Tables of Transform Pairs

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1 Tble of Trnform Pir 005 by Mrc Stoecklin mrc toecklin.net December, 005 verion.5 Student nd engineer in communiction nd mthemtic re confronted with trnformtion uch the -Trnform, the ourier trnform, or the plce trnform. Often it i quite hrd to quickly find the pproprite trnform in book or the Internet, much le to hve good overview of trnformtion pir nd correponding propertie. In thi document I preent hndy collection of the mot common trnform pir nd propertie of the continuou-time frequency ourier trnform (πf), continuou-time pultion ourier trnform (), -Trnform, dicrete-time ourier trnform DTT, nd plce trnform rrnged in tble nd ordered by ubject. The propertie of ech trnformtion re indicted in the firt prt of ech topic where pecific trnform pir re lited fterwrd. Plee note tht, before including trnformtion pir in the tble, I verified their correctne. However, it i till poible tht there might be ome mitke due to typo. I d be grteful to everyone for dropping me line nd indicting me erroneou formul. Some ueful convention nd formul Sinc function inc (x) in(x) x Convolution f g(t) = + f(τ)g (t τ)dτ Prevl theorem + f(t)g (t)dt = + (f)g (f)df + f(t) dt = + (f) df Rel prt Re{f(t)} = [f(t) + f (t)] Imginry prt Im{f(t)} = [f(t) f (t)] Sine / Coine Geometric equence in (x) = ejx e jx j k=0 xk = x Generl ce : co (x) = ejx +e jx n k=0 xk = xn+ x n k=m xk = xm x n+ x

2 Mrc Stoecklin : TABES O TRANSORM PAIRS Tble of Continuou-time requency ourier Trnform Pir f(t) = { (f)} = R + f(t)ejπft df (f) = {f(t)} = R + f(t)e jπft dt f(t) f( t) f (t) (f) ( f) ( f) f(t) i purely rel f(t) i purely imginry even/ymmetry odd/ntiymmetry f(t) = f ( t) f(t) = f ( t) time hifting f(t t 0 ) f(t)e jπf 0t time cling f (f) f f f(t) + bg(t) f(t)g(t) f(t) g(t) δ(t) δ(t t 0 ) e jπf 0t (f) = ( f) even/ymmetry (f) = ( f) odd/ntiymmetry (f) i purely rel (f) i purely imginry (f)e jπft 0 (f f 0 ) frequency hifting f (f) frequency cling (f) + bg(t) (f) G(f) (f)g(f) e jπft 0 δ(f) δ(f f 0 ) e t > 0 e πt e jπt in (πf 0 t + φ) co (πf 0 t + φ) f(t) in (πf 0 t) f(t) co (πf 0 t) in (t) co (t) rect ` t T = [ T,+ T ](t) = < t T : 0 t > T tring ` < t t t T T = T : 0 t > T < t 0 u(t) = [0,+ ] (t) = : 0 t < 0 < t 0 gn (t) = : t < 0 inc (Bt) inc (Bt) d n dt n f(t) t n f(t) +4π f e πf e jπ( 4 f ) +t j ˆe jφ δ (f + f 0 ) e jφ δ (f f 0 ) ˆe jφ δ (f + f 0 ) + e jφ δ (f f 0 ) j [ (f + f 0) (f f 0 )] [ (f + f 0) + (f f 0 )] ˆδ(f) 4 δ `f π δ `f + π ˆδ(f) 4 + δ `f π + δ `f + π T inc T f T inc T 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 Stoecklin : TABES O TRANSORM PAIRS 3 Tble of Continuou-time Pultion ourier Trnform Pir x(t) = {X()} = R + x(t)ejt d X() = {x(t)} = R + x(t)e jt dt x(t) x( t) x (t) X() X( ) X ( ) x(t) i purely rel x(t) i purely imginry even/ymmetry odd/ntiymmetry x(t) = x ( t) x(t) = x ( t) time hifting x(t t 0 ) x(t)e j 0t time cling x (f) x f x (t) + bx (t) x (t)x (t) x (t) x (t) δ(t) δ(t t 0 ) e j 0t e t > 0 e t u(t) R{} > 0 e t u( t) R{} > 0 e t σ in ( 0 t + φ) co ( 0 t + φ) x(t) in ( 0 t) x(t) co ( 0 t) in ( 0 t) co ( 0 t) rect ` t T = [ T,+ T ](t) = < t T : 0 t > T tring ` < t t t T T = T : 0 t > T < t 0 u(t) = [0,+ ] (t) = : 0 t < 0 < t 0 gn (t) = : t < 0 inc (T t) inc (T t) X(f) = X ( ) even/ymmetry X(f) = X ( ) odd/ntiymmetry X() i purely rel X() i purely imginry X()e jt 0 X( 0 ) frequency hifting X ` X() frequency cling X () + bx () π X () X () X ()X () e jt 0 πδ() πδ( 0 ) + +j j σ πe σ ˆe jφ jπ δ ( + 0 ) e jφ δ ( 0 ) ˆe jφ π 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 )] T inc T T inc T πδ(f) + j j T rect ` πt = T [ πt,+πt ](f) T tring ` πt d n dt n f(t) t n f(t) t (j) n X() j n d n df n X() jπgn()

