2 Frequency-Domain Analysis

Transcription

1 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 o oher sysems analyss. Since he communicaions engineer is concerned primarily wih signal bandwidhs and signal locaions in he frequency domain, raher han wih ransien analysis, he essenially seady-sae approach of he (complex exponenial) ourier series and ransforms is used raher han he Laplace ransform.. Mahemaical Background.. Euler s formula: e jθ = cos θ + j sin θ. () cos (x) = Re { e jx} = ( e jx + e jx) () sin (x) = Im { e jx} = j ( e jx e jx). (3) (.. We can use cos x = e jx + e jx) and sin x = j many rigonomeric ideniies. See Example.4. ( e jx e jx) o derive Example.3. Use he Euler s formula o show ha d dx sin x = cos x. Example.4. Use he Euler s formula o show ha cos (x) = (cos(x) + ). 6

2 .5. Similar echnique gives (a) cos( x) = cos(x), (b) cos ( x ) π = sin(x), (c) sin x = ( cos (x)) (d) sin(x) cos(x) = sin(x), and (e) he produc-o-sum formula cos(x) cos(y) = (cos(x + y) + cos(x y)). (4). Coninuous-Time ourier Transform Definiion.6. The (direc) ourier ransform of a signal g() is defined by + G(f) = g()e jπf d (5) This provides he frequency-domain descripion of g(). Conversion back o he ime domain is achieved via he inverse (ourier) ransform: g () = G (f) e jπf df (6) We may combine (5) and (6) ino one compac formula: G (f) e jπf df = g () G (f) = g () e jπf d. (7) We may simply wrie G = {g} and g = {G}. Noe ha he area under he curve of a funcion in one domain is he same as is value a in anoher domain: G() = g()d and g() = 7 G(f)df. (8)

3 .7. In some references 5, he (direc) ourier ransform of a signal g() is defined by In which case, we have π Ĝ(ω) = + Ĝ (ω) e jω dω = g () Ĝ (ω) = g()e jω d (9) g () e jω d () In MATLAB, hese calculaions are carried ou via he commands fourier and ifourier. Noe ha Ĝ() = g()d and g() = π Ĝ(ω)dω. The relaionship beween G(f) in (5) and Ĝ(ω) in (9) is given by G(f) = Ĝ(ω) ω=πf () Ĝ(ω) = G(f) f= ω π () Before we inroduce our firs bu crucial ransform pair in Example.3 which will involve recangular funcion, we wan o inroduce he indicaor funcion which gives compac represenaion of he recangular funcion. We will see laer ha he ransform of he recangular funcion gives a sinc funcion. Therefore, we will also inroduce he sinc funcion as well. Definiion.8. An indicaor funcion gives only wo values: or. I is usually wrien in he form [some condiion(s) involving ]. Is value a a paricular is one if and only if he condiion(s) inside is saisfied for ha. or example, {, a a, [ a] =, oherwise. 5 MATLAB uses his definiion. 8

4 Alernaively, we can use a se o specify he values of a which he indicaor funcion gives he value : {, A, A () =, / A. In paricular, he se A can be some inervals: {, a a, [ a,a] () =, oherwise, and [ a,b] () = {, a b,, oherwise. Example.9. Carefully skech he funcion g() = [ 5] Definiion.. Recangular pulse [3, Ex. p 45]: () = [.5] = [.5,.5] () This is a pulse of uni heigh and uni widh, cenered a he origin. Hence, i is also known as he uni gae funcion rec () [5, p 78]. In [3], he values of he pulse () a.5 and.5 are no specified. However, in [5], hese values are defined o be.5. In MATLAB, he funcion recangularpulse() can be used o produce 6 he uni gae funcion above. More generally, we can produce a recangular pulse whose rising edge is a a and falling edge is a b via recangularpulse(a,b,). ( ) T = [ ] T = [ T, T ] () Observe ha T is he widh of he pulse. 6 Noe ha recangularpulse(-.5) and recangularpulse(.5) give.5 in MATLAB. 9

5 sinc funcion Definiion.. The sinc funcion sinc(x) (sin x)/x (3) is ploed in igure. cos sin / / - sinc igure : Sinc funcion This funcion plays an imporan role in signal processing. I is also known as he filering or inerpolaing funcion. The full name of he funcion is sine cardinal 7. Using L Hôpial s rule, we find lim x sinc(x) =. sinc(x) is he produc of an oscillaing signal sin(x) (of period π) and a monoonically decreasing funcion /x. Therefore, sinc(x) exhibis sinusoidal oscillaions of period π, wih ampliude decreasing coninuously as /x. Is zero crossings are a all non-zero ineger muliples of π. 7 which corresponds o he Lain name sinus cardinalis. I was inroduced by Woodward in his 95 paper Informaion heory and inverse probabiliy in elecommunicaion [], in which he noed ha i occurs so ofen in ourier analysis and is applicaions ha i does seem o meri some noaion of is own

