seoul national university 1. discrete-time signals and systems

420.461: Digital Signal Processing copyright@Yong-Hwan Lee

Seoul National UniversitySchool of Electrical Engineering

1. Discrete time signals and systems

1. Discrete-time signals and systems

• Discrete - time signals: sequences

– Discrete-time signals are represented as sequences of numbers

– A sequence is a function whose domain is the set of integers

– Define the nth number in the sequence by , a set of numbers can be represented

by

– Note that

discrete-time signal analog signal

where T denote the sampling period.

– A delayed or shifted version of :

where is an integer

[ ]x n

[ ]{ },x x n n= −∞ < < ∞

[ ] ( )ax n x nT=

x

x n

n

x −1x 0

x 1

x 2

[ ]x n

[ ] [ ]0y n x n n= −

0n




– An unit sample sequence is defined by

which is also referred to as a discrete-time impulse or simply as an impulse

– An arbitrary sequence can be represented as a sum of weighted and delayed

impulses

Ex)

– The unit step sequence is defined by-3 -2 -1 0 1 2 3

4

5 6

m1

m−3

m4

[ ] [ ] [ ]k

x n x k n kδ∞

=−∞

= −∑

[ ] [ ] [ ] [ ]3 1 43 1 4x n m n m n m mδ δ δ−= + + − + −

[ ] 0; 01; 0

nn

nδ

=⎧= ⎨ ≠⎩

[ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]0

1, 00, 0

1 ...

1k

nu n

n

u n n n n k

n u n u n

δ δ δ

δ

∞

=

≥⎧= ⎨ <⎩

⇒ = + − + = −

⇒ = − −

∑-1 0 1 2 3




• Let be an exponential sequence given by

– For complex and

– Complex exponential sequence

where is called the frequency of the complex sinusoid and is called the phase

– Since

complex exponential sequences with frequencies , where k is an integer,

are indistinguishable from one another

– A sinusoidal sequence can be represented by

– Since

we consider in a frequency interval of length

or

[ ]x n

[ ] [ ]nx n A u nα=

[ ]( ) ( )

0

0cos sin

n j nj

no

x n A e e

A n j n

ωφ α

α ω φ ω φ

=

= + + +⎡ ⎤⎣ ⎦

0je ωα α= ,jA A e φ=

( )1α =

[ ] ( ) ( )0 0cos sinx n A n j A nω φ ω φ= + + +

0ω φ

[ ] ( ) 0 02 2 ,oj n j n j nj nx n Ae Ae e Aeω π ω ωπ+= = =

( )0 2 kω π+

[ ] 0cos( ), x n A n nω φ= + ∀

[ ] ( ) ( )0 0cos 2 cos ,x n A k n A nω π φ ω φ= + + = +⎡ ⎤⎣ ⎦

2 ,π0π ω π− < ≤ 00 2 .ω π≤ <

ω




– A sequence is called periodic with an integer period of N, if

– In order for a sinusoidal sequence to be

it requires for

– A sinusoidal sequence is not necessarily periodic with a period of and may not

be periodic at all, depending on the value of

Ex

• High and low frequency concept: A continuous time sinusoidal signal

oscillates more rapidly as increases. However, oscillates

more rapidly as increases from 0 toward .

But, as increases from to , the oscillation becomes slower due to symmetry with respect to .

[ ] [ ], x x x n N n= + ∀

( ) ( )0 0 cos cosA n A n Nω φ ω φ+ = + +⎡ ⎤⎣ ⎦

0 2 .N kω π=

0

0

1 2 3 631 2 No integer

kN k

N k N

πω ππ

ω π

= ⇒ = ⋅ =

= ⇒ = ⇒

0

2πω

0ω

[ ] ( )0cosx n A t φ= Ω +

0Ω [ ] ( )0cosx n A nω φ= +

0ω π

0ω 2πππ

15 cos cos

8 8A n A n

πφ π φ+ = +⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Ex :




• Discrete-time systems

– Ideal delay system

– Moving averager

– Memoryless system

A system is referred to as memoryless if the output depends only on the input

– Linear system : principle of superposition

Ex: accumulator

x n y n{}T ⋅ [ ] [ ]{ }y n T x n=

[ ] [ ] [ ] [ ]{ } [ ]1

0

1 11 ... 1N

k

y n x n x n x n N x n kN N

−

=

= + − + + − + = −∑

[ ] [ ], d dy n x n n n= − is a positive integer

[ ]y n [ ]x n

[ ] [ ]2 ,y n x n n= ∀Ex :

[ ] [ ]{ } [ ]{ } [ ]{ }1 2 1 2

T ax n bx n aT x n bT x n=+ +

[ ] [ ]n

k

y n x k=−∞

= ∑




– Time-invariant system (TIS): A time shift or delay of the input sequence results in a

corresponding shift in the output sequence

– Causality: A system is causal if the output sequence at depends only on the

input sequences for

Ex: non causal

causal

– Stability: A system is stable in the BIBO sense iff every bounded input sequence

produces a bounded output sequence

[ ] [ ]{ } [ ] [ ][ ] [ ]

[ ] [ ]

1 1

2

d d

n

k

y n T x n n y n n y n

y n x n

y n x k=−∞

= − ⇒ − =

= →

= →∑

Ex : TIS

TIS

[ ] [ ][ ] [ ]

[ ] ( ) [ ]1 1

0 0 1

:

o

y n x Mn

y x Mn x Mn n

y n n x m n n y n

=

= = −

− = − ≠ →⎡ ⎤⎣ ⎦

compressor

notTIS

[ ] [ ] [ ][ ] [ ] [ ]

1

1

y n x n x n

y n x n x n

= + − ⇒

= − − ⇒

0, 0n n n≤ ∀on n=

[ ] [ ] , X y

x n B y n B n≤ < ∞ ⇒ ≤ < ∞ ∀

[ ] [ ] 0 01 0

n

k

ny n u k

n n=−∞

<⎧= = ⎨ + ≥⎩∑




• Linear time-invariant systems

– Let be the response of the system to the input , i.e.,

– Time invariance implies

– Discrete-time convolution

– To compute first reflect about the origin to obtain and then shift

the origin to

[ ]kn n [ ]n kδ − [ ] [ ]{ }kh n T n kδ= −

[ ] [ ]{ }

[ ] [ ] [ ] [ ]{ }

[ ] [ ]

k k

kk

y n T x n

T x k n k x k T n k

x k h n

δ δ∞ ∞

=−∞ =−∞

∞

=−∞

=

⎧ ⎫= − = −⎨ ⎬

⎩ ⎭

=

∑ ∑

∑

[ ] [ ] [ ] [ ] [ ] kn h n n k h n k h nδ δ→ ⇒ − ⇒ − =

[ ] [ ] [ ] [ ] [ ] k

y n x k h n k x n h n∞

=−∞

= − Δ ∗∑[ ],h n k− [ ]h k [ ]h k−

;k n=

0 1 -2 -1 0 0 1 2 3 4 5

h k h k− h k6−




[ ] [ ] [ ]

[ ] [ ][ ]

1, 0 10, otherwise

0; 0

n

h n u n u n N

n N

x n a u n

n y n

= − −

≤ ≤ −⎧= ⎨⎩

=

< =

Ex

[ ]1

0

10 1; 1

nnk

k

an N y n aa

+

=

−≤ ≤ − = =

−∑

n N− −1b g n 0 k

[ ]1 1

1

1

1; 1

11

n N nnk

k n N

Nn N

a an N y n aa

aaa

− + +

= − +

− +

−> − = =

−

⎛ ⎞−= ⎜ ⎟−⎝ ⎠

∑

n0

y n




• Properties of LTI system

– Commutative

– Cascaded connection

Letting we have

[ ] [ ] [ ]( ) [ ] [ ] [ ] [ ]1 2 1 2x n h n h n x n h n x n h n∗ + = ∗ + ∗

[ ]x n [ ]v n [ ]y nh1 h2

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ]k

k

x n h n h n x n

y n x k h n k

x n k h k

∞

=−∞

∞

=−∞

∗ = ∗

⇒ = −

= −

∑

∑

[ ] [ ] [ ]1 2 ,h n h n h nΔ ∗

[ ] [ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

1

2

1 2

v n x n h n

y n v n h n

x n h n h n

x n h n

= ∗

= ∗

= ∗ ∗

= ∗




– A Linear system is stable iff the impulse response is absolutely summable, i,e.,

Proof

Sufficient condition:

If is bounded, i.e.,

then,

Necessary condition:

Show that if , a bounded input can cause an unbounded output.

Consider a bounded sequence given by

( ) k

S h k∞

=−∞

Δ < ∞∑

[ ] [ ] [ ] [ ] [ ]k k

y n h k x n k h k x n k∞ ∞

=−∞ =−∞

= − ≤ −∑ ∑[ ] Xx n B≤[ ]x n

[ ] [ ]Xk

y n B h k∞

=−∞

≤ ∑

[ ] [ ] k

h k y n∞

=−∞

< ∞ ⇒ <∞∑

S = ∞

[ ][ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ]

2*, 0

00 , 0

k k

h nh n h kh nx n y x k h k S

h kh n

∞ ∞

=−∞ =−∞

⎧ −≠⎪ −= ⇒ = − = = = ∞⎨

⎪ =⎩

∑ ∑




Ex

Note: The property of convolution can be used to analyze the time-invariant system

[ ] [ ]

0

11

1

1

n

n

n

h n a u n

S aa

a S

a S

∞

=

=

⇒ = =−

< ⇒ <∞

≥ ⇒ =∞⇒

∑

unstable

Accumulator Backwarddifference

x n y n x n

[ ] [ ] [ ] [ ]( )[ ] [ ][ ]

1

1

h n u n n n

u n u n

n

δ δ

δ

= ∗ − −

= − −

=

y none sampledelay

Forwarddifference

x n

δ n−1 δ δn n+ −1

Backwarddifference

[ ] [ ] [ ] [ ]( )[ ] [ ]

1 1

1

h n n n n

n n

δ δ δ

δ δ

= − ∗ + −

= − −

⇒




• Linear constant coefficients difference equations

– An Nth order linear const. difference eq. is represented by

When the right side term is equal to zero, it is called the homogeneous difference eqn.

Ex: Accumulator

Ex: Moving average

[ ] [ ]0 0

N M

k kk k

a y n k b x n k= =

− = −∑ ∑

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

1

1

1

n n

k k

y n x n x n x k x n y n

y n y n x n

−

=−∞ =−∞

= = + = + −

⇒ − − =

∑ ∑

D

[ ]x n [ ]y n+

[ ]1y n −

[ ] [ ] [ ]( )

[ ] [ ]

[ ] [ ] [ ]( ) [ ]

1

0

1

1

1

M

k

h n u n u n MM

y n x n kM

h n n n M u nM

δ δ

−

=

= − −

⇒ = −

= − − ∗

∑

M-sampledelay

x n y n∑1

M _

[ ] [ ] [ ] [ ]( )11y n y n x n x n MM

− − = − −




Ex[ ] [ ] [ ]

[ ] [ ] [ ]

[ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ][ ]

[ ]

2

1 1

1 1

1 1 2

1

1

y -1 .0;

0

1 0 0:

, 01;

2 1 1

3:

, 1

n n n n

n

y n ay n x n

x n K n cn

y ac K

y ay a c aK

y n a c a K n y n a c Ka u nn

y a y x a c

y a a c a c

y n a c n

δ

+ +

− −

− − −

+

= − +

= = ←

≥

⇒ = +

= + = +

= ⋅ + ≥ ⇒ = ⋅ +

< −

⎡ ⎤− = − − − = ⋅⎣ ⎦− = ⋅ =

= ⋅ ≤ −

Consider and auxiliary condFor

For




• Frequency-domain representation of DTS

Let

eigen-value eigen-function of the system

[ ]

[ ] [ ] [ ]

[ ]

j n

j n k

k

j n j k

k

x n e

y n h k e

e h k e

ω

ω

ω ω

∞−

=−∞

∞−

=−∞

=

⇒ =

=

∑

∑

( ) [ ]

( ) ( )( ) ( )

j

H

j j k

k

j jR

j ej

H e h k e

H e jH e

H e eω

ω ω

ω ω

θω

∞−

=−∞

Δ

= +

=

∑

[ ] ( )j j ny n H e eω ω=




Ex: Ideal delay

Since for an ideal delay system,

– For

the output of the system is

← principle of superposition

Ex:

[ ] [ ][ ]

[ ] [ ] ( )0 0 0

0

,j n

j n n j n j nj n j

y n x n n

x n e

y n e e e H e e

ω

ω ω ωω ω− − −

= −

=

= = ⇒ =

For

[ ] [ ]0h n n nδ= −

( ) [ ] ( )

( ){ }

00

0

1j nj j n j

n

j

H e n n e e H e

Arg H e n

ωω ω ω

ω

δ

ω

∞−−

=−∞

= − = ⇒ =

= −

∑

[ ] ,kj nk

k

x n e ωα=∑

[ ] ( )k kj j nk

ky n H e eω ωα=∑

[ ] ( ) ( )

[ ] ( ) ( )[ ] ( ) ( ) ( )

[ ] ( ) ( ) ( )

0 0

0 0 0

0

0

cos2

2

cos , =arg

j n j nj j

j j j nj j n j

j j j j

j j

Ax n A n e e e e

Ay n H e e e H e e e

h n H e H e H e e

y n A H e n H e

ω ωφ φ

ω ω ωφ ω φ

ω ω ω θ

ω ω

ω φ

ω φ θ θ

−−

− −−

− ∗ − −

= + = +

⎡ ⎤⇒ = +⎣ ⎦

= =

⎡ ⎤⇒ = + + ⎣ ⎦

If is real,

where




Ex: Delay

– Discrete-time LTI → periodicity of

Ex: Ideal frequency selective filters (LPF)

( )[ ] ( )( )

00 0

0 0

1;

cos

jH e n

y n A n n

ω θ ω

ω φ

= = −

⇒ = − +

2π

( )( ) [ ] ( )

[ ] ( )

( )

2 2

2

j k j k n

k

jj n

k

j

H e h n e

h n e H e

H e

ω π ω π

ωω

ω π

∞+ − +

=−∞

∞−

=−∞

⇒ =

⎡ ⎤= = ⎣ ⎦

⇒

∑

∑is periodic with a period of

− −π ω 0 −π ω 0 π 2 0π ωr 2π 2 0π ω+ω

H e jωc h




Ex: [ ] 1 21 2

11

0 otherwise

M n MM Mh n

⎧ − ≤ ≤⎪ + += ⎨⎪⎩

−π−π

−π

πθ

( )( )

( ) ( ) ( )

( )

( )

2

1

21

2 1 1 2 1 2

2 1

1 2

1

1 2

1 / 2 1 / 2 1 / 2

1 2

1 2

1 2 _

11

11 1

1 11

sin1 21 sin

2

Mj j n

n M

j Mj M

j

j M M j M M j M M

j

j M M

H e eM M

e eM M e

e e eM M e

M M

eM M

ω ω

ωω

ω

ω ω ω

ω

ω

ω

ω

−

=−

− ++

−

− + − + + − + +

−

− −

⇒ =+ +

−=

+ + −

−=

+ + −

+ +

=+

∑




• Representation of sequences by Fourier transform

– A stable sequence can be represented by a Fourier integral of the form

where

Since is periodic with period , it is of the form of Fourier series for the

continuous-variable periodic function .

