2 Frequency-Domain Analysis

Transcription

1 2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, o time and requency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so perhaps than to other systems analysts. Since the communications engineer is concerned primarily with signal bandwidths and signal locations in the requency domain, rather than with transient analysis, the essentially steady-state approach o the (complex exponential) ourier series and transorms is used rather than the Laplace transorm. 2. Mathematical Background 2.. Euler s ormula: e jθ = cos θ + j sin θ. () cos (x) = Re { e jx} = 2 ( e jx + e jx) (2) sin (x) = Im { e jx} = 2j ( e jx e jx). (3) ( 2.2. We can use cos x = 2 e jx + e jx) and sin x = 2j many trigonometric identities. See Example 2.4. ( e jx e jx) to derive Example 2.3. Use the Euler s ormula to show that d dx sin x = cos x. Example 2.4. Use the Euler s ormula to show that cos 2 (x) = 2 (cos(2x) + ). 5

2 2.5. Similar technique gives (a) cos( x) = cos(x), (b) cos ( x 2) π = sin(x), (c) sin 2 x = 2 ( cos (2x)) (d) sin(x) cos(x) = 2 sin(2x), and (e) the product-to-sum ormula cos(x) cos(y) = (cos(x + y) + cos(x y)). (4) Continuous-Time ourier Transorm Deinition 2.6. The (direct) ourier transorm o a signal g(t) is deined by + G() = g(t)e j2πt dt (5) This provides the requency-domain description o g(t). Conversion back to the time domain is achieved via the inverse (ourier) transorm: g (t) = G () e j2πt d (6) We may combine (5) and (6) into one compact ormula: G () e j2πt d = g (t) G () = g (t) e j2πt dt. (7) We may simply write G = {g} and g = {G}. Note that the area under the curve o a unction in one domain is the same as its value at in another domain: G() = g(t)dt and g() = 6 G()d. (8)

3 Time Domain direct transorm j2 t j 2 t g t G e d G g t e dt inverse transorm Capital letter is used to represent corresponding signal in the requency domain. j2 t j 2 t g t G e d G g t e dt j2 t Complex exponential: e cos2 t jsin2 t sinusoids The relationship on the let is simply a decomposition o the signal into a linear combination o (potentially ininitely many) components at dierent requencies. j2 t j 2 t g t G e d G g t e dt rom the decomposition point o view, the value o at a particular requency is simply the weight (scaling/coeicient) which tells how much component there is in. requency Domain By the orthogonality among complex exponential unctions, the value o at a particular requency can be calculated by the ormula above. This coeicient considered as a unction o requency is the ourier transorm o our signal. igure 2: ourier coeicients rom the decomposition in the time domain. 7

4 2.7. In some reerences 5, the (direct) ourier transorm o a signal g(t) is deined by In which case, we have 2π Ĝ(ω) = + Ĝ (ω) e jωt dω = g (t) Ĝ (ω) = g(t)e jωt dt (9) g (t) e jωt dt () In MATLAB, these calculations are carried out via the commands ourier and iourier. Note that Ĝ() = g(t)dt and g() = 2π Ĝ(ω)dω. The relationship between G() in (5) and Ĝ(ω) in (9) is given by G() = Ĝ(ω) ω=2π () Ĝ(ω) = G() = ω 2π (2) Beore we introduce our irst but crucial transorm pair in Example 2.3 which will involve rectangular unction, we want to introduce the indicator unction which gives compact representation o the rectangular unction. We will see later that the transorm o the rectangular unction gives a sinc unction. Thereore, we will also introduce the sinc unction as well. Deinition 2.8. An indicator unction gives only two values: or. It is usually written in the orm [some condition(s) involving t]. Its value at a particular t is one i and only i the condition(s) inside is satisied or that t. 5 MATLAB uses this deinition. 8