6 Definiion.. Normalized sinc funcion: In MATLAB and in many sandard exbooks such as [3, p 37], [4, eq..64], and [], he funcion sinc(x) is defined as Normalized sinc funcion sin(πx) πx. (4) Is zero crossings are a non-zero ineger values of is argumen as shown in igure 3. sinc sin igure 3: Normalized sinc funcion Is zero crossings are a non-zero ineger values of is argumen. The normalized par of he name is added o disinguish i from (3) which is unnormalized..3. Recangular funcion and sinc funcion as a ourier ransform pair: [ a] sin(πf a) πf = a sinc(πf a) (5) The righ half of igure 4 illusraes (5). By seing a = T /, we have [ T T sinc(πt f). In paricular, when T =, we have ] rec () sinc(πf). The ourier ransform of he uni gae funcion is he normalized sinc funcion.

7 ( π f) sinc f f ω π f f f π f f f ω f T T T sinc T T ω T π T Tsinc ( π T f ) T T ω f igure 4: ourier ransform of sinc and recangular funcions Observe ha (a) The heigh of he sinc funcion s peak is he same as he area under he recangular funcion. This follows from (8). Pracice Problems (b) The firs zero crossing of he sinc funcion occurs a /(widh of he recangular funcion). Example.4. - f - f 79

8 Definiion.5. The (Dirac) dela funcion or (uni) impulse funcion is denoed by δ(). I is usually depiced as a verical arrow a he origin. Noe ha δ() is no 8 a rue funcion; i is undefined a =. We define δ() as a generalized funcion which saisfies he sampling propery (or sifing propery) for any funcion g() which is coninuous a =. g()δ()d = g() (6) In his way, he dela funcion has no mahemaical or physical meaning unless i appears under he operaion of inegraion. Inuiively we may visualize δ() as an infiniely all, infiniely narrow recangular pulse of uni area: lim ε ε [ ] ε..6. Properies of δ(): δ() = for. δ( T ) = for T. A δ()d = A(). (a) δ()d =. (b) {} δ()d =. (c) x δ()d = [, )(x). Hence, we may hink of δ() as he derivaive of he uni sep funcion U() = [, ) (x) [3, Defn 3.3 p 6]. g()δ( c)d = g(c) for g coninuous a c. In fac, for any ε >, Convoluion 9 propery: c+ε c ε (δ g)() = (g δ)() = g()δ( c)d = g(c). where we assume ha g is coninuous a. g(τ)δ( τ)dτ = g() (7) 8 The δ-funcion is a disribuion, no a funcion. In spie of ha, i s always called δ-funcion. 9 See Definiion.37. 3

9 76 Time Manipulaion Consider a funcion of ime. Time shifing: When, is righ-shifed (delayed) by. When, is lef-shifed (advanced) by. Summary: is righ-shifed by. Time scaling: When, is expanded in ime by a facor of When, is compressed in ime by a facor of Summary: When, is scaled horizonally by a facor of. Noe ha he signal remains anchors a. In oher words, he signal a remains unchanged. Time inversion (or folding): is he mirror image of abou he verical axis. [Lahi & Ding, 9, Secion.3, p. 8-3] Time Manipulaion or may consider i as : irs scale horizonally by a facor of. Then, righ-shif by : irs righ-shif by. Then scale horizonally by a facor of. m m c m a c m m c b c m 77 a m b m

10 Example: Plo and find he area under he curve of -3 5 Area under he graph is The graph is compressed horizonally. 78 Noe: sill he same heigh Area under he graph is Example: Plo and find he area under he curve of -3 5 This poin corresponds o he argumen of being. The same poin will happen in when This poin corresponds o he argumen of being. The same poin will happen in when This poin corresponds o he argumen of being. The same poin will happen in when 79

11 Plo and area under he curve of -3 5 Area under he graph is lipped horizonally Area under he graph is 8 () Area = (a) Area = smaller Area = (a) Area = Area = smaller Area = 8 Here, I use

12 acoring a consan a ou of he δ-funcion means scaling i by a : In paricular, δ(a) = δ(). (8) a δ(ω) = = δ(f) (9) π and δ(ω ω ) = δ(πf πf ) = π δ(f f ), () where ω = πf and ω = πf. Example.7. δ ()d = and δ ()d =. Example.8. δ ()d = Example.9. δ(). Example.. e jπf δ (f f ). Example.. e jω Example.. e j4π πδ (ω ω ). 4

13 Example.3. cos(π).4. cos(πf ) (δ (f f ) + δ (f + f )). Example.5. cos() Example.6. cos (π) + cos (4π) ourier ransform: Example Example.7. cos () 3 cos (3) + 5 cos (5) cos 3 cos 3 cos 5 f Example.8. cos (π) 5