– Note that Eq (A) is the inverse of Eq. (B)

[ ] ( )2

1 - - - ( )2

j j nx n X e e dπ ω ω

πω

π= ∫ A

( ) [ ]

( ) ( )

( ) ( )1

- - - ( )

j

X

j j n

n

j ej

j jR

X e x n e

X e e

X e jX e

ω

ω ω

θω

ω ω

∞−

=−∞

=

= ←

= + ←

∑ B

polar cord. form

rectangular cord. form

( )jX e ω 2π

( )jX e ω

[ ] ( ) [ ]

[ ] ( ) [ ]

( ) ( )( ) [ ]

2

2

1 12 2

12

sin1 2

j j n j n j n

n

j n m

n

j n m

x n X e e x m e e d

x m e d x n

n me d n m

n m

π πω ω ω ω

π π

π ω

π

π ω

π

ωπ π

ωπ

πω δ

π π

∞−

− −=−∞

∞−

=−∞

−

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

= =

⎛ ⎞−= = −⎜ ⎟

−⎝ ⎠

∑∫ ∫

∑ ∫

∫∵




– If is absolutely summable (i.e., stable sequence), exists.

– Furthermore, can be showen to converge uniformly to a continuous function of

⇒ Any stable sequence (or system) have a finite and continuous frequency response:

(Sufficient condition for existence of )

– A finite-length sequence is absolutely summable

Ex:

– If a sequence is not absolutely summable, but square summable ; i.e.,

it can be represented by a Fourier transform with mean-square convergence, i.e.,

( ) [ ]

[ ] [ ]

j j n

n

j

n n

X e x n e

x n e x n

ω ω

ω

∞−

=−∞

∞ ∞−

=−∞ =−∞

=

≤ ≤ < ∞

∑

∑ ∑

[ ]x n ( )jX e ω

[ ]x n .ω

( )jH e ω

[ ] [ ]

( ) [ ]

1 , 11

n

j n j n n j n

n n

jj

x n a u n

X e a u n e a e

ae aae

ω ω ω

ωω

∞ ∞− −

=−∞ =−∞

−−

=

= =

= = <−

∑ ∑

if

[ ] 2

n

x n∞

=−∞

< ∞∑

( ) [ ]2

lim 0M

j j n

M n MX e x n e d

π ω ω

πω−

−→∞=−

− =∑∫




Ex: Ideal LPF

non causal and not absolutely summable

is discontinuous at

But is mean-square summable

Ex: : neither absolutely nor square summable

Since

When

is not finite for all

( )

( )

1

0

1 sin , 2

c

c

cjLP

c

j n cLP

H e

nh n e d nn

ω

ω ω

ω

ω ω

ω ω π

πωωπ π−

⎧ <⎪= ⎨< <⎪⎩

⇒ = = ∀

⇒

∫

( )jLpH e ω

cω ω=

[ ] 1, x n n= ∀

( ) ( )2 2j

k

X e kω πδ ω π∞

=−∞

⇒ = +∑( ) ( )

[ ] ( )0

0

0

2 2

1 22

j

k

j n

j n

X e k

x n e d

e

ω

π ω

π

ω

πδ ω ω π

πδ ω ω ωπ

∞

=−∞

−

= − +

⇒ = −

= ⇒

∑

∫[ ]0 0, 1x nω = =

( )jX e ωϖ




– Theory of generalized functions

– Conjugate symmetric sequence:

– Conjugate anti-symmetric sequence:

where

( ) ( )

[ ] ( ) ( )

002 2

2 2m

j n

n k

j n jm m m

m k m

e k

x n a e X e a k

ω ω

ω ω

πδ ω ω π

π δ ω ω π

∞ ∞−

=−∞ =−∞

∞

=−∞

= − +

⇒ = ⇒ = − +

∑ ∑

∑ ∑∑

[ ] [ ]*e ex n x n= −

[ ] [ ]*o ox n x n= − −

[ ] [ ] [ ]

[ ] [ ] [ ]( )

[ ] [ ] [ ]( )

1212

e o

e

o

x n x n x n

x n x n x n

x n x n x n

∗

∗

⇒ = +

= + −

= − −

[ ] [ ][ ] [ ]

:

: e e

o o

x n x n

x n x n

= −

= − −

even sequence

odd sequence

( ) ( ) ( ) ( ) ( )j j j j je o R IX e X e X e X e jX eω ω ω ω ω⇒ = + = +

( ) ( ) ( )( )( ) ( ) ( )( )

1212

j j je

j j jo

X e X e X e

X e X e X e

ω ω ω

ω ω ω

∗

∗ −

= +

= −




– Note that

Ex:

( ) ( ) ( ) ( ) j j j je oX e X e X e X eω ω ω ω∗ − ∗ −= = −and

[ ]{ } ( ) ( )j jx n X e X eω ω∗↔ =Re[ ]{ } ( )[ ]{ } ( )[ ] ( )[ ] ( )

Re

Im

je

jo

je R

jo I

x n X e

j x n X e

x n X e

x n jX e

ω

ω

ω

ω

↔

↔

↔

↔

( )

( ) ( )

( ) ( ) ( )( ) ( )( ) ( )

2

1

1 11

11

1 1 cos1 2 cos

sintan1 cos

jj

j jj

j jR R

j jI I

j jX X

X e aae

X e X eae

X e a X ea a

X e X e

ae ea

ωω

ω ωω

ω ω

ω ω

ω ω

ωω

ωθ θω

−

∗−

−

− −

= <−

= =−

= − =+ −

= −

−= = −

−




• Fourier transform theorems

– Linearity:

– Delay:

– Time reversal:

If is real,

– Differentiation:

– Parseval’s theorem:

[ ]{ } ( )[ ] ( ){ }

[ ] ( )

1

j

j

F j

F x n X e

x n F x e

X n X e

ω

ω

ω

−

=

=

⇒ ←⎯→

[ ] [ ] ( ) ( )1 2 1 2F j jax n bx n aX e bX eω ω+ ←⎯→ +

[ ] ( )( )

0

0 0

0

( )

j nF j

j n j nF

x n n X e e

e X e

ωω

ω ω ω

−

−

− ←⎯→

←⎯→

[ ] ( )F jx n X e ω−− ←⎯→

[ ] ( )F jx n X e ω∗ +− ←⎯→[ ]x n

[ ] ( )F jdnx n j X ed

ω

ω←⎯→

[ ] ( )2

21 2

j

n

E x n X e dπ ω

πω

π

∞

−=−∞

Δ =∑ ∫




– Convolution:

periodic convolution

⇒ Multiplication of two sequences is equivalent to periodic convolution of

corresponding Fourier Transforms.

Ex:

[ ] ( ) [ ] ( )[ ] [ ] [ ] [ ] [ ] ( ) ( ) ( )

, F Fj j

F j j j

k

x n X e h n H e

y n x k h n k x n h n Y e X e H e

ω ω

ω ω ω∞

=−∞

←⎯→ ←⎯→

⇒ = − = ∗ ←⎯→ =∑

[ ] [ ] [ ] ( ) ( ) ( )( )12

jF j jy n x n h n Y e X e H e dπ ω ξω ξ

πξ

π−

−= ←⎯→ = ∫

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] ( )

1 11 12 4

1 111 1 14 41 1 1 1 14 2 1 1 12 2 2

j j

j

j j j

y n y n x n x n

e en n h n h n H e

e e e

ω ω

ω

ω ω ωδ δ

− −

− − −

− − = − −

−⇒ − − = − − ⇒ = = −

− − −

[ ] [ ] [ ] [ ]11 1 1 1; 1 1

1 2 4 2

n nFn

ja u n h n u n u nae ω α

−

−⎛ ⎞ ⎛ ⎞←⎯→ < ⇒ = − −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠




• Discrete-time Random Signals

– Consider an LTI system with impulse response

When is a WSS discrete-time random process,

– If stationary,

[ ]h n

[ ] [ ] [ ]k

y n h n k x k∞

=−∞

= −∑[ ]x n

[ ] [ ]{ } [ ] [ ]{ } , X Yn E X n n E y nμ μΔ Δ

[ ]X Xnμ μ=

[ ] [ ] [ ]

[ ] [ ]{ } [ ] [ ]{ }

[ ]

( )0

Yk

k k

Xk

jX

n E h n k x k

h n k E x k h k E x n k

h k

H e

μ

μ

μ

∞

=−∞

∞ ∞

=−∞ =−∞

∞

=−∞

⎧ ⎫= −⎨ ⎬

⎩ ⎭

= − = −

=

=

∑

∑ ∑

∑




– Autocorrelation of [ ]y n[ ] [ ] [ ]{ }

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]{ }

[ ] [ ] [ ] [ ]

,yy

k l

k l

xx yyk l

n n m E y n y n m

E h k h l x n k x n l m

h k h l E x n k x m m l

h k h l m k l m

φ

φ φ

∞ ∞

=−∞ =−∞

+ = +

⎧ ⎫= − − +⎨ ⎬

⎩ ⎭= − + −

= + − =

∑ ∑

∑∑

∑∑

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

.

:

yy xxi k

hk

yy h xxi

l k i

m m i h k h i k

h n i h k h i k

m i m i

φ φ

φ

φ φ φ

∞ ∞

=−∞ =−∞

∞

=−∞

∞

=−∞

− Δ

= − +

Δ +

⇒ = −

∑ ∑

∑

∑

Let Then,

Autocorrelation sequence of

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )

2

2

22

1 1: 0 02 2

j j j j jyy h xx xx

j j j jh

j j jyy yy xx

e e e H e e

e H e H e H e

E y n e d H e e d

ω ω ω ω ω

ω ω ω ω

π πω ω ω

π πφ ω ω

π π

∗

− −

⇒Φ =Φ Φ = Φ

Φ = =

= = Φ = Φ ≥∫ ∫

where

Note

( )jH e ω

ω a ω b



2. Periodic sampling of continuous-time signals

2. Periodic Sampling of Continuous-time Signals• Impulse sampling: ideal continuous-to–discrete (C/D) converter

where is the sampling period and is called the sampling frequency

( )cx t ( )sx t

( ) ( )s t t nTδ= −∑

[ ] ( ),cx n x nT n= −∞ < < ∞

( ) ( ) ( )

( ) ( )

s cn

cn

x t x t t nT

x nT t nT

δ

δ

∞

=−∞∞

=−∞

⇐= −

= −

∑

∑

shifting property of the impulse function

1

s

Tf

⎛ ⎞=⎜ ⎟⎝ ⎠

sf




– Fourier transform of

( ) ( ) ( )

( )1

1

s cn

cn

cn

X f X f F t nT

nX f fT T

nX fT T

δ

δ

∞

=−∞

∞

=−∞

∞

=∞

⎧ ⎫= ∗ −⎨ ⎬

⎩ ⎭⎛ ⎞= ∗ −⎜ ⎟⎝ ⎠

⎛ ⎞= − ⇐⎜ ⎟⎝ ⎠

∑

∑

∑ periodic function

( )sx t

( ) ( ){ } ( ) ( )

( )

2

2

j fts s c

n

j fnTc

n

X f F x t x nT t nT e dt

x nT e

π

π

δ∞

−

=∞

∞−

=∞

= = −

=

∑∫

∑

− f s W− 0 W sf W− f s

( )cX f

W− W

( )sX f

2

s sf W W f W− ≤ ⇔ ≤⇒ Aliasing

= 2 / 2s sT fπ πΩ =




• Natural Sampling

( )cx t ( )sx t

( )p t0 τ T T +τ

2 1( ) ; sinsj f ntn n

n

np t p e p cT T

π τ∞

=−∞

= =∑

( )sX f

sf− sf0

( ) ( ) ( )

( ) 2

( )

s

s c

j f nts c n

n

n cn

x t x t p t

X f F x t p e

np X fT

π∞

=−∞

∞

=−∞

=

⎧ ⎫⇒ = ⎨ ⎬

⎩ ⎭⎛ ⎞= −⎜ ⎟⎝ ⎠

∑

∑




• Sample and hold (S/H)

( )cx t ( )sx t

0 τ

( )s t( )p t

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

1

1

s cn

s cn

n

x t x t t nT p t

nX f X f f p fT T

np f X fT T

δ

δ

∞

=−∞

∞

=−∞

∞

=−∞

⎡ ⎤= − ∗⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⇒ = ∗ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞= −⎜ ⎟⎝ ⎠

∑

∑

∑ ( )sX f

sf− sf0

1( ) sincP f fT T

τ⎛ ⎞= ⎜ ⎟⎝ ⎠




• Nyquist Sampling Theorem

– If is strictly band limited, i.e.,

and the sampling frequency is chosen such that

then

Since

if the sample values are specified for all time, is uniquely determined

by using the Fourier series.

⇒ is uniquely determined by

( )cx t ( ) 0 X f f W= >for1 2 ,sf WT

= =

( ) ( ) 2j fnTs c

n

X f x nT e π∞

−

=−∞

= ∑

( ) ( )1 , ,2c sX f X f f WW

= ≤

( )cx nT ( )cX f

( )cx t ( ), 0, 1, 2, .cx nT n = ± ± ⋅ ⋅ ⋅




• Reconstruction of a band limited signal from its samples: Interpolation formula

⇒ A band-limited signal of finite energy can be completely recovered by its samples

taken at a rate of 2W/sec

– Ideal reconstruction filter

( ) ( )

[ ]

[ ] ( )

[ ] ( )( )

[ ]

2

2

2

12

12sin /

/

sin /( ); ( )/

j ftc c

fjW j ftWW

n

W j f t nT

Wn

n

n

x t X f e df

x n e e dfW

x n e dfW

t nT Tx n

t nT T

t Tx n h t nT h tt T

π

ππ

π

ππ

ππ

∞

−∞

∞ −

−=−∞

∞−

−=−∞

∞

=−∞

∞

=−∞

=

=

=

−=

−

= − =

∫

∑∫

∑ ∫

∑

∑

( )H f

f−

12T

12T

sin /( )/t Th t

t Tπ

π=




• Poisson sum formula

A periodic signal with period T is represented by the Fourier series,( )sx t

( )

( )

( ) ( )

22

2

2 22

2

2

1

1

1 1

1

;s

s

s s

Tj f nt

Tn s

Tj f nt j f nt

Tnn

j f nts n s

n

n

x x t e dtT

t nT e x e dtT T

t x e fT

T

x

π

π π

π

δ δ τ

−

−

∞ ∞−

−=−∞ =−∞

∞

=−∞

=

⎛ ⎞− = = =⎜

⎝ ⎠

=

⎟

=

∫

∫

∑

∑ ∑ ∵

where

Ex :

( ) ( )x t t nTδ= −∑ ( )y th tb g

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

2 2

2

21

1 1

1

s s

s

s

n n

j f nt

n

j f n t j f nt

j f nts

n n

sn n

y t h t t nT h t nT

y t h t eT

h e d H

h t nT H nf

nT

e

e

T

fT

π

π τ π

π

δ

τ τ

∞ ∞

=−∞ =−∞

∞

=−∞

∞ ∞−

=−

∞ ∞

=−∞ =−

∞ =−∞

∞

= ∗ − = −

⎡ ⎤= ∗ ⎢ ⎥⎣ ⎦

−

= =

=⇒

∑ ∑

∑

∑

∑

∑∫

∑




• Discrete-time processing of continuous-time signals

low passreconstruction filter

– The freq-domain representation is easier than the time-domain representation

x tcb g x n y n y tr b g

T T

C D/ h n D C/

discrete - timesystem

[ ] ( ) ( )

( ) [ ]( )

( )

( ) ( ) ( )

( )

1 2 ;

sin;

,

0 , otherwise

jc c

k

rn

j Tr r

j T

kx n x nT X e XT T T

t nTTy t y n

t nTT

Y H Y e

TY eT

ω ω π

π

π

π

∞

=−∞

∞

=−∞

Ω

Ω

⎛ ⎞= ⇒ = −⎜ ⎟⎝ ⎠

−

=−

⇒ Ω = Ω

⎧ Ω <⎪= ⎨⎪⎩

∑

∑




• LTI discrete-time systems

– If is band limited and the sampling rate is larger than the Nyquist rate,

the effective frequency response is

( ) ( ) ( )j j jY e H e X eω ω ω=

( ) ( ) ( )r eff cY H XΩ = Ω Ω

( ) ( ),0 , otherwise

j T

eff

H eH T

πΩ⎧ Ω <⎪Ω = ⎨⎪⎩

( )cX Ω

( ) ( ) ( ) ( )( ) ( )

( ) ( )

;

1 2

,

0 , otherwise

j T j Tr r

j Tr c

k

j Tc

Y j H H e X e T

kH H e XT T

H e XT

ω

π

π

Ω Ω

∞Ω

=−∞

Ω

⇒ Ω = Ω =Ω

⎛ ⎞= Ω Ω −⎜ ⎟⎝ ⎠

⎧ Ω Ω <⎪= ⎨⎪⎩

∑




• Example of signal reconstruction




Ex: Ideal LPF

Ex: Ideal band-limited differentiator

( ) ( ) ( )

( )

( )

[ ] 2

,

0 , otherwise

,

cos , 0cos sin

0 , 0

c c c

eff

j

dy t x t H jdt

jH T

jH eT

n nn n nh n nTn T n

ω

π

ω ω π

ππ π π

π

= ⇒ Ω = Ω

⎧ Ω Ω <⎪⇒ Ω = ⎨⎪⎩

= <

⎧ ≠− ⎪⇒ = = ⎨⎪ =⎩

( )1, 0, otherwise

ceff

TH

ω⎧ Ω <Ω = ⎨

⎩




– Impulse invariance

⇒ When , the discrete-time system is said to be an impulse-invariance

version of the continuous-time system.