5 Alternatively, we can use a set to speciy the values o t at which the indicator unction gives the value : {, t A, A (t) =, t / A. In particular, the set A can be some intervals: {, a t a, [ a,a] (t) =, otherwise, and [ a,b] (t) = {, a t b,, otherwise. Example 2.9. Careully sketch the unction g(t) = [ t 5] Deinition 2.. Rectangular pulse [3, Ex 2.2 p 45]: (t) = [ t.5] = [.5,.5] (t) This is a pulse o unit height and unit width, centered at the origin. Hence, it is also known as the unit gate unction rect (t) [5, p 78]. In [3], the values o the pulse (t) at.5 and.5 are not speciied. However, in [5], these values are deined to be.5. In MATLAB, the unction rectangularpulse(t) can be used to produce 6 the unit gate unction above. More generally, we can produce a rectangular pulse whose rising edge is at a and alling edge is at b via rectangularpulse(a,b,t). ( ) t T = [ ] t T 2 = [ T 2, T 2 ] (t) Observe that T is the width o the pulse. 6 Note that rectangularpulse(-.5) and rectangularpulse(.5) give.5 in MATLAB. 9

6 sinc unction Deinition 2.. The sinc unction sinc(x) (sin x)/x (3) is plotted in igure 3. cos sin / / - sinc igure 3: Sinc unction This unction plays an important role in signal processing. It is also known as the iltering or interpolating unction. The ull name o the unction is sine cardinal 7. Using L Hôpital s rule, we ind lim x sinc(x) =. sinc(x) is the product o an oscillating signal sin(x) (o period 2π) and a monotonically decreasing unction /x. Thereore, sinc(x) exhibits sinusoidal oscillations o period 2π, with amplitude decreasing continuously as /x. Its zero crossings are at all non-zero integer multiples o π. 7 which corresponds to the Latin name sinus cardinalis. It was introduced by Woodward in his 952 paper Inormation theory and inverse probability in telecommunication [2], in which he noted that it occurs so oten in ourier analysis and its applications that it does seem to merit some notation o its own

7 Deinition 2.2. Normalized sinc unction: In MATLAB and in many standard textbooks such as [3, p 37], [4, eq. 2.64], and [2], the unction sinc(x) is deined as Normalized sinc unction sin(πx) πx. (4) Its zero crossings are at non-zero integer values o its argument as shown in igure 4. sinc sin igure 4: Normalized sinc unction Its zero crossings are at non-zero integer values o its argument. The normalized part o the name is added to distinguish it rom (3) which is unnormalized Rectangular unction and sinc unction as a ourier transorm pair: [ t a] sin(2π a) π = 2a sinc(2π a) (5) The right hal o igure 5 illustrates (5). By setting a = T /2, we have [ t T T sinc(πt ). In particular, when T =, we have 2 ] rect (t) sinc(π). The ourier transorm o the unit gate unction is the normalized sinc unction.

8 ( π t) sinc 2 2 t ω 2 π 2 2 2π 2 ω T 2 T T 2 t sinc T T ω 2 T 2π T Tsinc ( π T ) T T ω igure 5: ourier transorm o sinc and rectangular unctions Observe that (a) The height o the sinc unction s peak is the same as the area under the rectangular unction. This ollows rom (8). Practice Problems (b) The irst zero crossing o the sinc unction occurs at /(width o the rectangular unction). Example t -2 2 t 79 2

9 Delta unction () (Dirac) delta unction or (unit) impulse unction Usually depicted as a vertical arrow at the origin Not a true unction Undeined at = [Deinition 2.5] Intuitively we may visualize () as an ininitely tall, ininitely narrow rectangular pulse o unit area 2 Area = Area = Area = smaller A() (Dirac) delta unction or (unit) impulse unction Usually depicted as a vertical arrow at the origin Not a true unction Undeined at = Intuitively we may visualize A() as an ininitely tall, ininitely narrow rectangular pulse o area A 2 Area = A Area = A smaller Area = A

