Fourier transform of continuous-time signals

Transcript

1 Fourier ransform of coninuous-ime signals Specral represenaion of non-periodic signals Fourier ransform: aperiodic signals repeiion of a finie-duraion signal x()> periodic signals. x x T x kt x kt k k () () δ () () δ( ) ( ) Periodic signal (T )> non-periodic signal x() T () x() x

2 Non-periodic signal & periodic signal, period T. Non-periodic Δ, T x() pt (), oherwise repeiion T Periodic 3 T () x() x 4

3 analyze a non-periodic signal in he frequency domain: using he frequency analysis of he corresponden periodic signal and compue he limi for T. 5 The periodic signal is non band-limied. 6 3

4 The produc T c k & he envelope () envelope for T c k Relaion beween hem? 7 General case he Fourier coefficiens of he periodic signal : T jk T () ck x e d T Equal on [-T/, T/] T jk T () ck x e d T 8 4

5 Fourier ransform ouside [-T/,T/] he non-periodic signal jk () ck x e d T Wih he funcion: j ( ) ( ) x e d ( ) π ck k, T T (The envelope of T c k ) 9 Square wave:differen values of T 5

6 Remarks The envelope is no affeced by T. Increase T specral componens are closer. T disance he discree specral represenaion becomes coninuous. he periodic signal non-periodic. Fourier pair Definiion j j x() ( ) e d ( ) x( ) e d π Inverse Fourier Transform Fourier Transform (specrum) 6

7 Remarks periodic signals : specral lines ( ) π ck k, T T non-periodic signals specra are coninuous 3 CTFT for signals in he class L for signals in L, he Fourier ransform is no necessarily from L Reconsrucion heorem! 4 7

8 x () p () σ () σ ( ) τ τ ( ) sin τ p τ () d < he Fourier ransform is convergen (he signal x() L ) bu () L. he reconsrucion of he signal from is 5 specrum is no obvious. Reconsrucion heorem If he signal x() belongs o L and has bounded variaion on he enire real axis hen is Fourier ransform can be invered using : x R () { x () }( ) e j lim F d R R 6 8

9 . Lineariy If x() and y() L and have he Fourier ransform () and Y() hen for any complex consans a and b he signal ax()+by() L and has he Fourier ransform a()+by(). Homework: Prove i. () + by( ) a ( ) by ( ) ax + 7. Time Shifing Time shifing -> modulaion in frequency (muliplicaion wih a complex exponenial). j x e Proof ( ) ( ) τ F j j { ( )} ( ) ( τ ) ( τ + ) j x x e d x e dτ e ( ). 8 9

10 Remarks Fourier ransform: complex funcion. The Fourier ransform H() of he impulse response h() of a sysem: frequency response. frequency dependence of he magniude of H() magniude characerisic of he sysem H() frequency dependence of he argumen of H() phase characerisic of he sysem arg{h()} 9 3. Modulaion Modulaion in ime -> shifing in frequency. Proof F e ( ) ( ) j e x j { () x e } x() j () ( ).. j j e e d j () ( ) x e d ( ) x

11 Dualiy operaion in ime anoher operaion in frequency : modulaion shifing (3 rd propery) nd operaion in ime firs operaion in frequency. ime shifing modulaion ( nd propery) This behavior is named dualiy. 4. Time Scaling If x() L is scaled version x(/a) L and he specrum of x(/a) is a frequency scaled version of he specrum of x(). he scaling is an auo-dual operaion. ( ) x a. a a

12 Proof F x j { x( a )} x( a ) e d x( τ ) a a ( a ). τ a a j τ e a dτ a ; a 3 Example: he square wave specrum is ime-scaled varian, a: a/ p τ () sinτ sin τ sin τ p ( ) () τ pτ () sin τ sin p p τ τ τ 4

13 ime compression frequency dilaion ime dilaion frequency compression 5 CTFT of he consan disribuion F () πδ ( ) 6 3

14 Proof he consan disribuion can be approximaed: lim p ( ) ( ) We know ha pτ lim pτ τ () e j τ τ () () d lim τ j e sinτ d j e d lim τ sinτ sinτ, τ τ, 7 The area under he graphical represenaion of he specrum: sinτ sin sin τ u + 4 τ u A d du du Si u u ( ) π So: (), e j d, and: A π j () e d πδ ( ) () πδ ( ) F 8 4