14 Example.9. cos (π) cos (4π).3. Conjugae symmery : If g() is real-valued, hen G( f) = (G(f)) (a) Even ampliude symmery: G ( f) = G (f) (b) Odd phase symmery: G ( f) = G (f) Observe ha if we know G(f) for all f posiive, we also know G(f) for all f negaive. Inerpreaion: Only half of he specrum conains all of he informaion. Posiive-frequency par of he specrum conains all he necessary informaion. The negaive-frequency half of he specrum can be deermined by simply complex conjugaing he posiive-frequency half of he specrum. urhermore, (a) If g() is real and even, hen so is G(f). (b) If g() is real and odd, hen G(f) is pure imaginary and odd. Hermiian symmery in [3, p 48] and [9, p 7 ]. 6

15 .3. Shifing properies Time-shif: g ( ) e jπf G (f) Noe ha e jπf =. So, he specrum of g ( ) looks exacly he same as he specrum of g() (unless you also look a heir phases). requency-shif (or modulaion): e jπf g () G (f f ).3. Le g(), g (), and g () denoe signals wih G(f), G (f), and G (f) denoing heir respecive ourier ransforms. (a) Superposiion heorem (lineariy): a g () + a g () a G (f) + a G (f). (b) Scale-change heorem (scaling propery [5, p 88]; reciprocal spreading [3, p 46]): ( ) f g(a) a G. () a The funcion g(a) represens he funcion g() compressed in ime by a facor a (when a > ). The funcion G(f/a) represens he funcion G(f) expanded in frequency by he same facor a. 7

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17 The scaling propery says ha if we squeeze a funcion in, is ourier ransform sreches ou in f, i is no possible o arbirarily concenrae a funcion and is ourier ransform simulaneously, generally speaking, he more concenraed g() is, he more spread ou is ourier ransform G(f) mus be. This rade-off can be formalized in he form of an uncerainy principle. See also.45 and.46. Inuiively, we undersand ha compression in ime by a facor a means ha he signal is varying more rapidly by he same facor. To synhesize such a signal, he frequencies of is sinusoidal componens mus be increased by he facor a, implying ha is frequency specrum is expanded by he facor a. Similarly, a signal expanded in ime varies more slowly; hence, he frequencies of is componens are lowered, implying ha is frequency specrum is compressed. (c) Dualiy heorem (Symmery Propery [5, p 86]): G() g( f). In words, for any resul or relaionship beween g() and G(f), here exiss a dual resul or relaionship, obained by inerchanging he roles of g() and G(f) in he original resul (along wih some minor modificaions arising because of a sign change). In paricular, if he ourier ransform of g() is G(f), hen he ourier ransform of G(f) wih f replaced by is he original imedomain signal wih replaced by f. If we use he ω-definiion (9), we ge a similar relaionship wih an exra facor of π: Ĝ() πg( ω). 8

18 Example.33. Le s ry o use he scale-change heorem o double-check he ourier ransform of a simple funcion. Consider he funcion x() = g(a) where g() = e jπf. Noe ha g() is simply a complex exponenial funcion a frequency f. rom Example., is ourier ransform G(f) is simply δ(f f ). (a) rom x() = g(a) = e jπf (a), by grouping he facor a wih f, we ge x() = e jπ(af ). Therefore, x() is a complex exponenial funcion a frequency af. As in Example., is ourier ransform is X(f) = δ(f af ). (b) Alernaively, we can also apply he ( scale-change ) heorem. rom x() = g(a), we know ha X(f) = a G f a. Plugging in G(f) = δ(f f ), we ge ) X(f) = ( f a δ a f = a δ ( ) a (f af ). Now, recall, from.6 ha, facoring a consan α ou of he δ-funcion means scaling i by α. Here, he consan is α = a. Therefore, X(f) = a δ (f af ) = δ (f af ). a Exercise.34. Similar o Example.33, one can also ry o apply he scale-change heorem o show ha x() = cos(πaf ) (δ(f af ) + δ(f + af )). Example.35. rom Example.3, we know ha By he dualiy heorem, we have [ a] a sinc (πaf) () a sinc(πa) [ f a], 9

19 which is he same as sinc(πf ) Boh ransform pairs are illusraed in igure 5. f [ f f ]. (3) Dualiy Theorem igure 5: Dualiy heorem: recangular and sinc funcions Example.36. Le s ry o derive he ime-shif propery from he frequencyshif propery. We sar wih an arbirary funcion g(). Nex we will define anoher funcion x() by seing X(f) o be g(f). Noe ha f here is jus a dummy variable; we can also wrie X() = g(). Applying he dualiy heorem o he ransform pair x() X(f), we ge anoher ransform pair X() x( f). The LHS is g(); herefore, he RHS mus be G(f). This implies G(f) = x( f). Nex, recall he frequency-shif propery: The dualiy heorem hen gives e jπc x () X (f c). X ( c) ejπc f x ( f).

