Tables in Signals and Systems

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1 ables in Signals and Systems Magnus Lundberg Revised October 999 Contents I Continuous-time Fourier series I-A Properties of Fourier series I-B Fourier series table II Continuous-time Fourier transform 4 II-A Properties of the Fourier transform II-B Fourier transform table III Discrete-time Fourier series 7 III-AProperties of discrete-time Fourier series III-BFourier series table IV Discrete-time Fourier transform 9 IV-AProperties of the discrete-time Fourier transform IV-B Discrete-time Fourier transform table V Sampling and reconstruction VI Z-transform VI-AProperties of the Z-transform VI-B Z-transform table he major part of this collection of tables was originally developed at the Div. of Signal Processing, Luleå University of echnology. It has been revised by Magnus Lundberg in October 999

2 ABLES I SIGALS AD SYSEMS, OC. 999 Definitions sinc t) = sin t) t Ω o = 0 I. Continuous-time Fourier series A. Properties of Fourier series Periodic signal Fourier serie coefficient x t) = a k e jkωot a k = x t) e jkωot dt o o xt) yt) } Periodic with period 0 Ax t) By t) a k b k Aa k Bb k x t t 0 ) a k e jk/ 0)t 0 e jm/ 0)t x t) x t) x t) a k M a k a k x αt), α > 0 Periodic with period 0/α) x τ) y t τ) dτ 0 x t) y t) d dt x t) t x t) x τ) dτ Bounded and periodic only if a 0 = 0) a k 0 a k b k a l b k l l= jk a k 0 jk / 0 ) a k If x t) is real valued then a k = a k R {a k } = R {a k } I {a k } = I {a k } a k = a k arg {a k } = arg {a k } x e t) = E{xt)} R{a k } x o t) = O{xt)} ji{a k } a k e jkω 0t a k e jkω 0t = R{a k } coskω 0 t) I{a k } sinkω 0 t) Parsevals relation for periodic signals xt) dt = a k 0 0

3 ABLES I SIGALS AD SYSEMS, OC B. Fourier series table xt) a k or the Fourier series expansion a) δt n ) a k =, all k b) a 0 =, a k = 0 otherwise), 0 > 0 c) e jωot a =, a k = 0 otherwise d) cos Ω o t a = a =, a k = 0 otherwise e) sin Ω o t a = a = j, a k = 0 otherwise f) {, t < 0, < t o a k = period 0 g) {, 0 < t <, < t < 0 h) { t, 0 < t < t = t, < t < 0 Ω o sinc kωo = sin kω o k 4 sin t sin 3t sin 5t 3 5 ) 4 cos t cos 3t cos 5t ) 3 5 i) t, < t < sin t sin t sin 3t 3 ) j) t, 0 < t < sin t k) sin t, < t < { 0, 0 < t < a l), 0, a < t < a a < t < 4 cos t 3 sin t sin 3t 3 ) cos 4t 3 5 cos 6t 5 7 ) a sin a cos t sin a cos t sin 3a cos 3t 3 )

4 ABLES I SIGALS AD SYSEMS, OC A. Properties of the Fourier transform xt) = xt) = xt) yt) axt) byt) II. Continuous-time Fourier transform on-periodic signal XjΩ)e jωt dω XjΩ) = Fourier transform xt)e jωt dt Alternativly with frequency f instead of angular frequency Ω. X f f)e jft df X f f) = XjΩ) Y jω) axjω) by jω) xt)e jft dt = Xω) {Ω=f} xt t 0 ) e jωt 0 XjΩ) e jω 0t xt) XjΩ Ω 0 )) x t) x t) xat) xt) yt) xt)yt) d dt xt) t xt)dt txt) xt) x e t) = E{xt)} x o t) = O{xt)} fu) = X j Ω)) Xj Ω)) ) Ω a X a XjΩ)Y jω) XjΩ) Y jω) jωxjω) XjΩ) X0)δΩ) jω j d dω XjΩ) If xt) is real valued then XjΩ) = X j Ω)) R{XjΩ)} = R{Xj Ω))} I{XjΩ)} = I{Xj Ω))} XjΩ) = Xj Ω)) arg{xjω)} = arg{xj Ω))} R{XjΩ)} ji{xjω)} Duality gv)e juv F gt) fjω) dv, F ft) gj Ω)) Parsevals relation for non-periodic signals xt) dt = XjΩ) dω

