Tables of Common Transform Pairs

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1 ble of Common rnform Pir 0 by Mrc Ph. Stoecklin mrc toecklin.net verion v.5.3 Engineer nd tudent in communiction nd mthemtic re confronted with tion uch the -rnform, the ourier, or the plce. Often it i quite hrd to quickly find the pproprite in book or the Internet, much le to hve comprehenive overview of tion pir nd correponding propertie. In thi document I compiled hndy collection of the mot common pir nd propertie of the continuou-time frequency ourier (πf), continuou-time pultion ourier (), -rnform, dicrete-time ourier D, nd plce. Plee note tht, before including tion pir in the tble, I verified it correctne. Neverthele, it i till poible tht you my find error or typo. I m very grteful to everyone dropping me line nd pointing out ny concern or typo. Nottion, Convention, nd Ueful ormul Imginry unit j = Complex conjugte = + jb = jb Rel prt Re {f(t)} = [f(t) + f (t)] Imginry prt Im {f(t)} = [f(t) f (t)] j { Dirc delt/unit impule δ[n] =, n = 0 0, n 0 Heviide tep/unit tep u[n] = {, n 0 0, n < 0 Sine/Coine Sinc function in (x) = ejx e jx j inc (x) in(x) x co (x) = ejx +e jx (unnormlied) { Rectngulr function rect( t ) = if t 0 if t > ringulr function tring ( ) t = rect( t ) rect( t ) = t t 0 t > Convolution continuou-time: (f g)(t) = + f(τ) g (t τ)dτ Prevl theorem Geometric erie dicrete-time: generl ttement: continuou-time: dicrete-time: k=0 xk = x (u v)[n] = m= u[m] v [n m] + f(t)g (t)dt = + (f)g (f)df + f(t) dt = + (f) df + n= x[n] = +π π π X(ej ) d n k=0 xk = xn+ x in generl: n k=m xk = xm x n+ x

2 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v.5.3 ble of Continuou-time requency ourier rnform Pir f(t) = { (f)} = + f(t)ejπft df (f) = {f(t)} = + f(t)e jπft dt time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry f(t) f( t) f (t) f ( t) f(t) i purely rel f(t) i purely imginry f(t) = f ( t) f(t) = f ( t) (f) ( f) frequency reverl ( f) revered conjugtion (f) complex conjugtion (f) = ( f) even/ymmetry (f) = ( f) odd/ntiymmetry (f) i purely rel (f) i purely imginry time hifting f(t t 0 ) time cling linerity time multipliction delt function f(t)e jπf 0t f (f) f f f(t) + bg(t) f(t)g(t) f(t) g(t) δ(t) hifted delt function δ(t t 0 ) e jπf 0t two-ided exponentil decy e t > 0 e πt e jπt ine in (πf 0 t + φ) coine co (πf 0 t + φ) ine modultion f(t) in (πf 0 t) coine modultion f(t) co (πf 0 t) qured ine qured coine in (t) co (t) (f)e jπft 0 (f ( f 0 ) frequency hifting f (f) frequency cling (f) + bg(t) (f) G(f) (f)g(f) frequency multipliction e jπft 0 δ(f) delt function δ(f f 0 ) hifted delt function +4π f e πf e jπ( 4 f ) j [ e jφ δ (f + f 0 ) e jφ δ (f f 0 ) ] [ e jφ δ (f + f 0 ) + e jφ δ (f f 0 ) ] j [ (f + f 0) (f f 0 )] [ (f + f 0) + (f f 0 )] [ ] 4 δ(f) δ f π δ f + π [ ] 4 δ(f) + δ f π + δ f + π rectngulr rect ( t t = 0 t > tringulr tring ( t = t t 0 t > t 0 tep u(t) = [0,+ ] (t) = 0 t < 0 t 0 ignum gn (t) = t < 0 inc qured inc n-th time derivtive n-th frequency derivtive inc (Bt) inc (Bt) d n dt n f(t) t n f(t) +t inc f inc f jπf + δ(f) jπf B rect ( f B B tring ( f B (jπf) n (f) ( jπ) n dn df n (f) πe π f ) = B [ ) B,+ B ](f)

