Fourier series (FS) Fourier transform (FT) PDF

Table of Fourier Transform Pairs

Fourier Transform F(w). Definition of Inverse Fourier Transform cos( t t p t rect t. A. 2. 2. )2(. ) cos( w t p wt.

Fourier Transform of a Cosine Example: Fo

Amplitude of combined cosine and sine. Phase. Relative proportions of sine and cosine. The Fourier Transform: Examples Properties

Untitled

For Fourier transform we want to consider L oo

Lecture 11 The Fourier transform

the Fourier transform of a signal f is the function by allowing impulses in F(f) we can define the Fourier transform of a step ... pulsed cosine: f(t) =.

a) Determine the function f(t) whose Fourier transform is shown in

Figure 11: The function ƒ(t) = [1 – cos (Wt)]/t for W = 1 odd symmetry. Problem 3.2.1 a) Find the Fourier transform for of the raised cosine pulse ...

Fourier series (FS) Fourier transform (FT)

Cosine. 2A cos(?0t + B) a1 = AejBa?1 = Ae?jB. Parseval sin(Wt) ?t. ?( ?. 2W. ) Parseval. ? ? ... Discrete-time Fourier transform (DTFT) x[n] =.

Solutions to Exercises

An Introduction to Laplace Transforms and Fourier Series. Exercises 2.8. 1. If F(t) = cos (at) then F'(t) = -asin(at). The derivative formula thus.

Chapter 4: Frequency Domain and Fourier Transforms

Frequency domain analysis and Fourier transforms are a cornerstone of signal cos(?0t) ?[?(? ? ?0) + ?(? + ?0)] x(t)=1. 2??(?) sin(Wt).

Fourier Series and Fourier Transform

Q2: Show that the inverse Fourier transform of X(jw) = 2??(w) + ??(w ? 4?) + ??(w + 4?) is x(t) = 1+cos 4?t. Q3: Calculate the Fourier transform of te?

Chapter 1 The Fourier Transform

Mar 1 2010 cos(?t)dt = 2 sin(??) ?. = 2? sinc ?. Thus sinc ? is the Fourier transform of the box function. The inverse. Fourier transform is. ? ?. ? ...

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

X(j!) +¼X(0)±(!)

ConvolutionR1

¡1h(¿)x(t¡¿)d¿ H(j!)X(j!)

Multiplicationx(t)y(t)1

2¼R

¡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

Decaying stepu(t)e¡at,a >01

a+j!

Rectangular pulse ¦(

2T)2sin(!T)

Sinc (normalized)

sin(Wt) ¼t 2W)

ParsevalR1

¡1jx(t)j2dt=1

2¼R

¡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

2+!20(Refsg>0)

Causal Sine sin(w0t)u(t)!0

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] Fourier series (FS) Fourier transform (FT)

Fourier series (FS)

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

Di®erentiation

IntegrationRt

¡1x(t)dt,a0= 0ak

ConvolutionR

Multiplicationx(t)y(t)P1

Cosine 2Acos(!0t+B)a1=AejB;a¡1=Ae¡jB

Parseval

Tjx(t)j2dt=P1

Fourier transform (FT)

2¼Z

X(j!)ej!td! X(j!) =Z

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

Di®erentiation

IntegrationRt

¡1x(¿)d¿1

X(j!) +¼X(0)±(!)

ConvolutionR1

¡1h(¿)x(t¡¿)d¿ H(j!)X(j!)

Multiplicationx(t)y(t)1

2¼R

¡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

Unit stepu(t)1

Decaying stepu(t)e¡at,a >01

Rectangular pulse ¦(

2T)2sin(!T)

Sinc (normalized)

ParsevalR1

¡1jx(t)j2dt=1

2¼R

¡1jX(j!)j2d!

22 February 2004 1

Discrete-time Fourier transform (DTFT)

2¼Z

2¼X(ej!)ej!nd! X(ej!) =1X

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

Multiplicationx[n]y[n]1

2¼R

2¼X(ejµ)Y(ej(!¡µ))dµ

Delta±[n] 1

One 1 2¼P1

Exponentej!0n2¼P1

Cosine cos[w0n]¼P1

Sine sin[w0t]¼

Decaying stepu[n]an,jaj<11

1¡ae¡j!

Rectangular pulse ¦

N[n]sin[!(N+1

Sinc (normalized)

ParsevalP1

2¼R

2¼jX(ej!)j2

Discrete Fourier transform (DFT)

N¡1X

Property/signal Time domain Transform domain

Linearityax[n] +by[n]aX(k) +bY(k)

Time shiftingx[n¡n0]modNe¡j(2¼

ConvolutionPN¡1

Multiplicationx[n]y[n]1

N¡1

ParsevalPN¡1

N¡1