transform_pairs

Tables of Common Transform Pairs 2012 by Marc Ph. Stoecklin | marc a stoecklin.net | http://www.stoecklin.net/ | 2012-12-20 | version v1.5.3 Engineers and students in communications and mathematics are confronted with transformations such as the z-Transform, the Fourier transform, or the Laplace transform. Often it is quite hard to quickly nd the appropriate transform in a book or the Internet, much less to have a comprehensive overview of transformation pairs and corresponding properties. In this document I compiled a handy collection of the most common transform pairs and properties of the .continuous-time frequency Fourier transform (2f), .continuous-time pulsation Fourier transform (!), .z-Transform , .discrete-time Fourier transform DTFT , and .Laplace transform . Please note that, before including a transformation pair in the table, I veried its correctness. Nev- ertheless, it is still possible that you may nd errors or typos. I am very grateful to everyone dropping me a line and pointing out any concerns or typos. Notation, Conventions, and Useful Formulas Imaginary unit j2=1 Complex conjugate z=a+jb !z=ajb Real part <eff(t)g=1 2[f(t) +f(t)] Imaginary part =mff(t)g=1 2j[f(t)f(t)] Dirac delta/Unit impulse [n] =( 1; n = 0 0; n6= 0 Heaviside step/Unit step u[n] =( 1; n0 0; n< 0 Sine/Cosine sin ( x) =ejxejx 2jcos (x) =ejx+ejx 2 Sinc function sinc ( x)sin(x) x(unnormalized) Rectangular function rect(t T) =( 1 ifjtj6T 2 0 ifjtj>T 2 Triangular function triangt T
= rect(t T)rect(t T) =8 >< >:1jtj Tjtj6T 0jtj>T Convolution continuous-time: ( fg)(t) =R+1 1f()g(t)d discrete-time: ( uv)[n] =P1 m=1u[m]v[nm] Parseval theorem general statement:R+1 1f(t)g(t)dt=R+1 1F(f)G(f)df continuous-time:R+1 1jf(t)j2dt=R+1 1jF(f)j2df discrete-time:P+1 n=1jx[n]j2=1 2R+ jX(ej!)j2d! Geometric seriesP1 k=0xk=1 1xPn k=0xk=1xn+1 1x in general:Pn k=mxk=xmxn+1 1x 1 Marc Ph. Stoecklin |TABLES OF TRANSFORM PAIRS | v1.5.3 2 Table of Continuous-time Frequency Fourier Transform Pairs f(t) =F1fF(f)g=R+1 1f(t)ej2ftdfF(= =)F(f) =Fff(t)g=R+1 1f(t)ej2ftdt transform f(t)F(= =)F(f) time reversal f(t)F(= =)F(f) frequency reversal complex conjugation f(t)F(= =)F(f) reversed conjugation reversed conjugation f(t)F(= =)F(f) complex conjugation f(t) is purely realF(= =)F(f) =F(f) even/symmetry f(t) is purely imaginaryF(= =)F(f) =F(f) odd/antisymmetry even/symmetry f(t) =f(t)F(= =)F(f) is purely real odd/antisymmetry f(t) =f(t)F(= =)F(f) is purely imaginary time shifting f(tt0)F(= =)F(f)ej2ft0 f(t)ej2f0tF(= =)F(ff0) frequency shifting time scaling f(af)F(= =)1 jajF f a 1 jajf f aF(= =)F(af) frequency scaling linearity af(t) +bg(t)F(= =)aF(f) +bG(t) time multiplication f(t)g(t)F(= =)F(f)G(f) frequency convolution