Representing a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
|
|
- Victor Woods
- 5 years ago
- Views:
Transcription
1 Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion as a linear combinaion of impulses and he response as a linear combinaion of impulse responses he ourier series represens a signal as a linear combinaion of complex sinusoids M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al Lineariy and Superposiion If an exciaion can be expressed as a sum of complex sinusoids he response of an LI sysem can be expressed as he sum of responses o complex sinusoids. Real and Complex Sinusoids cos( x) = e jx + e jx sin( x) = e jx e jx j M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 3 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 4 Jean Bapise Joseph ourier Concepual Overview he ourier series represens a signal as a sum of sinusoids. he bes approximaion o he dashed-line signal below using only a consan is he solid Consan line. (A consan is a 0.6 cosine of zero frequency.) x().6 Exa() Approximaion of x() by a consan 3//768-5/6/ M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 5 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 6
2 Concepual Overview he bes approximaion o he dashed-line signal using a consan plus one real sinusoid of he same fundamenal frequency as he dashed-line signal is he solid line. Sinusoid Concepual Overview he bes approximaion o he dashed-line signal using a consan plus one sinusoid of he same fundamenal frequency as he dashed-line signal plus anoher sinusoid of wice he fundamenal frequency of he dashed-line signal is he solid line x().6 Exa() Approximaion of x() hrough sinusoid M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 7 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 8 Concepual Overview he bes approximaion o he dashed-line signal using a consan plus hree sinusoids is he solid line. In his case (bu no in general), he hird sinusoid has zero ampliude. his means ha no sinusoid of hree imes he fundamenal frequency improves he approximaion. Concepual Overview he bes approximaion o he dashed-line signal using a consan plus four sinusoids is he solid line. his is a good approximaion which ges beer wih he addiion of more sinusoids a higher ineger muliples of he fundamenal frequency. Sinusoid x() Exa() Approximaion of x() hrough 3 sinusoids M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 9 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 0 Coninuous-ime ourier Series Definiion Orhogonaliy he ourier series represenaion of a signal x() over a ime 0 < < 0 + is = x = e j π / where [] is he harmonic funcion and is he harmonic number. he harmonic funcion can be found from he signal using he princple of orhogonaliy. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al
3 Using Euler's ideniy 0 + e j π / j πq / (,e ) = cos π q + j sin π q d If = q, e j π / j πq / (,e ) = cos( 0) + j sin( 0) d = d =. If q, he inegral e j π / j πq / (,e ) = cos π q + j sin π q d 0 is over a non-zero ineger number of cycles of a cosine and a sine and is herefore zero. Orhogonaliy herefore e j π / and e j πq / are orhogonal if and q are no equal. Now muliply he ourier series expression x j πq / by e (q an ineger) xe j πq / = e = j π( q) / = = and inegrae boh sides over he inerval 0 < xe j πq / j π( q) / d = e = d. 0 Orhogonaliy e j π / M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 3 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 4 Orhogonaliy Coninuous-ime ourier Series Definiion Summarizing = x e j π / and = = 0 + xe j π / d. he signal and is harmonic funcion form a ourier series where is he represenaion ime and, pair x c x herefore, he fundamenal period of he CS represenaion of x. If is also period of x, he CS represenaion of x is valid for all ime. his is, by far, he mos common use of he CS in engineering applicaions. If is no a period of x, he CS represenaion is generally valid only in he inerval 0 < M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 5 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 6 CS of a Real uncion I can be shown ha he coninuous-ime ourier series (CS) harmonic funcion of any real-valued funcion x ha = c * x [ ]. has he propery One implicaion of his fac is ha, for real-valued funcions, he magniude of he harmonic funcion is an even funcion and he phase is an odd funcion. he rigonomeric CS he fac ha, for a real-valued funcion x = c * x [ ] also leads o he definiion of an alernae form of he CS, he so-called rigonomeric form. x = a x [ 0] + { a x cos( π / ) + b x sin( π / )} where a x b x = = = xcos( π / )d xsin( π / )d M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 7 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 8 3
