Definition of Inverse Fourier Transform Р ¥ ¥- = w w p w de F tf tj )( 2 1 )( wt tSa ) 2 ( 2 Bt Sa B p )( B rect w )( ttri ) 2 (2 w Sa ) 2 () 2 cos( t t p t rect
Amplitude of combined cosine and sine Phase Relative proportions of sine and cosine The Fourier Transform: Examples, Properties, Common Pairs Example:
Transform domain Linearity T akbk Multiplication x(t)y(t) ∑∞ m=−∞ ambk− m Cosine 2A cos(ω0t + B) 1 a+jω Rectangular pulse Π( t 2T ) 2 sin(ωT ) ω Sinc (normalized) sin(Wt) πt Π( ω 2W ) Discrete-time Fourier transform (DTFT)
f(t) cos (wt) dt For the sake of symmetry, we may define the Fourier cosine transform c by f
Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform Р wt tSa ) 2 ( 2 Bt Sa B p )( B rect w )( ttri ) 2 (2 w Sa ) 2 () 2 cos( t t p t rect
Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform ò wt tSa ) 2 ( 2 Bt Sa B p )( B rect w )( ttri ) 2 (2 w Sa ) 2 () 2 cos( t t p t rect
There are two similar functions used to describe the functional form sin(x)/x One is the sinc() function, and the other is the Sa() function We will only use the
Let's define a function F(m) that incorporates both cosine and sine series coefficients, with the sine series distinguished by making it the imaginary component:
of sine and cosine functions (fourier analysis) ▫ Fourier General form of solution: y=A cos wt+ B sin wt ; ω= k m properties of a signal (fourier transform),
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Introduction to Signal Processing Summer semester 2003/4 Transform tables
Fourier series (FS)
x(t) =1X k=¡1a kejk!0tak=1 T Z T x(t)e¡jk!0tdt
Property/signal Time domain Transform domain
LinearityAx(t) +By(t)Aak+Bbk
Time shiftingx(t¡¿)e¡jk!0¿ak
Time reversalx(¡t)a¡k
Time scaling
x(at),a >0 (periodicT a )ak
Di®erentiation
d dt x(t)jk!0ak
IntegrationRt
¡1x(t)dt,a0= 0ak
jk! 0
ConvolutionR
Multiplicationx(t)y(t)P1
m=¡1ambk¡m
Cosine 2Acos(!0t+B)a1=AejB;a¡1=Ae¡jB
Parseval
1 T R
Tjx(t)j2dt=P1
k=¡1jakj2
Fourier transform (FT)
x(t) =1
2¼Z
1 ¡1
X(j!)ej!td! X(j!) =Z
1 ¡1 x(t)e¡j!tdt
Property/signal Time domain Transform domain
Linearityax(t) +by(t)aX(j!) +bY(j!)
Time shiftingx(t¡¿)e¡j!¿X(j!)
Time scalingx(at)1
jajX(j!=a)
Di®erentiation
d dt x(t)j!X(j!)
IntegrationRt
¡1x(¿)d¿1
j!
X(j!) +¼X(0)±(!)
ConvolutionR1
¡1h(¿)x(t¡¿)d¿ H(j!)X(j!)
Multiplicationx(t)y(t)1
2¼R
1
¡1X(ju)Y(j!¡ju)du
Delta±(t) 1
One 1 2¼±(!)
Exponentej!0t2¼±(!¡!0)
Cosine cos(w0t)¼[±(!¡!0) +±(!+!0)]
Sine sin(w0t)¼
j [±(!¡!0)¡±(!+!0)]
Unit stepu(t)1
j!
Decaying stepu(t)e¡at,a >01
a+j!
Rectangular pulse ¦(
t
2T)2sin(!T)
Sinc (normalized)
sin(Wt) ¼t 2W)
ParsevalR1
¡1jx(t)j2dt=1
2¼R
1
¡1jX(j!)j2d!
22 February 2004 1
Introduction to Signal Processing Summer semester 2003/4 Transform tables
Discrete-time Fourier transform (DTFT)
x[n] =1
2¼Z
2¼X(ej!)ej!nd! X(ej!) =1X
n=¡1x[n]e¡j!n
Property/signal Time domain Transform domain
Linearityax[n] +by[n]aX(ej!) +bY(ej!)
Time shiftingx[n¡n0]e¡j!n0X(ej!)