Ex:

The resulting discrete-time frequency response is aliased

( ) ( ); jcH e H H

Tω ω

ω π= = Ω <⎛ ⎞⎜ ⎟⎝ ⎠

[ ] ( )ch n Th nT=

( ) ( ) ( )

( ) ,

( )1 2

1

j j Ts T

jc

k

jc

X X ekH e HT T T

H eT

e

HT

Xω

ω

ω

ω

ω π

ω ω π

∞

=−

Ω

Ω∞

=

⎛ ⎞= − ⇐⎜ ⎟⎝ ⎠⎛ ⎞⇒ = <⎜ ⎟⎝ ⎠

Ω = =∑∵

[ ] ( ) [ ]

( )

0

0

0 0

0

1

( ) ( ) ( )

( ) 1 1

( )

s tc c

s nTc

jcs T s T j

c

Ah t Ae u t H ss s

h n Th nT Ae u n

AT ATH z H e He z e e Th t

ωω

ω− −

= ⇒ =−

= =

⎛ ⎞= = ≠ ⎜ ⎟− − ⎝ ⎠∵

or

is not strictly band - limited




• Continuous-time processing of discrete-time signals

( ) [ ]( )

( ) ( )

( ) [ ]( )

( ) ( ) ( ) ( )

sin; ( ) ,

sin; ;

j Tc c

n

c c c cn

t nTTx t x n X T X e

t nT TTt nTTy t y n Y H X

t nT TT

ππ

π

ππ

π

∞Ω

=−∞

∞

=−∞

−

= Ω = Ω ≤−

−

= Ω = Ω Ω Ω <−

∑

∑

x tc b gx n y n( )cy t

T T

C D/( )ch tD C/

( ) ( ) ( ); /j Tc effH e H H TπΩ = Ω = Ω Ω ≤⇒

( )( ) ( ) ( ) ( )

1 ,jc

j jc c

Y e YT T

H X H e X e

ω

ω ω

ω ω π⎛ ⎞= <⎜ ⎟⎝ ⎠

= Ω Ω =




Ex: Non-integer delay ( ) ,j jH e eω ω ω π− Δ= <

[ ] [ ] [ ] [ ]( )

( ) ( )[ ] ( )

[ ]( )

( )

[ ] ( )( ) [ ]

2

,

,

sin

sin s

j f Tc

c c

c

k

t nT

k

y n x n h n n

H f e

y t x t T

y n x nT T

t T kTTx k

t T kTT

n kx k h n

n k

π

δ

π

π

ππ

− Δ

∞

=−∞

=

∞

=−∞

Δ = − Δ ⇒ = − Δ

Δ =

= − Δ

⇒ = − Δ

− Δ −

=− Δ −

− − Δ= ⇒ =

− − Δ

∑

∑

If is an integer

If is not an integer

( )( )

in

1 . ., ;2

nn

e g

ππ

− Δ− Δ

Δ =

( )2c cTy t x t⎛ ⎞= −⎜ ⎟

⎝ ⎠

0 T 2T 3TT−




– Ex: Moving average with non-integer delay

When M=5,

for ( )( )

[ ] cos 0.25

[ ] 0.308cos 0.25 2.5

x n n

y n n

π

π

=

⇒ = −⎡ ⎤⎣ ⎦

[ ]1 , 0

10, otherwise

n Mh n M

⎧ ≤ ≤⎪= +⎨⎪⎩

( ) 2

( 1)sin1 2 , 1 sin

2

Mjj

M

H e eM

ωω

ω

ω πω−

+

⇒ = <+

[ ] 2My n w n⎡ ⎤= −⎢ ⎥⎣ ⎦




• Changing the sampling rate

– Decimation by M (sampling rate compressor)

Let Then,

[ ] [ ] [ ] ( )

( )

( )

1 2

c d c

jc

k

x n x nT x n x nM x nMT

kX e XT T T

ω ω π∞

=−∞

=

⎛

⇒ Δ =

⎞= −⎜ ⎟⎝ ⎠

∑

( )

1

0

'

21

'

0

1 1 2 2

1

1 2'1 2

M

ci k

iM jM

i

jd c

c

i

kX e XT T T

X i kMMT

kXM T M

MT M

M

T

T T

X e

ω

ω π

ω

ω π

π π

ω π

∞

=−∞

∞

=−

− ∞

= =−∞

∞

−−

=

⎡ − ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞

=

⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= − ⇐ = +⎜ ⎟

⎜ ⎟⎝ ⎠

⎝ ⎠

∑ ∑

∑

∑

∑

.T MT′ =

x nM↓

[ ] [ ]dx n x nM=




– Decimation by M (cont.)

To avoid aliasing, a LPF is

required before down-sampling

Example: When M=3:

[ ]x nx nT

LPF

[ ]dh n'T MT=

[ ]dx nM↓




– Interpolation by L (sampling rate expander)

[ ] ( );

i cTx n x nT TL

nxL

′ ′= =

⎡ ⎤= ⎢ ⎥⎣ ⎦

[ ]ex nx nT ' /T T L=L↑ LPF

[ ]ix n

[ ]ih n

[ ]

[ ] [ ]

( ) [ ] [ ]

[ ] ( )

, 0, , 2 ,...

0 , otherwise

e

k

j j ne

k k

j Lk j L

k

nx n L Lx n L

x k n kL

X e x k n kL e

x k e X e

ω ω

ω ω

δ

δ

∞

=−∞

∞ ∞−

=−∞ =−∞

∞−

=−∞

⎧ ⎡ ⎤ = ± ±⎪ ⎢ ⎥= ⎣ ⎦⎨⎪⎩

= −

⎛ ⎞⇒ = −⎜ ⎟

⎝ ⎠

= =

∑

∑ ∑

∑

W− WΩ

/ Lπ0

2πω = ′ΩT

π−π2π− 2πTω =Ω

( )jX e ω

( )cX Ω

( )jeX e ω

/ Lπ−3 / Lπ− 3 / Lπω = ′ΩT

/ Lπ/ Lπ−2π−

( )jiX e ω

L

π π

H eijωc h

L

π−

L

π− 2π2π−




– D/C conversion

In practice, we use an approximate LPF.

– Linear interpolation

[ ]

[ ] [ ] [ ] '

sin 1 00 , 2 ...

( )

i

i i c ck

nnLh n n n L L

Ln nTx n x k h n k x x nT xL L

π

π=⎧

= ⇒ ⎨ = ± ±⎩

⎡ ⎤ ⎛ ⎞= − = = = ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠∑

[ ]

[ ] [ ] [ ]

[ ] [ ]

( )

2

1 ,

0, otherwise

sin1 2sin

2

ek

m

j

nn Lh n L

x n x k h n k

x m h n mL

L

H eL

ω

ω

ω

⎧− <⎪= ⎨

⎪⎩= −

= −

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

∑

∑

π5

25π 4

5π

H eijωc h

H e Ljωc h: = 5




– Changing the sampling rate by a non-integer ratio

[ ]ex nx n [ ]ix n

TL

LPFLPFL↑[ ]dx n

M↓MTL

TLT

L

T

[ ]ex nx n [ ]dx n[ ]ix nM↓LPFL↑

LRM

=

min ,

G L

c L Mπ πω

=

⎛ ⎞= ⎜ ⎟⎝ ⎠




– Changing the sampling rate by a non-integer ratio (cont.)LRM

=




• Multirate signal processing

– Interchange the filtering and the down sampling process

[ ]x n [ ]ax n [ ]y nM↓ ( )H z

[ ]x n( )MH z

[ ]bx n [ ]y nM↓

( ) ( ) ( )j j M jbX e H e X eω ω ω=

( )

( )( )

( )

( ) ( )

21

0

212

0

21

0

1

1

1

iM jj Mb

i

iM j j iM

i

iM jj M

i

j ja

Y e X eM

X e H eM

H e X eM

H e X e

ω πω

ω πω π

ω πω

ω ω

−−

=

−−−

=

−−

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞

= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎝ ⎠

=

∑

∑

∑

( )21

0

1 ( )iM jj M

ai

X e X eM

ω πω

−−

=

= ∑




• Multirate signal processing

– Interchange the filtering and the down sampling process

– Interchange the filtering and the interpolation process

( ) ( ) ( )j j M jbX e H e X eω ω ω=

( ) ( )( )

( ) ( ) ( )

2 21 12

0 0

21

0

1 1

1

i iM Mj j j ij M Mb

i i

iM jj j jMa

i

Y e X e X e H eM M

H e X e H e X eM

ω π ω πω πω

ω πω ω ω

− −− −−

= =

−−

=

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞

= =⎜ ⎟⎝ ⎠

∑ ∑

∑

( )21

0

1 ( )iM jj M

ai

X e X eM

ω πω

−−

=

= ∑

[ ]x n ( )H z [ ]ax n [ ]y nL↑

[ ]x nL↑

[ ]bx n [ ]y n( )LH z

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

j j L j L j La

j j L j j L j j L j Lb b

Y e X e X e H e

X e X e Y e H e X e X e H e

ω ω ω ω

ω ω ω ω ω ω ω

= =

= ⇒ = =

[ ]x nM↓

[ ]ax n [ ]y n( )H z

[ ]x n ( )MH z [ ]bx n [ ]y nM↓




– Poly phase implementation of interpolation filters

Note that only every Lth sample of is nonzero

Consider decomposition of an impulse response by

By successively delaying these subsequences, we can reconstruct by

[ ]x nL↑

[ ]u n [ ]y n( )H z

[ ]u n

z

...

[ ]h n [ ]0e n[ ]h n [ ]0h n

[ ]1h n +

[ ]2h n +

[ ]1h n M+ −

[ ]1e n

[ ]2e n

[ ]1Me n−

L↓ L↑

z

z

L↑

L↑

L↓

L↓

1z−

2z−

+

L↓ L↑ ( )1Mz− −

...

...

[ ] [ ] [ ]k ke n h nL k h nL= + =

[ ] [ ], , 0, 1, 2,0 , otherwisek

h n k n mL mh n

+ = = ± ± ⋅ ⋅ ⋅⎧= ⎨⎩

[ ] [ ]1

0

L

kk

h n h n k−

=

= −∑

[ ]h n

[ ]h n




z

[ ]h n [ ]0e n

[ ]1h n +

[ ]1h n M+ −

[ ]1e n

[ ]1Me n−

L↓ L↑

z

L↑L↓

L↓ L↑

...

...

1z−

1z−

+

+

+

:

1z−

[ ]x n

[ ]y n

( )0LE z

1z−...

( )1L

ME z−

+( )1LE z

:

( )0E z +[ ]x n L↑

+L↑

L↑

( )1E z

( )1LE z−

1z −

1z −

[ ]y n

⇒

⇒




• Digital processing of analog signals

– Quantization error:

– Assuming that the error sequence is uncorrelated with and it is uniformly

distributed over

– For a (K+1)- bit quantizer with full-scale value

SQNR is increases approximately 6dB for each bit added to the quantizer.

Anti-aliasing

filter

( )cx t ( )xa tS&H

Compensatedreconstructure

filterD/ADiscrete-time

systemA/Dy n⎡ ⎤⎣ ⎦x n⎡ ⎤

⎣ ⎦x n⎡ ⎤⎣ ⎦ ( )y t

[ ]x nQ

[ ]x n

[ ]q n

⇔[ ]x n [ ]x n

↑+

[ ] [ ] [ ]ˆ2 2

q n x n x nΔ Δ− < = − ≤

[ ]q n [ ]x n

, ,2 2Δ Δ⎡ ⎤−⎢ ⎥⎣ ⎦ 2

2 22

2

1 12q q dqσ

Δ

Δ−

Δ= ⋅ =

Δ∫

,mX2 2

2 212

km

qXσ

−

=

2

210 log 6.02 10.8 20logX m

q X

XKσσ σ

= = + −SQNR

⇒

Digital Signal Processing Yong-Hwan Lee


Transform analysis of L.T.I. systems• Frequency response of LTI systems

– Ideal frequency-selective filters

x n y nh n

[ ] [ ] [ ] ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )j j

k

j ArgH e ArgX ej j j j j

y n x k h n k Y z H z X z

Y e H e X e H e X e eω ω

ω ω ω ω ω +

= − ⇔ =

⇒ = =

∑

[ ] j nx n e ω=

( )

( ) ( )[ ] [ ] [ ]

[ ]

c

1,

0, otherwise

1 :

sin

cjlp

j jhp lp

hp lp

c

H e

H e H e

h n n h n

nnn

ω

ω ω

ω ω

ω ω π

δ

ωδπ

⎧ ≤⎪= ⎨< ≤⎪⎩

⇒ = −

⇒ = −

= − ⇐

ideal high - pass filter

not computationally realizable

↑complex gain (or eigenvalue)

eigenfunction



– Ideal delay

– Low-pass filter with linear phase (or delay)

– Group delay: A measure of the nonlinearity of the phase

Example

⇒ The time delay of the envelope of narrowband signal centered at is

given by the negative of the slope of the phase at

( )

[ ] ( )( )

c

,

0 ,

sin,

dj ncj

p

c dhp d

d

eH e

n nh n n

n n

ωω ω ω

ω ω π

ωπ

−⎧ ≤⎪= ⎨≤ ≤⎪⎩

−⇒ = ←

−Regardless of it is still noncausal

( ) ( )( )Arg jdI H ed

ωωω

= −

( )( ) 0Arg jdH e nω φ ω≈ − −

[ ] [ ][ ] [ ] ( )( )