5 ABLES I SIGALS AD SYSEMS, OC B. Fourier transform table he table is valid for R{α} > 0 and R{β} > 0 a) u t ) ) u t b) c) { sin W t t xt) XjΩ) Xf) = W sincw t t, t < d) e αt ut) sin Ω/ Ω/ sin f f = sincf ) u Ω W ) u Ω W ) u f W ) ) u f W [ sin Ω/4 Ω/4 jω α e) e α t α Ω α f) [ e αt e βt] ut) β α g) te αt ut) h) t n n )! e αt ut) jω α)jω β) ] sinc f/) jf α α f) α jf α)jf β) jω α) jf α) jω α) n jf α) n i) e αt) j) e αt sinω o t)ut) α e Ω/α) Ω o jω α) Ω o α e f/α) Ω o jf α) Ω o e αt sinω o t)u t) k) e αt cosω o t)ut) e αt cosω o t)u t) l) cos Ω o t) [ u t ) )] u t Ω o α jω) Ω o α jω jω α) Ω o α jω α jω) Ω o [ sinω Ωo ) Ω Ω o ) sinω Ω o) ) ] Ω Ω o ) ) Ω o α jf) Ω o α jf jf α) Ω o α jf α jf) Ω o [ sin f fo ) f f o ) sin f f ] o) f f o )

6 ABLES I SIGALS AD SYSEMS, OC Generalized Fourier transform power signals) xt) XjΩ) Xf) a) δt) b) δt t 0 ) e jωt 0 e jft 0 c) δt n ) d) ut) δω) jω e) sgnt) = t t f) t δω n) jω jsgnω) δf n ) δf) jf jf jsignf) g) K KδΩ) Kδf) h) tut) jδ Ω) j Ω 4 δ f) 4 f ) j n i) t n j) n δ n) Ω) δ n) f) j) cos Ω o t [δω Ω o ) δω Ω o )] k) sin Ω o t l) c n e j nt j [δω Ω o) δω Ω o )] c n δ Ω n ) [δf f o) δf f o )] j [δf f o) δf f o )] c n δ f n m) e jωot δω Ω o ) δf f o ) ) n) Periodic square wave {, t 0, < t o period o A k Ω)δ Ω kω o ) A k Ω) = sin kω o k A k f)δ f kf o ) A k f) = sin kf o k

7 ABLES I SIGALS AD SYSEMS, OC A. Properties of discrete-time Fourier series x[n] = k=<> III. Discrete-time Fourier series Periodic signal a k e jk/)n a k = Fourier serie coefficient n=<> x[n]e jk/)n x[n] y[n] } Periodic with period a k b k } Periodic with period Ax[n] By[n] Aa k Bb k x[n n 0 ] a k e jk/)n 0 e jm/)n x[n] x [n] x[ n] x m) [n] = r=<> x[n]y[n] { x[n/m], If n is a multiple av m x[r]y[n r] x[n] x[n ] a k M a k a k m a k, a k b k period m a l b k l l=<> e j/k ) a k n x[n] x[k] Bounded and periodic only if a 0 = 0 e jk/ a k If x[n] is real valued then a k = a k R{a k } = R{a k } I{a k } = I{a k ]} a k = a k arg{a k } = arg{a k } x e [n] = E{x[n]} R{a k } x o [n] = O{x[n]} ji{a k } Parsevals relation for periodic signals x[n] = a k n=<> k=<>

8 ABLES I SIGALS AD SYSEMS, OC B. Fourier series table x[n] δn k) a k a k =,for all k a k = {, k=0,±,±,... e jωon ω o = m {, k=m,m±,m±,... a k = ω o = irrational : he signal is non-periodic ω o = m cos ω o n a k = {, k=±m,±m±,±m±,... sin ω o n ω o = irrational : he signal is non-periodic ω o = m a k = j, j, k=m,m±,m±,... k= m, m±, m±,... ω o = irrational : he signalen is non-periodic {, n 0, < n period a k = sin k ) sin k, k 0,±,±,..., k=0,±,±,...