15 An immediae consequence: a new represenaive sring for he Dirac disribuion: lim τ sinτ δ π ( ) 9 5. Complex Conjugaion complex conjugaion in ime -> reversal and complex conjugaion in frequency. Proof F * x * F x * () ( ) * j j { ()} () () ( ) * x x* e d x e d ( ) F * () ( ) * 3 5

16 6. Time Reversal Time reversal -> reversal in frequency. Homework. Prove i. x F ( ) ( ) 3 7. Signal s Derivaion Time differeniaion -> muliplicaion wih j in frequency. x' F () j ( ) 3 6

17 Proof: F Inegraing by pars: F he signal is in L : j { x' () } x' ( ) e d j j { x' () } x( ) e + j x() e d j () e lim x() lim x ± ± So: x' F () j ( ) Signal s Inegraion For x() L wih () (no DC componen), is inegral L Time inegraion -> muliplicaion wih / j in frequency ( ) F x( τ) dτ for ( ) j 34 7

18 Proof We have: y() x( τ d )τ Apply for y() he differeniaion propery: F ( ) ( ) ( ) ( ) ( ) ( ) y ' x jy Y j Y defined in : So: x ( ) F ( τ ) ( ) dτ j Signals convoluion convoluion heorem he convoluion of wo signals from L belongs o L. convoluion of wo signals in ime -> produc in frequency. F ( ) ( ) ( ) ( ) x y Y 36 8

19 F x j { x() y() } ( x y)( ) e d x( τ ) y( τ ) x Proof: jτ j ( ) ( ) ( τ τ τ ) jτ e x e ddτ x( τ ) e dτ y( u) F () y() ( ) Y ( ) τ u j dτ e d ju e du. 37 Example. Triangle s specrum convoluion of wo recangular pulses, same duraion a riangle p p τ τ pτ τ () () τ () pτ () τ τ sin sin τ τ τ sin p () τ τ τ τ τ (convoluion heorem) 38 9

20 39. Specrum s Derivaion The derivaive of he specrum is he Fourier ransform of he signal jx(). () x ( ) d j d ( ) d d x d d ()( j) e x j () d j e d x() ( e ) j d d d 4

21 . CTFT of Real Signals. Properies. (The Specrum of he Even and Odd Pars of a Real Signal) The specrum of a real and even signal is real and even. The specrum of a real and odd signal is imaginary and odd. { } E( ) { } O( ) ( ) { } () Im ( ) x Re and e x j j o 4 Proof he real signal x() wih specrum (), complex : jφ( ) ( ) ( ) e Re{ ( ) } + jim{ ( ) } Polar form Is complex conjugae real ( ) ( ) ( ) x Caresian form ( ) * ( ) * { ( )} Im ( ) * jφ e Re j { } Polar form Caresian form 4

22 For real signals: By idenificaion: () ( ) ( ) ( ) * * x x ( ) ( ) Φ ( ) Φ( ) ( ) ; ; { } { ( ) } { ( ) } { ( ) } Re Re ; Im Im. Magniude and real par of specrum are even funcions Phase and imaginary par of specrum are odd funcions 43 Example - odd real signal τ sin τ τ j j τ τ x() p p e e τ τ + The specrum of a real and odd signal is imaginary and odd 44

23 p τ () τ sin ime shifing p τ τ e τ j τ sin and p τ τ + e τ j τ sin τ sin τ τ j j cos x () τ e e j Euler s relaion sin(u) -cos (u) 45. A Parseval like heorem for signals from L F { ( )}() () ( ) F { ()}( ) x y d x y d equivalen form: Fourier ransform of he signal x() wih he variable ( ) y( ) d x( ) Y ( ) d Signal x() wih variable 46 3

24 3. Relaion Fourier Transform of a non-periodic signal & exponenial Fourier series coefficiens of he periodic signal (see previous slides) ( ) π ck k, T T 47 Example T () T + x pt p T 48 4