0

0 0

cos ,

cosd d

x n s n n

y n s n n n n

ω

ω φ

=

≈ − − −

For an input

[ ]s n 0ω

0 .ω

[ ] [ ]( ) 1 ,d d

id d

j n j njid

h n n n

H e e eω ωω

δ

ω π− −

= −

⇒ = = <



• Linear constant coefficient difference equation

Example

[ ] [ ]

( ) ( )

( ) ( )( )

( )

( )

0 0

0 0

0

0

10

1

10

1

1 zeros: ; poles:

1

N M

k kk k

N Mk k

k kk k

Mk

kkN

kk

k

M

kk

k kN

kk

a y n k b x n k

a z Y z b z X z

b zY zH z

X z a z

b c zz c z d

a d z

= =

− −

= =

−

=

−

=

−

=

−

=

− = −

⇒ =

⇒ Δ = ⇐

−= ⇒ = =

−

∑ ∑

∑ ∑

∑

∑

∏

∏

It does specifythe impulse response of an

not uniquel

LTI s

y

ystem

( ) ( ) ( )( )

[ ] [ ] [ ] [ ] [ ] [ ]

21 1 2

1 21 1

1 1 2 1 3 zeros: 1, 1; poles: ,1 31 3 2 411 14 82 4

1 31 2 2 1 24 8

z Y zz zH z z zX zz zz z

y n y n y n x n x n x n

− − −

− −− −

+ + += = = ⇒ = − − = −⎛ ⎞⎛ ⎞ + −− +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⇒ + − − − = + − + −



– Causal system

is a right-sided sequence

⇒ The region of convergence of should be outside the outermost pole

– Stable system: absolutely summable

– Example

To be causal, ROC ⇒

To be stable, ROC ⇒

– In order for an LTI system to be both stable and causal, the ROC must be outside the outermost pole and include the unit circle, i.e., all poles inside the unit circle

– Depending upon the choice of ROC, the same difference equation results in a different

impulse response

[ ]h n

[ ] [ ] 1n

n nh n h n z z

∞−

=−∞

< ∞⇔ <∞ =

⇒

∑ ∑The ROC of H(z) includes the unit ci

for

rcle

[ ] [ ] [ ] [ ]

( )( )1 2 1 1

5 1 22

1 15 11 1 1 22 2

y n y n y n x n

H zz z z z− − − −

− − + − =

⇒ = =⎛ ⎞− + − −⎜ ⎟⎝ ⎠

12

2

2z >

1 2 2

z< <

( )cH z



– Inverse system

Let be

To hold this equation, the ROC of Hi(z) and H(z) must overlap

If H(z) is causal, the ROC is

⇒ The ROC of should overlap with

Example 1:

Example 2:

( )H zi ( ) ( )1H zi H z

=

( ) ( ) ( ) [ ] ( ) [ ] [ ] 1 i iG z H z H z g n h n h n nδ= ⇒ = ∗ =

max kkz d>

( )iH z max kkz d>

( )

( )

[ ] [ ] [ ]

1

1

1

1

1

1 0.5 : z 0.91 0.9

1 0.9 : z 0.5 ( z 0.9)1 0.50.5 0.9 0.5 1

i

n ni

zH zz

zH zz

h n u n u n

−

−

−

−

−

−= >

−−

⇒ = ⇒ > >−

⇒ = − ⋅ −

with ROC

ROC to be overlapped with

( )

( )

[ ] [ ] ( ) [ ][ ] [ ] [ ]2 2; 2 2 1 8 2 1iz h n u n u n> = − ⋅ + ⋅ ⋅ − ⇒For an ROC unstable & causal

1

1

1

1

11

1

0.5 z 0.91 0.9

2 1.82 1 2

2; 2 2 1 1 8 2

i

n ni

n n

zH zz

zH zz

z h n u n u n

−

−

−

−

−

−

−= >

−− +

⇒ =−

< = ⋅ − − − ⋅ ⋅ − ⇒

with ROC:

For an ROC stable & noncausal



If H(z) is causal with zeros at , will be causal iff the Roc of is

If is stable, the RoC of must include the unit circle,

H(z) and its inverse are stable and causal iff both poles and zeros of H(z) are

inside the limit circle. ⇒ minimum-phase system

– Impulse response of for rational system functions

If causal,

Example: A first-order IIR filter

( )1

0 1

,1

M N Nk

k k

AH z B z M N

d z

−−

−= =

= + ≥−

∑ ∑

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]0 1

0 1

, M N N

nk k

k

M N Nn

k kk

h n B n A d u n M N

y n B x n A d u n x n

δ−

= =

−

= =

= − + ≥

⇒ = − + ∗

∑ ∑

∑ ∑

[ ] [ ] [ ] ( )

[ ] [ ]1

11 ; : 1

1 n

y n ay n x n H z z aaz

a h n a u n

−− − = ⇒ = >−

< ⇒ =

ROC

For stability,

max 1kkc <

max kkz c>

( )H zi( )H zi

( )H zi

IIR termFIR term

kc

( )H zi



FIR system

Example

0

0

0

( )

, 00, otherwise

Mk

kk

Mn

kkM

kk

H z b z

b n Mh n b n k

y n b x n k

δ

−

=

=

=

⎧⎪⎡ ⎤⎡ ⎤ ⎨⎣ ⎦ ⎣ ⎦ ⎪⎩

⎡ ⎤⎡ ⎤⎣ ⎦ ⎣ ⎦

=

≤ ≤⇒ = − =

⇒ = −

∑

∑

∑

[ ]

( )( )

[ ] [ ]

[ ] [ ] [ ] [ ]

11

10

21

0

1

, 00 , otherwise

11

, 0,1,2,...,

1 1

n

MMMn n

n

kjM

k

Mk

k

M

a n Mh n

a zH z a zaz

z ae k M

y n a x n k

y n ay n x n a x n M

π

− ++−

−=

+

=

+

⎧ ≤ ≤= ⎨⎩

−⇒ = =

−

= =

⇒ = −

− − = − − −

∑

∑

Zeros :

or

7M =



• Frequency response for rational systems

– Gain (or attenuation)

– Phase response

– Group delay

( ) ( )( )

( )

( )

00 1

0

0 1

0 1

0

1

1

1

1

1

j

MMjj k

kkj k k

N Nz ej k j

k kk k

Mj

kj k

Nj

kk

c eb ebH e H zaa e d e

c ebH ea d e

ω

ωω

ω

ω ω

ω

ω

ω

−−

= ==

− −

= =

−

=

−

=

−= = =

−

−⇒ =

−

∑ ∏

∑ ∏

∏

∏

( )10

010 10 10

1 10

20log

20log 20 log 1 20 log 1 ( )

j

M Mj j

k kk k

H e

b c e d e dBa

ω

ω ω− −

= =

=

= + − − −∑ ∑

Gain

( )( )( )

( )( )

*2

20 1

*0

1

1 1

1 1

Mj j

k kj k

Nj j

k kk

c e c ebH ea d e d e

ω ω

ω

ω ω

−

=

−

=

− −⎛ ⎞

= ⎜ ⎟⎝ ⎠ − −

∏

∏

( ) ( ) ( ) ( )0

1 10

Arg ( ) Arg Arg 1 Arg 1 ; g x ArM M

j j jk k

k k

bH e c e d ea

ω ω ω π π− −

= =

−⎛ ⎞

= + − − −⎜ <⎝

≤⎟⎠∑ ∑

( ) ( ) ( )1 1

grd ( ) Arg 1 Arg 1M M

j j jk k

k k

d dH e c e d ed d

ω ω ω

ω ω− −

= =

= − − + −∑ ∑



• Frequency response of a single pole or zero

– Single zero

( ) ( ) ( )( ) ( )( )

( )

( )( ) ( )( )

( )( ) ( )( )( )( )

1

2

2

1

2

2

1 1 ;

1 1 1

1 2 cos

sinrg tan

1 cos

grd rg

cos

1 2 cos

j j j

j j j j j j j

j

j j

H z az H e ae a re

H e re e re e re e

r r

rA H e

rdH e A H e

dr rr r

ω ω θ

ω θ ω θ ω θ ω

ω

ω ω

ω θ

θ ωθ ω

ωθ ωθ ω

− −

− − −

−

= − ⇒ = − =

⇒ = − = − −

= + − −

−⇒ =

− −

⎡ ⎤⇒ = ⎣ ⎦

− −=

+ − −2v

φ 3

3vω 1vθ

`

( )

( )

( )( )

33

1

3 1 3 rg

j jj

j

j

z aH zz

ve reH e ve v

A H e

ω θω

ω

ω φ φ φ ω

−=

−⇒ = = =

= − = −

0

0.5

θ

θ π

θ π

=

=

=

0.9r =



– Example: Real single zero

( )( ) ( )

( )

( )( )( )( ) ( )( )

1

2

2

1

2

2

1

1 ;

= ,

1

1 2 cossinrg tan

1 cos

grd rg

cos 1 2 cos

j j j

j j

j

j j

H z az

H e ae a re

a r

H e re

r rrA H er

dH e A H ed

r rr r

ω ω θ

ω ω

ω

ω ω

θ π

ωωω

ωωω

−

−

−

−

= −

⇒ = − =

= −

⇒ = +

= + −

⇒ =+

⎡ ⎤⇒ = ⎣ ⎦

+=

+ +

If



– Second-order IIR system

( ) ( )( )[ ] [ ] [ ] [ ]

[ ] ( ) [ ]

1 2 21 1

2

1 11 2 cos1 1

2 cos 1 2

sin 1sin

j j

n

H zr z r zre z re z

y n r y n r y n x n

r nh n u n

θ θ θ

θ

θθ

− −− −= =

− +− −

⇔ − − + − =

+⇒ =

1vω 3v

2vθ−

( )

( )( ) ( )( )

( )( )

23

1 2 1 2

1 1

1

sin sinrg tan tan

1 cos 1 cos

j

j

vH e

v v v v

r rA H e

r r

ω

ω θ ω θ ωθ ω θ ω

− −

= =⋅

− += − −

− − − +

( ) 1 2

11 0.9 2 0.81

H zz z− −

=− +



– Third-order IIR system

( ) ( )( )( )( )

1 1 2

1 1 2

1.0376

0.6257

0.05634 1 1 1.0166

1 0.683 1 1.4461 0.7957

1, (59.45 )

0.683, 0.892 (35.85 )

jz

jp

z z zH z

z z z

z e

z e

− − −

− − −

±

±

+ − +=

− − +

⇒ = −

⇒ =



• Relationship between magnitude and phase

– In general, magnitude information phase information

– In cases of rational system functions, there is some constraint between magnitude and

phase

– The square of the magnitude frequency response is the evaluation of z-transform on

the unit circle

– Question: Can we know of H(z) from C(z) ?

( ) ( ) ( ) ( )

( )( )

( )

( )( )

2

*

1

0 01 1

10 0

1 1

1

1 11;

1 1

j

j j j

z e

M M

k kk kN N

k kk k

H e H e H e H z Hz

c z c zb bH z Ha z ad z d z

ω

ω ω ω∗ ∗

=

∗−

∗= =∗

∗−

= =

⎛ ⎞= = ⎜ ⎟⎝ ⎠

− −⎛ ⎞= =⎜ ⎟⎝ ⎠− −

∏ ∏

∏ ∏

( ) ( ) ( )*

* 1 C z H z Hz

Δ

No

No⎯⎯→←⎯⎯

( )

( ) ( )1 1

: ; 1 : ;

k k

k k

H z d c

H d cz

− −∗ ∗ ∗∗

⎛ ⎞⎜ ⎟⎝ ⎠

pole zero conjugate reciprocal pairs

pole zero



– If H(z) is assumed to be causal & stable, all its poles are in the unit circle

⇒ Poles of H(z) can be identified uniquely

Example System with the same C(z)

( ) ( )( )

( ) ( )( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

1 1

11 14 4

1 1

21 14 4

1 2

*1 1 1 *

*2 2 2 1 2*

2 1 1 0.5

1 0.8 1 0.8

1 1 2

1 0.8 1 0.8

1

1

j j

j j

j j

z zH z

e z e z

z zH z

e z e z

H e H e

C z H z Hz

C z H z H C z C zz

π π

π π

ω ω

− −

−− −

− −

−− −

− +=⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

− +=⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

⇒ =

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞= ⇒ =⎜ ⎟⎝ ⎠

−05. −2



• All pass system

– A stable system function of the form

has a frequency-response magnitude independent of

– In general, the transfer function of an all-pass system is given by

– All-pass system can be used for compensating group delay distortion.

– Causal all-pass system has positive group delay (but not continuous phase) property

( )1

11apz aH z

az

− ∗−=

−

( ) ( )( )( )( )

1 11

1 1 11 1

; 0, 1, 11 1 1

cr MMk kk

ap k kk kk k k

z e z ez dH z A A d ed z e z e z

∗− −−

− − ∗ −= =

− −−= > < <

− − −∏ ∏

( ) ( )

( )( ) ( )( )

1

1

sinrg 2 tan

1 cos

11 1

j jj j j

ap ap

ap

j

j

j

rA H e

e a a eH e e H eae ae

r

ω ωω ω ω

ω ω

ω ω θω

ω θ−

− ∗ ∗−

− −

− −= = ⇒

−−

−

=− −

= −−

.ω



– Example

( )1

1

0.91 0.9zH z

z

−

=∓

∓

0.9

0.9p

p

z

z

=

= −



– Example

4π

1 20.5.75−43

−

0.8



• Minimum-phase system

– It has its poles and zeros all inside the unit circle.

⇒ Any minimum-phase system is causal and stable.

⇒ is a causal, stable, and min-phase system

– Cascade of both systems will not introduce any delay

– Energy concentration

For an arbitrary stable and causal system , let be the min-phase

function s.t.,

Then, for any sequence and K,

⇒ The signal energy is transported with the smallest possible delay using a

minimum-phase system

Example If

( )1H z−

( ) ( )1 1 joH z H z e− = ⋅

( )jF e ω

( ) ( ) , .j jG e F eω ω ω= ∀

[ ]x n

[ ] [ ]( ) [ ] [ ]( )2 2

0 0

K K

k kk k

x n g n x n f n= =

∗ ≥ ∗∑ ∑

[ ] [ ] [ ] [ ]2 2

0 0

, K K

k k

x n n g k f kδ= =

= ≥∑ ∑

( )jG e ω

( )H z



– The min-phase function has its energy maximally concentrated near the time origin.

proof

Let

The energy of

Assume vanishes for

But is not necessarily zero for

– cannot be uniquely determined from , because cascading all-pass filters

does not affect the magnitude of the frequency response.

