DSP Formulas

EE-556 Formula Sheet Some useful Mathematical formulas and identities:

e j! = cos(! ) + j sin(! ) ,

cos(! ) =

e j! + e " j! , 2

sin(! ) =

e j! " e " j! 2j

sin(2! ) = 2 sin(! ) cos(! ) , cos(2! ) = 2 cos 2 (! ) " 1 = 1 " 2 sin 2 (! ) 1 cos(" ) cos(! ) = [cos(" # ! ) + cos(" + ! )] 2 1 sin(" ) sin( ! ) = [cos(" # ! ) # cos(" + ! )] 2 1 sin(! ) cos( " ) = [sin(! # " ) + sin(! + " ) ] 2 sin(! x) sinc( x) = !x n

"a

Geometric series:

k

k =m

=

a m ! a n +1 1! a

Fourier series: "

x(t ) = a0 +

! [an cos(n$0t ) + bn sin(n$0t )]= n =1

a0 =

1 T0

! x(t )dt ,

T0

an =

2 T0

+"

!c e n

jn$0t

! x(t ) cos(n" t )dt , 0

T0

Fourier transform and inverse Fourier transform: +"

X ($ ) =

!

x(t )e # j$t dt ,

#"

x(t ) =

1 2%

+"

! X ($ )e

#"

Some useful Fourier transform pairs:

1 $ 2"# (! ) u (t ) $ "# (! ) +

1 j!

" (t ) ! 1 e j!0t % 2"# (! $ !0 )

&t # & (T # rect$ ! = Tsinc$ ! %T " % 2' " cos(!0t ) % # [" (! + !0 ) + " (! $ !0 )] sin(!0 t ) % j# [" (! + !0 ) $ " (! $ !0 )]

,

n = #"

j $t

dt

bn =

2 T0

! x(t ) sin(n" t )dt , 0

T0

cn =

1 T0

! x(t )e

T0

" jn#0t

dt

Energy and power of discrete-time signals: +"

Energy: E x =

!

+N

1 1 2 x[n] , N-periodic: Px = N $# 2 N + 1 N n=" N

2

x[n] , Power: non-periodic Px = lim

n = #"

!

! x[n]

2

n =# N "

+"

Discrete-time linear convolution:

y[n] = x[n] $ h[n] =

! x[k ]h[n # k ]

k = #"

Discrete-time Fourier Transform (DTFT):

X ( e j$ ) =

+"

! x[n]e

# j $n

n = #"

DTFT Properties: Symmetry relations: Sequence

DTFT

x[n]

X ( e j! )

x[!n]

X ( e " j! )

x * [ ! n]

X * ( e j! )

Re{x[n]}

j Im{x[n]}

1 X ( e j! ) + X * ( e " j! ) 2 1 X as (e j! ) = X (e j! ) " X * (e " j! ) 2

[ [

X cs (e j! ) =

xcs [n]

X re (e j! )

xas [n]

jX im (e j! )

] ]

Note: Subscript “cs” and “as” mean conjugate symmetric and conjugate anti-symmetric signals, respectively. DTFT of commonly used sequences: Sequence

DTFT

! [n]

1

µ[n] (Unit Step Function)

+"

1 1 # e # j%

+

!$& (% + 2$k )

k = #"

+"

e j! 0 n

! 2$& (% # %

0

+ 2$k)

k = #"

a n µ[n] , ( a < 1 ) (n + 1)a n µ[n] , ( a < 1 )

h[n] =

sin("c n) , ( "! < n < +! ) !n

1 1 " ae " j! 1 & # $ ! % 1 ' ae ' j( "

2

$!1 0 % ' % 'c H ( e j' ) = # !"0 'c < ' < &

DTFT Theorems: Theorem

Sequence

DTFT

g[n]

G ( e j! )

h[n]

H ( e j! )

Linearity

"g[n] + !h[n]

# G ( e j! ) + " H ( e j! )

Time Reversal

g[ ! n]

G ( e " j! )