25 he specrum of he signal x(): T cos ( ) j Applying he propery 3: c x k kt cos j coskπ j T k kπ 49 ) finie energy signals x() L L The Fourier ransform of a signal from L L is from L he energy of a signal in he frequency / ime domain. (Parseval or Rayleigh relaion) () energy densiy. using he L norm: ( ) π () d x d ( ) π x() 5 5

26 Proof If x() L L x*(-) L L. Their convoluion belongs o L. y( ) x( ) x * ( ) So, i has Fourier ransform, Y(). from he convoluion heorem : Y * * ( ) ( ) ( ( ) ) ( ) ( ) ( ) 5 We have: for : y y π j * () Y ( ) e d x( τ) x ( τ ) dτ π * () ( ) d x() τ x ()τ τ d So: ( ) π ( ) d y finie In consequence he funcion () belongs o L. 5 6

27 ) finie energy signals x() L \L he Fourier ransform of a finie energy signal: Fourier ransform in L space convergence in mean square τ j I { x() }( ) l.i.m. x( ) e d τ τ The L norm of he Fourier ransform : I { x() }( ) lim x() τ τ τ e j d 53 Truncaion x() by muliplicaion wih p τ () approximaion of x() L L Two approximaions. The beer one - longer suppor. The oher -an approximaion of he firs. The error ends o zero if he wo duraions end o infiniy. 54 7

28 he approximaion error I { x() p () }( ) I { x() p () }( ) limi for τ τ y ( ) L L ( ) x( ) p ( ) p ( ) τ τ () pτ () x() pτ () π x() d + π x() π x d τ,, τ > τ lim τ τ, τ >τ τ x τ τ τ j j () e d x() e d τ τ τ Parseval τ τ j () I () () x e d x p { τ } in mean square. 55 Plâncherel s Theorem The Fourier ransform definiion of a finie energy signal already given can be found under he name of Plâncherel s Theorem: 56 8

29 Plâncherel s Theorem i) Plâncherel s heorem shows ha he Fourier ransform of any finie energy signal belongs o L. ii) The Fourier ransform on L is a paricular case of he Fourier ransform on L. All he properies of he Fourier ransform on L are verified by he Fourier ransform on L. The Parseval s relaion - proved for signals in L L. I is no verified by signals in L L iii) The Parseval s relaion can be generalized on L, in he form: x π (), y() ( ), Y ( ) 57 he definiion of he scalar produc on L : If he wo signals are equal : Parseval s relaion. x π * * () y () d ( ) Y ( ) d x() d ( ) d π 58 9

30 4. Specrum s Convoluion The convoluion of he Fourier ransforms () and Y() gives he Fourier ransform of he produc x() y() muliplied by π. ( ) ( ) πf { ( ) ( )}( ) Y x y The convoluion of wo finie energy signals is of finie energy. Convolving wo finie energy specra () and Y() a finie energy specrum Z() 59 ( ) ( ) ( ) ( ) ( ) Z Y u Y u du j( u) π ( u) y() e d du π Z π ju j ( ) π y( ) ( u) e du e d Z ( ) π x() y() e j d Inverse Fourier ransform of x() he Fourier ransform of he produc x() y(). 6 3

31 5. Dualiy The inverse Fourier ransform : For : ( ) ( ) πx j () ( ) e d Applying wo imes he Fourier ransform a reversed varian of he original signal weighed by π. j πx e d, dualiy πx Fourier ransform of (). ( ) I { I { x( ) }( ) }( ) 6 πx double change of variables: j ( ) ( ) e d and j ( ) ( ) { ( )}( ) πx e d I, anoher form of dualiy. 5. Dualiy Using he wo forms of dualiy we can compue he specrum of a signal. 6 3

32 Sar from a known pair (x(), ()) Wha is he specrum of he signal ()? Change he variable and consans of ime wih variable and consans of frequency and vice versa obain he corresponding pair ((), πx(-)). 63 The Fourier Transform of signals Temporal window he specrum p τ sin τ () σ( + τ) σ( τ). In his case, x () p () and ( ). τ sin τ Changing he variables and consans τ sin ( ) ( ). π x π p () 64 3