[ ]y n

( ) ( )( ) ,

jj

j

F eH e

G e

ωω

ω ωΔ ∀ be stable, causal and all pass

x n z nG e jωc h H e jωc h

y n

min-phase all-passphase shifter

( ) 1jH e ω =

[ ]{ } ( )

[ ]{ } ( ) ( )

0

0

0

0

22 20

0 2

22 2

0 2

1 2,

1

jy

j jz y

E E y n Y e dT

E E z n H e Y e d E

ωω

ω

ωω ω

ω

πω ωω

ωω

−

−

Δ = =

Δ = =

∫

∫[ ]y n [ ]2

00 , . .,

K

yn

n n K i e E y n=

< < = ∑a n d[ ]z n n K>

[ ]2

0

K

z yn

E E z n=

= ≥ ∑( )H z ( )G z



– Any rational rational transfer function can be expressed as

Assume that has all zeros and poles inside the unit circle except one

zero outside the unit circle at

where is minimum phase

has poles and zeros of inside the unit circle plus zeros that are

conjugate reciprocals of the zeros of outside the unit circle

contains all the zeros of outside the unit circle

Example

( ) ( ) ( )m in apH z H z H z=

*

1 , 1z cc

= <

( )H z

( ) 1 *1

1 *1

1 1

( )( )

( ) (1 )1

H z H z z c

z cH z czcz

−

−−

−=

= −

−⋅ − ⋅

−minimum phase

all pass

( )

m in (

1 11

(

1 1

)

1

)

1 111 3 3 331 1 11 1 12 2 3

apH z H z

z zzH zz z z

− −−

− − −

+ ++= = ⋅ ⋅

+ + +

( )1H z

( )minH z

( )apH z ( )H z

( )H z

( )H z



– The amplitude characteristics of a minimum-phase filter is fully determined by the

phase characteristics and vice versa

Proof

Since has its poles and zeros inside the unit circle,

has a causal and well behaved inverse Fourier transform, i.e.,

We can see that and form a Hilbert transform pair.

Since is the logarithm of a real-valued symmetric function of frequency, it

is real and symmetric

Thus we can compute

( ) ( ) j jG e G eω ω⇒ We can compute from

( )jX e ω ( )jG e ωφ

( )jX e ω

[ ] [ ]

( ) [ ] [ ]

[ ] [ ] [ ] [ ]0

0 2 cos

0 0 2 , 1

,

j

nX e x x n n

g x g n x n n

x n x n n

ω ω∞

=

⇒

⇒ +

⇒ =

= − ∀

= ≥

∑ =

and

( ) jG e ωφ

( )G z ( )ln 1jG e zω > is analytic for

( )ln jG e ω⇒

( ) [ ] ( ) ( )

( ) ( ) [ ] ( ) [ ]0

0 0 cos ;

si

ln

l nn

j n j jG

n

j jG

n n

j

j

g n e X e j e

X e g n n e g n

G e

G e n

ω ω ω

ω

ω

ω ω

φ

ω φ ω

∞−

=

∞ ∞

= =

=

= = −

= +∑

∑ ∑where =



– Frequency-response compensation

If is a min-phase system, perfect compensation is generally possible

The frequency response magnitude is exactly compensated with some delay

Example

s n s nc

H zdb g H zcb g( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

min

min

1

d ap

c d c ap

H z H z H z

H z G z H z H z H zH z

=

= ⇒ = =Let

( )dH z

( )( )( )( )( )( )

( )( )( )( )

0.6 1 0.6 1 0.8 1 0.8 1

2 0.6 1 0.6 1min

0.8 1 0.8 1

0.8 1 0.8 1

( ) 1 0.9 1 0.9 1 1.25 1 1.25

( ) 1.25 1 0.9 1 0.9

1 0.8 1 0.8

1 1.25 1 1.25( )

1 0

j j j j

j j

j j

j j

ap

H z e z e z e z e z

H z e z e z

e z e z

e z e zH z

π π π π

π π

π π

π π

− − − − − −

− − −

− − −

− − −

= − − − −

⇒ = − −

⋅ − −

− −⇒ =

−( )( )0.8 1 0.8 1.8 1 0.8j je z e zπ π− − −−



( )H z ( )apH zmin ( )H z



In general, min-energy delay property

– Maximum phase system is a stable system whose poles and zeros are all outside the

unit circle

stable smallest

⇒ all poles contribute to with terms of the form

⇒ anti causal!

[ ] [ ]m in0 0h h≤

( ) ( )

( ) ( )

[ ] [ ] [ ] [ ]

[ ] [ ]

1

m in 1

1

m in 1

m in m in

m in

, 11

1lim lim

10 0 0 0 1

0 0

z z

z aH z H z aaz

a zH z H zz a

h h h ah aa

h h

−

−

−

−→ ∞ → ∞

−= <

−−

=−

⇒ = − ⇒ = − <

⇒ ≤

( )( )

( )1

1

1, 1

M

kk

k kN

kk

z cH z c d

z d

=

=

−= > >

−

∏

∏

R o c : kz d⇒ <

[ ]h n ( ) [ ]1nkd u n− − −



Note that FIR max-phase sequence can be made causal by introducing a finite

delay

– Property of min-phase systems

minimum phase-lag

negative for

minimum group-delay

minimum energy delay property

( ) [ ] ( )( ) ( )

1

m ax m in

m in

0 11

kk

k

ap

z cH z h cc z

H z H z

−∗

∗

−= −

−

=

∏ ∏

( ) ( ) ( )minarg arg argj j japH e H e H eω ω ω⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 ω π≤ ≤

[ ] [ ]m in0 0h h≤



4. Structure for discrete-time systems

Structure for discrete-time systems• Characterization of an LTI system can be represented by difference equation, impulse

response or system function

Example:

• For description of systems for implementation, use the block diagram and/or signal flow

graph method

( )

[ ] [ ] [ ]

10 1

1

10 1

,1

1

n n

b b zH z z aaz

h n b a u n b a u n

−

−

−

+= >

−⇒ = + −

⇒⇒⇒

Impulse response :Infinite duration impulse response Not possible to implement the system by discrete time convolutionThe

[ ] [ ] [ ] [ ]0 1 1 1y n ay n b x n b x n= − + + −

output can be calculated by a recursive computation algorithm




• Basic block diagram symbols

Example

y n

x n

z n x n y n= +

x n a y n ax n=

x n y n x n= −1z−1

[ ] [ ] [ ] [ ]1 21 2y n a y n a y n bx n= − + − +

z−1

z−1

x n b y n

y n −1a1

a2




–

< direct form I>

z−1

z−1

::

z−1

z−1

:

z−1 z−1

x n b0

b1

b2

bM −1

bM

x n −1

x n − 2

x n M− +1

x n M−

H z1b g H z2 b gy n

a1

a2

aN −1

aN

y n −1

y n N− +1

y n N−

[ ] [ ] [ ]1 0

N M

k kk k

y n a y n k b x n k= =

= − + −∑ ∑




– By interchanging the cascaded blocks,

< direct form II or canonic direct form >

( )( ) ( )

( ) ( )( ) ( )

1 2

2 1

Y zH z

X z

H z H z

H z H z

=

=

=

z − 1

:

:z − 1

:

b0

1a

2a

Na

[ ]1nω −

[ ]2nω −

[ ]n Nω −

y n

1Nb −

z − 1z − 1

z − 1

x n

1Na −

Nb

[ ]nω

:

1b

z − 1

z −1

z −1

:

:

z −1

b0

b1

b2

bN −1

bN

a N

a N −1

a 2

a1

[ ]x n [ ]y n

⇒




• Signal flow graph representation

Example

node jj nω ⎡ ⎤⎣ ⎦

branch j k,

node kk nω ⎡ ⎤⎣ ⎦

x n ω 1 n c

b

d

ey nω 2 n

a

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ] [ ]

1 2 2

2 1

2

n x n a n b n

n c n

y n dx n e n

ω ω ω

ω ω

ω

= + +

=

= +




Example

z−1

x n ω nb0

ω n −1

y n

b1a

x n ω n y nb0

b1

z−1

a

[ ] [ ] [ ][ ] [ ] [ ]

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )

0 1

1 1

1 10 1 0 1

1 10 1

10 1

1

1

1

1

1

1

n x n a n

y n b n b n

W z X z aW z z X z az W z

Y z b W z bW z z Y z b b z W z

Y z X zW z

b b z azY z b b zH zX z az

ω ω

ω ω− −

− −

− −

−

−

= + −

= + −

⇒ = + ⇒ = +

= + ⇒ = +

⇒ = =+ +

+⇒ = =

+




• Basic structures for IIR systems

– Direct form I

– Direct form II

( )1 2

1 2

1 21 0.75 0.125

z zH zz z

− −

− −

+ +=

− +x n

z−1

z−1

1

2

1

075.

z−1

z−1

−0125.

y n

x nz−1

2

z−1075.

y n

−0125.




– Cascade form

( )( ) ( )( )

( ) ( )( )

1 2

1 2

1 1 * 1

1 1

1 1 * 1

1 1

1 1 1

1 1 1

M M

k k kk kN N

k k kk k

g z h z h zH z A

c z d z d z

− − −

= =

− − −

= =

− − −=

− − −

∏ ∏

∏ ∏

z−1

z−1

− d12 h1

2

d d1 1+ *c h − +h h1 1*c h

d d2 2+ *c h − d22 h2

2

( )*2 2h h− + c1 −g1

z−1

z−1

z−1

( ) 2* 1 21 k k kd d z d z− −− + +

0 25. 05.

z−1 z−1

x n y n

( ) ( )( )( )( )

1 1

1 1

1 1

1 0.5 1 0.25z z

H zz z

− −

− −

+ +=

− −Example




– Parallel form

( ) ( )( )( )

1 21

1 1 * -10 1 1

1 2

11 1 1

2,

pN N Nk kk k

kk k kk k k

p

B e zAH z g zc z d z d z

N N NM N N M N

−−

− −= = =

−= + +

− − −

= +≥ = −

∑ ∑ ∑

21

0 11 2

1 1 21

Nk k

k k k

e e za z a z

−

− −=

+− −

∑

y nx n e01

e11

a11z −1

z −1

a21

g0

: :

z −1

z −1

g1

g2

g N p

z −1

A1

c1

:




Example

x n y n

8

−7

z−1

z−180 75.

−01125.

z−1

z−1

x n y n8 18

−0 25.05.

0 25.

( )1 2

1 1

1

1 1

1 1

1 21 0.75 0.125

7 881 0.75 0.125

18 2581 0.5 1 0.25

z zH zz z

zz z

z z

− −

− −

−

− −

− −

+ +=

− +− +

= +− +

= + −− −




• Transposed forms

– Signal flow graph methods provide procedures for transforming graphs into different forms

– Transposition of a flow graph (flow graph reversal) is obtained by reversing the directions

of all branches, while keeping the branch transnittances and reversing the roles of the

input and output

Examplex n y n

z −1

a

y n x n

z −1

a

x n y n

z −1

a

( ) 1

11

H zaz−

=−

The difference is the change in ordering

Transposed form




Example Transposed form of a second-order structure

Transposed direct form II

z−1

y nx n ω n

a1 b1

a2 b2

z−1

0b

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

1 2

0 1 2

1 2

1 2

n a n a n x n

y n b n b n b n

ω ω ω

ω ω ω

= − + − +

= + − + −

b2a2

b0

b1

z −1

a1

z −1

y n x n

x n y nb0 v n0

z −1

z −1

b1 a1

b2 a2




• Basic network structure for FIR systems

– Direct form

Causal FIR systems

In the transposed form,

[ ] [ ] [ ]0

, 00 , otherwise

Mn

kk

b n My n b x n k h n

=

≤ ≤⎧= − ⇒ = ⎨

⎩∑

x n z−1 z−1 z−1

h 0 h 1 h 2

y n

[ ]h M

x n

z−1y n z−1

h 1 h 2 h Mh 0

z−1 z−1 z−1

h M h M −1

y n

h 0x n

z−1

⋅ ⋅ ⋅

[ ]2h M −

Transversal filter or tapped delay line




– Cascade form

( ) [ ] ( ) ( )1 20 1 2

0 1; 1 / 2

sMMn

k k k sk k

H z h n z b b z b z M M− − −

= =

= = + + = +⎢ ⎥⎣ ⎦∑ ∏

z−1

z−1

z−1

z−1

b02b01

b11 b12

b21 b22

z−1

z−1

0 sMb

1 sMb

2 sMb

[ ]x n [ ]y n




– Structure for linear phase FIR systems

Causal linear phase FIR systems

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

0

12

0 / 2 1

1 12 2

0 0

for 0

2 2

2 2

M

k

MM

k k M

M M

k k

h M n h n h M n h n n M

y n h k x n k

M Mh k x n k h x n h k x n k

M Mh k x n k h x n h M k x n M k

=

−

= = +

− −

= =

− = − = − ≤ ≤

⇒ = −

⎡ ⎤ ⎡ ⎤= − + − + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= − + − + − − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑

∑ ∑

∑ ∑

or

[ ] [ ]

[ ] [ ] [ ] [ ]( )1

2

0

2 2

M

k

h M n h n

M My n h k x n k x n M k h x n−

=

− =

⎡ ⎤ ⎡ ⎤= − + − + + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑

When (type - I system),

z−1 z−1 z−1

z−1 z−1z−1

h 0 h 1 h 2 h M2

1−LNMOQP h M

2LNMOQP

y n




Symmetry condition

⇒ zeros of occur mirror image pairs

If is real, zeros of occur in complex conjugate pairs.

z3

z4

z3*

z1*

z1

z2 1

2z

1

1z

1

1z*

( )H z

[ ]h n ( )H z

( ) [ ]( )( )( )( )

{ }

1 1 2 1 2 1 2 3 4

2

2 3 1 12 1 1

0 1 1 1 1

1 1 1, 2Re , 2Re , 2

H z h z az z bz z cz dz cz z

a z b z c z d zz z z

− − − − − − − − −= + + + + + + + + +

⎧ ⎫= + = = − + = + +⎨ ⎬

⎩ ⎭where



5. Filter design techniques

Filter Design Techniques• Design procedure

– Specify the desired frequency Response

– Approximate the specifications

– Realize the filter ( ) ,jeffH e H

Tω ω ω π⎛ ⎞= <⎜ ⎟

⎝ ⎠




• Design of discrete-time IIR filters from continuous-time filters

– Filter design by impulse invariance

The discrete-time filter specifications are transformed to continuous-time filter

specifications by

Design a suitable continuous-time filter

Transform to the desired discrete-time filter

[ ] ( )d c dh n T h nT=

C D/ h n D C/xa τb g x n y n ya τb g

TTdT

( ) 2jc

k d d

H e H kT T

ω ω π∞

=−∞

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑

( ) ( ) 1, , 0,2

jc c

d d

H e H H f fT T

ω ω ω π⎛ ⎞

= ≤ = ≥⎜ ⎟⎝ ⎠

if

( )2d

fTω π⇔ =Ω

( )cH s

( )H z




For

By sampling

A pole at in the s-plane is transformed to a pole at in the z-plane.

If is stable, the real part of

⇒

⇒ pole is inside the unit circle

⇒ is also stable

Although the poles in the s-plane are mapped in the z-plane in a simple manner,

the zeros are not.

( ),ch t

[ ] [ ] [ ] ( ) [ ]

( )1 1

11 1

k d k d

k d

N N ns nT s Td c d d k d k

k k

Nd ks T

k

h n T h nT T A e u n T A e u n

T AH ze z

= =

−=

= = =

⇒ =−

∑ ∑

∑

ks s= k ds Tz e=

( )cH s 0ks <

1k ds Te <

( )H z

( ) ( ) 11

, 0,

0 , 0

kN

s tNkk

kc ck k

A e tAH s h ts s t

==

⎧ ≥⎪= = ⎨− ⎪ >⎩

∑∑




– Butterworth (BW) filter

The magnitude response of a BW filter is

where is the 3dB frequency

( ) ( ) ( )2 2

2 1 1

1

12

Nc

c

N

c

H s H ssf

H fff π

= ⇐ − =⎛ ⎞

+ ⎜ ⎟⎝

⎞⎜ ⎟⎝ ⎠

⎛+

⎠

cf

( )2 12 ; 0,1,2, ,2 1

j k NN

k cs e k Nπ

+ −⎧ ⎫⇒ =Ω = −⎨ ⎬⎩ ⎭

poles are




Example Design a continuous-time Butterworth (BW) filter s.t.