10 Deinition 2.5. The (Dirac) delta unction or (unit) impulse unction is denoted by δ(t). It is usually depicted as a vertical arrow at the origin. Note that δ(t) is not 8 a true unction; it is undeined at t =. We deine δ(t) as a generalized unction which satisies the sampling property (or siting property) or any unction g(t) which is continuous at t =. g(t)δ(t)dt = g() (6) In this way, the delta unction has no mathematical or physical meaning unless it appears under the operation o integration. Intuitively we may visualize δ(t) as an ininitely tall, ininitely narrow rectangular pulse o unit area: lim ε ε [ t 2] ε Properties o δ(t): δ(t) = or t. δ(t T ) = or t T. A δ(t)dt = A(). (a) δ(t)dt =. (b) {} δ(t)dt =. (c) x δ(t)dt = [, )(x). Hence, we may think o δ(t) as the derivative o the unit step unction U(t) = [, ) (x) [3, Den 3.3 p 26]. g(t)δ(t c)dt = g(c) or g continuous at c. In act, or any ε >, Convolution 9 property: c+ε c ε (δ g)(t) = (g δ)(t) = g(t)δ(t c)dt = g(c). where we assume that g is continuous at t. g(τ)δ(t τ)dτ = g(t) (7) 8 The δ-unction is a distribution, not a unction. In spite o that, it s always called δ-unction. 9 See Deinition

11 actoring a constant a out o the δ-unction means scaling it by a : In particular, δ(at) = δ(t). (8) a δ(ω) = = δ() (9) 2π and δ(ω ω ) = δ(2π 2π ) = 2π δ( ), (2) where ω = 2π and ω = 2π. Example δ (t)dt = and 2 δ (t)dt =. Example δ (t)dt = Example 2.9. δ(t). Example 2.2. e j2π t δ ( ). Example 2.2. e jω t Example e j4πt 2πδ (ω ω ). 4

12 Example cos(2πt) cos(2π t) 2 (δ ( ) + δ ( + )). Example cos(2t) Example cos (2πt) + cos (4πt) ourier transorm: Example Example cos (t) 3 cos (3t) + 5 cos (5t) t cos 3 cos 3 cos t Example cos 2 (2πt) 5

13 Example cos (2πt) cos (4πt) 2.3. Conjugate symmetry : I g(t) is real-valued, then G( ) = (G()) (a) Even amplitude symmetry: G ( ) = G () (b) Odd phase symmetry: G ( ) = G () Observe that i we know G() or all positive, we also know G() or all negative. Interpretation: Only hal o the spectrum contains all o the inormation. Positive-requency part o the spectrum contains all the necessary inormation. The negative-requency hal o the spectrum can be determined by simply complex conjugating the positive-requency hal o the spectrum. urthermore, (a) I g(t) is real and even, then so is G(). (b) I g(t) is real and odd, then G() is pure imaginary and odd. Hermitian symmetry in [3, p 48] and [9, p 7 ]. 6

14 2.3. Shiting properties Time-shit: g (t t ) e j2πt G () Note that e j2πt =. So, the spectrum o g (t t ) looks exactly the same as the spectrum o g(t) (unless you also look at their phases). requency-shit (or modulation): e j2πt g (t) G ( ) Let g(t), g (t), and g 2 (t) denote signals with G(), G (), and G 2 () denoting their respective ourier transorms. (a) Superposition theorem (linearity): a g (t) + a 2 g 2 (t) a G () + a 2 G 2 (). (b) Scale-change theorem (scaling property [5, p 88]; reciprocal spreading [3, p 46]): ( ) g(at) a G. (2) a The unction g(at) represents the unction g(t) compressed in time by a actor a (when a > ). The unction G(/a) represents the unction G() expanded in requency by the same actor a. 7