4 he rigonomeric CS CS Example # Since boh he complex and rigonomeric forms of he CS represen a signal, here mus be relaionships beween he harmonic funcions. hose relaionships are a x b x a x [ 0] = [ 0] b x [ 0] = 0 = + c * x = j( c * x ) [ 0] = a x [ 0] = a x j b x = c * x = a x + j b x, =,, 3,, =,, 3, M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 9 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 0 CS Example # CS Example # Le a signal be defined by x = 0 ms which is 0. = cos( 400π) and le M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al CS Example # CS Example #3 Le x = / ( 3 / 4)cos( 0π) + / sin 30π and le = 00 ms. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 3 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 4 4
5 CS Example #3 CS Example #3 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 5 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 6 CS Example #3 Lineariy of he CS hese relaions hold only if he harmonic funcions of all he componen funcions are based on he same represenaion ime. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 7 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 8 CS Example #4 Le he signal be a 50% duy-cycle square wave wih an ampliude of one and a fundamenal period 0 =. x = rec δ CS Example #4 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 9 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 30 5
6 CS Example #4 CS Example #4 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 3 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 3 CS Example #4 A graph of he magniude and phase of he harmonic funcion as a funcion of harmonic number is a good way of illusraing i. he Sinc uncion Le x = Arec ( / w) δ 0, w < 0. hen x = Arec ( / w) δ 0 = A sin ( πw / 0 ) he mahemaical form sin( π x) arises frequenly enough π x o be given is own name "sinc". ha is sinc = 0 π π sin π. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 33 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 34 Le x CS Example #5 = cos( 400π) and le = 7.5 ms which is.5 fundamenal periods of his signal. CS Example #5 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 35 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 36 6
7 CS Example #5 CS Example #5 he CS represenaion of his cosine is he signal below, which is an odd funcion, and he disconinuiies mae he represenaion have significan higher harmonic conen. his is a very inelegan represenaion. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 37 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 38 CS of Even and Odd uncions or an even funcion, he complex CS harmonic funcion is purely real and he sine harmonic funcion a x zero. b x is or an odd funcion, he complex CS harmonic funcion is purely imaginary and he cosine harmonic funcion is zero. Convergence of he CS or coninuous signals, convergence is exac a every poin. A Coninuous Signal Parial CS Sums x N N = e j π / 0 = N M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 39 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 40 Convergence of he CS Parial CS Sums Convergence of he CS or disconinuous signals, convergence is exac a every poin of coninuiy. Disconinuous Signal A poins of disconinuiy he ourier series represenaion converges o he mid-poin of he disconinuiy. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 4 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 4 7
8 Numerical Compuaion of he CS Numerical Compuaion of he CS How could we find he CS of a signal which has no nown funcional descripion? Numerically. = x( ) e j π / d Unnown N ( n+) s n=0 n s x( n s )e j π n s / d Samples from x() M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 43 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 44 Numerical Compuaion of he CS Numerical Compuaion of he CS I can be shown (Web Appendix ) ha, for harmonic numbers << N ( / N )D x n s where D ( ), << N N - ( x( n s )) = x n s n=0 e - j πn/n he Discree ourier ransform D N ( x( n s )) = x n s n=0 e j πn/n is an inrinsic funcion in mos modern high-level compuer languages. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 45 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 46 CS Properies Le a signal x() have a fundamenal period 0 x and le a signal y() have a fundamenal period 0 y. Le he CS harmonic funcions, each using a common period as he represenaion ime, be [] andc y []. hen he following properies apply. CS Properies ime Shifing x 0 e j π 0 / Lineariy α x + β y α + β c y M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 47 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 48 8
