Tables of Common Transform Pairs


 Frederick McDaniel
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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 continuoutime frequency ourier (πf), continuoutime pultion ourier (), rnform, dicretetime 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 continuoutime: (f g)(t) = + f(τ) g (t τ)dτ Prevl theorem Geometric erie dicretetime: generl ttement: continuoutime: dicretetime: 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 Continuoutime 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 twoided 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 nth time derivtive nth 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 Continuoutime 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 twoided 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 nth time derivtive nth 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 nth power t n n! n+ exponentil decy e t twoided 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 nth 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 nth 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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