frequency convolution f(t)g(t)F(= =)F(f)G(f) frequency multiplication delta function (t)F(= =) 1 shifted delta function (tt0)F(= =)ej2ft0 1F(= =)(f) delta function ej2f0tF(= =)(ff0) shifted delta function two-sided exponential decay eajtja>0F(= =)2a a2+42f2 et2F(= =)ef2 ejt2F(= =)ej(1 4f2) sine sin (2 f0t+)F(= =)j 2 ej(f+f0)ej(ff0) cosine cos (2 f0t+)F(= =)1 2 ej(f+f0) +ej(ff0) sine modulation f(t) sin (2f0t)F(= =)j 2[F(f+f0)F(ff0)] cosine modulation f(t) cos (2f0t)F(= =)1 2[F(f+f0) +F(ff0)] squared sine sin2(t)F(= =)1 4 2(f) f1 
 f+1 
 squared cosine cos2(t)F(= =)1 4 2(f) + f1 
+ f+1 
 rectangular rectt T
=8 < :1jtj6T 2 0jtj>T 2F(= =)TsincTf triangular triangt T
=8 < :1jtj Tjtj6T 0jtj>TF(= =)Tsinc2Tf step u(t) = 1 [0;+1](t) =8 < :1t>0 0t<0F(= =)1 j2f+(f) signum sgn ( t) =8 < :1t>0 1t<0F(= =)1 jf sinc sinc ( Bt)F(= =)1 Brect f B =1 B1[B 2;+B 2](f) squared sinc sinc2(Bt)F(= =)1 Btriang f B n-th time derivativedn dtnf(t)F(= =) (j2f)nF(f) n-th frequency derivative tnf(t)F(= =)1 (j2)ndn dfnF(f) 1 1+t2F(= =)e2jfj Marc Ph. Stoecklin |TABLES OF TRANSFORM PAIRS | v1.5.3 3 Table of Continuous-time Pulsation Fourier Transform Pairs x(t) =F1 !fX(!)g=R+1 1x(t)ej!td!F!(= =)X(!) =F!fx(t)g=R+1 1x(t)ej!tdt transform x(t)F!(= =)X(!) time reversal x(t)F!(= =)X(!) frequency reversal complex conjugation x(t)F!(= =)X(!) reversed conjugation reversed conjugation x(t)F!(= =)X(!) complex conjugation x(t) is purely realF!(= =)X(f) =X(!) even/symmetry x(t) is purely imaginaryF!(= =)X(f) =X(!) odd/antisymmetry even/symmetry x(t) =x(t)F!(= =)X(!) is purely real odd/antisymmetry x(t) =x(t)F!(= =)X(!) is purely imaginary time shifting x(tt0)F!(= =)X(!)ej!t0 x(t)ej!0tF!(= =)X(!!0) frequency shifting time scaling x(af)F!(= =)1 jajX! a
1 jajx f aF!(= =)X(a!) frequency scaling linearity ax1(t) +bx2(t)F!(= =)aX1(!) +bX2(!) time multiplication x1(t)x2(t)F!(= =)1 2X1(!)X2(!) frequency convolution frequency convolution x1(t)x2(t)F!(= =)X1(!)X2(!) frequency multiplication delta function (t)F!(= =) 1 shifted delta function (tt0)F!(= =)ej!t0 1F!(= =) 2(!) delta function ej!0tF!(= =) 2(!!0) shifted delta function two-sided exponential decay eajtja>0F!(= =)2a a2+!2 exponential decay eatu(t)<fag>0F!(= =)1 a+j! reversed exponential decay eatu(t)<fag>0F!(= =)1 aj! et2 22F!(= =)p 2e2!2 2 sine sin ( !0t+)F!(= =)j ej(!+!0)ej(!!0) cosine cos ( !0t+)F!(= =) ej(!+!0) +ej(!!0) sine modulation x(t) sin (!0t)F!(= =)j 2[X(!+!0)X(!!0)] cosine modulation x(t) cos (!0t)F!(= =)1 2[X(!+!0) +X(!!0)] squared sine sin2(!0t)F!(= =)2[2(f)(!!0)(!+!0)] squared cosine cos2(!0t)F!(= =)2[2(!) +(!!0) +(!+!0)] rectangular rectt T
=8 < :1jtj6T 2 0jtj>T 2F!(= =)Tsinc !T 2 triangular triangt T
