dsp formulas
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EE-556 Formula Sheet Some useful Mathematical formulas and identities:
)sin()cos( !!!je
j+= ,
2)cos(
!!
!jj
ee"
+= ,
j
eejj
2)sin(
!!
!"
"=
)cos()sin(2)2sin( !!! = , )(sin211)(cos2)2cos( 22 !!! "="=
[ ])cos()cos(2
1)cos()cos( !"!"!" ++#=
[ ])cos()cos(2
1)sin()sin( !"!"!" +##=
[ ]1
sin( )cos( ) sin( ) sin( )2
! " ! " ! "= # + +
x
xx
) sin()(sinc
!
!=
Geometric series: a
aaa
nmn
mk
k
!
!=
+
=
"1
1
Fourier series:
[ ] !!+"
#"=
"
=
=++=
n
tjnn
n
nn ectnbtnaatx 0
1
000 )sin()cos()($$$ ,
!=
0
)(1
00
T
dttxT
a , !=
0
)cos()(2
00T
ndttntx
Ta " , !=
0
)sin()(2
00T
ndttntx
Tb " , !
"=
0
0)(1
0 T
tjnn dtetxT
c#
Fourier transform and inverse Fourier transform:
!+"
"#
#= dtetxX tj$$ )()( , !
+"
"#
= dteXtx tj$$%
)(2
1)(
Some useful Fourier transform pairs:
)(21 !"#$
!!"#
jtu
1)()( +$
1)( !t"
)(2 00 !!"#
!$%
tje
!"
#$%
&=!
"
#$%
&
'
(
2sincrect
TT
T
t
[ ])()()cos( 000 !!"!!"#! $++%t [ ])()()sin( 000 !!"!!"#! $$+% jt
Energy and power of discrete-time signals:
Energy: !+"
#"=
=
n
xnxE2][ , Power: non-periodic !
+
"=#$ +
=
N
NnN
xnx
NP
2][
12
1lim , N-periodic: !
"#=
=
Nn
xnx
NP
2][
1
Discrete-time linear convolution: !+"
#"=
#=$=
k
knhkxnhnxny ][][][][][
Discrete-time Fourier Transform (DTFT): !+"
#"=
#=
n
njjenxeX
$$ ][)(
DTFT Properties: Symmetry relations:
Sequence DTFT
][nx )( !jeX
][ nx ! )( !jeX"
][*nx ! )(* !j
eX
]}[Re{ nx [ ])()(2
1)( * !!! jjj
cs eXeXeX"+=
]}[Im{ nxj [ ])()(2
1)( * !!! jjj
as eXeXeX"
"=
][nxcs
)( !jre eX
][nxas
)( !jim ejX
Note: Subscript “cs” and “as” mean conjugate symmetric and conjugate anti-symmetric signals, respectively. DTFT of commonly used sequences:
Sequence DTFT ][n! 1
][nµ (Unit Step Function) !+"
#"=
#++
#k
jk
e)2(
1
1$%$&
%
nje 0! !
+"
#"=
+#k
k)2(2 0 $%%$&
][nanµ , ( 1<a )
!jae
""1
1
][)1( nannµ+ , ( 1<a )
2
1
1!"
#$%
&
' ' (jae
n
nnh
c
!
" )sin(][ = , ( +!<<"! n )
!"
!#$
<<
%%=
&''
'''
c
cjeH
0
01)(
DTFT Theorems:
Theorem Sequence DTFT
][ng ][nh
)( !jeG
)( !jeH
Linearity ][][ nhng !" + )()( !! "# jjeHeG +
Time Reversal ][ ng ! )( !jeG"
Time Shifting ][ 0nng ! )(0 !! jnjeGe
"
Frequency Shifting ][0 ngenj! )(
)( 0!!"jeG
Differentiation in Frequency ][nng !
!
d
edGj
j )(
Convolution ][][ nhng ! )()( !! jjeHeG
Modulation ][][ nhng !"
"
"
#
#
$%$ $#
deHeG jj )()(2
1 )(
Parseval’s Identity !"+
#
+$
#$=
=
%
%
&& &%
deHeGnhng jj
n
)()(2
1][][ **
Discrete Fourier Transform (DFT): !"
=
"=
1
0
2
][][
N
n
nkNj
enxkX
#
, 10 !"" Nk
Inverse DFT: 21
0
1[ ] [ ]
N j nkN
k
x n X k eN
!"
=
= # , 10 !"" Nn
Circular Convolution: !"
=
#"$=
1
0
][][][
N
m
Nc mnhmxny , 10 !"" Nn
Symmetry properties of DFT:
Length-N Sequence N-Point DFT
][][][ njxnxnx imre += ][][][ kjXkXkX imre +=
][*nx ][
*N
kX !"#
][*
Nnx !"# ][
*kX
][nxre
[ ]][][2
1][
*Ncs
kXkXkX !"#+=
][njxim [ ]][][2
1][
*Nas
kXkXkX !"##=
][nxcs
][kXre
][nxas
][kjX im
DFT Theorems:
Theorem Sequence DTFT
][ng ][nh
][kG ][kH
Linearity ][][ nhng !" + ][][ kHkG !" +
Circular Time Shifting ][ 0 Nnng !"# 0
2
[ ]j knNe G k!
"
Circular Frequency Shifting 0
2
[ ]j nkNe g n!
][ 0 NkkG !"#
Duality ][nG ][ NkNg !"#
N-Point Circular Convolution !"
=
#"$1
0
][][
N
m
Nmnhmg ][][ kHkG
Modulation ][][ nhng !"
=
#"$1
0
][][1N
m
NmnHmG
N
Parseval’s Identity !!"
=
"
=
=
1
0
21
0
2][
1][
N
n
N
n
kGN
ng
Z-Transform: !+"
#"=
#=
n
nznxzX ][)( Inverse Z-Transform: !
"=
C
n dzzzGj
nx 1)(2
1][
#
Z-Transform Properties:
Property Sequence z-Transform Region of Convergence
][ng ][nh
)(zG )(zH
gR
hR
Conjugation ][*ng )( **
zG gR
Time Reversal ][ ng ! ( )zG 1 gR1
Linearity ][][ nhng !" + )()( zHzG !" + Includes hg RR !
Time Shifting ][ 0nng ! )(0 zGzn! gR except possibly the point 0=z or !
Multiplication by an exponential sequence ][ng
n! ( )!zG gR!
Differentiation in the z-domain
][nng dz
zdGz
)(! gR except possibly the point 0=z or !
Convolution ][*][ nhng )()( zHzG Includes hg RR !
Some useful z-transform pairs:
Sequence z-Transform ROC ][n! 1 All values of z
][nµ 1
1
1
!! z
1>z
][nnµ!
11
1
!! z"
!>z
][nnnµ!
( )211
1!
!
! z
z
"
" !>z
][)cos( 0 nnrn µ! 221
0
10
)cos(2(1
)cos((1!!
!
+!
!
zrzr
zr
"
" rz >
][)sin( 0 nnrn µ! 221
0
10
)cos(2(1
)sin((!!
!
+! zrzr
zr
"
" rz >