Time Shifting

g[n ! n0 ]

e " j!n0 G (e j! )

Frequency Shifting

e j! 0 n g [ n ]

G (e j ( ! "!0 ) )

Differentiation in Frequency

ng[n]

Convolution

g[n] ! h[n]

j

G ( e j! ) H ( e j! ) 1 2#

g[n]h[n]

Modulation

+$

1

*

n = #$

N "1

Discrete Fourier Transform (DFT):

X [k ] =

!

x[n]e

"#

! G (e

j$

) H (e j (% "$ ) )d$

"#

+%

" g[n]h [n] = 2% ! G(e

Parseval’s Identity

dG (e j! ) d!

j&

) H * (e j& )d&

#%

"j

2# nk N

,

0 " k " N !1

n =0

1 x[n] = N

Inverse DFT:

N "1

# X [k ]e

j

2! nk N

,

0 " n " N !1

N],

0 " n " N !1

k =0

N "1

yc [ n] =

Circular Convolution:

! x[m]h[$n " m#

m =0

Symmetry properties of DFT: Length-N Sequence

N-Point DFT

x[n] = xre [n] + jxim [n]

X [k ] = X re [k ] + jX im [k ]

x * [ n]

X *["# k ! N ]

x*["# n! N ]

X *[ k ]

xcs [n]

1 X [k ] + X *["# k ! N ] 2 1 X as [k ] = X [k ] # X * ["# k ! N ] 2 X re [k ]

xas [n]

jX im [k ]

xre [n] jxim [n]

X cs [k ] =

[

]

[

]

DFT Theorems: Theorem

Sequence

DTFT

g[n]

h[n]

G[k ] H [k ]

Linearity

"g[n] + !h[n]

"G[k ] + !H [k ]

Circular Time Shifting

g[# n " n0 ! N ]

e

Circular Frequency Shifting

e

j

2! nk0 N

2! kn0 N

G[k ]

G[# k " k 0 ! N ]

g[ n]

G[n]

Duality

"j

Ng["# k ! N ]

N "1

! g[m]h[$n " m#

N-Point Circular Convolution

G[k ]H [k ]

N]

m =0

1 N

g[n]h[n]

Modulation

N "1

!

Parseval’s Identity

n =0

+"

Z-Transform:

X ( z) =

! x[n]z

#n

2

g[ n] =

1 N

Inverse Z-Transform:

n = #"

N "1

! G[m]H [$n " m#

N]

m =0

N "1

! G[k ]

2

n =0

x[n] =

1 G ( z ) z n "1dz 2#j

!

C

Z-Transform Properties: Property

Sequence

z-Transform

Region of Convergence

g[n]

G (z ) H (z )

Rg

h[n]

Rh

Conjugation

g [ n]

G (z )

Rg

Time Reversal

g[ ! n]

G (1 z )

1 Rg

Linearity

"g[n] + !h[n]

"G ( z ) + ! H ( z )

Includes Rg ! Rh

Time Shifting

g[n ! n0 ]

z ! n0 G ( z )

Rg except possibly the point z = 0 or !

Multiplication by an exponential sequence

! n g[n]

G (z ! )

! Rg

Differentiation in the z-domain

ng[n]

Rg except possibly the point z = 0 or !

Convolution

g[n] * h[n]

dG ( z ) dz G( z) H ( z)

*

*

!z

*

Includes Rg ! Rh

Some useful z-transform pairs: Sequence

z-Transform

ROC

! [n]

1 1

All values of z

µ[n]

! n µ[n] n! n µ[n] r n cos(!0 n) µ[n] r n sin(!0 n) µ[n]

1 ! z !1 1 1 ! "z !1

"z !1 !1 2

(1 ! "z ) 1 ! (r cos("0 ) z !1 1 ! (2r cos("0 ) z !1 + r 2 z ! 2 (r sin("0 ) z !1 1 ! (2r cos("0 ) z !1 + r 2 z ! 2

z >1 z >! z >! z >r z >r