=8 < :1jtj Tjtj6T 0jtj>TF!(= =)Tsinc2 !T 2 step u(t) = 1 [0;+1](t) =8 < :1t>0 0t<0F!(= =)(f) +1 j! signum sgn ( t) =8 < :1t>0 1t<0F!(= =)2 j! sinc sinc ( Tt)F!(= =)1 Trect! 2T
=1 T1[T;+T](f) squared sinc sinc2(Tt)F!(= =)1 Ttriang! 2T
n-th time derivativedn dtnf(t)F!(= =) (j!)nX(!) n-th frequency derivative tnf(t)F!(= =)jndn dfnX(!) time inverse1 tF!(= =) jsgn(!) Marc Ph. Stoecklin |TABLES OF TRANSFORM PAIRS | v1.5.3 4 Table of z-Transform Pairs x[n] =Z1fX(z)g=1 2jH X(z)zn1dzZ(= =)X(z) =Zfx[n]g=P+1 n=1x[n]znROC transform x[n]Z(= =)X(z) Rx time reversal x[n]Z(= =)X(1 z)1 Rx complex conjugation x[n]Z(= =)X(z) Rx reversed conjugation x[n]Z(= =)X(1 z)1 Rx real part <efx[n]gZ(= =)1 2[X(z) +X(z)] Rx imaginary part =mfx[n]gZ(= =)1 2j[X(z)X(z)] Rx time shifting x[nn0]Z(= =)zn0X(z) Rx scaling inZ anx[n]Z(= =)Xz a
jajRx downsampling by N x[Nn];N2N0Z(= =)1 NPN1 k=0X Wk Nz1 N WN=ej2! NRx linearity ax1[n] +bx2[n]Z(= =)aX1(z) +bX2(z) Rx\Ry time multiplication x1[n]x2[n]Z(= =)1 2jH X1(u)X2z u
u1du R x\Ry frequency convolution x1[n]x2[n]Z(= =)X1(z)X2(t) Rx\Ry delta function [n]Z(= =) 1 8z shifted delta function [nn0]Z(= =)zn0 8z step u[n]Z(= =)z z1jzj>1 u[n1]Z(= =)z z1jzj<1 ramp nu[n]Z(= =)z (z1)2 jzj>1 n2u[n]Z(= =)z(z+1) (z1)3 jzj>1 n2u[n1]Z(= =)z(z+1) (z1)3 jzj<1 n3u[n]Z(= =)z(z2+4z+1) (z1)4 jzj>1 n3u[n1]Z(= =)z(z2+4z+1) (z1)4 jzj<1 (1)nZ(= =)z z+1jzj<1 exponential anu[n]Z(= =)z zajzj>jaj anu[n1]Z(= =)z zajzj<jaj an1u[n1]Z(= =)1 zajzj>jaj nanu[n]Z(= =)az (za)2 jzj>jaj n2anu[n]Z(= =)az(z+a (za)3 jzj>jaj eanu[n]Z(= =)z zea jzj>jeaj exp. interval( ann= 0;:::;N1 0 otherwiseZ(= =)1aNzN 1az1 jzj>0 sine sin ( !0n)u[n]Z(= =)zsin(!0) z22 cos(!0)z+1jzj>1 cosine cos ( !0n)u[n]Z(= =)z(zcos(!0)) z22 cos(!0)z+1jzj>1 ansin (!0n)u[n]Z(= =)zasin(!0) z22acos(!0)z+a2 jzj>a ancos (!0n)u[n]Z(= =)z(zacos(!0)) z22acos(!0)z+a2 jzj>a dierentiation in Z nx[n]Z(= =) zdX(z) dzRx integration inZx[n] nZ(= =) Rz 0X(z) zdz R xQm i=1(ni+1) amm!amu[n]Z(= =)z (za)m+1 Note: z z1=1 1z1 Marc Ph. Stoecklin |TABLES OF TRANSFORM PAIRS | v1.5.3 5 Table of Common Discrete Time Fourier Transform (DTFT) Pairs x[n] =1 2R+ X(ej!)ej!nd!DTFT(= =)X(ej!) =P+1 n=1x[n]ej!n transform x[n]DTFT(= =)X(ej!) time reversal x[n]DTFT(= =)X(ej!) complex conjugation x[n]DTFT(= =)X(ej!) reversed conjugation x[n]DTFT(= =)X(ej!) x[n] is purely realDTFT(= =)X(ej!) =X(ej!) even/symmetry x[n] is purely imaginaryDTFT(= =)X(ej!) =X(ej!) odd/antisymmetry even/symmetry x[n] =x[n]DTFT(= =)X(ej!) is purely real odd/antisymmetry x[n] =x[n]DTFT(= =)X(ej!) is purely imaginary time shifting x[nn0]DTFT(= =)X(ej!)ej!n 0 x[n]ej!0nDTFT(= =)X(ej(!!0)) frequency shifting downsampling by N x[Nn]N2N0DTFT(= =)1 NPN1 k=0X(ej!2k N) upsampling by N8 < :xn N n=kN 0otherwiseDTFT(= =)X(ejN!) linearity ax1[n] +bx2[n]DTFT(= =)aX1(ej!) +bX2(ej!) time multiplication x1[n]x2[n]DTFT(= =)X1(ej!)X2(ej!) = frequency convolution 1 2R+ X1(ej(!))X2(ej)d frequency convolution x1[n]x2[n]DTFT(= =)X1(ej!)X2(ej!) frequency multiplication delta function [n]DTFT(= =) 1 shifted delta function [nn0]DTFT(= =)ej!n 0 1DTFT(= =) ~(!) delta function ej!0nDTFT(= =) ~(!!0) shifted delta function sine sin ( !0n+)DTFT(= =)j 2[ej~(!