9 CS Properies CS Properies requency Shifing (Harmonic Number Shifing) e j π 0 / x 0 A shif in frequency (harmonic number) corresponds o muliplicaion of he ime funcion by a complex exponenial. ime Reversal x ime Scaling Le z = x( a), a > 0 c z = If a is an ineger, = [ / a], / a an ineger Case. = 0 x / a = 0z for z Case. = 0 x for z c z 0, oherwise M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 49 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 50 CS Properies CS Properies ime Scaling (coninued) Change of Represenaion ime c x,m = [ / m], / m an ineger Wih = 0 x, x Wih = m 0 x, x,m 0, oherwise (m is any posiive ineger) M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 5 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 5 CS Properies Change of Represenaion ime CS Properies d d ime Differeniaion ( x ) jπ / M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 53 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 54 9
10 CS Properies CS Properies ime Inegraion Case. [ 0] = 0 x( λ)dλ jπ /, 0 Case. [ 0] 0 x( λ)dλ is no periodic Case Case Muliplicaion - Convoluion Dualiy xy c x c y (he harmonic funcions [] and c y [] mus be based on he same represenaion ime.) x y c x c y he symbol indicaes periodic convoluion. Periodic convoluion is defined mahemaically by x y = x( τ )y( τ )dτ x y = x ap y where x ap is any single period of x M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 55 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 56 CS Properies CS Properies Conjugaion Parseval s heorem x * c * x x d = = he average power of a periodic signal is he sum of he average powers in is harmonic componens. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 57 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 58 Some Common CS Pairs CS Examples, arbirary ( / 0 ), / m an ineger δ δ 0 e j πq / 0 m 0 0, oherwise δ mq m 0 ( ) m 0 ( j / ) ( δ [ + mq] δ [ mq] ) ( ) m 0 ( / ) ( δ [ mq] + δ [ + mq] ) δ 0 m 0 ( w / 0 )sinc( w / m 0 )δ m δ m sin πq / 0 cos πq / 0 rec / w ri( / w) δ 0 ( w / 0 )sinc w / m 0 m 0 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 59 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 60 0
11 CS Examples x = sin( π / 0.0) rec ( / 0.0) δ 0.0 x = sin( 00π) rec ( 00) δ 0.0 ind he CS harmonic funcion of x wih = 0 ms. sin( πq / 0 ) m 0 ( j / ) ( δ [ + mq] δ [ mq] ) sin( 00π) j / 0.0 ( δ [ + ] δ [ ] ) rec ( / w) δ 0 m 0 ( w / 0 )sinc( w / m 0 )δ m rec ( 00) δ 0.0 / 0.0 sinc( / ) Using xy c x c y, sin( 00π) rec ( 00) δ 0.0 j / 0.0 ( δ [ + ] δ [ ] ) ( / )sinc( / ) sin( 00π) rec ( 00) δ 0.0 j3 sinc ( ( / ) sinc( ( ) / ) ) CS Examples ind he CS harmonic funcion of x wih = 0 8. = ( / ) xe j π / d [ 0] = 0 8 ( )d = 35 / = 0 8 ( )e j π 08 d = e j π 08 d e j π e j π 08 = jπ 0 8 jπ 0 8 d = e j π 0 8 jπ 0 8 e j π ( jπ 0 8 ) 0 = e j π e j π + jπ ( jπ) 0 6 = 35 e j π jπe j π ( jπ) /, = 0 = 35 j π, M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 6 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 6 LI Sysems wih Periodic Exciaion LI Sysems wih Periodic Exciaion he differenial equaion describing an RC lowpass filer is RC + v ou = v in v ou If he exciaion v in CS, v ou, v in is periodic i can be expressed as a = c in e = j π / he equaion for he h harmonic alone is RC + v ou, = v in, = c in e j π / If he exciaion is periodic, he response is also, wih he same fundamenal period. herefore he response can be expressed as a CS also. j π / = c ou e v ou, hen he equaion for he h harmonic becomes ( jπ RC / )c ou e j π / + c ou e j π / = c in e Noice ha wha was once a differenial equaion is now an algebraic equaion. j π / M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 63 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 64 LI Sysems wih Periodic Exciaion LI Sysems wih Periodic Exciaion Solving he h-harmonic equaion, = c c ou = in jπ RC / + hen he response can be wrien as c v ou = c ou e j π / = in jπ RC / + e j π / = c in he raio c ou is he harmonic response of he sysem. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 65 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 66
