Guest Lectures for Dr. MacFarlane s EE3350 Part Deux
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1 Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A in engineering. 1. Dirac Dela δ() is he Dirac dela. I is even, has uni area, and has infinie ampliude. I doesn generally make sense ouside of an inegral. In paricular, he argumen of he dela should be linear in he variable of inegraion. Basic properies: 0 < 0 δ() = δ( ), δ() = 1, δ() = = 0 0 > 0 ake a recangle of widh A and heigh 1/A, cenered on he origin, as shown in Figure 1.1. In he limi as A 0, we ge our dela. 1/A A -A/2 A/2 Figure 1.1: Dirac Dela Progression 1
2 Our sifing propery diagram is shown in Figure 1.2. f() Figure 1.2: Sifing Propery We can ignore regions no immediaely over he dela: 0 0 f ()δ() d = f () 0 d f ()δ() d + f () 0 d Now, because f () is coninuous in he near viciniy of = 0, we can ake i as approximaely consan for a small region. hen: f ()δ() d = } {{ } f ()δ() d = f (0) δ() d = f (0) 1 Similarly, Also, as illusraed in Figure 1.3, f ()δ( ) d = f () δ(τ) dτ = u(), u(τ) dτ = u(), δ () d d δ() 2
3 u() u() ~δ() ~δ () Figure 1.3: Dela, is Derivaive, and is Firs wo Inegrals he Kronecker dela is: 1 n = 0 δn] = 0 n 0 2. LI and ransfer Funcions We like o use he dela o characerize linear sysems. ake x() as he inpu, and y() as he oupu, as shown in Figure 2.1. he impulse response h() will be inroduced shorly. Sysem/D.U.. δ() δn] h() hn] h() hn] x() xn] Sysem/D.U.. h() hn] y() yn] Figure 2.1: Sysem, Inpu, Oupu, Impulse Response 3
4 ime-invariance implies ha x( a) yields y( a) (i.e., boh have heir ime origin shifed by he same amoun). If his holds rue for any x and a, hen he sysem is ime-invarian. If x 1 () is inpu, and we obain an oupu of y 1 (), and dio for x 2 /y 2, hen lineariy implies ha ax 1 () + bx 2 () should yield ay 1 () + by 2 (). If his holds rue for any x, a, and b, hen he sysem is linear. Sysems ha are boh linear and ime-invarian ge he acronym LI, since we refer o hem so ofen. If x() = δ() is inpu ino a coninuous linear sysem, he oupu is h(), he impulse response. If xn] = δn] is inpu ino a discree linear sysem, he oupu is hn], he impulse response. If an arbiary x() is inpu ino he linear sysem, i is convolved wih he impulse response o obain he oupu. So, for he discree case, we have: yn] = xn] hn] = xk]hn k] = xn p]hp] (p = n k) and for he coninuous case, we have: y() = x() h() = k= x(τ)h( τ) dτ = p= x ( τ ) h ( τ ) dτ ( τ = τ ) In he above, i s no required o choose a differen dummy variable name when you do a change of variables, bu i ofen helps wih clariy and checking your work. I won be doing any discree examples, bu, if he coninuous example (inegral) looks difficul, you re in luck Convoluion Examples 3.1 Boxcar and Sine f() g() = sin ω 2π/ω 1/(b-a) =a =b Figure 3.1: Uni-area Boxcar As shown in Figures 3.1 and 3.2, Convolve f () and g(): f (τ) g( τ) dτ = = b f () = 1 u( a) u( b)], b a b g() = sin ω, ω > 0 b g( τ) a b a dτ = a 2 ω (b a) sin sin ω (b a) 2 sin ω ( τ) dτ = b a ω 4 ω (a + b) 2 Figure 3.2: Sine Wave > a 1 cos ω ( b) cos ω ( a)] ω (b a) ] ] ω (b a) ω (a + b) = sinc sin ω 2 2
5 Firs noe ha when b = a, he phase offse on he final sinusoid disappears: sinc bω sin ω. So he symmery of he boxcar abou he origin leads o a lack of a phase offse. his is a special case of (a + b) /2 being a muliple of 2π/ω, or, equivalenly, he cener of he boxcar being a muliple of he period of he sinusoid. his could be inerpreed as he sinusoid being shifed by he cener of he boxcar, bu, when ha s a muliple of 2π radians, i is unnoiceable. I is ineresing o noe wha happens in he limi as a and b become close: lim ( f g) = sin ω ( b) b a he sinusoid has jus been shifed o he righ by b unis of ime. his is precisely wha would have happened if f had insead been a Dirac dela a b: f () δ( b). 3.2 riangle and One-Sided Exponenial f() g()=u() e-/ 1/ 1/a -a a e-1/ (36.8%) Figure 3.3: Uni-Area riangle As shown in Figures 3.3 and 3.4, + 1 a 0 a 2 a f () = a a 2 a 0 oherwise Figure 3.4: Uni-Area One-Sided Exponenial, a > 0 Convolve f () and g(): g() = 1 e / u(), > 0 f (τ) g( τ) dτ = 1 a 2 = e / a 2 = e / a 2 0 = e / a 2 (a + τ) e ( τ)/ u( τ) dτ + 1 a 2 a min(0,) (a + τ) e τ/ dτ + e / min( a,) a 2 ] (a + τ ) e τ/ min(0,) + e / min( a,) a 2 a 0 min(a,) min(0,) (a τ) e ( τ)/ u( τ) dτ (a τ) e τ/ dτ (a τ + ) e τ/ ] min(a,) 2 + e a/ + e a/ a 2 + e a/ + (a + ) e / a 0 e a/ + (a + ) e / 0 a 0 a min(0,) 5