The magnitude response of a BW filter is

where is the frequency

By solving

1

1 1− δ

δ 2

ω p ω s

1

2

0.108750.177830.20.3

p

s

δδω π

ω π

===

=

( )

( )

0.89125 1, 0 0.1

10.17783 0.152

c

c

H f f

H f f

≤ ≤ ≤ ≤

≤ ≤ ≤

( ) ( ) ( )2 2

2 1 1

1

12

Nc

c

N

c

H s H ssf

H fff π

= ⇐ − =⎛ ⎞

+ ⎜ ⎟⎝

⎞⎜ ⎟⎝ ⎠

⎛+

⎠

2 2

2 2

0.1 110.89125

0.15 110.17783

N

c

N

c

f

f

⎧ ⎛ ⎞ ⎛ ⎞+ =⎪ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎪ ⎝ ⎠

⎨⎛ ⎞⎪ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎩

5.8858 60.11216c

N Nf

⇒ = ⇒ ==

cf 3dB

1 dT ω= = ΩLet so that

( )2 12

0.182 0.6792 0.497 0.497

0.679 0.182

j k NN

k c

js f e j

j

π

π+ −

− ±⎛ ⎞⎜ ⎟= ⇒ − ±⎜ ⎟⎜ ⎟− ±⎝ ⎠

poles are12

0.11 1.258928

0.111918c

c

ff

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⇒ =

0.7032




This method is useful only for band-limited filters

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

2 2 2

1 2

3 4

5 6

0.120930.3645 0.4945 0.99455 0.4945 1.35855 0.4945

0.182 0.679 0.182 0.679

0.497 0.497 0.497 0.497

0.679 0.182 0.679 0.182

cH ss s s

A As j s j

A As j s j

A As j s j

=+ + + + ⋅ + +

= ++ + + −

+ ++ + + −

+ ++ + + −

( )0.182 0.6791

2

1

0.8336 0.7782 0.6280

0.6487 0.5235 0.69489

jz e j

j z

− −= = −

= − ⇒ =

( ) ( ) ( )1 2

0.182 0.679 1 0.182 0.679 11 1j j

A AH ze e z e e z− − − + −

⇒ = + + ⋅ ⋅ ⋅− −

( )1

1 2

1

1 2

1

1 2

0.2871 0.44661 1.2974 0.6949

2.1428 1.14551 1.0691 0.3699

1.8557 0.630321 0.9972 0.2570

zH zz z

zz z

zz z

−

− −

−

− −

−

− −

−=

− +− +

+− +

−+

− +




– Bilinear transformation

To avoid aliasing, we employ the bilinear transform

∞ −∞

2πf

−π πω

( )1

1

2 11c

d

zH z HT z

−

−

⎡ ⎤−⎛ ⎞= ⎢ ⎥⎜ ⎟+⎝ ⎠⎣ ⎦

1

1

12 2

12

12

2 1

1

21, 0

1

d

dd

dd

d

sz s jzs

T zfT s

T j fT

T j fT

z

π

σ π

σ π

σ π

σ

−

−

⎛ ⎞ +⇒ = ⇐ = +

−

+ +=

− −

⇒ <

−= ⎜ ⎟+⎝ ⎠

<if

1

1

2 11d

zsT z

−

−

−⎛ ⎞= ⎜ ⎟+⎝ ⎠




The axis of the s-plane maps onto the unit circle of the z-plane

For

or

2j fπ

2 ,1 1 1 1 1

jd d

d d

s j fj fT j fTz z ej fT j fT

ω

ππ ππ π

=+ +

= ⇒ = ⇒ =− −

( )

2

2

2 1 21

2 sin2 2

2 cos2

2 tan2

j

jd

j

jd

d

es j fT e

e j

T e

jT

ω

ω

ω

ω

σ π

ω

ω

ω

−

−

−

−

⎛ ⎞−= = +⎜ ⎟+⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

=

-1

1 20 tan tan2 2

=2tan2

d d

d

fT T

T

ω ωσπ

ω

⎛ ⎞⇒ = = Ω =⎜ ⎟

⎝ ⎠Ω⎛ ⎞

⎜ ⎟⎝ ⎠

and or

or

ωπ

−π

Ω 2πfb gor

ω

π

= FHGIKJ

=

−

−

22

2

1

1

tan

tan

ΩT

T f

d

db g

σ

jΩ

−π

Re

Im




Hc

(Ω)

Ωp

Ωs

Ω

ω p ω s πωω




Example Bilinear transformation of a BW filter

( )2 2

2

2

2 2

1 0.6498 11 1.25890.89125

1

1.01905 1 1 31.62200.17783

N

c Nc

c

N

c

H⎛ ⎞ ⎛ ⎞Ω = ⇒ + = =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎛ ⎞ ⎝ ⎠Ω

+ ⎜ ⎟Ω⎝ ⎠

⎛ ⎞ ⎛ ⎞+ = =⎜ ⎟ ⎜ ⎟Ω ⎝ ⎠⎝ ⎠

( )( )

0 0.2

0.3

2 0.20 0.2

0.891

0 tan 0.6498; 12

2 0.30.3 tan 1.01905

25 1,

0.17783,

2

dd

j

j

d

T

H e

H e

T

T

ω

ω

ω π

π π

πω π

ππ

ω

ω

≤ ≤

≤ ≤ ⇒ ≤Ω ≤

≤ ≤

≤ ≤ ≤

= =

≤ Ω⇒≤ ⇒ ≤Ω

2 21 11 10.17783 1.89125 2.0729 5.30267

1.01905 0.39092log0.6498

6 0.76622c

N

N

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞− −⎢ ⎥ ⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭⇒ = ≅ ≅

⎛ ⎞⎜ ⎟⎝ ⎠

⇒ = ⇒Ω =




( ) ( )( )

( )

2 2

2

0.202380.39965 0.5871 1.08365 0.5871

11.48025 0.5871

cH ss s

s

⇒ =+ + + +

×+ +

( ) ( )( )( )

( )

61

1 2 1 2

1 2

0.0007378 1

1 1.2686 0.7051 1 1.0106 0.3583

11 0.9044 0.2155

zH z

z z z z

z z

−

− − − −

− −

+⇒ =

− + − +

×− +

12

312

512

0.76622 , 0.7401 0.1983

0.76622 0.5418 0.5418

0.76622 0.1983 0.7401

j

j

j

e j

e j

e j

π

π

π

±

±

±

⎛⋅ = ±⎜

⎜⎜⇒ ⋅ = ±⎜⎜ ⋅ = ±⎜⎝

Poles are




The magnitude of N-th order Butterworth filter can be represented by

Note that this BW filter is maximally flat, , and periodic with a period

of 2π.

Maximally flat ⇔ The first (2N-1) derivatives of are zero at

It is not easy to directly find the poles and zeros in the z-plane from

since the magnitude-squared function should be factorized into to

determine

( ) 2

21 2; tan tan

2 2 2tan

21tan

2

j c c DN

d

c

TH eT

ω ω ωω

ω

Ω= = ⇐Ω =

⎛ ⎞⎜ ⎟

+ ⎜ ⎟⎜ ⎟⎝ ⎠

( ) 12

cjH e ω =

0ω =

( ) 2jH e ω

( ) ( )1H z H z−

( ).H z




• Design of type-I Chebyshev filters

– is the N-th order Chebyshev polynomial defined by

( ) 2

2 2

1

1 Nc

HVε

Ω =⎛ ⎞Ω+ ⎜ ⎟Ω⎝ ⎠

( )NV x

( ) ( )1cos cosNV x N x−=

( )( ) ( )( ) ( )( ) ( )

( ) ( ) ( )( )

( )

0

11

1 22

2 33

1 1

2

1

cos0 1

cos cos

cos 2cos 2 1

2 1 4 3

2

0 1 0 1

1 cos

N N N

N

N

V x

V x x x

V x x x

V x x x x x x

V x xV x V x

x V x

x x V x

−

−

+ −

−

= =

= =

= = −

= − − = −

= −

⇒ ≤ ≤ ⇒ ≤ ≤

> ⇒ ⇒ ∼is imaginary hyperbolic cosine ( ) NV x⇒ increases monotonically

11 ε−

Ω p Ωs




– Property of

forms an orthogonal set over

For each N,

For even N,

For odd N,

The zeros of lie in

The range of is between –1 and 1 for

( )NV x

( ){ }NV x

( ) ( )1

21

0, 1 / 2, 0

1 , 0N M

M NV x V x dx M N

x M Nππ

−

≠⎧⎪= = ≠⎨

− ⎪ = =⎩∫

( )1 1NV =

( ) ( ) ( )( ) ( )

/ 21 1; 0 1 NN N

N N

V V

V x V x

− = = −

= −

( ) ( )( ) ( )

1 1; 0 0N N

N N

V V

V x V x

− = − =

= − −

1 1x− ≤ ≤( ){ }NV x

( ){ }NV x 1 1x− ≤ ≤

1 1x− ≤ ≤




Ripple factor:

Since

Rule: choose and

( )

( )

2

2

1 0 1, 11

1,

c

c

H

H

εΩ

⇒ ≤ ≤ ≤ Ω ≤Ω +

Ω> Ω

Ω

For

decreases monotonically

, Nε cΩ

11 ε−

Ω p Ωs2

2

1 =1-1+

ε δε

⇒ ripple amplitude( )1 1NV =

( ) ( ) ( )2 2

1 1 1 1 tan / 2

pj T j T

N

H e H eV k T

ω ω

ε ε ω= ⇐ =

+ + ⎡ ⎤⎣ ⎦




– Consider the first zeros of

Letting

The poles of the Chebyshev filter lie on an ellipse in the s-plane

( )2 21 0NV xε+ =

( )1

1cos cosh 0 & sin sinh

2 1 / 2 , 0,1,2, ,2 11 1 sinh

Nu Nv Nu Nv

u M N M N

vN

επ

ε−

⇒ = = ±

⇒ = + = −

=

2 2

2 2 1; sinh cosh

s jv v

σ σΩ+ = = + Ω

( ) ( )1

1

cos cos( ) cos

cos cos cos

cos cosh sin sinh/

N

r u jv x x r

V x N x Nr

Nu Nv j Nu Nvj ε

−

−

≡ + = ⇒ =

= =

= −= ±

1 1 1 11 21 1, , 1

2 2N N N Na bα α α α α ε ε

− − − −⎛ ⎞ ⎛ ⎞= − = + = + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

( )( )sinh cos cos

cosh sin sinp c

p c

v a

v b

σ θ θ

θ θ

= = Ω

Ω = = Ω

3N =




• Design of Type II Chebyshev Filters

• Elliptic filter

– The error is distributed over the entire frequency band ⇒ Equal ripple in all band

⇒ min order of the filter

: Chebyshev rational function

of degree N

( ) 2

12

1

1c

cN

HVε

−Ω =Ω⎡ ⎤⎛ ⎞+ ⎜ ⎟⎢ ⎥Ω⎝ ⎠⎣ ⎦

( ) 2

2 2

1

1c

Nc

HUε

Ω =⎛ ⎞Ω+ ⎜ ⎟Ω⎝ ⎠

( )NU Ω




• Example

– Filter specifications

Passband

Stopband

This implies that 1 20.01, 0.001, 0.4 0.6p sδ δ ω π ω π= = = =and

( ) 0.001; 0.6jH e ω π ω π≤ ≤ ≤

( )0.99 1.01; 0.4jH e ω ω π≤ ≤ ≤




Butterworth filter with N=14 Chev-type I filter with N=8




Chev-type II filter with N=8 Elliptic filter with N=6

1

1

2 11d

zsT z

−

−

−⎛ ⎞= ⎜ ⎟+⎝ ⎠




– Poles and zeros of the designed filters




• Design of FIR filters by windowing

– FIR filter design directly approximates the desired frequency response of the discrete-

time system

– When the desired frequency response is

consider the design of a causal FIR filter that approximates the ideal response

– The truncation of the desired impulse response results in the Gibbs phenomenon

– Rectangular Windowing

where is the windowing function given by

Note that is a smeared version of

( ) [ ] [ ] ( )1 2

j j n jd d d d

hH e h n e h n H e d

πω ω ω

πω

π

∞−

−=−∞

= =∑ ∫;

[ ] [ ] [ ] [ ] [ ], 0

0 , otherwised

d r

h n n Mh n h n h n nω

≤ ≤⎧= ⇒ =⎨⎩

[ ]r nω

[ ] 1, 00, otherwiser

n Mnω

≤ ≤⎧= ⎨⎩

( ) ( ) ( )( )12

jj jd rH e H e e d

π ω θω θ

πω θ

π−

−⇒ = ∫

( )jH e ω ( )jdH e ω




0

0

H edjωc h

ω ω

( )jrW e ω

– Effect of windowing

( )( )

( )

0

1

2

11

1sin

2sin

2

Mj j n

rn

j M

j

Mj

W e e

ee

M

e

ω ω

ω

ω

ωω

ω

−

=

− +

−

−

=

−=

−+

=

∑

ω ωr

jec h

21

πM +

ΔWMm =

+4

1π

peak side lobe

design paramters




• Common windowing functions

– Bartlett (triangular)

– Hanning

– Hamming

– Blackman

[ ]

2 , 02

22 ,2

0, otherwise

B

n MnM

n Mw n n MM

⎧ ≤ ≤⎪⎪⎪= − ≤ ≤⎨⎪⎪⎪⎩

[ ]20.5 1 cos , 0

0 , otherwisehan

n n Mw n M

π⎧ ⎛ ⎞− ≤ ≤⎪ ⎜ ⎟= ⎝ ⎠⎨⎪⎩

[ ]20.54 0.46cos , 0

0 , otherwiseham

n n Mw n M

π⎧ − ≤ ≤⎪= ⎨⎪⎩

[ ]2 40.42 0.5cos 0.08cos , 0

0BL

n n n Mw n M M

π π⎧ ⎛ ⎞− + ≤ ≤⎪ ⎜ ⎟= ⎝ ⎠⎨⎪⎩




Trade off between the main lobe width and the peak of the side lobe

Peak side lobe Main-lobe Peak approx.

width error

13 4 /( 1) 21 25 8 / 25

31

MM

ππ

− + −− −−

RectangularBartlettHanning 8 / 44

41 8 / 5357 12 /

MM

M

πππ

−− −− −

Hamming Blackman 74

Rectangular window




Hanning window

Bartlett window Hamming window

Blackman window




– Kaiser window

where is the zeroth-order modified Bessel function of the first kind defined by

As β = 0 → rectangular window case

If the window is tapered more (i.e., β is increased), the main lobe width becomes

wider and the peak of the side lobe becomes lower

[ ]( )

2

0

0

1

,0 , 2

0 ,otherwise

nIMw n n M

I

αβα

αβ

⎧ ⎡ ⎤−⎡ ⎤⎪ ⎢ ⎥− ⎢ ⎥⎣ ⎦⎪ ⎢ ⎥⎣ ⎦= ⎨ ≤ ≤ =⎪⎪⎩

( )0I ⋅

( ) ( )sin0

1 2 cos for large 2 4

jxI x e d x xx

π θ

π

πθπ π−

⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠

∫




Increasing M while holding β constant

results in the main lobe width decreased

but does not affect the peak of the side

lobe.