12 Exending he CS Coninuous-ime ourier Mehods he CS is a good analysis ool for sysems wih periodic exciaion bu he CS canno represen an aperiodic signal for all ime he coninuous-ime ourier ransform (C) can represen an aperiodic signal for all ime M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 67 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 68 CS-o-C ransiion Consider a periodic pulse-rain signal x wih duy cycle w / 0 CS-o-C ransiion for 50% and 0% duy Below are graphs of he magniude of cycles. As he period increases he sinc funcion widens and is magniude falls. As he period approaches infiniy, he CS harmonic funcion becomes an infiniely-wide sinc funcion wih zero ampliude. Is CS harmonic funcion is = Aw 0 sinc w As he period 0 is increased, holding w consan, he duy cycle is decreased. When he period becomes infinie (and he duy cycle becomes zero) x 0 is no longer periodic. w = 0 w = 0 0 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 69 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 70 CS-o-C ransiion his infiniy-and-zero problem can be solved by normalizing he CS harmonic funcion. Define a new modified CS harmonic funcion 0 = Awsinc( wf 0 ) and graph i versus f 0 insead of versus. f 0 = / 0 CS-o-C ransiion In he limi as he period approaches infiniy, he modified CS harmonic funcion approaches a funcion of coninuous frequency f (f 0 ). M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 7 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 7
13 CS-o-C ransiion Definiion of he C X( f ) = x X( jω ) = x orward f form Inverse = x e j π f d x = - X( f ) orward ω form Inverse = x e jω d x = - X( jω ) = X f = Commonly-used noaion: x X( f ) or x X( jω ) π e + j π f df X( jω )e + jω dω M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 73 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 74 Some Remarable Implicaions of he ourier ransform he C expresses a finie-ampliude, real-valued, aperiodic signal which can also, in general, be ime-limied, as a summaion (an inegral) of an infinie coninuum of weighed, infiniesimalampliude, complex sinusoids, each of which is unlimied in ime. (ime limied means having non-zero values only for a finie ime. ) requency Conen Lowpass Bandpass Highpass M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 75 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 76 e α u / jω + α e α u / jω + α n! n e α u jω + α e α sin( ω 0 )u jω + α δ, α > 0 e α u( ), α > 0 e α u( ), α > 0 n e α u( ) n+ ω 0 + ω 0 e α jω + α cos( ω 0 )u jω + α Some C Pairs + ω 0, α < 0, α < 0 / jω + α / jω + α n! jω + α, α > 0 e α sin( ω 0 )u( ) jω + α n+, α < 0 ω 0 + ω 0, α > 0 e α cos( ω 0 )u( ) jω + α jω + α α e α, α > 0 ω + α + ω 0, α < 0, α < 0 Convergence and he Generalized ourier ransform Le x = A. hen from he definiion of he C, X f = Ae j π f d = A e j π f d his inegral does no converge so, sricly speaing, he C does no exis. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 77 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 78 3
14 Convergence and he Generalized ourier ransform Bu consider a similar funcion, x σ = Ae σ, σ > 0 Is C inegral X σ ( f ) = Ae σ e j π f d does converge. Convergence and he Generalized ourier ransform σ Carrying ou he inegral, X σ ( f ) = A σ + π f Now le σ approach zero. If f 0 hen lim funcion is A σ 0 A. σ σ = 0. he area under his + π f σ σ + π f df which is A, independen of he value of σ. So, in he limi as σ approaches zero, he C has an area of A and is zero unless f = 0. his exacly defines an impulse of srengh A. herefore A Aδ ( f ). M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 79 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 80 Convergence and he Generalized ourier ransform Convergence and he Generalized ourier ransform By a similar process i can be shown ha cos π f 0 δ f f 0 and sin π f 0 j δ f + f 0 + δ ( f + f 0 ) δ ( f f 0 ) hese C s which involve impulses are called generalized ourier ransforms (probably because he impulse is a generalized funcion). M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 8 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 8 Negaive requency his signal is obviously a sinusoid. How is i described mahemaically? I could be described by x Bu i could also be described by x = Acos( π / 0 ) = Acos( π f 0 ) = Acos π ( f 0 ) x() could also be described by Negaive requency x = A e j π f0 + e j π f 0 or x = A cos( π f 0 ) + A cos( π ( f 0 )), A + A = A and probably in a few oher differen-looing ways. So who is o say wheher he frequency is posiive or negaive? or he purposes of signal analysis, i does no maer. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 83 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 84 4