+!0+ 2k)e+j~(!!0+ 2k)] cosine cos ( !0n+)DTFT(= =)1 2[ej~(!+!0+ 2k) +e+j~(!!0+ 2k)] rectangular rectn M
=8 < :1jnj6M 0 otherwiseDTFT(= =)sin[!(M+1 2)] sin(!=2) step u[n]DTFT(= =)1 1ej!+1 2~(!) decaying step anu[n] (jaj<1)DTFT(= =)1 1aej! special decaying step ( n+ 1)anu[n] (jaj<1)DTFT(= =)1 (1aej!)2 sincsin(!cn) n=!c sinc (!cn)DTFT(= =) ~rect ! !c =8 < :1j!j<!c 0!c<j!j< MA rectn M1 2
=8 < :1 06n6M 0 otherwiseDTFT(= =)sin[!(M+1)=2] sin(!=2)ej!M= 2 MA rect n M11 2 =8 < :1 06n6M1 0 otherwiseDTFT(= =)sin[!M= 2] sin(!=2)ej!(M1)=2 derivation nx[n]DTFT(= =)jd d!X(ej!) dierence x[n]x[n1]DTFT(= =) (1ej!)X(ej!) ansin[!0(n+1)] sin!0u[n]jaj<1DTFT(= =)1 12acos(!0ej!)+a2ej2! Note: ~(!) =+1X k=1(!+ 2k) ~rect(!) =+1X k=1rect(!+ 2k) Parseval: +1X n=1jx[n]j2=1 2Z+ jX(ej!)j2d! Marc Ph. Stoecklin |TABLES OF TRANSFORM PAIRS | v1.5.3 6 Table of Laplace Transform Pairs f(t) =L1fF(s)g=1 2jlimT!1Rc+jT cjTF(s)estdsL(= =)F(s) =Lff(t)g=R+1 1f(t)estdt transform f(t)L(= =)F(s) complex conjugation f(t)L(= =)F(s) time shifting f(ta)t>a>0L(= =)aasF(s) eatf(t)L(= =)F(s+a) frequency shifting time scaling f(at)L(= =)1 jajF(s a) linearity af1(t) +bf2(t)L(= =)aF1(s) +bF2(s) time multiplication f1(t)f2(t)L(= =)F1(s)F2(s) frequency convolution time convolution f1(t)f2(t)L(= =)F1(s)F2(s) frequency product delta function (t)L(= =) 1 shifted delta function (ta)L(= =)easexponential decay unit step u(t)L(= =)1 s ramp tu(t)L(= =)1 s2 parabola t2u(t)L(= =)2 s3 n-th power tnL(= =)n! sn+1 exponential decay eatL(= =)1 s+a two-sided exponential decay eajtjL(= =)2a a2s2 teatL(= =)1 (s+a)2 (1at)eatL(= =)s (s+a)2 exponential approach 1 eatL(= =)a s(s+a) sine sin ( !t)L(= =)! s2+!2 cosine cos ( !t)L(= =)s s2+!2 hyperbolic sine sinh ( !t)L(= =)! s2!2 hyperbolic cosine cosh ( !t)L(= =)s s2!2 exponentially decaying sine eatsin (!t)L(= =)! (s+a)2+!2 exponentially decaying cosine eatcos (!t)L(= =)s+a (s+a)2+!2 frequency dierentiation tf(t)L(= =) F0(s) frequencyn-th dierentiation tnf(t)L(= =) (1)nF(n)(s) time dierentiation f0(t) =d dtf(t)L(= =)sF(s)f(0) time 2nd dierentiation f00(t) =d2 dt2f(t)L(= =)s2F(s)sf(0)f0(0) timen-th dierentiation f(n)(t) =dn dtnf(t)L(= =)snF(s)sn1f(0):::f(n1)(0) time integrationRt 0f()d= (uf)(t)L(= =)1 sF(s) frequency integration1 tf(t)L(= =)R1 sF(u)du time inverse f1(t)L(= =)F(s)f1 s time dierentiation fn(t)L(= =)F(s) sn+f1(0) sn+f2(0) sn1+:::+fn(0) s