log dx a u = log a e du

Transcription

1 Formuls from Trigonometry: sin A cos A = cosa ± B = cos A cos B sin A sin B sin A = sin A cos A tn A = tn A tn A sina ± B = sin A cos B ± cos A sin B tn A±tn B tna ± B = tn A tn B cos A = cos A sin A sin A = ± cos A cos A = ± cos A tn A = sin A cos A sin A = cos A cos A = cos A sin A sin B = sin A B cos A B sin A sin B = cos A B sin A B cos A cos B = cos A B cos A B cos A cos B = sin A B sin B A sin A sin B = {cosa B cosa B} cos A cos B = {cosa B cosa B} sin A cos B = {sina B sina B} cosθ = sinθ π/ Differentition Formuls: v uv = u uv x x x y Chin rule: = y u x u x u cos u = sin u x x sin u =, π < x sin u < π x tn u =, π < x tn u < π x ln u = u x u u u u u x Integrtion Formuls: Integrtion by prts: u = ln u u u u = u ln, > 0, cos u u = sin u u v = uv v u sin u u = u sin u 4 = u sin u cos u cos u u = u sin u 4 = u sin u cos u u u = ln u u u u = lnu u e x sin bx x = ex sin bx b cos bx b x sin x x = sin x x = x sin x sin x 4 x cos x x cos x x = x cos x tn x x = tn x ln x x = x ln x x x x 3 sin x x x u v = vu/x uv/x v u sin u = cos u tn u = x x sec u u x x cos u = x eu = e u u x u u log x u = log e u, 0, u x e u u = e u sin u u = cos u tn u u = ln cos u tn u u = tn u u u u = tn u u u = sin u x, 0 < cos u < π u u = lnu u e x cos bx x = ex cos bxb sin bx b x sin x x = x sin x x cos x x = cos x x = x xe x x = ex x ln x x = x cos x sin x 4 x ln x x sin x x 3 cos x

2 Rules for Exponents: Tylor Series: Euler s Formul: bc = b c b c = c b c b b c = bc b = = b e x = x x! x3 3! x4 4! = cos x = x! x4 4! x6 6! = n=0 sin x = x3 3! x5 5! x7 7! = cos θ = ejθ e jθ e jθ = cos θ j sin θ Rectngulr n Polr Form of Complex Number: z = jb = re jθ n=0 n=0 x n n! n x n n! n x n n! sin θ = ejθ e jθ j Phsors: r = z = b r = zz θ = rctn b = Re{z} = r cos θ = Re{z} = z z b = Im{z} = r sin θ b = Im{z} = z z j Complex Signl: zt = Ae jω0tφ = Ae jφ e jω 0t Rel Signl: xt = Re{zt} = A cosω 0 t φ Phsor Representtion: X = Ae jφ Phsor Aition: Let x t = A cosw 0 t φ, x t = A cosw 0 t φ, n xt = x t x t. Then xt = A cosw 0 t φ n: the phsor representtion for xt is X = Ae jφ = A e jφ A e jφ. Continuous-Time Unit Impulse n Unit Step: δt t = xtδt t = x0 ut = t δt t xtδt t 0 t = xt 0

3 Discrete-Time Unit Impulse n Unit Step: δ[n] = {, n = 0, 0, otherwise. u[n] = {, n 0, 0, n < 0. Complex Exponentil Signls: e jω 0t e jω 0n Distinct signls for istinct w 0 Ienticl signls for vlues of w 0 seprte by multiples of π Perioic for ny choice of w 0 Perioic only if w 0 /π = m/n Q Funmentl frequency w 0 Funmentl frequency w 0 /m Funmentl perio: Funmentl perio: w 0 = 0: unefine w 0 = 0: one w 0 0: π/w 0 w 0 0: N = πm/w 0 Perioicity of Discrete-Time Sinusois: cosω 0 n, sinω 0 n, n e jω0n re perioic if n only if w 0 is rtio of two integers. π If perioic, then write in reuce form: w 0 π = m no common fctors between m n N N N: Funmentl Perio m: In ech perio, the grph ppers to go through m cycles. Summtion Formuls: N α k = αn αn α k=n k k = k=0, α k=0, < n n k=0 k=0 k k = { n n n n } Time Domin Representtion of Discrete-Time Signls: k =, < k = n, x[n] = x[ ]δ[n ] x[ ]δ[n ] x[0]δ[n] x[]δ[n ] x[]δ[n ] = x[k]δ[n k]. k= Systems: System H is liner if H{x [n] bx [n]} = H{x [n]} bh{x [n]}. System H is time invrint if H{x[n n 0 ]} = y[n n 0 ]. Impulse response: for LTI system H, h[n] = H{δ[n]}. System H is cusl if the current output oes not epen on future inputs. LTI system H is cusl iff h[n] = 0 n < 0. System H is stble if every boune input signl prouces boune output signl. 3