3 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of Continuou-time Pultion ourier rnform Pir x(t) = {X()} = + x(t)ejt d X() = {x(t)} = + x(t)e jt dt time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry x(t) x( t) x (t) x ( t) x(t) i purely rel x(t) i purely imginry x(t) = x ( t) x(t) = x ( t) X() X( ) frequency reverl X ( ) revered conjugtion X () complex conjugtion X(f) = X ( ) even/ymmetry X(f) = X ( ) odd/ntiymmetry X() i purely rel X() i purely imginry time hifting x(t t 0 ) time cling x(t)e j 0t x (f) x f X()e jt 0 X( 0 ) frequency hifting X X() frequency cling linerity time multipliction x (t) + bx (t) x (t)x (t) x (t) x (t) X () + bx () π X () X () X ()X () frequency multipliction delt function δ(t) hifted delt function δ(t t 0 ) e j 0t two-ided exponentil decy e t > 0 exponentil decy e t u(t) R{} > 0 revered exponentil decy e t u( t) R{} > 0 e t σ ine in ( 0 t + φ) coine co ( 0 t + φ) ine modultion x(t) in ( 0 t) coine modultion x(t) co ( 0 t) qured ine in ( 0 t) qured coine co ( 0 t) rectngulr rect ( t t = 0 t > tringulr tring ( t = t t 0 t > t 0 tep u(t) = [0,+ ] (t) = 0 t < 0 t 0 ignum gn (t) = t < 0 inc inc ( t) qured inc inc ( t) e jt 0 πδ() delt function πδ( 0 ) hifted delt function + +j j σ πe σ [ jπ e jφ δ ( + 0 ) e jφ δ ( 0 ) ] [ π e jφ δ ( + 0 ) + e jφ δ ( 0 ) ] j [X ( + 0) X ( 0 )] [X ( + 0) + X ( 0 )] π [δ(f) δ ( 0 ) δ ( + 0 )] π [δ() + δ ( 0 ) + δ ( + 0 )] inc inc πδ(f) + j j rect π = [ π,+π ](f) tring π n-th time derivtive n-th frequency derivtive time invere d n dt n f(t) t n f(t) t (j) n X() j n d n df n X() jπgn()

4 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of -rnform Pir x[n] = {X()} = πj X() n d X() = {x[n]} = + n= x[n] n ROC time reverl complex conjugtion revered conjugtion rel prt imginry prt x[n] x[ n] x [n] x [ n] Re{x[n]} Im{x[n]} X() R x X( ) R x X ( ) R x X ( ) R x [X() + X ( )] R x j [X() X ( )] R x time hifting x[n n 0 ] n 0X() R x cling in n x[n] X R x downmpling by N x[nn], N N 0 N N k=0 X WN k N W N = e j N R x linerity time multipliction x [n] + bx [n] x [n]x [n] x [n] x [n] X () + bx () R x R y ( X (u)x ) πj u u du R x R y X ()X (t) R x R y delt function δ[n] hifted delt function δ[n n 0 ] n 0 tep rmp u[n] u[ n ] nu[n] n u[n] n u[ n ] n 3 u[n] n 3 u[ n ] ( ) n > < ( ) > (+) ( ) 3 > (+) ( ) 3 < ( +4+) ( ) 4 > ( +4+) ( ) 4 < + < exponentil exp. intervl n u[n] n u[ n ] n u[n ] n n u[n] n n u[n] e n u[n] { n n = 0,..., N 0 otherwie > < > ( ) > (+ ( ) 3 > e > e N N > 0 ine coine in ( 0 n) u[n] co ( 0 n) u[n] n in ( 0 n) u[n] n co ( 0 n) u[n] in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ > > > > differentition in integrtion in mi= (n i+) m m! nx[n] x[n] n m u[n] dx() R d x X() 0 d R x ( ) m+ Note: =

