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
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 umber: 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 = xt)δt) t = x0) ut) = t δt) t xt)δt t 0 ) t = xt 0 )
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/ 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: = 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 ) : Funmentl Perio m: In ech perio, the grph ppers to go through m cycles. Summtion Formuls: α k = α α α k= 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 is liner if {x [n] bx [n]} = {x [n]} b{x [n]}. System is time invrint if {x[n n 0 ]} = y[n n 0 ]. Impulse response: for LTI system, h[n] = {δ[n]}. System is cusl if the current output oes not epen on future inputs. LTI system is cusl iff h[n] = 0 n < 0. System is stble if every boune input signl prouces boune output signl. 3
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[n] = h [n] h [n] x[ n] y[ n] e jω ) = e jω ) e jω ) z) = z) z) Prllel x [ n] Σ y[ n] h[n] = h [n] h [n] e jω ) = e jω ) e jω ) z) = z) z) egtive Feeck x [ n] Σ F y[ n] G h[n] : o generl expression e jω ) = z) = F e jω ) F e jω )Ge jω ) F z) F z)gz) ROC of z ): If is cusl iscrete-time LTI system, then the ROC of z) is exterior. A iscrete-time LTI system is stble if n only if the ROC of z) contins the unit-circle of the z-plne. This is ssuming tht z) is rtio of two polynomils in z ) 4
Bsic Discrete-Time Fourier Trnsform Pirs: k=) e jω 0n cos ω 0 n sin ω 0 n x[n] = Signl k e jkπ/)n π π π π j Fourier trnsform k δ ω πk ) 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 < x[n] = 0, < n / n x[n ] = x[n] δ[n k] n u[n], < x[n] = π π e jω {, n sin[ω )] 0, n > sinω/) sin W n πn, 0 < W < π Xejω ) = k δ ω πk ) δ ω πk ) {, 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 5
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 6
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 7
Properties of the z-trnsform: Xz) = n= x[n]z n x[n] = Xz)z 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 z)x 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 8