9 ABLES I SIGALS AD SYSEMS, OC IV. Discrete-time Fourier transform A. Properties of the discrete-time Fourier transform on-periodic signal x[n] = Xe jω )e jωn dω Xe jω ) = } x[n] y[n] Xe jω ) Y e jω ) Fourier transform x[n]e jωn } Periodic with period ax[n] by[n] axe jω ) by e jω ) x[n n 0 ] e jωn 0 Xe jω ) e jω 0n x[n] Xe jω ω 0) ) x [n] X e j ω) ) x[ n] Xe j ω) ) { x m) [n] = x[n/m], n multiple of m 0, n not multiple av m Xejmω) ) x[n] y[n] Xe jω )Y e jω ) x[n]y[n] Xe jθ )Y e jω θ) )dθ x[n] x[n ] e jω ) Xe jω ) n x[k] nx[n] x[n] e j ω Xejω ) X0) j d dω Xejω ) If x[n] is real valued then Xe jω ) = X e j ω) ) R{Xe jω )} = R{Xe j ω) )} I{Xe jω )} = I{Xe j ω) )} Xe jω ) = Xe j ω) ) δω k) arg{xe jω )} = arg{xe j ω) )} x e [n] = E{x[n]} R{Xe jω )} x o [n] = O{x[n]} ji{xe jω )} Parsevals relation for non-periodic signals x[n] = Xe jω ) dω

10 ABLES I SIGALS AD SYSEMS, OC B. Discrete-time Fourier transform table x[n] Xe jω ) δ[n] δ[n n 0 ] e jωn 0 δn k) e jωon cos ω o n sin ω o n u[n] a n un), a < n )a n u[n], a < n m )! a n u[n], a < n!m )! a a n, a < {, n 0, < n period {, n j δ ω k ) δ ω k) δ ω ω o k) [δω ω o k) δω ω o k)] [δω ω o k) δω ω o k)] e jω ae jω ae jω ) ae jω ) m a acosω δω k) a k δ ω k ) sin ω ) 0, n > sin ω { sin W n = W n sincw n 0 < W < {, ω W 0, W < ω period

11 ABLES I SIGALS AD SYSEMS, OC. 999 V. Sampling and reconstruction he sampling theorem: Let xt) with transform X c jω) be a bandlimited signal such that X c jω) = 0, Ω > Ω M. hen xt) is uniquely described by the samples xn ), n = 0, ± ±... if Ω s > Ω M where Ω s = = f s Given xn ), if the sampling theorem is satisfied, it is possible with an ideal reconstruction filter to exactly reconstruct xt). Discrete-time processing of continuous-time signals Sampling: x d t) = xn )δt n ) X d jω) = xn )e jωn ormalization in time gives x[n] = xn ) Xe jω ) = x[n]e jωn = xn )e jωn where Ω = ω = Ω f s or f = q = f f s Poissons summation formula: X d jω) = X c jω kω s )) Xe jω ) = X c j Ω k )) If the sampling theorem is satisfied then X d jω) = XjΩ), < Ω < or Ideal reconstruction: X d f) = Xf), < f < x r t) = xn )ht n ) where ht) = Ω c sincω ct HjΩ) = {, ω Ωc

12 ABLES I SIGALS AD SYSEMS, OC. 999 A. Properties of the Z-transform VI. Z-transform signal Z-transform ROC x[n] Xz) = x[n]z n R x ax[n] by[n] axz) by z) Contains R x R y x[n n 0 ] z n 0 Xz) R x, except possible addition or deletion of the origin or z0 nx[n] Xz/z 0) z 0 R x x [n] X z ) R x x [ n] X /z ) /R x x[n] y[n] Xz)Y z) Contains R x R y nx[n] z d dz Xz) R x, except possible addition or deletion of the origin or R{x[n]} [Xz) X z )] Contains R x I{x[n]} j [Xz) X z )] Contains R x Initial value theorem x[n] = 0, n < 0 lim z Xz) = x[0]

13 ABLES I SIGALS AD SYSEMS, OC B. Z-transform table x[n] Xz) ROC δ[n] All z δ[n n 0 ] z n 0 All z, except 0n 0 > 0) or n 0 < 0) u[n] u[ n ] a n u[n] a n u[ n ] na n u[n] na 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] { a n, 0 n z z > z z < az z > a az z < a az az ) z > a az az ) z < a [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 a z az z > 0