4 Convolution: y[n] = x[n] h[n] = x[k]h[n k] = x[n k]h[k] Convolution with δ[n]: x[n] δ[n] = x[n] x[n] δ[n n 0 ] = x[n n 0 ]. LTI System Interconnections: Series/Csce H h[n] = h [n] h [n] x[ n] H H y[ n] He jω = H e jω H e jω Hz = H zh z Prllel H H x [ n] Σ y[ n] H h[n] = h [n] h [n] He jω = H e jω H e jω Hz = H z H z Negtive Feeck H x [ n] Σ F y[ n] G h[n] : No generl expression He jω = Hz = F e jω F e jω Ge jω F z F zgz ROC of Hz: If H is cusl iscrete-time LTI system, then the ROC of Hz is exterior. A iscrete-time LTI system H is stble if n only if the ROC of Hz contins the unit-circle of the z-plne. This is ssuming tht Hz is rtio of two polynomils in z 4

5 Smpling: Ω: nlog frequency ω: igitl frequency ω = ΩT s = Ω π Ω s = Ω f s Ω = ω T s = ω Ω s π = ωf s x[n] = xnt s T s : smpling intervl Ω s = π T s : smpling frequency rins f s = T s : smpling frequency Hz Ω N = Ω s = π T s : Nyquist rte rins f N = f s = T s : Nyquist rte Hz If XΩ = F{xt} = 0 for Ω > Ω M, then we sy tht the signl xt is bnlimite to Ω M. To voi lising, you must smple xt with smpling frequency Ω s > Ω M. In other wors, you must smple t frequency tht is t lest twice the highest frequency in the signl. Another wy of sying this is: to voi lising, the highest frequency in the signl must be less thn the Nyquist frequency: Ω M < Ω N. When you smple xt to get x[n], the nlog frequencies ±Ω N mp to the igitl frequencies ±π. To convert nlog Herzin frequency to nlog rin frequency, multiply by π. To convert nlog rin frequency to nlog Herzin frequency, ivie by π. To convert nlog Herzin frequency to igitl Herzin frequency, multiply by T s. To convert nlog Herzin frequency to igitl rin frequency, multiply by πt s. To convert nlog rin frequency to igitl rin frequency, multiply by T s. To convert nlog rin frequency to igitl Herzin frequency, multiply by T s π. To convert igitl rin frequency to normlize igitl frequency, ivie by π. 5

6 Common Winow Functions for FIR filter esign: Rectngulr: w[n] =, M n M, Hnn: w[n] = Hmming: Blckmn: [ cos πn M w[n] = cos w[n] = cos ], M n M, πn M πn 0.08 cos M, M n M, πn, M n M. M Min Properties of the Winow Functions: Type of Min Lobe Reltive Sielobe Minimum Stopbn Trnsition Winow With ML Level A sl Attenution Bnwith ω Rectngulr 4π/M 3.3 B 0.9 B 0.9π/M Hnn 8π/M 3.5 B 43.9 B 3.π/M Hmming 8π/M 4.7 B 54.5 B 3.3π/M Blckmn π/m 58. B 75.3 B 5.56π/M Design Steps:. Convert the minimum stopbn ttenution spec δ s to B using the formul α s = 0 log 0 δ s.. Look in column 4 Minimum Stopbn Attenution of the tble bove to etermine which winow functions cn provie t lest α s B of stopbn ttenution. 3. Let ω = ω s ω p. Use the lst column of the tble to figure out which winow function w[n] cn meet the stopbn spec with the smllest vlue M. - To o this, set ω equl to the formul in the lst column of the tble n solve for M. - The orer of your filter will be M. - The length of h[n] will be M. 4. Let ω c = ω p ω s. 5. Let h LP [n] = sin ω cn. πn 6. For the winow function w[n] tht meets the stopbn spec with the smllest M, compute the centere impulse response w[n]h LP [n], M n M. 7. Shift it right by M to mke it cusl: h[n] = w[n M]h LP [n M], 0 n M. 6