20 Replacing X() by g() and x( f) by G(f), we finally ge he ime-shif propery. Definiion.37. The convoluion of wo signals, g () and g (), is a new funcion of ime, g(). We wrie g = g g. I is defined as he inegral of he produc of he wo funcions afer one is reversed and shifed: g() = (g g )() (4) = + g (µ)g ( µ)dµ = + g ( µ)g (µ)dµ. (5) Noe ha is a parameer as far as he inegraion is concerned. The inegrand is formed from g and g by hree operaions: (a) ime reversal o obain g ( µ), (b) ime shifing o obain g ( (µ )) = g ( µ), and (c) muliplicaion of g (µ) and g ( µ) o form he inegrand. In some references, (4) is expressed as g() = g () g (). Example.38. We can ge a riangle from convoluion of wo recangular waves. In paricular, [ a] [ a] = (a ) [ a]..39. Convoluion properies involving he δ-funcion:

21 .4. Convoluion heorems: (a) Convoluion-in-ime rule: g g G G. (6) (b) Convoluion-in-frequency rule: g g G G. (7) Example.4. We can use he convoluion heorem o prove he frequencyshif propery in rom he convoluion heorem, we have g G G if g is band-limied o B, hen g is band-limied o B.43. Parseval s heorem (Rayleigh s energy heorem, Plancherel formula) for ourier ransform: + g() d = + G(f) df. (8) The LHS of (8) is called he (oal) energy of g(). On he RHS, G(f) is called he energy specral densiy of g(). By inegraing he energy specral densiy over all frequency, we obain he signal s oal energy. The energy conained in he frequency band B can be found from he inegral B G(f) df. More generally, ourier ransform preserves he inner produc [, Theorem.]: g, g = g ()g ()d = G (f)g (f)df = G, G.

22 Example.44. Perform he following inegraion graphically wih he help of ourier ransform properies: (a) sinc () d. (b) sinc () d. 3

23 .45. (Heisenberg) Uncerainy Principle [, ]: Suppose g is a funcion which saisfies he normalizing condiion g = g() d = which auomaically implies ha G = G(f) df =. Then ( ) ( g() d ) f G(f) df 6π, (9) and equaliy holds if and only if g() = Ae B where B > and A = B/π. In fac, we have ( ) ( g( ) d for every, f. ) f G(f f ) df 6π, The proof relies on Cauchy-Schwarz inequaliy. or any funcion h, define is dispersion h as h() d h(). Then, we can d apply (9) o he funcion g() = h()/ h and ge h H 6π..46. A signal canno be simulaneously ime-limied and band-limied. Proof. Suppose g() is simulaneously () ime-limied o T and () bandlimied o B. Pick any posiive number T s and posiive ineger K such ha f s = T s > B and K > T T s. The sampled signal g Ts () is given by g Ts () = k g[k]δ ( kt s ) = K k= K g[k]δ ( kt s ) where g[k] = g (kt s ). Now, because we sample he signal faser han he Nyquis rae, we can reconsruc he signal g by producing g Ts h r where he LP h r is given by H r (ω) = T s [ω < πf c ] wih he resricion ha B < f c < T s B. In frequency domain, we have G(ω) = K k= K g[k]e jkωt s H r (ω). 4

24 Consider ω inside he inerval I = (πb, πf c ). Then, ω>πb = G(ω) ω<πf c = T s K k= K g (kt s ) e jkωt s K z=ejωts = T s k= K g (kt s ) z k (3) Because z, we can divide (3) by z K and hen he las erm becomes a polynomial of he form a K z K + a K z K + + a z + a. By fundamenal heorem of algebra, his polynomial has only finiely many roos ha is here are only finiely many values of z = e jωt s which saisfies (3). Because here are uncounably many values of ω in he inerval I and hence uncounably many values of z = e jωt s which saisfy (3), we have a conradicion..47. The observaion in.46 raises concerns abou he signal and filer models used in he sudy of communicaion sysems. Since a signal canno be boh bandlimied and imelimied, we should eiher abandon bandlimied signals (and ideal filers) or else accep signal models ha exis for all ime. On he one hand, we recognize ha any real signal is imelimied, having saring and ending imes. On he oher hand, he conceps of bandlimied specra and ideal filers are oo useful and appealing o be dismissed enirely. The resoluion of our dilemma is really no so difficul, requiring bu a small compromise. Alhough a sricly imelimied signal is no sricly bandlimied, is specrum may be negligibly small above some upper frequency limi B. Likewise, a sricly bandlimied signal may be negligibly small ouside a cerain ime inerval. Therefore, we will ofen assume ha signals are essenially boh bandlimied and imelimied for mos pracical purposes. 5