Signal Processing and Linear Systems I


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1 Sanford Universiy Summer Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 1
2 Time Domain Analysis of Coninuous Time Sysems Zeroinpu and zerosae responses of a sysem Impulse response Exended lineariy Response of a linear imeinvarian (LTI) sysem Superposiion inegral Convoluion EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 2
3 Sysem Equaion The Sysem Equaion relaes he oupus of a sysem o is inpus. Example from las ime: he sysem described by he block diagram x Z y a has a sysem equaion y + ay = x. In addiion, he iniial condiions mus be given o uniquely specifiy a soluion. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 3
4 Soluions for he Sysem Equaion Solving he sysem equaion ells us he oupu for a given inpu. The oupu consiss of wo componens: The zeroinpu response, which is wha he sysem does wih no inpu a all. This is due o iniial condiions, such as energy sored in capaciors and inducors. x() = H y() EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 4
5 The zerosae response, which is he oupu of he sysem wih all iniial condiions zero. x() H y() If H is a linear sysem, is zeroinpu response is zero. Homogeneiy saes if y = F (ax), hen y = af (x). If a = hen a zero inpu requires a zero oupu. x() = y() = H EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 5
6 Example: Solve for he volage across he capacior y() for an arbirary inpu volage x(), given an iniial value y() = Y. i() R x() + C + y() From Kirchhoff s volage law x() = Ri() + y() Using i() = Cy () RCy () + y() = x(). This is a firs order LCCODE, which is linear wih zero iniial condiions. Firs we solve for he homogeneous soluion by seing he righ side (he inpu) o zero RCy () + y() =. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 6
7 The soluion o his is y() = Ae /RC which can be verified by direc subsiuion. To solve for he oal response, we le he undeermined coefficien be a funcion of ime y() = A()e /RC. Subsiuing his ino he differenial equaion [ RC A ()e /RC 1 ] RC A()e /RC + A()e /RC = x() Simplying [ 1 A () = x() which can be inegraed from = o ge A() = x(τ) RC e/rc ] [ ] 1 RC eτ/rc dτ + A() EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 7
8 Then y() = A()e /RC = e /RC x(τ) = x(τ) [ ] 1 RC eτ/rc dτ + A()e /RC ] [ 1 RC e ( τ)/rc dτ + A()e /RC A =, y() = Y, so his gives A() = Y y() = [ 1 x(τ) ] dτ RC e ( τ)/rc } {{ } zero sae response + Y } e /RC {{ }. zero inpu response EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 8
9 Impulse Response The impulse response of a linear sysem h(, τ) is he oupu of he sysem a ime o an impulse a ime τ. This can be wrien as h(, τ) = H(δ( τ)) Care is required in inerpreing his expression! δ() δ( τ) H h(, ) h(, τ) τ EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 9
10 Noe: Be aware of poenial confusion here: When you wrie h(, τ) = H(δ( τ)) he variable serves differen roles on each side of he equaion. on he lef is a specific value for ime, he ime a which he oupu is being sampled. on he righ is varying over all real numbers, i is no he same as on he lef. The oupu a ime specific ime on he lef in general depends on he inpu a all imes on he righ (he enire inpu waveform). EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 1
11 Assume he inpu impulse is a τ =, h(, ) = H(δ()). We wan o know he impulse response a ime = 2. I doesn make any sense o se = 2, and wrie h(2, ) = H(δ(2)) No! Firs, δ(2) is somehing like zero, so H() would be zero. Second, he value of h(2, ) depends on he enire inpu waveform, no jus he value a = 2. δ() H δ(2) h(, ) h(2, ) 2 2 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 11
12 Compare o an equaion such as y () + 2y() = x() which holds for each, so ha y (1) + 2y(1) = x(1). EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 12
13 If H is ime invarian, delaying he inpu and oupu boh by a ime τ should produce he same response h(, τ) = h( τ, τ τ) = h( τ, ). Hence h is only a funcion of τ. We suppress he second argumen, and define he impulse response of a linear imeinvarian (LTI) sysem H o be h() = H(δ()) δ() h() τ δ( τ) H h( τ) EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 13
14 RC Circui example i() R x() + C + y() The soluion for an inpu x() and iniial y() = Y is y() = x(τ) [ ] 1 RC e ( τ)/rc dτ + Y e /RC The zerosae response is (Y = ) is y() = x(τ) [ ] 1 RC e ( τ)/rc dτ EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 14
15 The impulse response is hen h() = δ(τ) = 1 RC e /RC [ ] 1 RC e ( τ)/rc dτ for, and zero oherwise. We inegrae from o include he impulse. This impulse response looks like: 1 RC 1 RC e /RC RC 2RC EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 15