15 Negaive requency Consider an experimen in which we muliply wo sinusoidal = cos( π f ) and x = cos( 00π) o form can be expressed using a rigonomeric signals x x = x x. x ideniy as x = ( / ) cos( π ( f 00)) + cos( π ( f +00)) Now imagine ha we coninuously change f from a frequency above00 o a frequency below 00. f 00 becomes negaive. More C Pairs he generalizaion of he C allows us o exend he able of C pairs o some very useful funcions. δ δ f sgn / jπ f u ( / )δ f ri δ 0 rec sinc f sinc sinc, f 0 = / 0 0 δ 0 δ ( f f 0 ) + δ ( f + f 0 ) sin( π f 0 ) sinc f f 0 δ f0 f cos π f 0 / +/ jπ f rec ( f ) ri( f ), 0 = / f 0 δ ( f + f 0 ) δ ( f f 0 ) δ f0 f j / M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 85 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 86 If C Properies ( x ) = X( f ) or X( jω ) and ( y ) = Y( f ) or Y( jω ) hen he following properies can be proven. Lineariy α x α x + β y + β y α X f α X jω + β Y( f ) + β Y( jω ) ime Shifing x( 0 ) X( f )e j π f 0 x( 0 ) X( jω )e jω 0 C Properies M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 87 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 88 C Properies C Properies requency Shifing xe + j π f 0 X( f f 0 ) xe + jω 0 X( ω ω 0 ) ime Scaling requency Scaling x( a) a X f a x( a) a X j ω a a x a a x a X( af ) X( jaω ) M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 89 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 90 5
16 he Uncerainy Principle he ime and frequency scaling properies indicae ha if a signal is expanded in one domain i is compressed in he oher domain. his is called he uncerainy principle of ourier analysis. ransform of a Conjugae C Properies x * x * X * f X * jω e π π / e e π f e π( f ) Muliplicaion Convoluion Dualiy x y X( f )Y( f ) x y X( jω )Y( jω ) xy X( f ) Y( f ) xy ( / π ) X( jω ) Y( jω ) M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 9 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 9 C Properies C Properies In he frequency domain, he cascade connecion muliplies he frequency responses insead of convolving he impulse responses. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 93 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 94 C Properies C Properies ime Differeniaion Modulaion ransforms of Periodic Signals d ( x ) d d ( x ) d xcos π f 0 xcos ω 0 jπ f X( f ) jω X( jω ) X f f 0 X j ω ω 0 x = Xe j π f = x = Xe jω = + X( f + f 0 ) ( ) + X( j( ω + ω 0 )) X( f ) = Xδ f f 0 = X( jω ) = π Xδ ω ω 0 = M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 95 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 96 6
17 Parseval s heorem Inegral Definiion of an Impulse Dualiy C Properies d x = X f x ( ) d = π e j π xy X x f X j π x ω df X( jω ) df dy = δ x and X( ) and X( j) x f π x( ω ) oal - Area Inegral Inegraion C Properies X( 0) = xe j π f d x( 0) = X( f )e + j π f df X( 0) = xe jω d x( 0) = X( jω )e + jω dω π f 0 0 ω 0 0 = xd = X( f )df = xd = π X( jω )dω x( λ)dλ X( f ) jπ f + X( 0)δ ( f ) x( λ)dλ X( jω ) jω + π X( 0)δ ( ω ) M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 97 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 98 C Properies C Properies X( 0) = x( )d x( 0) = X( f )df M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 99 M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 00 Numerical Compuaion of he C I can be shown (Web Appendix G) ha he D can be used o approximae samples from he C. If he signal is a causal energy signal and N samples are aen x from i over a finie ime beginning a = 0, a a rae f s hen he relaionship beween he C of x he samples aen from i is X f s / N and he D of s e jπ /N sinc( / N ) X D or hose harmonic numbers for which << N s X D X f s / N As he sampling rae and number of samples are increased, his approximaion is improved. M. J. Robers - All Righs Reserved. Edied by Dr. Rober Al 0 7
Chapter 4 The Fourier Series and Fourier Transform
Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are
More informationSystem Processes input signal (excitation) and produces output signal (response)
Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response 1. Types of signals 2. Going from analog o digial world 3. An example
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information6.003 Homework #9 Solutions
6.003 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 3 0 a 0 5 a k a k 0 πk j3 e 0 e j πk 0 jπk πk e 0
More information6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
More informationOutline Chapter 2: Signals and Systems
Ouline Chaper 2: Signals and Sysems Signals Basics abou Signal Descripion Fourier Transform Harmonic Decomposiion of Periodic Waveforms (Fourier Analysis) Definiion and Properies of Fourier Transform Imporan
More informationSignals and Systems Review. 8/25/15 M. J. Roberts - All Rights Reserved 1
Signals and Sysems Review 8/25/15 M. J. Robers - All Righs Reserved 1 g Coninuous-Time Sinusoids ( ) = Acos( 2π / T 0 +θ ) = Acos( 2π f 0 +θ ) = Acos( ω 0 +θ ) Ampliude Period Phase Shif Cyclic Radian
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More information9/9/99 (T.F. Weiss) Signals and systems This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.