7 Bsic Discrete-Time Fourier Trnsform Pirs: k=n e jω 0n cos ω 0 n sin ω 0 n x[n] = Signl k e jkπ/nn π π π π j Fourier trnsform k δ ω πk N l= l= π l= l= δω ω 0 πl {δω ω 0 πl δω ω 0 πl} {δω ω 0 πl δω ω 0 πl} δω πl Perioic squre wve {, n < N x[n] = 0, N < n N/ n x[n N] = x[n] δ[n kn] n u[n], < x[n] = π π N e jω {, n N sin[ωn ] 0, n > N sinω/ sin W n πn, 0 < W < π Xejω = k δ ω πk N δ ω πk N {, 0 ω W 0, W < ω π Xe jω is π-perioic δ[n] u[n] δ[n n 0 ] e jωn 0 e jω πδω πk n n u[n], < nr! n!r! n u[n], < e jω e jω r 7

8 Properties of the Discrete-Time Fourier Trnsform: Xe jω = n= x[n]e jωn x[n] = Xe jω e jωn ω π π Property Aperioic signl Fourier trnsform Linerity x[n] by[n] Xe jω by e jω Time Shifting x[n n 0 ] e jωn 0 Xe jω Frequency Shifting e jω0n x[n] Xe jω ω0 Conjugtion x [n] X e jω Time Reversl x[ n] { Xe jω x[n/k], if n = multiple of k Time Expnsion x k [n] = Xe 0, if n multiple of k Convolution x[n] y[n] Xe jω Y e jω Multipliction x[n]y[n] Xe jθ Y e jω θ θ π Differencing in Time x[n] x[n ] e jω Xe jω n Accumultion x[k] e jω Xejω Differentition in Frequency Conjugte Symmetry for Rel Signls nx[n] x[n] rel Symmetry for Rel n x[n] rel n even Even Signls Symmetry for Rel n x[n] rel n o O Signls Even-O Decomposi- x e [n] = Ev{x[n]} [x[n] rel] Re{Xe jω } tion for Rel Signls x o [n] = O{x[n]} [x[n] rel] jim{xe jω } π j ω Xejω Xe jω = X e jω Re{Xe jω } = Re{Xe jω } Im{Xe jω } = Im{Xe jω } Xe jω = Xe jω Xe jω = Xe jω Xe jω rel n even Xe jω purely imginry n o Prsevl s Reltion for Aperioic Signls x[n] = Xe jω ω π n= π 8

9 Common z-trnsform Pirs: Signl Trnsform ROC δ[n] All z u[n] u[ n ] z z > z z < δ[n m] z m All z, except 0 if m > 0 or if m < 0 α n u[n] αz z > α α n u[ n ] αz z < α nα n u[n] nα n u[ n ] [cos ω 0 n]u[n] [sin ω 0 n]u[n] [r n cos ω 0 n]u[n] [r n sin ω 0 n]u[n] αz αz z > α αz αz z < α [cos ω 0 ]z [ cos ω 0 ]z z z > [sin ω 0 ]z [ cos ω 0 ]z z z > [r cos ω 0 ]z [r cos ω 0 ]z r z z > r [r sin ω 0 ]z [r cos ω 0 ]z r z z > r 9

10 Properties of the z-trnsform: Xz = n= x[n]z n x[n] = Xzz n z πj C Property Signl z-trnsform ROC x[n] Xz R x [n] X z R x [n] X z R Linerity x [n] bx [n] X z bx z At lest R R Time Shifting x[n n 0 ] z n 0 Xz R, except possibly z = 0 z-domin Scling e jω0n x[n] Xe jω 0 z R z0 n z x[n] X z 0 z 0 R n x[n] X z R Time Reversl x[ n] { Xz R x[r], n = rk Time Expnsion x k [n] = 0, n rk r Z Xzk R k Conjugtion x [n] X z R Convolution x [n] x [n] X zx z At lest R R First Difference x[n] x[n ] z Xz At lest R { z > 0} Accumultion z-domin Differentition n x[k] Xz At lest R { z > } z nx[n] z z Xz R 0