Fourier Series Summary (From Salivahanan et al, 2002)

Transcription

1 Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t<, ith the fundaental frequency (i.e. f(t) f(t + / )) can be represented as a linear cobination of sinusoidal functions ith periods, here N +. represented by sine and cosine: f ( t) a + a cos + sin represented by coplex exponential: ( t) b ( t) f t j c e ( ) t ()

2 a and b can be resolved as a b / f / () t cos( t) dt,,,,... / f / () t sin( t) dt,,,... represented by coplex exponential: c c / f / () jt t e dt,,,,... / f / () jt t e dt,,,,... () c and c - are coplex conjugate pairs

3 The integral can also be perfored ithin [, / ] c c More concise relation beteen {a, b } and c : a / f jt () t e dt,,,,... / f jt () t e dt,,,,... [ c ], b I[ c ] Re In () and (), f(t) and c (- < t<, Z) for a transfor pair, here () is referred to as a forard transfor, and () is an inverse transfor. Fourier series specifies Fourier transfor in situation of periodic signals.

4 Frequency doain: e say that the periodic signal f(t) is transfored to the frequency doain specified by ( Z) by Fourier series. c is the frequency coponent of f(t) ith respect to the frequency. The frequency doain (or spectru) or a periodic continuous signal is discrete. In contrast, the doain hich the signal is defined is referred to as the tie doain or space doain. The frequency coponents c is a coplex nuber. Hence, the transfor doain can be divided into to parts: agnitude and phase. aplitude : phase : c c

5 Exaple: f(t) t ( ) ( ) ( ) j j t j e t d te c t t t j t j 5 5 ) ( 5 () / / 5 t j t j dt te dt e t f c (only hen > )

6 hen, c 5 td( t) c - is c s coplex conjugate, so ( ) ( t) agnitude in the transfor doain (in this case, the agnitude is an even function that is syetric to the y-axis) C 5/ c j 5 5 t t 5 5/ 5/4 3

7 phase in the transfor doain: (it is an odd function) C / 3 3 / The frequency doain of a Fourier transfor contains both the agnitude and phase spectra. High frequency ( is large): corresponds to the signal coponents that highly vary ith tie. Lo frequency ( is sall) corresponds to the signal coponents that sloly vary ith tie.

8 Parserval s Theore for Fourier series Reeber that ( ) ( ) [ ] ( ) ( ) t t f b t t f a t f a t f t b t a a t f )sin ( )cos ( ) ( ) ( sin cos ) ( [ ] ( ) ( ) + + / / / / / / / / )sin ( )cos ( ) ( 4 ) ( dt t t f b dt t t f a dt t f a t f () ( ),...,,, cos / / dt t t f a () ( ),...,, sin / / dt t t f b

9 We have In general, Parserval s theore eans energy preservation. poer of a periodic signal: poer in the frequency doain: [ ] ( ) + + / / ) ( c b a a t f [ ] / / ) ( t f c n

10 Continuous Fourier Transfor Fourier series transfors a periodic continuous signal into the frequency doain. What ill happen hen the continuous signal is not periodic? Consider the period of a signal ith the fundaental frequency being : (note that T specifies the fundaental period instead of the tie step defined before) A non-periodic signal can be conceptually thought of as a periodic signal hose fundaental period T is infinite long, T. In this case, the fundaental period. T f

11 Reeber that the spectru (in the frequency doain) of a periodic continuous signal is discrete, specified by ( Z). The interval beteen adjacent frequencies is. When, e can iage that the frequency becoes continuous: c / j d t () jt f t e dt f () t e dt / (here the interval becoes d) The suation in the inverse transfor becoes integral. Thus, d jt jt () t f () t e dt e f f t j c e ( ) t

12 Continuous Fourier Transfor: The forard transfor: F The inverse transfor: f jt e dt ( j) f ( t) jt j e d () t F( ) The continuous Fourier transfor has very siilar forard and inverse fors. The only difference is that the forard uses j and the inverse uses j in the coplex exponential basis. This suggests that the roles of tie and frequency can be exchanged, and soe properties are syetric to each other. Both tie and frequency doains in continuous Fourier transfor are continuous. The frequency avefor is also referred to as the spectru.