9/9/99 (T.F. Weiss) Lecure #: Inroducion o signals Moivaion: To describe signals, boh man-made and naurally occurring. Ouline: Classificaion ofsignals Building-block signals complex exponenials, impulses
More informationLaplace Transform and its Relation to Fourier Transform
Chaper 6 Laplace Transform and is Relaion o Fourier Transform (A Brief Summary) Gis of he Maer 2 Domains of Represenaion Represenaion of signals and sysems Time Domain Coninuous Discree Time Time () [n]
More informationSignals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 1 Solutions
8-90 Signals and Sysems Profs. Byron Yu and Pulki Grover Fall 07 Miderm Soluions Name: Andrew ID: Problem Score Max 0 8 4 6 5 0 6 0 7 8 9 0 6 Toal 00 Miderm Soluions. (0 poins) Deermine wheher he following
More informationADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding
More informationA complex discrete (or digital) signal x(n) is defined in a
Chaper Complex Signals A number of signal processing applicaions make use of complex signals. Some examples include he characerizaion of he Fourier ransform, blood velociy esimaions, and modulaion of signals
More informationQ1) [20 points] answer for the following questions (ON THIS SHEET):
Dr. Anas Al Tarabsheh The Hashemie Universiy Elecrical and Compuer Engineering Deparmen (Makeup Exam) Signals and Sysems Firs Semeser 011/01 Final Exam Dae: 1/06/01 Exam Duraion: hours Noe: means convoluion
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationIII-A. Fourier Series Expansion
Summer 28 Signals & Sysems S.F. Hsieh III-A. Fourier Series Expansion Inroducion. Divide and conquer Signals can be decomposed as linear combinaions of: (a) shifed impulses: (sifing propery) Why? x() x()δ(
More informationLecture 2: Optics / C2: Quantum Information and Laser Science
Lecure : Opics / C: Quanum Informaion and Laser Science Ocober 9, 8 1 Fourier analysis This branch of analysis is exremely useful in dealing wih linear sysems (e.g. Maxwell s equaions for he mos par),
More informationChapter One Fourier Series and Fourier Transform
Chaper One I. Fourier Series Represenaion of Periodic Signals -Trigonomeric Fourier Series: The rigonomeric Fourier series represenaion of a periodic signal x() x( + T0 ) wih fundamenal period T0 is given
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More information6.003: Signal Processing
6.003: Signal Processing Coninuous-Time Fourier Transform Definiion Examples Properies Relaion o Fourier Series Sepember 5, 08 Quiz Thursday, Ocober 4, from 3pm o 5pm. No lecure on Ocober 4. The exam is
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
More informationME 452 Fourier Series and Fourier Transform
ME 452 Fourier Series and Fourier ransform Fourier series From Joseph Fourier in 87 as a resul of his sudy on he flow of hea. If f() is almos any periodic funcion i can be wrien as an infinie sum of sines
More information6.003 Homework #13 Solutions
6.003 Homework #3 Soluions Problems. Transformaion Consider he following ransformaion from x() o y(): x() w () w () w 3 () + y() p() cos() where p() = δ( k). Deermine an expression for y() when x() = sin(/)/().
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationCommunication Systems, 5e
Communicaion Sysems, 5e Chaper : Signals and Specra A. Bruce Carlson Paul B. Crilly The McGraw-Hill Companies Chaper : Signals and Specra Line specra and ourier series Fourier ransorms Time and requency
More informationES.1803 Topic 22 Notes Jeremy Orloff
ES.83 Topic Noes Jeremy Orloff Fourier series inroducion: coninued. Goals. Be able o compue he Fourier coefficiens of even or odd periodic funcion using he simplified formulas.. Be able o wrie and graph
More information6.003: Signals and Systems. Fourier Representations
6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers. Fourier Series
More informationEE 224 Signals and Systems I Complex numbers sinusodal signals Complex exponentials e jωt phasor addition
EE 224 Signals and Sysems I Complex numbers sinusodal signals Complex exponenials e jω phasor addiion 1/28 Complex Numbers Recangular Polar y z r z θ x Good for addiion/subracion Good for muliplicaion/division
More informationKEEE313(03) Signals and Systems. Chang-Su Kim
KEEE313(03) Signals and Sysems Chang-Su Kim Course Informaion Course homepage hp://mcl.korea.ac.kr Lecurer Chang-Su Kim Office: Engineering Bldg, Rm 508 E-mail: changsukim@korea.ac.kr Tuor 허육 (yukheo@mcl.korea.ac.kr)
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More information6.003: Signals and Systems
6.3: Signals and Sysems Lecure 7 April 8, 6.3: Signals and Sysems C Fourier ransform C Fourier ransform Represening signals by heir frequency conen. X(j)= x()e j d ( analysis equaion) x()= π X(j)e j d
More informationLinear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
More informationWeb Appendix N - Derivations of the Properties of the LaplaceTransform