33 Symmeric riangular signal he specrum () (). T sin p T T ri T T () () ( ). and T sin ri x T () sin ( ) ( ). π π ri x

34 Decreasing causal exponenial () (). e x wih > σ ( ) ( ) ( ) j e j d e d e e j j j ( ) j j ( ) ( ) { } {} { } + + Φ arcg j arg arg j arg arg

35 ( ) + ( ) arcg Φ 69 Decreasing non-causal exponenial x () e σ( ) >. x wih () e σ( ) >. wih j ( ) ( ). ( ) ( ) arg ( ) + { } arg arcg. j 7 35

36 36 7 Symmeric decreasing exponenial () ; < <, e, e e x s ( ) ( ) ( ). x x x s + ( ) ( ) ( ) j j s 7

37 Gaussian signal e a π e a 4a, a >. The specrum of a Gaussian signal is Gaussian 73 The Fourier Transform of Disribuions ) The specrum of he Dirac s disribuion for any es funcion ϕ(): ϕ ()() δ d ϕ( ) or ϕ()( δ ) d ϕ( ) he Dirac s disribuion is even. j j () e () e d { () }( ) Hence, we have obained: δ ϕ δ I δ ( ) ( ) 74 37

38 ) The specrum of he consan () dualiy () πδ( ) πδ( ) c πcδ ( ) 75 3) The specrum of he uni sep σ() I u() sgn and v(). { u' ( ) }( ) I{ δ( ) }( ). { u ( ) }( ) I ' I { u() }( ) σ u + v j j Iσ +πδ j { () }( ) ( ) () () () 76 38

39 4) The specrum of sgn() I { sgn} I{ u( ) } j, > sgn,, < 77 5) The specrum of he signal /(π) sgn j j π sgn ( ) π sgn (dualiy) j, > j sgn, π j, < 78 39

40 79 6) Fourier Transform of he inegral of a signal having DC componen, () x () τ dτ ( ) j + π ( ) δ( ) Proof: y () x() τ dτ x( τ)( σ -τ)τ d ( ) Y( ) ( ) I{ σ ( ) }( ) ( ) + πδ( ) π δ j + j ( ) δ( ) ( ) δ( ) ( ) ( ) 8 4

41 7) The specrum of he complex exponenial ( ) ( ) πδ e j πδ Modulaion: ( - ) 8 8) The specrum of cos j j e + e cos π δ - + δ + ( ) ( ) 8 4

42 Specrum for a limied cosine of duraion τ ( ) ( ) x cos p τ ( ) I{ cos} I{ pτ () } π sinτ δ( ) +δ( +) π sin( ) τ sin( +) τ ( ) τ + ( ) τ ( +) τ

43 9) The specrum of sin ( ) ( ) j j e e πδ - πδ + sin j j ( ) ( ) sin jπ δ - δ + 85 Fourier ransform of periodic signals The periodic signal y() convoluion of is resricion a one period, x() and he periodic Dirac s disribuion () x() δ () y T δ δ jk () e ( ) T T k he Fourier ransform of periodic δ T () wih period T -proporional wih he periodic δ () wih period

44 δ jk () e δ( kt ) T T k k Bu: { () }( ) I δ( -kt )}( ) { I δt k δ j () and δ( - ) e So: δt () e T I jk k { δ ()}( ) T e k jkt Variable and consan changes and T δ ( ) k I jkt e { δt () }( ) 87 ( ) ( ) I{ δ ( ) }( ) ( ) δ ( ) Y T Y π T k ( ) ( k ) δ( - k ) We have he relaion Fourier coefficiens of he periodic signal y() wih he Fourier ransform of he non-periodic signal x(): The Fourier ransform of he periodic signal is: Y ( k ) c y k T y ( ) π c δ( - k ) k k 88 44

45 The effec of signal s runcaion sin x() p ( ) π sin sinτ p p ˆ τ π π () ( ) ( ) + sin uτ sin uτ π u π u π ( ) ( ) Si( ) ˆ p u du du y Si τ( + ) Si τ( ) π π ˆ converges in mean square o p : ( ) ( ) ( ) F{ () τ ()} F{ ()} lim... x p x τ τ( + ) τ( )