Given is fixed, is defined to be

the highest frequency such that

and to be the lowest frequency such that

Then β can be empirically determined by

δ pω

( ) 1jH e ω δ≥ −

sω

( )jH e ω δ≤

( )( ) ( )0.4

0.1102 8.7 , 50

0.5842 21 0.07886 21 21 500 , 21

8 ; 2.285

, 20logs p

A A

A A AA

AM

A

β

ωω ω ω δ

− >⎧⎪⎪= − + − ≤ ≤⎨⎪ <⎪⎩

−=

ΔΔ = − = −where




– Comparison of commonly used windows




Design example

( )

1 2

0.4 ; 0.6 0.52

0.20.01; 0.001 0.001

20log 600.1102 8.7 5.65326

8 60 8 36.219 37; 2.285 2.285 0.2

p sp s c

s p

A dBA

AM M

ω ωω π ω π ω π

ω ω ω π

δ δ δδ

β

ω π

+= = ⇒ = =

Δ = − =

= = ⇒ =⇒ = − =

⇒ = − =

− −= = = ⇒ =

Δ ×Type - II FIR

[ ] [ ] [ ]

( )( ) ( )

12 2

0

0

18.55.65326 118.5sin 0.5 18.5 , 0

18.5 5.653260, otherwise

dh n h n n

nIn n M

n I

ω

ππ

⇒ = ⋅

⎧ ⎡ ⎤⎛ ⎞−⎛ ⎞⎪ ⎢ ⎥−⎜ ⎟⎜ ⎟⎪ ⎢ ⎥⎝ ⎠⎪ ⎝ ⎠− ⎢ ⎥= ≤ ≤⎣ ⎦⎨ ⋅⎪ −⎪⎪⎩




• Linear system with generalized linear phase

– For casual system, zero phase is not achievable

Phase distortion exists

Linear phase results in a simple time shift

– System with linear phase

Ideal delay

From the inverse Fourier transform,

For an input

( )( )

( )

( )

,

1

rg

grd

j jid

jid

jid

jid

H e e

H e

A H e

H e

ω ωα

ω

ω

ω

ω π

αω

αω αω

−= <

⇒ =

⎡ ⎤ = −⎣ ⎦∂⎡ ⎤ = − =⎣ ⎦ ∂

[ ] ( )( )

sin,id

nh n n

nπ α

π α−

= −∞ < < ∞−

[ ],x n

[ ] [ ] ( )( )

[ ] ( )( )

sin

sin

k

ny n x n

n

n kx k

n k

π απ α

π απ α

∞

=−∞

−= ∗

−

− −=

− −∑




In particular, if

Ideal LPF with linear phase

If the impulse response is symmetric about i.e.,

[ ] [ ][ ] [ ] [ ] [ ]

, d id d

id d

n h n n n

y n x n h n x n n

α δ= = −

⇒ = ∗ = −

( ) [ ] ( )( )

, sin

0,

jcj c

p pc

e nH e h n

n

ωαω ω ω ω α

π αω ω π

−⎧ < −⎪= ⇒ =⎨ −< ≤⎪⎩

,dnα = ,n α=

[ ] [ ]2p ph n h nα − =




– Generalized linear phase

This is a necessary condition for to have a constant group delay

There are other systems having linear phase characteristics without this

symmetry condition

( ) ( )( )

[ ][ ] [ ] ( )sinsin

tan sin 0,cos cos n

h n nh n n

h n nωβ ωα

β ωα ω α β ωβ ωα ω−

− = = − ⇒ − + = ∀⎡ ⎤⎣ ⎦−∑ ∑∑

[ ]h n

[ ] [ ][ ] ( )

[ ] [ ]

0 or , 2 2

= /2 3 / 2, cos 0

2 2

h n h n

h n n

M h n h n

β π α α

β π π ω α

α α

= − =

− =⎡ ⎤⎣ ⎦⇒ = = − = −

∑This is satisfied when is an integer, and

Alternatively, if or

an integer and

( ) ( ) ( )

( ) ( ) ( ) ( )cos sinj j

jj jH e

A e jA

A e e

e

αω βω ω

ω ωβ ωα β ωα

− −

+

=

= − − ⇒ ( ) ( )

( ){ }grd

rg

j

j

H e

d A H ed

ω

ω

τ ω

ωα

⎡ ⎤= ⎣ ⎦

⎡ ⎤= − ⎣ ⎦

=( ) [ ]

[ ] [ ]cos sin

j j n

n

n n

H e h n e

h n n j h n n

ω ω

ω ω

∞−

=−∞

∞ ∞

=−∞ =−∞

=

= −

∑

∑ ∑




– Causal generalized linear phase systems

If causal,

Causal FIR systems of length (M+1) have generalized linear phase, if

If

If

[ ] ( )0

sin 0,n

h n nω α β ω∞

=

− + = ∀⎡ ⎤⎣ ⎦∑

[ ] [ ]

( ) ( )( )

2

, 0

0 , otherwise

j Mj j

e

je

h M n n Mh n

H e A e e

A e

ωω ω

ω ω

−

− ≤ ≤⎧= ⎨⎩

=then

where is a real, even periodic function of

[ ] [ ]

( ) ( ) ( )( )

2 2

, 0

0 , otherwise

j M j Mj j j

o o

jo

h M n n Mh n

H e jA e e A e e

A e

ω ω πω ω ω

ω ω

+− −

− − ≤ ≤⎧= ⎨⎩

= =then

where is a real, odd periodic function of

[ ] [ ] [ ] [ ] 2 or 2h n h n h n h nα α= − = − −




Type I FIR linear phase systems

where M is an ever integer

where

where

[ ] [ ] 0h n h M n n M= − ≤ ≤

( ) [ ] [ ] [ ]

[ ] [ ] ( )

[ ] ( )( )

[ ]

12

2

0 0 12

1 12 2

2

0 0

12

2

0

12

0

2

2

2

2 cos2 2

MMM Mjj j n j n j n

Mn n n

M MMj j M nj n

n n

MMjj M nj n

n

M

n

MH e h n e h n e h e h n e

Mh n e h e h M n e

Mh n e e h e

M Mh n n h e

ωω ω ω ω

ωωω

ωωω

ω

−−− − −

= = = +

− −− − −−

= =

−−− −−

=

−

=

⎡ ⎤= = + +⎢ ⎥⎣ ⎦

⎡ ⎤= + + −⎢ ⎥⎣ ⎦

⎡ ⎤= + + ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑ ∑

∑ ∑

∑

∑ [ ]2

2 2

0cos

MM Mj j

ka k k e

ωωω

− −

=

= ⋅∑

[ ] [ ]0 , 2 , 1,2,.... .2 2 2M M Ma h a k h k k⎡ ⎤ ⎡ ⎤= = − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( ) 2Mjj j

eH e A e eω

ω ω −⇒ =

( ) [ ]2

0

cos

M

je

k

A e a k kω ω=

=∑




Example

[ ]

( )4

2

0

1, 0 40, otherwise

5sin2

sin2

j j n j

n

nh n

H e e eω ω ω

ω

ω− −

=

≤ ≤⎧= ⎨⎩

⇒ = =∑




Type II linear phase systems

where is an odd integer

Example

[ ] [ ], 0h n h M n n M= − ≤ ≤

M

( ) [ ]0

12

2

1

1 12 cos2 2

Mj j n

n

MMj

k

H e h n e

Mh k k e

ω ω

ω

ω

−

=

+

−

=

⇒ =

+ ⎡ ⎤⎡ ⎤ ⎛ ⎞= − − ⋅⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

∑

∑

[ ]

( )52

1, 0 50, otherwise

sin3

sin2

jj

nh n

H e eωω ω

ω−

≤ ≤⎧= ⎨⎩

⇒ =




Type III FIR linear- phase systems

where M is an even integer

Example

[ ] [ ], 0h n h M n n M= − − ≤ ≤

( )2

2

1

2 sin2

MMjj

k

MH e j h k k eω

ω ω−

=

⎡ ⎤⇒ = − ⋅ ⋅⎢ ⎥⎣ ⎦∑

[ ] [ ] [ ]( ) 2

2

1 2sinj j j

h n n n

H e e j eω ω ω

δ δ

ω− −

= − −

= − = ⋅




Type IV FIR linear- phase systems

where is an odd integer

Example

[ ] [ ], 0h n h M n n M= − − ≤ ≤

( )1

22

1

1 12 sin2 2

MMjj

k

MH e j h k k eω

ω ω

+

−

=

+ ⎡ ⎤⎡ ⎤ ⎛ ⎞⇒ = − ⋅ − ⋅⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦∑

[ ] [ ] [ ]( ) ( )

2

1

1 1

2sin2

j j j

j

h n n n

H e e e

j e

ω ω ω

ω

δ δ

ω

−

−

= − −

⇒ = + = −

= ⋅




– Location of zeros for FIR linear-phase systems

If are symmetric,

If is a zero of then

That is, is also a zero of

When is real and is a zero of will also be a zero of

and so will

Therefore, when is real, there will be four conjugate reciprocal zeros of the

form

unless

( ) [ ]0

Mn

n

H z h n z−=

=∑[ ]{ }h n

( ) [ ] [ ] ( )1

0

M on k M M

n k M

H z h M n z h k z z H z− − − −

= =

= − = =∑ ∑

0jz re θ= ( ),H z

( ) ( )10 0 0 0MH z z H z− −= =

1 10

jz r e θ− − −=

[ ]h n( ).H z

0z ( ) 0, jH z z re θ∗ −= ( ),H z

( ) 1

0 .z−∗

( )( )( )( )1 1 1 1 1 11 1 1 1j j j jre z re z r e z r z zθ θ θ θ− − − − − − − −− − − −

[ ]h n

1.r =




If are anti-symmetric, we have

In particular, when

Thus, must have a zero at

It also must have a zero at when is even.

( ) [ ] [ ] ( )1

0

M on k M M

n k M

H z h M n z h k z z H z− − − −

= =

= − − = − = −∑ ∑

[ ]{ }h n

( ) ( ) ( ) ( ) ( )11, 1 1 1, 1 1 1 .Mz H H z H H+= = − = − − = − −and if

( )H z 1.z =

1z ω π= − ⇔ = M

type - II type - III type - IV




• Relation of FIR linear-phase systems to minimum-phase systems

– Any FIR linear-phase system can be represented by

where

and has only zeros on the unit circle.

⇒ The order of

( ) ( ) ( )1max min min, iM

iH z H z z M H z−−= = no.of zeros of

( )ucH z

( ) ( ) ( ) ( )min max ucH z H z H z H z=

0M

0 2 iM M M+is




Example: Decomposition of a linear-phase system

The overall system has linear phase( ) ( ) ( )min maxH z H z H z=

( )( )( )( )( ) ( )( )( )( )

2 0.6 1 0.6 1 0.8 1 0.8 1min

2 0.6 1 0.6 1 0.8 1 0.8 1max

( ) 1.25 1 0.9 1 0.9 1 0.8 1 0.8

( ) 0.9 1 1.1111 1 1.1111 1 1.25 1 1.25

j j j j

j j j j

H z e z e z e z e z

H z e z e z e z e z

π π π π

π π π π

− − − − − −

− − − − − −

= − − − −

⇒ = − − − −




– Causal and linear phase FIR filter

The impulse response of a desired filter is also symmetric;

The overall response

The resulting response has generalized linear phase and its magnitude is also an

even real function.

[ ] [ ] ( ) ( ) 20

0

Mjj je

w M n n Mw n W e W e e

ωω ω −− ≤ ≤⎧= ⇒ =⎨⎩

ωeven function of [ ] [ ]d dh n h M n= −

( ) ( ) 2Mjj j

d eH e H e eω

ω ω −=

( ) ( ) ( )( )

( ) ( )( )( )

( ) ( )( )

( )

( ) ( ) ( )( )

2 2

2

2

121

21

2

12

jj jd

MMj jjje e

Mjjje e

Mjje

jj je e e

H e H e W e d

H e e W e e d

H e W e e d

A e e

A e H e W e d

ω θω θ

ω θθω θθ

ωω θθ

ωω

π ω θω θ

π

θπ

θπ

θπ

θπ

−

−− −−

−−

−

−

−

=

=

=

=

=

∫

∫

∫

∫where




Example Linear-phase lowpass filter

[ ] [ ] [ ]dh n h n w n=[ ] [ ] [ ]dh n h n nω=

( ) [ ]2

21, 20 ,

sin2

2

c

c

Mj Mjj j ncd d

c

c

eH e h n e e d

Mn

Mn

ωω

ωω ω

ω

ω ω ωπω ω π

ω

π

−−

−

⎧⎪ <= ⇒ =⎨< ≤⎪⎩

⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=⎛ ⎞−⎜ ⎟⎝ ⎠

∫




– Filter design by Kaiser method

High-pass filter

• Design example:

( )

( ) ( )

[ ]

2

2

0 0

sin sin2 2

2 2

cj

Mhp j

c

Mjj jhp ep

c

hp

H ee

H e e H e

M Mn nh n

M Mn n

ωω

ωω ω

ω ω

ω ω π

π ω

π π

−

−

≤ ≤⎧⎪= ⎨≤ ≤⎪⎩

= −

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⇒ = −

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) ( )

1 2

0.4

0.35 , 0.5 0.02133.56

0.5842 21 0.07886 212.5974

33.56 8 23.73 242.285 0.15

s p

A

A A

M M

ω π ω π δ δ δ

β

π

= = = = =

⇒ =

⇒ = − + −

=−

= = ⇒ =×

If and




High-pass filter example

Actual error=0.0213>0.021

33.562.597424

A

Mβ=== ⇒ type I FIR




High-pass Kaiser high-pass filter example with M=25 (type-II)

• Use of a type-II Kaiser HPF

may not be appropriate




– FIR filter with generalized response

– Differentiator

Kaiser formulae were developed for frequency response with simple magnitude

discontinuities, but still works on differentiator

G1

G2

G4

G3

0 ω 1 ω 2 ω 3 π

4N =[ ] ( )

[ ] [ ] [ ]

11

1

sin2

20

kN

d k kk

N

d

Mnh n G G

Mn

Gh n h n n

ω

π

ω

+=

+

⎛ ⎞−⎜ ⎟⎝ ⎠⇒ = −

⎛ ⎞−⎜ ⎟⎝ ⎠

=

⇒ =

∑

where

( )

[ ]

2

2

cos sin2 2

2 2

Mjjd

d

H e j e

M Mn nh n M Mn n

ωω ω π ω π

π π

π

−= − < <

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⇒ = −⎛ ⎞− −⎜ ⎟⎝ ⎠

[ ] [ ] [ ][ ] [ ]

1, dM h n h n n

h M n h n

ω+ = ⋅

− = −

In the case of using a symmetric window of size results in




Kaiser window design of a differentiator

The amplitude error:

By not imposing constraint H(z) to have a zero at z=-1, it is possible to design a

filter having better approximation to the desired response, while using a less

number of filter taps.