15 The scaling property says that i we squeeze a unction in t, its ourier transorm stretches out in, it is not possible to arbitrarily concentrate a unction and its ourier transorm simultaneously, generally speaking, the more concentrated g(t) is, the more spread out its ourier transorm G() must be. This trade-o can be ormalized in the orm o an uncertainty principle. See also 2.45 and Intuitively, we understand that compression in time by a actor a means that the signal is varying more rapidly by the same actor. To synthesize such a signal, the requencies o its sinusoidal components must be increased by the actor a, implying that its requency spectrum is expanded by the actor a. Similarly, a signal expanded in time varies more slowly; hence, the requencies o its components are lowered, implying that its requency spectrum is compressed. (c) Duality theorem (Symmetry Property [5, p 86]): G(t) g( ). In words, or any result or relationship between g(t) and G(), there exists a dual result or relationship, obtained by interchanging the roles o g(t) and G() in the original result (along with some minor modiications arising because o a sign change). In particular, i the ourier transorm o g(t) is G(), then the ourier transorm o G() with replaced by t is the original timedomain signal with t replaced by. I we use the ω-deinition (9), we get a similar relationship with an extra actor o 2π: Ĝ(t) 2πg( ω). 8

16 Example Let s try to use the scale-change theorem to double-check the ourier transorm o a simple unction. Consider the unction x(t) = g(at) where g(t) = e j2π t. Note that g(t) is simply a complex exponential unction at requency. rom Example 2.2, its ourier transorm G() is simply δ( ). (a) rom x(t) = g(at) = e j2π (at), by grouping the actor a with, we get x(t) = e j2π(a )t. Thereore, x(t) is a complex exponential unction at requency a. As in Example 2.2, its ourier transorm is X() = δ( a ). (b) Alternatively, we can also apply the ( scale-change ) theorem. rom x(t) = g(at), we know that X() = a G a. Plugging in G() = δ( ), we get ) X() = ( a δ a = a δ ( ) a ( a ). Now, recall, rom 2.6 that, actoring a constant α out o the δ-unction means scaling it by α. Here, the constant is α = a. Thereore, X() = a δ ( a ) = δ ( a ). a Exercise Similar to Example 2.33, one can also try to apply the scale-change theorem to show that x(t) = cos(2πa t) 2 (δ( a ) + δ( + a )). Example rom Example 2.3, we know that By the duality theorem, we have [ t a] 2a sinc (2πa) (22) 2a sinc(2πat) [ a], 9

17 which is the same as sinc(2π t) Both transorm pairs are illustrated in igure 6. 2 [ ]. (23) Duality Theorem igure 6: Duality theorem: rectangular and sinc unctions Example Let s try to derive the time-shit property rom the requencyshit property. We start with an arbitrary unction g(t). Next we will deine another unction x(t) by setting X() to be g(). Note that here is just a dummy variable; we can also write X(t) = g(t). Applying the duality theorem to the transorm pair x(t) X(), we get another transorm pair X(t) x( ). The LHS is g(t); thereore, the RHS must be G(). This implies G() = x( ). Next, recall the requency-shit property: The duality theorem then gives e j2πct x (t) X ( c). X (t c) ej2πc x ( ). 2

18 Replacing X(t) by g(t) and x( ) by G(), we inally get the time-shit property. Deinition The convolution o two signals, g (t) and g 2 (t), is a new unction o time, g(t). We write g = g g 2. It is deined as the integral o the product o the two unctions ater one is reversed and shited: g(t) = (g g 2 )(t) (24) = + g (µ)g 2 (t µ)dµ = + g (t µ)g 2 (µ)dµ. (25) Note that t is a parameter as ar as the integration is concerned. The integrand is ormed rom g and g 2 by three operations: (a) time reversal to obtain g 2 ( µ), (b) time shiting to obtain g 2 ( (µ t)) = g 2 (t µ), and (c) multiplication o g (µ) and g 2 (t µ) to orm the integrand. In some reerences, (24) is expressed as g(t) = g (t) g 2 (t). Example We can get a triangle rom convolution o two rectangular waves. In particular, [ t a] [ t a] = (2a t ) [ t 2a] Convolution properties involving the δ-unction: 2