16 Lineariy and Exended Lineariy Lineariy: A sysem S is linear if i saisfies boh Homogeneiy: If y = Sx, and a is a consan hen ay = S(ax). Superposiion: If y 1 = Sx 1 and y 2 = Sx 2, hen y 1 + y 2 = S(x 1 + x 2 ). Combined Homogeneiy and Superposiion: If y 1 = Sx 1 and y 2 = Sx 2, and a and b are consans, ay 1 + by 2 = S(ax 1 + bx 2 ) EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 16
17 Exended Lineariy Summaion: If y n = Sx n for all n, an ineger from ( < n < ), and a n are consans ( ) a n y n = S a n x n n Summaion and he sysem operaor commue, and can be inerchanged. Inegraion (Simple Example) : If y = Sx, n a(τ)y( τ) dτ = S ( ) a(τ)x( τ)dτ Inegraion and he sysem operaor commue, and can be inerchanged. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 17
18 Oupu of an LTI Sysem We would like o deermine an expression for he oupu y() of an linear ime invarian sysem, given an inpu x() x H y We can wrie a signal x() as a sample of iself x() = x(τ)δ( τ) dτ This means ha x() can be wrien as a weighed inegral of δ funcions. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 18
19 Applying he sysem H o he inpu x(), y() = H (x()) ( = H ) x(τ)δ( τ)dτ If he sysem obeys exended lineariy we can inerchange he order of he sysem operaor and he inegraion y() = x(τ)h (δ( τ)) dτ. The impulse response is h(, τ) = H(δ( τ)). EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 19
20 Subsiuing for he impulse response gives y() = x(τ)h(, τ)dτ. This is a superposiion inegral. The values of x(τ)h(, τ)dτ are superimposed (added up) for each inpu ime τ. If H is ime invarian, his wrien more simply as y() = x(τ)h( τ)dτ. This is in he form of a convoluion inegral, which will be he subjec of he nex class. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 2
21 Graphically, his can be represened as: δ() Inpu Oupu h() δ( τ) h( τ) τ τ (x(τ)dτ)δ( τ) (x(τ)dτ)h( τ) x() τ τ y() x() x() = τ Z x(τ)δ( τ)dτ y() = Z x(τ)h( τ)dτ EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 21
22 RC Circui example, again The impulse response of he RC circui example is h() = 1 RC e /RC The response of his sysem o an inpu x() is hen y() = = x(τ)h( τ)dτ x(τ) [ ] 1 RC e ( τ)/rc dτ which is he zero sae soluion we found earlier. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 22
23 Example: High energy phoon deecors can be modeled as having a simple exponenial decay impulse response. 54 Doshi e al.: LSO PET deecor 154 ABLE I. Summary resuls from he various lighguide configuraion experiens. Coupler Energy resoluion FWHN % Ligh Ligh collecion efficiency % Average peakovalley raio Scinillaing Number of crysals Crysal clearly resolved irec LSO a ighguide a CV lens iber a iber aper Phoomuliplier Ligh Fibers Crysal nergy resoluion and ligh collecion efficiency were measured wih single ighguide elemens. uding he PMT socke conaining he dynode resisor chain ias nework, is 3 cm long, 3 cm wide, and 9.75 cm long. I. METHODS DETECTOR CHARACTERIZATION. Flood source hisogram A deecor module was uniformly irradiaed wih a 68 Ge oin source 2.6 Ci. The signals from he PSPMT were eaed and digiized as described above in Sec. II D. The wer energy hreshold was se o approximaely 1 kev ih he aid of he hreshold on he consan fracion disiminaor and no upper energy hreshold was applied.. Energy specra Phoon From: Doshi e al, Med Phys. 27(7), p1535 July 2 FIG. 5. A picure of he assembled deecor module consising of a 9 9 array of mm 3 LSO crysals coupled hrough a apered opical fiber bundle o a Hamamasu R59C8 PSPMT. These are used in posiiron emmision omography (PET) sysems. were defined. The deecors were hen configured in coincidence, 15 cm apar, and lismode daa was acquired by sepping a 1 mm diameer 22 Na poin source same as used in Sec. III C beween he deecors in.254 mm seps. The poin source was scanned across he fifh row of he deecor. For each opposing crysal pair, he couns were recorded as a funcion of he poin source posiion. A lower energy window of 1 kev was applied. The FWHM of he resuling disribuion for each crysal pair was deermined o give he inrinsic spaial resoluion of he deecors. EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 23
24 Inpu is a sequence of impulses (phoons). Oupu is superposiion of impulse responses (ligh). Inpu: Phoons Oupu: Ligh EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 24
25 Summary For an inpu x(), he oupu of an linear sysem is given by he superposiion inegral y() = x(τ)h(, τ) dτ If he sysem is also ime invarian, he resul is a convoluion inegral y() = x(τ)h( τ) dτ The response of an LTI sysem is compleely characerized by is impulse response h(). EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons 25
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