13 Many variations of fors of continuous F. T. Fro Kuhn 5

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15 Fro Kuhn 5

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17 What is the continuous Fourier transfor of a periodic signal?

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19 Dirac s delta function is also called unit ipulse function. The unit ipulse function δ(t) can be ultiplied by a real nuber r (or a coplex nuber c), say rδ(t) (or cδ(t)), to represent the delta function of different agnitudes (or agnitudes/angles). The continuous Fourier transfor of a periodic signal is a ipulse train. F(j) 5/ 5/ 5/4 3 Relationship to the discrete Fourier series: the agnitudes of the ipulse functions are proportional to those coputed fro the discrete Fourier series.

20 Periodic signal exaples

21 Discrete Tie Fourier Transfor (DTFT) Fro Fourier Series Tie doain is periodic frequency doain is discrete Reeber that the continuous Fourier transfor has siilar fors of forard and inverse transfors. Question: When tie doain is discrete, hat happens in the frequency doain? The frequency doain is periodic Discrete tie Fourier transfor: tie doain: a discrete-tie signal, x[-t], x[-t], x[], x[t], x[t], (T is the tie step)

22 Energy signal and poer signal

23 Forard transfor of DTFT: Property: Forard DTFT is periodic ith the period s /T. pf: Let r be any integers, then Since X(e jt ) is periodic, according the Fourier series principle, it can be expressed in the tie doain as a linear cobination of coplex exponentials, as the for shon in the forard DTFT. The coefficients of the linear cobination can then be coputed by finding the integral over a period: Inverse transfor of DTFT:

24 If e skip the tie step T (or siply T): forard DTFT: inverse DTFT: In this case, DTFT is periodic ith the period being. In digital signal processing, discrete-tie signals are of the ain interest, and DTFT is a ain tool for analyzing such a signal. Since the frequency spectru repeats periodically ith the period, e usually consider only a finite-duration [,] (or [, ]). High frequency region: The frequency nearing or. Lo frequency region: The frequency nearing.

25 Aliasing Effect (c.f. Shenoi, 6) Find the relationship beteen the continuous Fourier transfor X a (jω) of the continuous function x a (t) and the DTFT X(e jt ). Notation of continuous Fourier transfor: forard inverse Assue that the discrete-tie signal x(nt) is uniforly sapled fro the analog signal x a (t) ith the tie step T. Apply the inverse Fourier transfor, e have

26 Let s express this equation, hich involves integration fro Ω to Ω, as the su of integrals over successive intervals each equal to one period /T s. Hoever, each ter in this suation can be reduced to an integral over the range /T to /T by changing a variable fro Ω to Ω+r/T: Note that e jrn for all integers r and n. By changing the order of suation and integration, e have

27 Without loss of generality, e change the frequency variable Ω to, Coparing the above equation ith that of the inverse DTFT, We get the desired relationship (beteen DTFT and continuous Fourier transfor):

28 This shos that DTFT of the sequence x(nt) generated by sapling the continuous signal x a (t) ith a sapling period T is obtained by a periodic duplication of the continuous Fourier transfor X a (j) of x a (t) ith a period /T s and scaled by T. Because of the overlapping effect, ore coonly knon as aliasing, there is no ay of retrieving X a (j) fro X(e jt ); in other ords, e have lost the inforation contained in the analog function x a (t) hen e saple it. See figure explanation: (Note that hen x a (t) is a real-nuber signal, then the agnitude of its Fourier transfor is an even function, i.e., is syetric to the y-axis).

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30 Sapling Theore (c.f. Shenoi, 6) When can e reconstruct the continuous signal fro its unifor sapling? iff () the continuous signal x a (t) is band liited that is, if it is a function such that its Fourier transfor X a (j) for > b, here b is a frequency bound. () the sapling period T is chosen such that s > b (here s T). That is, to avoid the situation of aliasing, the sapling frequency shall be larger than tice of the highest frequency of the continuous signal. Nyquist sapling theore: f b ( b /) is called the Nyquist frequency, and f b is called the Nyquist rate. See figures for explanation.