M. J. Robers - 2/18/07 Web Appenix N - Derivaions of he Properies of he aplacetransform N.1 ineariy e z= x+ y where an are consans. Then = x+ y Zs an he lineariy propery is N.2 Time Shifing es = xe s +
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008
[E5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 008 EEE/ISE PART II MEng BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: :00 hours There are FOUR quesions
More informationChapter 9 - The Laplace Transform
Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC
More informationLaplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
More informationCommunication System Analysis
Communicaion Sysem Analysis Communicaion Sysems A naïve, absurd communicaion sysem 12/29/10 M. J. Robers - All Righs Reserved 2 Communicaion Sysems A beer communicaion sysem using elecromagneic waves o
More information2 Frequency-Domain Analysis
requency-domain Analysis Elecrical engineers live in he wo worlds, so o speak, of ime and frequency. requency-domain analysis is an exremely valuable ool o he communicaions engineer, more so perhaps han
More informationLecture #6: Continuous-Time Signals
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions
More informationFrom Complex Fourier Series to Fourier Transforms
Topic From Complex Fourier Series o Fourier Transforms. Inroducion In he previous lecure you saw ha complex Fourier Series and is coeciens were dened by as f ( = n= C ne in! where C n = T T = T = f (e
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationNotes 04 largely plagiarized by %khc
Noes 04 largely plagiarized by %khc Convoluion Recap Some ricks: x() () =x() x() (, 0 )=x(, 0 ) R ț x() u() = x( )d x() () =ẋ() This hen ells us ha an inegraor has impulse response h() =u(), and ha a differeniaor
More informationh[n] is the impulse response of the discrete-time system:
Definiion Examples Properies Memory Inveribiliy Causaliy Sabiliy Time Invariance Lineariy Sysems Fundamenals Overview Definiion of a Sysem x() h() y() x[n] h[n] Sysem: a process in which inpu signals are
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More information6.003: Signals and Systems. Relations among Fourier Representations
6.003: Signals and Sysems Relaions among Fourier Represenaions April 22, 200 Mid-erm Examinaion #3 W ednesday, April 28, 7:30-9:30pm. No reciaions on he day of he exam. Coverage: Lecures 20 Reciaions 20
More information2 Signals. 2.1 Elementary algebra on signals
2 Signals We usually use signals o represen quaniies ha vary wih ime. An example of a signal is he size of he sea swell a some locaion in False Bay: a any paricular ime he waves in he bay have an ampliude
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationAnswers to Exercises in Chapter 7 - Correlation Functions
M J Robers - //8 Answers o Exercises in Chaper 7 - Correlaion Funcions 7- (from Papoulis and Pillai) The random variable C is uniform in he inerval (,T ) Find R, ()= u( C), ()= C (Use R (, )= R,, < or
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6 Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is:
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationSignals and Systems Linear Time-Invariant (LTI) Systems
Signals and Sysems Linear Time-Invarian (LTI) Sysems Chang-Su Kim Discree-Time LTI Sysems Represening Signals in Terms of Impulses Sifing propery 0 x[ n] x[ k] [ n k] k x[ 2] [ n 2] x[ 1] [ n1] x[0] [
More informationThe Fourier Transform.
The Fourier Transform. Consider an energy signal x(). Is energy is = E x( ) d 2 x() x () T Such signal is neiher finie ime nor periodic. This means ha we canno define a "specrum" for i using Fourier series.
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationChapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis
Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...
More informatione 2t u(t) e 2t u(t) =?
EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More information6.003: Signals and Systems
6.003: Signals and Sysems Relaions among Fourier Represenaions November 5, 20 Mid-erm Examinaion #3 Wednesday, November 6, 7:30-9:30pm, No reciaions on he day of he exam. Coverage: Lecures 8 Reciaions
More informationContinuous Time Linear Time Invariant (LTI) Systems. Dr. Ali Hussein Muqaibel. Introduction
/9/ Coninuous Time Linear Time Invarian (LTI) Sysems Why LTI? Inroducion Many physical sysems. Easy o solve mahemaically Available informaion abou analysis and design. We can apply superposiion LTI Sysem
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationConvolution. Lecture #6 2CT.3 8. BME 333 Biomedical Signals and Systems - J.Schesser
Convoluion Lecure #6 C.3 8 Deiniion When we compue he ollowing inegral or τ and τ we say ha he we are convoluing wih g d his says: ae τ, lip i convolve in ime -τ, hen displace i in ime by seconds -τ, and
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mar Fowler Noe Se #1 C-T Signals: Circuis wih Periodic Sources 1/1 Solving Circuis wih Periodic Sources FS maes i easy o find he response of an RLC circui o a periodic source!
