Representing a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier

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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

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