6 his resul has a maximum a = a + ln ( 2e a/ 1 ), wih in he inerval from 0 o a. In he limi as a 0, an inermediae resul is: 1 e / > a 1 e / > 0 undefined a 0 undefined = 0 0 a > 0 0 > Oher han a he origin, his is effecively equal o g(). In oher words, f () δ() as a 0. Noe ha he limi in he middle case is dependen on he relaive way a and approach zero, so i is undefined. For example, on he assumpion ha approaches 0 much faser, hen i gives 1 /2, bu if we insead approach along he line = a, he limi changes o 0. his is no surprising, since some definiions of he uni sep are undefined a he origin. If we insead ake he limi as approaches 0, we obain: 0 a (a ) a 2 a 0 (a + ) a 2 0 a his is clearly jus f (). Wha s ineresing here is ha g(), which is clearly no even (for sricly posiive, anyway), appears o have he same effec as he even funcion δ() as approaches 0. I is no clear if his happens for every f (). 3.3 Windowed Ramp and Windowed Cosine f()=u(a- ) -3 2a 3 g() = u(- ) 1 + cos π 2 ( ) 3 2a 2 1/ a a -3 2a 2 Figure 3.5: runcaed Ramp, Approximaion of Uni Double 2 Figure 3.6: runcaed Cosine As shown in Figures 3.5 and 3.6, f () = 3 u(a ) 2a3 a > 0 ( g() = 1 + cos π ) u( ), 2 > 0 6
7 Convolve f () and g(): f ( τ) g(τ) dτ = 3 4a 3 ( ( τ) u(a τ ) 1 + cos πτ ) dτ We already accouned for he sep in g() by our choice of inegraion limis. We can accoun for he sep remaining in he inegrand by also forcing + a > τ > a. Boh ses of limis can be easily enforced by using he mehod we found before. Bu we also need o ensure ha + a > and > a, boh of which follow from ransiiviy. If ha is no obvious, we can equivalenly sae ha his amouns o clamping he original inegraion limis of ± o he range beween a and + a, raher han merely clamping from one side, as before. So: f ( τ) g(τ) dτ = 3 4a 3 = 3 4a 3 max(min(,+a), a) τ max(min(,+a), a) ( τ 2 ( ( τ) ) + π 2 ( cos πτ 1 + cos πτ ) dτ + π ( τ) sin πτ )] max(min(,+a), a) max(min(,+a), a) While his could be evaluaed enirely in erms of sep funcions, he resuls would no be erribly clear. Insead we ll consider ha here are really jus six possibiliies: 1. a + a 2. a + a 3. a + a 4. a + a 5. a + a 6. a + a he resuls are hen: ( ) 3 sin π 2a 3 π 2 aπ cos aπ aπ sin a + a 3 2a 3 1 8a 3 π 2 1 8a 3 π 2 3a 2 π π 2 ( + ) ( cos π(a+) 3a 2 π π 2 ( ) ( cos π(a ) )] + aπ sin π(a+) )] a + a a + a + aπ sin π(a ) a + a 0 a + a 0 a + a Noe ha his is only really 5 differen regions: for any given values of a and, eiher case 1 can occur or case 2 can, bu no boh, deermined enirely by wheher a or is bigger. In he limi as approaches 0, we obain for each respecive case: undefined a < 0 f () undefined undefined < a a a 0 < a 0 > a For much he same reasons as in he previous secion, cases 3 and 4 are undefined. Case 1 oscillaes close o he limi. Bu neverheless we have ha g() approaches a Dirac dela once again. 7
8 If we insead ake he limi as a approaches 0, we obain: g () < undefined < a < 0 > In case 2, he limi is only defined along a paricular pah, and is infinie oherwise. Regardless of wheher he limi in cases 3 and 4 exiss or no, hose are jus individual poins, so, like before, hey do no maer much. In his case, we have f () δ (), he uni double. he acion is o ake he derivaive of he funcion i is convolved wih. If ime o wrie: here s a proof (I hink?) ha he limi is, in fac, 0 in case 3; case 4 should be similar. Because f () and g() are cenered a zero, and no oher (unnamed) consans are meaningful, we can, wihou loss of generaliy, ake = 1. (o jusify his in a differen way, scale and a by whaever quaniy you had o use o ge down o 1.) Now change variables such ha = + r π cos ( ) θ + π 2 4 and a = r π sin ( ) θ + π 2 4. Choosing θ 0, π 2], r > 0 will allow us o le ( +, a) approach he origin, wih + bounded by ±a, in line wih he consrains defining case 3. hen we have for he new resul in his inerval (up o a consan facor): csc ( ) 3 θ + π cos (r cos θ) + r 2 sin θ cos θ r sin (r cos θ) (cos θ + sin θ) ] 2r 3 2 Now i is no sufficien simply o ake he limi as r approaches zero, because θ mus be allowed o vary arbirarily as a funcion of r as ha limi is aken: if any θ(r) failed o reach he same limi, he limi would no exis. In oher words, ha naïve approach would always ake a radial line ino he origin, which is insufficienly general. Now we wish o es he idea ha he limi is zero. For small enough r, his quaniy is nonnegaive for all θ, so we need o find he θ ha maximizes i for every r. his happens o simply occur when θ = 0, independenly of r, provided r is small enough: his gives an upper bound. hen, for any maximum error from zero, we merely need o show we can find an r 0 such ha for all posiive r < r 0, his quaniy will be he arge error. Looking a θ = 0 as our upper bound, we obain: 2 2 cos r r sin r r 3 Bu his is a monoonically-increasing odd-symmeric funcion abou he origin for small r, so i is inverible and we can solve for such an r 0, and so he limi converges in case 3. 8
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