31 Continuous signal Sapled signal The agnitude spectru of the continuous signal The spectru of the sapled signal

32 Particular discrete-tie signal exaples Unit pulse (or unit saple) function (or discrete-tie ipulse; ipulse) δ [] n n n

33 An arbitrary sequence x[n], n Z can be represented as a su of scaled, delayed ipulses. x [] n x[][ k δ n k] k

34 Unit step sequence u [] n n n <

35 Soe DTFT Transfor Pairs δ [ n] Tie doain shift : δ [ ] jn n n e exponential sequence : a n u ae [] n ( a < ) j k ( < n < ) δ + [ k]

36 DTFT Transfor Pairs (continue) u ae [] n + δ [ + k ] j k ( ) n n + a u[] n ( a < ) ( j ae ) e j n ( + k ) k δ x [] n [ ( M + ) / ] sin[ / ] n M sin jm e otherise /

37 DTFT Transfor Pairs (continue) [ ] ( ) ( ) k e k e n j k j δ δ φ φ φ cos ( ) < < e X n n c c j c sin ( ) [] j j p p p n e r e r r n u n r cos ) ( sin sin + < +

38 DTFT Theores Linearity x [n] X (e j ), x [n] X (e j ) iplies that a x [n] + a x [n] a X (e j ) + a X (e j ) Tie shifting x[n] X(e j ) iplies that x [ ] jn ( ) d j n n e X e d

39 DTFT Theores (continue) Frequency shifting iplies that e j x[n] X(e j ) [ ] j( n X e ) Tie reversal x[n] X(e j ) n x ( ) If the sequence is tie reversed, then x[ n] X(e j )

40 DTFT Theores (continue) Differentiation in frequency x[n] X(e j ) iplies that nx ( ) j [] dx e n j d Parseval s theore x[n] X(e j ) iplies that E n x [] ( ) j n X e d

41 Exaple Suppose e ish to find the Fourier transfor of x[n] a n u[n-5]. x x x [] [] ( ) n j n a u n X e [] n x [ n 5] Thus, X ae ( ) ( ) j j5 j e e X e [] [] ( ) 5 j n a x n, so X e j j a 5 e ae e j5 j5 ae j

42 Representation of Sequences by Discrete-tie Fourier Transfors (DTFT) (cf. Oppenhei et al. 999) Fourier Representation: representing a signal by coplex exponentials. x X jn Inverse Fourier [] ( ) j n X e ( ) j e x[] n n e e jn d transfor Fourier transfor (or forard Fourier transfor) A signal x[n] is represented as the Fourier integral of the coplex exponentials in the range of frequencies [, ]. The eight X(e j ) of the frequency applied in the integral can be deterined by the input signal x[n], and X(e j ) reveals ho uch of each frequency is required to synthesize x[n].

43 Representation of sequences by DTFT (continue) X ( ) ( ) ( j ) j j j X e e X e e The phase X(e j ) is not uniquely specified since any integer ultiple of ay be added to X(e j ) at any value of ithout affecting the result. Denote ARG[X(e j )] to be the phase value in [, ]. Since the frequency response of a LTI syste is the Fourier transfor of the ipulse response, the ipulse response can be obtained fro the frequency response by applying the inverse Fourier transfor integral: h [] ( ) j n H e e jn d

44 Existence of discrete-tie Fourier transfor pairs Whey they are transfor pairs? Consider the integral xˆ n x [ ] [ ] j( n e ) x e j e d jn d Since ( ) ( ) n sin n d ( n ) j e n n δ [ n ] [ ] δ [ n ] x[ n] xˆ n x

45 Conditions for the existence of discrete-tie Fourier transfor pairs Conditions for the existence of Fourier transfor pairs of a signal: Absolutely suable Mean-square convergence: li M n [] n < ( ) ( ) ( ) j j j [] jn e X e d, for X e x n e X M M In other ords, the error X(e j ) X M (e j ) ay not approach for each, but the total energy in the error does. x M n M

46 Conditions for the Existence of DTFT transfor Pairs (continue) Still soe other cases that are neither absolutely suable nor ean-square convergence, the Fourier transfor still exist: Eg., Fourier transfor of a constant, x[n] for all n, is an ipulse train: X ( ) j e δ( + r ) r The ipulse of the continuous case is a infinite heigh, zero idth, and unit area function. If soe properties are defined for the ipulse function, then the Fourier transfor pair involving ipulses can be ell defined too.