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationSpectral Analysis. Joseph Fourier The two representations of a signal are connected via the Fourier transform. Z x(t)exp( j2πft)dt
Specral Analysis Asignalx may be represened as a funcion of ime as x() or as a funcion of frequency X(f). This is due o relaionships developed by a French mahemaician, physicis, and Egypologis, Joseph
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More information4/9/2012. Signals and Systems KX5BQY EE235. Today s menu. System properties
Signals and Sysems hp://www.youube.com/v/iv6fo KX5BQY EE35 oday s menu Good weeend? Sysem properies iy Superposiion! Sysem properies iy: A Sysem is if i mees he following wo crieria: If { x( )} y( ) and
More informationSections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationk The function Ψ(x) is called wavelet function and shows band-pass behavior. The wavelet coefficients d a,b
Wavele Transform Wavele Transform The wavele ransform corresponds o he decomposiion of a quadraic inegrable funcion sx ε L 2 R in a family of scaled and ranslaed funcions Ψ,l, ψ, l 1/2 = ψ l The funcion
More informationThe complex Fourier series has an important limiting form when the period approaches infinity, i.e., T 0. 0 since it is proportional to 1/L, but
Fourier Transforms The complex Fourier series has an imporan limiing form when he period approaches infiniy, i.e., T or L. Suppose ha in his limi () k = nπ L remains large (ranging from o ) and (2) c n
More information( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.
Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/007-6.4 PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion
More information6.003 Homework #8 Solutions
6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d
More informationSignal and System (Chapter 3. Continuous-Time Systems)
Signal and Sysem (Chaper 3. Coninuous-Time Sysems) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 1 Dep. Elecronics and Informaion Eng. 1 Nodes, Branches, Loops A nework wih b
More informationB Signals and Systems I Solutions to Midterm Test 2. xt ()
34-33B Signals and Sysems I Soluions o Miderm es 34-33B Signals and Sysems I Soluions o Miderm es ednesday Marh 7, 7:PM-9:PM Examiner: Prof. Benoi Boule Deparmen of Elerial and Compuer Engineering MGill
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationThe Laplace Transform
The Laplace Transform Previous basis funcions: 1, x, cosx, sinx, exp(jw). New basis funcion for he LT => complex exponenial funcions LT provides a broader characerisics of CT signals and CT LTI sysems
More informationf t te e = possesses a Laplace transform. Exercises for Module-III (Transform Calculus)
Exercises for Module-III (Transform Calculus) ) Discuss he piecewise coninuiy of he following funcions: =,, +, > c) e,, = d) sin,, = ) Show ha he funcion ( ) sin ( ) f e e = possesses a Laplace ransform.
More informationFor example, the comb filter generated from. ( ) has a transfer function. e ) has L notches at ω = (2k+1)π/L and L peaks at ω = 2π k/l,
Comb Filers The simple filers discussed so far are characeried eiher by a single passband and/or a single sopband There are applicaions where filers wih muliple passbands and sopbands are required The
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationk B 2 Radiofrequency pulses and hardware
1 Exra MR Problems DC Medical Imaging course April, 214 he problems below are harder, more ime-consuming, and inended for hose wih a more mahemaical background. hey are enirely opional, bu hopefully will
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : 0. ND_NW_EE_Signal & Sysems_4068 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkaa Pana Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRICAL ENGINEERING
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationChapter 2 : Fourier Series. Chapter 3 : Fourier Series
Chaper 2 : Fourier Series.0 Inroducion Fourier Series : represenaion of periodic signals as weighed sums of harmonically relaed frequencies. If a signal x() is periodic signal, hen x() can be represened
More information6.003: Signal Processing
6.003: Signal Processing Working wih Signals Overview of Subjec Signals: Definiions, Examples, and Operaions Basis Funcions and Transforms Sepember 6, 2018 Welcome o 6.003 Piloing a new version of 6.003
More informationFourier Series & The Fourier Transform. Joseph Fourier, our hero. Lord Kelvin on Fourier s theorem. What do we want from the Fourier Transform?
ourier Series & The ourier Transfor Wha is he ourier Transfor? Wha do we wan fro he ourier Transfor? We desire a easure of he frequencies presen in a wave. This will lead o a definiion of he er, he specru.
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More information