Discrete-Time Fourier Transform Properties

Quote of the DayThe profound study of nature is the most fertile

source of mathematical discoveries.

Joseph Fourier

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 2

Absolute and Square Summability• Absolute summability is sufficient condition for DTFT• Some sequences may not be absolute summable but only

square summable

• To represent square summable sequences with DTFT– We can relax the uniform convergence condition– Convergence is in mean-squared sense

– Error does not converge to zero for every value of – The mean-squared value of the error over all does

n

2nx

nj

n

j enxeX

nj

n

j enxeX

0eXeXlim2j

Mj

M

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 3

Example: Ideal Lowpass Filter• The periodic DTFT of the ideal lowpass filter is

• The inverse can be written as

• Not causal • Not absolute summable but it has a DTFT?• The DTFT converges in the mean-squared sense• Role of Gibbs phenomenon

c

cjlp 0

1eH

n

nsinee

jn21

ejn2

1

de21

deeH21

nh

cnjnjnj

njnjjlplp

ccc

c

c

c

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 4

Example: Generalized DTFT• DTFT of • Not absolute summable • Not even square summable• But we define its DTFT as a pulse train

• Let’s place into inverse DTFT equation

1nx

r

j r22eX

1ede

der2221

deeX21

nx

n0jnj

nj

r

njj

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 5

Symmetric Sequence and Functions

Conjugate-symmetricConjugate-antisymmetric

Sequence

Function

nxnx *ee nxnx *

oo

nxnxnx oe nxnx21

nx *e nxnx

21

nx *o

j*e

je eXeX j*

oj

o eXeX

je

jo

j eXeXeX j*jje eXeX

21

eX j*jjo eXeX

21

eX

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 6

Symmetry Properties of DTFTSequence x[n] Discrete-Time Fourier Transform X(ej)

x*[n] X*(e-j)

x*[-n] X*(ej)

Re{x[n]} Xe(ej) (conjugate-symmetric part)

jIm{x[n]} Xo(ej) (conjugate-antisymmetric part)

xe[n] XR(ej)= Re{X(ej)}

xo[n] jXI(ej)= jIm{X(ej)}

Any real x[n] X(ej)=X*(e-j) (conjugate symmetric)

Any real x[n] XR(ej)=XR(e-j) (real part is even)

Any real x[n] XI(ej)=-XI(e-j) (imaginary part is odd)

Any real x[n] |X(ej)|=|X(e-j)| (magnitude is even)

Any real x[n] X(ej)=-X(e-j) (phase is odd)

xe[n] XR(ej)

xo[n] jXI(ej)

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 7

Example: Symmetry Properties• DTFT of the real sequence x[n]=anu[n]

• Some properties are

1a if ae11

eX jj

j1j

j

2

j

jI2

jI

jR2

jR

j*j

j

eXcosa1sina

tan eX

eXcosa2a1

1 eX

eXcosa2a1

sina eX

eXcosa2a1

cosa1 eX

eXae11

eX

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 8

Fourier Transform Theorems

Sequence DTFT

x[n]y[n]

X(ej)Y(ej)

ax[n]+by[n] aX(ej)+bY(ej)

x[n-nd]

x[-n] X(e-j)

nx[n]

x[n]y[n] X(ej)Y(ej)

x[n]y[n]

jnj eXe d

dedX

jj

deX21

nx : TheoremsParseval'2j

n

2

deYeX21

nynx : TheoremsParseval' j*j

n

*

deYeX21 jj

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 9

Fourier Transform PairsSequence DTFT

[n-no]

1

anu[n] |a|<1

u[n]

cos(on+)

onje

k

k22

jae11

kj k2

e11

n

nsin c

c

cj 0

1eX

otherwise

Mn0

0

1nx

2/Mje

2/sin2/1Msin

nj oe

k

o k22

k

oj

oj k2ek2e