47 Syetry Property of the Fourier Transfor Conjugate-syetric sequence: x e [n] x e *[ n] If a real sequence is conjugate syetric, then it is called an even sequence satisfying x e [n] x e [ n]. Conjugate-asyetric sequence: x o [n] x o *[ n] If a real sequence is conjugate antisyetric, then it is called an odd sequence satisfying x [n] x [ n]. Any sequence can be represented as a su of a conjugate-syetric and asyetric sequences, x[n] x e [n] + x o [n], here x e [n] (/)(x[n]+ x*[ n]) and x o [n] (/)(x[n] x*[ n]).

48 Syetry Property of the Fourier Transfor (continue) Siilarly, a Fourier transfor can be decoposed into a su of conjugate-syetric and antisyetric parts: X(e j ) X e (e j ) + X o (e j ), here X e (e j ) (/)[X(e j ) + X*(e j )] and X o (e j ) (/)[X(e j ) X*(e j )]

49 Syetry Property of the Fourier Transfor (continue) Fourier Transfor Pairs (if x[n] X(e j )) x*[n] X*(e j ) x*[ n] X*(e j ) Re{x[n]} X e (e j ) (conjugate-syetry part of X(e j )) ji{x[n]} X o (e j ) (conjugate anti-syetry part of X(e j )) x e [n] (conjugate-syetry part of x[n]) X R (e j ) Re{X(e j )} x o [n] (conjugate anti-syetry part of x[n]) jx I (e j ) ji{x(e j )}

50 Syetry Property of the Fourier Transfor (continue) Fourier Transfor Pairs (if x[n] X(e j )) Any real x e [n] X(e j ) X*(e j ) (Fourier transfor is conjugate syetric) Any real x e [n] X R (e j ) X R (e j ) (real part is even) Any real x e [n] X I (e j ) X I (e j ) (iaginary part is odd) Any real x e [n] X R (e j ) X R (e j ) (agnitude is even) Any real x e [n] X R (e j ) X R (e j ) (phase is odd) x o [n] (even part of real x[n]) X R (e j ) x o [n] (odd part of real x[n]) jx I (e j )

51 Exaple of Syetry Properties The Fourier transfor of the real sequence x[n] a n u[n] for a < is ( ) j X e j ae Its agnitude is an even function, and phase is odd.

52 Discrete Fourier Transfor (DFT) Currently, e have investigated three cases of Fourier transfor, Fourier series (for continuous periodic signal) Continuous Fourier transfor (for continuous signal) Discrete-tie Fourier transfor (for discrete-tie signal) All of the have infinite integral or suation in either tie or frequency doains.

53 There still have another type of Fourier transfor: Consider a discrete sequence that is periodic in the tie doain (eg., it can be obtained by a periodic expansion of a finiteduration sequence, ie., e iage that a finite-length sequence repeats, over and over again, in the tie doain). Then, in the frequency doain, the spectru shall be both periodic and discrete, ie, the frequency sequence is also ade up of a finite-length sequence, hich repeats over and over again in the frequency doain. Considering both the finite-length (or finite-duration) sequences in one period of the tie and frequency doains, leads to a transfor called discrete Fourier transfor.

54 Fro Kuhn 5

55 Fro Kuhn 5

56 Four types of Fourier Transfors Frequency doain nonperiodic Frequency doain periodic Tie doain nonperiodic Continuous Fourier transfor (both doains are continuous) DTFT (tie doain discrete, frequency doain continuous) Tie doain periodic Fourier series (tie doain continuous, frequency doain discrete) DFT/DTFS (tie doain discrete, frequency doain discrete, and both finite-duration)