5 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of Common Dicrete ime ourier rnform (D) Pir x[n] = +π π π X(ej )e jn d D X(e j ) = + n= x[n]e jn time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry x[n] x[ n] x [n] x [ n] x[n] i purely rel x[n] i purely imginry x[n] = x [ n] x[n] = x [ n] D X(e j ) D X(e j ) D X (e j ) D X (e j ) D X(e j ) = X (e j ) even/ymmetry D X(e j ) = X (e j ) odd/ntiymmetry D D X(e j ) i purely rel X(e j ) i purely imginry time hifting x[n n 0 ] x[n]e j 0n D X(e j )e jn 0 D X(e j( 0) ) frequency hifting D downmpling by N x[nn] N N 0 x [ ] n n = kn upmpling by N N 0 otherwie N D X(e jn ) N πk k=0 X(ej N ) linerity time multipliction x [n] + bx [n] x [n]x [n] x [n] x [n] D X (e j ) + bx (e j ) D X (e j ) X (e j ) = +π π π X (e j( σ) )X (e jσ )dσ D X (e j )X (e j ) frequency multipliction delt function δ[n] hifted delt function δ[n n 0 ] e j 0n D D e jn 0 D δ() delt function D δ( 0 ) hifted delt function ine in ( 0 n + φ) coine co ( 0 n + φ) D D j [e jφ δ ( πk) e +jφ δ ( 0 + πk)] [e jφ δ ( πk) + e +jφ δ ( 0 + πk)] rectngulr tep rect ( n n M M = 0 otherwie u[n] decying tep n u[n] ( < ) pecil decying tep (n + ) n u[n] ( < ) inc MA MA derivtion in( cn) πn D D D D = c π inc (cn) D rect ( n M 0 n M = 0 otherwie rect n M 0 n M = 0 otherwie nx[n] difference x[n] x[n ] n in[ 0 (n+)] u[n] < in 0 D D in[(m+ )] in(/) e j + δ() e j ( e j ) ( rect < = c c 0 c < < π in[(m+)/] e in(/) jm/ in[m/] in(/) e j(m )/ D j d d X(ej ) D ( e j )X(e j ) D co( 0 e j )+ e j Note: Prevl: δ() = + n= + k= x[n] = π δ( + πk) +π X(e j ) d π rect() = + k= rect( + πk)

6 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of plce rnform Pir f(t) = { ()} = πj lim c+j c j ()et d () = {f(t)} = + f(t)e t dt complex conjugtion f(t) f (t) () ( ) time hifting f(t ) t > 0 e t f(t) time cling f(t) linerity f (t) + bf (t) time multipliction f (t)f (t) time convolution f (t) f (t) () ( + ) frequency hifting ( ) () + b () () () () () frequency product delt function δ(t) hifted delt function δ(t ) e exponentil decy unit tep u(t) rmp tu(t) prbol t u(t) 3 n-th power t n n! n+ exponentil decy e t two-ided exponentil decy e t te t ( t)e t exponentil pproch e t ine coine hyperbolic ine hyperbolic coine exponentilly decying ine exponentilly decying coine in (t) co (t) inh (t) coh (t) e t in (t) e t co (t) + (+) (+) (+) + + (+) + + (+) + frequency differentition frequency n-th differentition tf(t) t n f(t) () ( ) n (n) () time differentition f (t) = d dt f(t) () f(0) time nd differentition f (t) = d dt f(t) () f(0) f (0) time n-th differentition f (n) (t) = dn dt n f(t) n () n f(0)... f (n ) (0) time integrtion frequency integrtion t 0 f(τ)dτ = (u f)(t) t f(t) () (u)du time invere time differentition f (t) f n (t) () f () n + f (0) n + f (0) n f n (0)

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