an inteval, o inteolation between data amle 83 Recontuction the continuou-time ignal Let ignal xt be the continuou-time ignal with limited bandwidth and the eult of FT i equal to X j If amling fequency of ignal xt i m, then the data amle will have ectum fequency X j x t
Pincile of the econtuction oce of continuou-time ignal can be exlained in fequency domain a following Signal x t xt when ignal x t ha eult of FT equal to X j Then, ectum fequency X j in the figue i equal to X j when it take the multilication between X j and H j, that i X j X j X j H j when H j i FT of ideal low-a filte by uing the multilication oety of ignal in fequency domain equal to the convolution ignal in time domain Thu, ignal x can be deived fom the invee FT of, t X j that i F - X j x t x t h t T c x nt tnt inc c t n Tc x nt inc c t nt n Thi equation will be called Nyquit-Shannon inteolation fomula Fom the eult, it can be een that the econtuction ignal deend on the etimation technique in an inteval uing Sinc function Howeve, o many etimation technique ae ovided in the actical uch a etimation technique zeo-ode hold filte, eay but doe not get a good ignal Thu, if we want to imove the coectne of etimation technique, we will ue highe-ode filte uch a fit-ode with imule eone, and ha FT a t, t T h t T 0, ele T in / H j F h t = T / 84 Aliaing Aliaing haen when the amling fequency i le than two time of the maximum fequency of continuou-time ignal, m Sectum fequency fom amling X j will ovela o aliaing It effect to the econtuction ignal fom data amle with fequency m Shae of thi ignal will be changed fom the oiginal ignal becaue ectum fequency of econtucted ignal wa changed fom the oiginal Uing fit-ode filte fo econtuction ignal may be conide a lineal inteolation technique that it i the ame a linking the line of each value of data equence
83 Dicete-time oceing Dicete-time time oceing i the oce of continuou-time-to-timdicete-time (C/D) to conveion 83 Continuou-time-to-dicete-time ignal conveion Fom figue, amling ignal x follow ignal conveion c t x oce to data equence when amling eiod T = T t xd n and T = T c x t x nt t nt n CtFT j nt tnt e j nt X = j F x t xc nt e n Conideing ectum fequency of x n can be deived fom X e = F x n x n e j jn d d d n Subtitution x n x nt in an equation; and then, we will have Fom equation, d c d = c X e x nt e j jn n j Xd e = X j T X j X j k T c k d
when / T, ectum fequency of x n i k Xd j Xc j T k T j Fom the figue, it can be een that ectum fequency Xd e which i fequency-caled veion, ha the eiod a = ad d X j / 83 Dicete-time-to-continuou-time ignal conveion Outut ignal of the ytem y t will have ectum fequency a c ( j ) Y j X j H e X j H j c c d c c when H ( e j ) and H j ha the elation a following d H c c jt Hd( e ), / j 0, ele That i, which i hown in the figue T 84 Samling dicete-time time ignal 84 Samling oce Delta imule tain i defined by n n kn k when N i the amling eiod and n i the delta function which i defined by, n 0 n 0, n 0 Fom the figue, it deict amling oce of dicete-time ignal when x n i the outut ignal fom thi oce x x n, n multile of N n 0, ele
Fom the oety of delta function: x k k k x k k k 0 0 0 Thu, data amle x n can be exeed in the math model a x n x n n x kn n kn That thei eult of FT i k j j j =F X e x n P e X e d Since, FT of ignal n i Pe n k N j =F k when / N i amling fequency, N j jk = k 0 X e X e N DtFT j jk = N x n x kn n kn X e X e N k k0 Fequency ectum Imule tain ignal Sectum of data amle Aliaing m 84 Recontuction ti dicete-time ti ignal The amled ignal will be econtucted to the oiginal ignal xn (when amling fequency ) fom data amle x m n by uing an ideal low-a filte when imule eone h n with the gain N and cut-off fequency and the FT ai i c Nin cn N h n inccn n N, c j H ( e ) N c 0, ele Outut ignal of ideal low-a filte x i the ame a oiginal n ignal xn when the cut-off fequency m c m
Theefoe, amling ignal oce eond to the amling j j theoem, X that i when ignal a e = X e x n xn x n Nc x n x n h n h n x kn inc c nkn k A we can ee, the econtuction ignal oce need to ue ideal low-a filte that it i difficult in actical Then, in geneally, we will aoximate ideal low-a filte uing inteolating technique to get the ignal x n which i exeed in the tem of x n x kn h n kn k when h n i an imule eone of dicete-time ignal of inteolating filte uch a zeo-ode hold filte o fit-ode hold filte 85 Multi-ate t amling In actical, ignal oceing of any ytem commonly ue amling ate uch a tanmitte and eceive ide ue diffeent amling ate Moeove, ignal oceing uing multi-ate amling ha a moe advantage uch a: Oveall amling can emloy the tanfomation ti amling ate with non-intege numbe Naowband filte can eay ceate by decimation oce and aoximation o inteolation technique Changing amling ate ued to the ytem that it equie adding bandwidth fo examle, time-diviion multilexing Noie ignal can be educed by taking oveamling to adjut the noie haing Uing oveamling ate can educe any tictne of filte fo examle, making lowe loe of the low-a filte in fequency-domain A/D and D/A conveion ocee have a highe accuacy and coectne 85 Decimation Decimation i the oce of deceaing the amling ate by a facto of M ie ie fom F to F / M Down amling : by a facto of M i achieved by dicading M - amle fo evey M amle Thi combined oeation of filteing (anti-aliaing) and down amling i called a Decimation Block diagam of Decimato
85 Inteolation Poce of inceaing the amling ate of the ignal by a facto of L ie fom F to L F U amling by a facto L -> ineting L - zeo between two amle Thi combined oeation of u amling and filteing (anti-aliaing) i called a Inteolation L = 3 Block diagam of Inteolation Examle : Conideing ignal xt 0 co 000 t co 8000 t Calculate minimum amling fequency which coeond to amling theoem Sol: Fom the oety of tigonometic: co co co co A B AB AB Then, ignal can be ewitten a, 0 co 000 co 8000 5co 6000 5co 0000 xt t t t t A a eult, the maximum m amling fequency enc of ignal xt i 0000 / Thu, the minimum amling fequency o m ad Nyquit ate i m 0000 0000 ad / Examle : Signal x i deived fom amling ignal with t xt fequency / when and x t co t and x t xcnt t nt n when / T i the amling fequency (a) Detemine g t which wa atified xt co co / t gt (b) Poof: gnt 0 fo n 0,,, (c) Poof: taking ignal x into ideal low-a filte with cut-off t fequency /, then it will get ignal yt coco t/ Sol: (a) Fom the oety of tigonometic, xt t t t That i, g t in t in co co coin in (b) Subtitution /T T and t nt in the equation of g t g nt nt n T in in in in That it ha the ame value a g nt 0 fo n 0,,, (c) Fom (a) and (b); x t tntco nt cognt n tntco nt co n Then, it wa aed ideal low-a filte with cut-off fequency /, then it will get ignal y t co co t /