4 Mrc Stoecklin : TABES O TRANSORM PAIRS 4 Tble of -Trnform Pir x[n] = {X()} = H πj X() n d X() = {x[n]} = P + n= x[n] n ROC 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 n x[n] X ` R x P downmpling by N x[nn] N N 0 N N k=0 X WN k N W N = e j N R x x [n] + bx [n] x [n]x [n] x [n] x [n] δ[n] δ[n n 0 ] X () + bx () R x R y H ` X (u)x πj u du R u x R y X ()X (t) R x R y n 0 u[n] u[ n ] nu[n] n u[n] n 3 u[n] ( ) n 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 in ( 0 n) u[n] co ( 0 n) u[n] n in ( 0 n) u[n] n co ( 0 n) u[n] > < ( ) > (+) ( ) 3 > ( +4+) ( ) 4 > + < > < > ( ) > (+ ( ) 3 > e > e N N > 0 in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ > > > > Q mi= (n i+) m m! nx[n] x[n] n m u[n] d X() Rx d R X() 0 d R x ( ) m+ Plee note : =

5 Mrc Stoecklin : TABES O TRANSORM PAIRS 5 Tble of Dicrete Time ourier Trnform (DTT) Pir x[n] = R +π π π X(ej )e jn d x[n] x[ n] x [n] x[n] i purely rel x[n] i purely imginry even/ymmetry odd/ntiymmetry x[n] = x [ n] x[n] = x [ n] DT T X(e j ) = P + n= x[n]e jn DT T X(e j ) DT T X(e j ) DT T X (e j ) DT T X(e j ) = X (e j ) even/ymmetry DT T X(e j ) = X (e j ) odd/ntiymmetry DT T DT T X(e j ) i purely rel X(e j ) i purely imginry DT T time hifting x[n n 0 ] X(e j )e jn 0 x[n]e j DT T 0n X(e j( 0) ) frequency hifting DT T P downmpling by N x[nn] N N 0 N πk N k=0 X(ej N ) < x ˆ n n = kn DT T upmpling by N N X(e jn ) : 0 otherwie MA : x [n] + bx [n] x [n]x [n] x [n] x [n] δ[n] δ[n n 0 ] e j 0n u[n] n u[n] ( < ) (n + ) n u[n] in ( 0 n + φ) co ( 0 n + φ) in( cn) = n c inc ( cn) Window : rect ` < n n M M = : 0 otherwie MA : rect ` n M < 0 n M = : 0 otherwie rect n M < 0 n M = : 0 otherwie nx[n] x[n] x[n ] n in[ 0 (n+)] u[n] < in 0 DT T X (e j ) + bx (e j ) DT T X (e j ) X (e j ) = π DT T X (e j )X (e j ) DT T DT T e jn 0 DT T P δ() = + k= δ( + πk) DT T δ( 0 ) = P + k= δ( 0 + πk) DT T DT T DT T DT T DT T DT T DT T DT T DT T e j + δ() e j ( e j ) R +π π X (e j( σ) )X (e jσ )dσ j [e jφ δ ( πk) e +jφ δ ( 0 + πk)] [e jφ δ ( πk) + e +jφ δ ( 0 + πk)] < rect < = c c : 0 c < < π in[(m+ )] in(/) in[(m+)/] e in(/) jm/ in[m/] in(/) e j(m )/ DT T j d d X(ej ) DT T ( e j )X(e j ) DT T co( 0 e j )+ e j Some remrk Prevl : δ() = + X n= + X k= x[n] = π δ( + πk) +π X(e j ) d π rect() = + X k= rect( + πk)

6 Mrc Stoecklin : TABES O TRANSORM PAIRS 6 Tble of plce Trnform Pir f(t) = { ()} = R c+j πj c j ()et d () = {f(t)} = R + f(t)e t dt f(t) () f(t ) t > 0 e t f(t) f(t) > 0 f (t) + bf (t) f (t)f (t) f (t) f (t) () ( + ) ( ) () + b () () () () () δ(t) t e t te t e t e t ` e t in (t) co (t) inh (t) coh (t) e t in (t) e t co (t) + (+) (+) (+) + + (+) + t n t n f(t) n! +n+ ( ) n (n) () In generl : f (t) = d dt f(t) f (t) = d dt f(t) f (n) (t) = dn dt n f(t) R t 0 f(u)du t f(t) f (t) f n (t) () f(0) () f(0) f (0) n () n f(0)... f (n ) (0) R () (u)du () f () n + f (0) n + f (0) n f n (0)

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