Time reversalx[¡n]X(e¡j!)
ConvolutionP1
m=¡1x[m]y[n¡m]X(ej!)Y(ej!)
Multiplicationx[n]y[n]1
2¼R
2¼X(ejµ)Y(ej(!¡µ))dµ
Delta±[n] 1
One 1 2¼P1
m=¡1±(!¡2¼m)
Exponentej!0n2¼P1
m=¡1±(!¡!0¡2¼m)
Cosine cos[w0n]¼P1
m=¡1[±(!¡!0¡2¼m) +±(!+!0¡2¼m)]
Sine sin[w0t]¼
j P 1
Decaying stepu[n]an,jaj<11
1¡ae¡j!
Rectangular pulse ¦
N[n]sin[!(N+1
2 sin(!=2)
Sinc (normalized)
sin[Wn] ¼n P 1 m=¡1¦(!¡2¼m 2W)
ParsevalP1
n=¡1jx[n]j2=1
2¼R
2¼jX(ej!)j2
Discrete Fourier transform (DFT)
x[n] =1 N
N¡1X
k=0X(k)ej2¼ N nkX(k) =N¡1X n=0x[n]e¡j2¼ N nk
Property/signal Time domain Transform domain
Linearityax[n] +by[n]aX(k) +bY(k)
Time shiftingx[n¡n0]modNe¡j(2¼
N n0k)X(k)
ConvolutionPN¡1
m=0x[m]modNy[n¡m]modNX(k)Y(k)
Multiplicationx[n]y[n]1
N P
N¡1
l=0X(l)modNY(k¡l)modN
ParsevalPN¡1
n=0jx[n]j2=1 N P
N¡1
k=0jX(k)j2
22 February 2004 2
Introduction to Signal Processing Summer semester 2003/4 Transform tables
Laplace transform
x(t) =1
2¼jZ
¾+j1
¾¡j1X(s)estds X(s) =Z
1 ¡1 x(t)e¡stdt
Property/signal Time domain Transform domain
Linearityax(t) +by(t)aX(s) +bY(s)
Time shiftingx(t¡¿)e¡s¿X(s)
time scalingx(at)1 jajX(s=a)
Di®erentiation
d dt x(t)sX(s)
IntegrationRt
¡1x(¿)d¿1
s X(s)
ConvolutionR1
¡1x(¿)y(t¡¿)d¿ X(s)Y(s)
Delta±(t) 1
Unit stepu(t)1
s (Refsg>0)
Decaying stepe¡atu(t)1
s+a(Refsg>¡a)
Decaying step¡e¡atu(¡t)1
s+a(Refsg<¡a)
Causal Cosine cos(w0t)u(t)s
s
2+!20(Refsg>0)
Causal Sine sin(w0t)u(t)!0
s
2+!20(Refsg>0)
Z transform
x[n] =1
2¼jI
X(z)zn¡1dz X(z) =1X
n=¡1x[n]z¡n
Property/signal Time domain Transform domain
Linearityax[n] +by[n]aX(z) +bY(z)
Time shiftingx[n¡n0]z¡n0X(z)
time reversalx[¡n]X(z¡1)
ConvolutionP1
m=¡1x[m]y[n¡m]X(z)Y(z)
Delta±[n] 1
Unit stepu[n]1
1¡z¡1(jzj>1)
Decaying stepanu[n]1
1¡az¡1(jzj> a)
Decaying step¡anu[¡n¡1]1
1¡az¡1(jzj< a)
22 February 2004 3
Introduction to Signal Processing Summer semester 2003/4 Transform tables
General
Description Equation
Rectangular pulse in continuous-time ¦(x) =8
>:1jxj<1 2 1 2 jxj=1 2
0elsewhere
Rectangular pulse in discrete-time ¦
N[n] =(1jnj ·N
0elsewhere
Unit step in continuous-timeu(x) =8
>:1x >0 1 2 x= 0
0elsewhere
Unit step in discrete-timeu[n] =(1n¸0
0elsewhere
Sinc in continuous-time sinc(x) =
sin(¼x) ¼x Cosine of sum of angles cos(a+b) = cos(a)cos(b)¡sin(a)sin(b) Sine of sum of angles sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
22 February 2004 4
quotesdbs_dbs6.pdfusesText_12
[PDF] Table of Fourier Transform Pairs