19 2.4. Convolution theorems: (a) Convolution-in-time rule: g g 2 G G 2. (26) (b) Convolution-in-requency rule: g g 2 G G 2. (27) Example 2.4. We can use the convolution theorem to prove the requencyshit property in rom the convolution theorem, we have g 2 G G i g is band-limited to B, then g 2 is band-limited to 2B Parseval s theorem (Rayleigh s energy theorem, Plancherel ormula) or ourier transorm: + g(t) 2 dt = + G() 2 d. (28) The LHS o (28) is called the (total) energy o g(t). On the RHS, G() 2 is called the energy spectral density o g(t). By integrating the energy spectral density over all requency, we obtain the signal s total energy. The energy contained in the requency band B can be ound rom the integral B G() 2 d. More generally, ourier transorm preserves the inner product [2, Theorem 2.2]: g, g 2 = g (t)g 2(t)dt = G ()G 2()d = G, G 2. 22

20 Example Perorm the ollowing integration graphically with the help o ourier transorm properties: (a) sinc (t) dt. (b) sinc 2 (t) dt. 23

21 2.45. (Heisenberg) Uncertainty Principle [2, ]: Suppose g is a unction which satisies the normalizing condition g 2 2 = g(t) 2 dt = which automatically implies that G 2 2 = G() 2 d =. Then ( ) ( t 2 g(t) 2 dt ) 2 G() 2 d 6π2, (29) and equality holds i and only i g(t) = Ae Bt2 where B > and A 2 = 2B/π. In act, we have ( ) ( t 2 g(t t ) 2 dt or every t,. ) 2 G( ) 2 d 6π 2, The proo relies on Cauchy-Schwarz inequality. t or any unction h, deine its dispersion h as 2 h(t) 2 dt h(t) 2. Then, we can dt apply (29) to the unction g(t) = h(t)/ h 2 and get h H 6π A signal cannot be simultaneously time-limited and band-limited. Proo. Suppose g(t) is simultaneously () time-limited to T and (2) bandlimited to B. Pick any positive number T s and positive integer K such that s = T s > 2B and K > T T s. The sampled signal g Ts (t) is given by g Ts (t) = k g[k]δ (t kt s ) = K k= K g[k]δ (t kt s ) where g[k] = g (kt s ). Now, because we sample the signal aster than the Nyquist rate, we can reconstruct the signal g by producing g Ts h r where the LP h r is given by H r (ω) = T s [ω < 2π c ] with the restriction that B < c < T s B. In requency domain, we have G(ω) = K k= K g[k]e jkωt s H r (ω). 24

22 Consider ω inside the interval I = (2πB, 2π c ). Then, ω>2πb = G(ω) ω<2π 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 then the last term becomes a polynomial o the orm a 2K z 2K + a 2K z 2K + + a z + a. By undamental theorem o algebra, this polynomial has only initely many roots that is there are only initely many values o z = e jωt s which satisies (3). Because there are uncountably many values o ω in the interval I and hence uncountably many values o z = e jωt s which satisy (3), we have a contradiction The observation in 2.46 raises concerns about the signal and ilter models used in the study o communication systems. Since a signal cannot be both bandlimited and timelimited, we should either abandon bandlimited signals (and ideal ilters) or else accept signal models that exist or all time. On the one hand, we recognize that any real signal is timelimited, having starting and ending times. On the other hand, the concepts o bandlimited spectra and ideal ilters are too useul and appealing to be dismissed entirely. The resolution o our dilemma is really not so diicult, requiring but a small compromise. Although a strictly timelimited signal is not strictly bandlimited, its spectrum may be negligibly small above some upper requency limit B. Likewise, a strictly bandlimited signal may be negligibly small outside a certain time interval t t t 2. Thereore, we will oten assume that signals are essentially both bandlimited and timelimited or most practical purposes. 25