46 The effec of he specrum s runcaion on he reconsruced signal sinτ Recangular pulse: x() pτ () ( ) ; sinτ Truncaed specrum from o : xˆ? p sin sinτ p ( ) and p ( ) τ π sin sinτ xˆ () pτ () p ( ) π sin p τ () Si ( + ) Si ( ) τ τ π π π Dualiy: () ˆ ( ) ( ) ( ) ( ) sin τ sin τ Si ( + τ) Si ( τ) π p ( ) p ( ) π π π So, xˆ 9 () Si ( + τ) Si ( τ) π π -- Truncaion in ime > Gibbs phenomenon in frequency Truncaion in frequency > Gibbs phenomenon in ime 9 46

47 Repariion Differen energy concenraion measures. The repariion of a random variable is described by is probabiliy densiy funcion f (x) : f ( x) and f ( x) dx i) Mean ii) Power μ E dx E{ } x f ( x) dx; { } xf ( x) ; iii) Variance { } ( ) μ ( ) { } ( μ ) Var E x f x dx iv) Sandard deviaion ( ) σ Var. 93 Example: Gaussian (normal) repariion ( x μ ) σ f ( ) x e πσ μ -mean σ -sandard deviaion πσ μ,σ e ( x μ ) σ x e dx π dx 94 47

48 Signal energy s disribuion in ime The energy of a signal x() : W x() d x() energy disribuion funcion, in ime. W - Average ime c - he energy of he signal is concenraed wih he dispersion of σ ime spreading x c x () () d d ( ) c x( ) σ x () d d 95 Signal energy s disribuion in frequency The energy of signal x(), specrum (): W ( ) d π ( ) energy disribuion funcion, in frequency. W Average frequency c -he energy of he signal is concenraed wih dispersion of σ frequency spreading, ( ) d c ( ) d σ ( ) c ( ) ( ) d d 96 48

49 The Heisenberg-Gabor uncerainy principle If σ and σ can be defined, hen for any signal we have: σσ The sign equal appears if and only if is a Gaussian signal. x( ) -here are no signals wih perfec concenraion of energy in he ime-frequency plane 97 Example: Gaussian signal a π 4a x () e ( ) e a c ; σ ; c σ a 4a The produc σσ. The Heisenberg-Gabor inequaliy is saisfied wih he equal sign

50 The energy in he (ime) inerval [ σ σ ] 3 a a π W6 σ W6 σ e d % a W 3 a The energy in he bandwidh π W π a a W 3 a π a W6 σ e d 3 a,3σ 3 3 3,3, a a 6σ.9974 ; 99.74% 3 Signal duraion T ; is bandwidh B 3 a a produc duraion-bandwidh TB 9 for 99.74%energy 99 Remarks: i)inerpreaion of Heisenberg-Gabor inequaliy σσ If he signal duraion σ increases bandwidh σ decreases. Example: he ime-scaling propery. For a fixed duraion, he specral sandard deviaion is C σ σ σ Beween all he signals wih he same duraion, he Gaussian one has minimum bandwidh. Reciprocically, beween all he signals wih he same bandwidh, he Gaussian one has minimum duraion. The Gaussian signal is ideal for elecommunicaions ransmission: a an imposed bandwidh i offers he highes ransmission speed. Someimes, he values σ and σ can be compued. 5

51 ii) The signal () x e σ () ( ) + j ( ) x () d ; W x () d ( ) ( ) C e d ; 3 8 σ () σ ( ) d d arcg d ( ) (even funcion) σ C ; + d σ can' be defined + ( ) 5

52 For : W 995, W B he duraionbandwidh produc is 3. A he same duraion he recangular pulse has a smaller bandwidh han he exponenial. 3 Special problems regarding signals i) Band-limied signals The band-limied signals have infinie duraion. They respec he Bernsein s heorem. A band-limied signal bounded by M has all he derivaives bounded : ( k x ) () M k M - signal wih slow variaion. 4 5

53 5 ii) Causal Signals and he Paley-Wiener Theorem The signal x() is causal if and only if he inegral: ( ) log I d + is convergen. The specrum can be zero, in a counable se of poins, having a null Lebesque measure. The causal signals are non band-limied. 6 53