( ) ( )( ) ( )

; 0

1

jo

jo

E A e

A e H z z

ω

ω

ω ω ω π= − ≤ ≤

= ±where is the amplitude of the filter and has zeros at




Example: Impulse response of Kaiser-windowed differentiatore




• Optimum approximation of FIR filters

– Design a filter that best approximates the desired response for a given M

– Rectangular window provides the best-mean squared approximation:

minimizes

Do not permit individual control errors in

different bands

[ ] [ ], 0

0 , otherwisedh n n M

h n≤ ≤⎧

= ⎨⎩

[ ][ ] [ ][ ]

; 0

; otherwiswd

d

h n h n n Me n

h n

− ≤ ≤⎧⎪⎨⎪⎩

( ) ( )

[ ]

22

2

12

j jd

n

H e H e d

e n

π ω ω

πε ω

π −

∞

=−∞

= −

=

∫

∑




• Consider the design of a type-I FIR filter with zero phase, i.e.,

– Corresponding frequency response

is a real, even and periodic function of

A causal system can be obtained by delaying it by L samples

– From a polynomial approximation

can be represented as an L-th degree trigonometric polynomial

– Define an error function by

where is a weighting function

[ ] [ ] [ ]2e eh n h n L h L n− = −

( ) [ ] [ ] [ ] ( )0

0 2 cosL L

j j ne e e e

n L n

A e h n e h h n nω ω ω−

=− =

= = +∑ ∑

( ) ( ) ( )1cos cos ; cos cosn nn T T x n xω ω −= =

( )jA e ω

[ ] [ ]e eh n h n= −

( )W ω

ω

( ) ( ) ( ) ( ) 2j jdE W H e H eω ωω ω= −

( ) ( ) ( )cos

0 0

cos ; L L

j k kk kx

k k

A e a P x P x a xωω

ω=

= =

= = =∑ ∑




– Example

Maximum weighted absolute approximation error

is

( )1, 0

0,

pjd

s

H e ωω ω

ω ω π

≤ ≤⎧= ⎨

≤ ≤⎩

2δ δ=

( ) 1

2

0

1,

1

; ;p

s

W KKω ω

ω ω

δωπ δ

≤ ≤

≤

⎧⎪= =⎨⎪⎩ ≤




– Minimax or Chevyshev criterion

Find a set of impulse response that minimizes

– Alternative theorem:

Let denote a closed subset comprising the disjoint union of closed subsets of real

axis x. Then

is an m-th order polynomial

For desired function and positive function which are continuous

on , and weighted error defined by

a necessary and sufficient condition that be a unique r-th order polynomial

minimizing is that has at least alternations

At least values in such that and

( )PE x

{ }0 , : p sF ω ω πω ω ω= ≤ ≤ ≤ ≤[ ]{ }( );0min max ( )

eh n n L FE

ωω

≤ ≤ ∈

( )PW x( )PD x

2r +

δ

( )maxP

Px FE E x

∈=

( )0

mk

kk

P x a x=

=∑

PF

( ) ( ) ( ) ( )P P PE x W x D x P x= −⎡ ⎤⎣ ⎦

1 2 2rx x x +< < <( ) ( )1 , 1,2, , 2P i P iE x E x E i r+= − = ± = +

ix2r + PF




– Example: Consider 5-th order polynomials that

approximate unity for and zero for

Assume that for these two regions

The optimum 5-th order polynomial has at least 7

alternations of the error in the region in

has 3 alternations in and

2 alternations in

also has 5 alternations; 3 in

and 2 in


optimum polynomial

( )1P x

( )3P x

( ) 1PW x =

0.1 1x≤ ≤

0.1 1x≤ ≤1 0.1x− ≤ ≤ −

( )2P x

( )jP x

PF

1 0.1x− ≤ ≤ −

0.1 1 x≤ ≤ ⇒

1 0.1x− ≤ ≤ −

1 0.1x− ≤ ≤ −

0.1 1x≤ ≤




• Optimum type-I lowpass filters

– Consider polynomial defined by

– Assume that

– Equi-ripple approximation

A set of coefficients is designed to make have at least

alternations on

{ }ka ( )2L +

PF

( )1/ ; cos cos 1

cos0; 1 cos cos

pP

s

KW

ω ωω

ω ω

≤ ≤⎧= ⎨

− ≤ ≤⎩

( )1; cos cos 1

cos0; 1 cos cos

pP

s

Dω ω

ωω ω

≤ ≤⎧= ⎨

− ≤ ≤⎩

( ) ( )0

cos cosL

kk

kP aω ω

=

=∑ 7L =

( )P x

{ }: 0 , P p sF ω ω ω πω ω= ≤ ≤ ≤ ≤

( )cosPE ω

( ) ( ) ( ) ( )cos cos cos cosP P PE W D Pω ω ω ω= −⎡ ⎤⎣ ⎦




• Optimum type-I lowpass filters (cont.)

– Redraw in terms of

7L =

( )jA e ω cosx ω=





– Max. possible number of alternations of the error

is since an L-th degree polynomial can

have at most points with zero slope

Has zeros at and roots of

-th order polynomial

– Alternations always occur at

– The filter will be equi-ripple except possibly at

( )1L −

( )1L −

p sω ω ω ω= =and

( ) ( )

( ) ( )

1

0

1

10

cos sin cos

sin 1 cos

Lk

kk

Lk

kk

P ka

k a

ω ω ωω

ω ω

−

=

−

+=

∂= −

∂

= − +

∑

∑

( )3L +

0 ω ω π= =or

0, ω ω π= =




• Parks-McClellan algorithm

– The alternation theorem indicates that the optimum filter will satisfies

– can be rewritten as

– Solution for can be found by means of polynomial interpolation

( ) [ ] [ ] [ ] ( )0 0 0

0 2 cos cosL L L L

j j n k ke e e e k k

n L n k k

A e h n e h h n n a a xω ω ω ω−

=− = = =

= = + = =∑ ∑ ∑ ∑

( )ijdH e ω

ka δand

( )jeA e ω

( )( )

( )

( )

( )

( )( )

1

2

2

21 1 1

02

2 2 2 1

22

2

1

2 2

2

2

11

11

11

L

L

L

LL

L L L

jd

jd

jd

L

WH e

H e W

x x x

ax x x a

x x xH e

W

ω

ω

ω

ω

δ

ω

ω

+ +

+ + ++

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎡⎢ ⎥ ⎢ ⎥− ⎢⎢ ⎥ ⎢ ⎥ ⎢=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎥ −⎢ ⎥⎣ ⎦

⎢ ⎥⎣ ⎦

⎤⎥⎥

⎢ ⎥⎢ ⎥

⎦

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1

1

1 ; 1,2, , 2

1 /

i i

i i

ij ji d e

ij jd e i

W H e A e i L

H e A e W

ω ω

ω ω

ω δ

δ ω

+

+

⎡ ⎤− = − = +⎣ ⎦

⇒ = + −




• Parks-McClellan algorithm (cont.)

– For a given set of extremal frequencies, the P-M algorithm determines

– For given , the error function has magnitude at frequencies

– has values for and for

– An L-th order trigonometric polynimial can be obtained by interpolating ( )iE ω

( ) ( )( ) ( )

1

21

11cos , , 1

k

L

k k Li k ii k

kj

k d kk

b xx C xx

ex

H dW

ω δω

ω =≠

+ +

+= −−

−= = − =∏

( ) ( )

1

11

1

cos

Lk k

j k ke L

k

k k

d Cx xA e P

dx x

ω ω

+

=+

=

−= =

−

∑

∑

2

1

1L

ki k ii k

bx x

+

=≠

=−∏

( )jeA e ω

iω

sω ω π≤ ≤δ±

ka δand

( )( )( )

2

112

1

;1

k

Lj

k dk

kLk

k k

b H e

bW

ω

δ

ω

+

=++

=

=−

∑

∑

( )jeA e ω

1 Kδ±

δ± 2L +

0 pω ω≤ ≤





Example: 7 L =

( )jeA e ω

ka δand

1 Kδ±




• Characteristics of optimum FIR filters

– Size of the filter tap( )( )

1 210log 132.324 s p

Mδ δω ω

− −=

−




• Filter design examples by Parks-McClellan algorithm

26M = 27M =

0.4 ; 0.6p sω π ω π= =

2 31 210 ; 10δ δ −= =




• Filter design example (cont.)

– Compensation of zero-order holder

28 14M L= ⇒ =

( ) ( )/ 2 ; 0

sin / 20;

pjd

s

H e ω

ω ω ωω

ω ω π

⎧ ≤ ≤⎪= ⎨⎪ ≤ ≤⎩

2 31 210 ; 10 ; 0.4 ; 0.6p sδ δ ω π ω π−= = = =




• Optimum bandpass filter design

– Local extrema may occur in the transition

regions

– The approximation need not be equi-ripple

in the approximation regions

( )

( )

0; 0 0.31; 0.35 0.30; 0.7

1; 0 0.31; 0.35 0.30.2; 0.7

jd

j

H e

W e

ω

ω

ω ππ ω ππ ω π


≤ ≤⎧⎪= ≤ ≤⎨⎪ ≤ ≤⎩

≤ ≤⎧⎪= ≤ ≤⎨⎪ ≤ ≤⎩

74 37M L= ⇒ =




• Optimum approximation of FIR filters

– Design a filter that best approximates the desired response for a given M

– Rectangular window provides the best-mean squared approximation:

minimizes

Do not permit individual control errors in

different bands

[ ] [ ], 0

0 , otherwisedh n n M

h n≤ ≤⎧

= ⎨⎩

[ ][ ] [ ][ ]

; 0

; otherwiswd

d

h n h n n Me n

h n

− ≤ ≤⎧⎪⎨⎪⎩

( ) ( )

[ ]

22

2

12

j jd

n

H e H e d

e n

π ω ω

πε ω

π −

∞

=−∞

= −

=

∫

∑




• Consider the design of a type-I FIR filter with zero phase, i.e.,

– Corresponding frequency response

is a real, even and periodic function of

A causal system can be obtained by delaying it by L samples

– From a polynomial approximation

can be represented as an L-th degree trigonometric polynomial

– Define an error function by

where is a weighting function

[ ] [ ] [ ]2e eh n h n L h L n− = −

( ) [ ] [ ] [ ] ( )0

0 2 cosL L

j j ne e e e

n L n

A e h n e h h n nω ω ω−

=− =

= = +∑ ∑

( ) ( ) ( )1cos cos ; cos cosn nn T T x n xω ω −= =

( )jA e ω

[ ] [ ]e eh n h n= −

( )W ω

ω

( ) ( ) ( ) ( ) 2j jdE W H e H eω ωω ω= −

( ) ( ) ( )cos

0 0

cos ; L L

j k kk kx

k k

A e a P x P x a xωω

ω=

= =

= = =∑ ∑




– Example

Maximum weighted absolute approximation error

is

( )1, 0

0,

pjd

s

H e ωω ω

ω ω π

≤ ≤⎧= ⎨

≤ ≤⎩

2δ δ=

( ) 1

2

0

1,

1

; ;p

s

W KKω ω

ω ω

δωπ δ

≤ ≤

≤

⎧⎪= =⎨⎪⎩ ≤




– Minimax or Chevyshev criterion

Find a set of impulse response that minimizes

– Alternative theorem:

Let denote a closed subset comprising the disjoint union of closed subsets of real

axis x. Then

is an m-th order polynomial

For desired function and positive function which are continuous

on , and weighted error defined by

a necessary and sufficient condition that be a unique r-th order polynomial

minimizing is that has at least alternations

At least values in such that and

( )PE x

{ }0 , : p sF ω ω πω ω ω= ≤ ≤ ≤ ≤[ ]{ }( );0min max ( )

eh n n L FE

ωω

≤ ≤ ∈

( )PW x( )PD x

2r +

δ

( )maxP

Px FE E x

∈=

( )0

mk

kk

P x a x=

=∑

PF

( ) ( ) ( ) ( )P P PE x W x D x P x= −⎡ ⎤⎣ ⎦

1 2 2rx x x +< < <( ) ( )1 , 1,2, , 2P i P iE x E x E i r+= − = ± = +

ix2r + PF




– Example: Consider 5-th order polynomials that

approximate unity for and zero for

Assume that for these two regions

The optimum 5-th order polynomial has at least 7

alternations of the error in the region in


2 alternations in

also has 5 alternations; 3 in

and 2 in


optimum polynomial

( )1P x

( )3P x

( ) 1PW x =

0.1 1x≤ ≤

0.1 1x≤ ≤1 0.1x− ≤ ≤ −

( )2P x

( )jP x

PF

1 0.1x− ≤ ≤ −

0.1 1 x≤ ≤ ⇒

1 0.1x− ≤ ≤ −

1 0.1x− ≤ ≤ −

0.1 1x≤ ≤




• Optimum type-I lowpass filters

– Consider polynomial defined by

– Assume that

– Equi-ripple approximation

A set of coefficients is designed to make have at least

alternations on

{ }ka ( )2L +

PF

( )1/ ; cos cos 1

cos0; 1 cos cos

pP

s

KW

ω ωω

ω ω

≤ ≤⎧= ⎨

− ≤ ≤⎩

( )1; cos cos 1

cos0; 1 cos cos

pP

s

Dω ω

ωω ω

≤ ≤⎧= ⎨

− ≤ ≤⎩

( ) ( )0

cos cosL

kk

kP aω ω

=

=∑ 7L =

( )P x

{ }: 0 , P p sF ω ω ω πω ω= ≤ ≤ ≤ ≤

( )cosPE ω

( ) ( ) ( ) ( )cos cos cos cosP P PE W D Pω ω ω ω= −⎡ ⎤⎣ ⎦





– Redraw in terms of

7L =

( )jA e ω cosx ω=





– Max. possible number of alternations of the error

is since an L-th degree polynomial can

have at most points with zero slope

Has zeros at and roots of

-th order polynomial

– Alternations always occur at

– The filter will be equi-ripple except possibly at

( )1L −

( )1L −

p sω ω ω ω= =and

( ) ( )

( ) ( )

1

0

1

10

cos sin cos

sin 1 cos

Lk

kk

Lk

kk

P ka

k a

ω ω ωω

ω ω

−

=

−

+=

∂= −

∂

= − +

∑

∑

( )3L +

0 ω ω π= =or

0, ω ω π= =




• Parks-McClellan algorithm

– The alternation theorem indicates that the optimum filter will satisfies

– can be rewritten as

– Solution for can be found by means of polynomial interpolation

( ) [ ] [ ] [ ] ( )0 0 0

0 2 cos cosL L L L

j j n k ke e e e k k

n L n k k

A e h n e h h n n a a xω ω ω ω−

=− = = =

= = + = =∑ ∑ ∑ ∑

( )ijdH e ω

ka δand

( )jeA e ω

( )( )

( )

( )

( )

( )( )

1

2

2

21 1 1

02

2 2 2 1

22

2

1

2 2

2

2

11

11

11

L

L

L

LL

L L L

jd

jd

jd

L

WH e

H e W

x x x

ax x x a

x x xH e

W

ω

ω

ω

ω

δ

ω

ω

+ +

+ + ++

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎡⎢ ⎥ ⎢ ⎥− ⎢⎢ ⎥ ⎢ ⎥ ⎢=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎥ −⎢ ⎥⎣ ⎦

⎢ ⎥⎣ ⎦

⎤⎥⎥

⎢ ⎥⎢ ⎥

⎦

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1

1

1 ; 1,2, , 2

1 /

i i

i i

ij ji d e

ij jd e i

W H e A e i L

H e A e W

ω ω

ω ω

ω δ

δ ω

+

+

⎡ ⎤− = − = +⎣ ⎦

⇒ = + −





– For a given set of extremal frequencies, the P-M algorithm determines

– For given , the error function has magnitude at frequencies

– has values for and for

– An L-th order trigonometric polynimial can be obtained by interpolating ( )iE ω

( ) ( )( ) ( )

1

21

11cos , , 1

k

L

k k Li k ii k

kj

k d kk

b xx C xx

ex

H dW

ω δω

ω =≠

+ +

+= −−

−= = − =∏

( ) ( )

1

11

1

cos

Lk k

j k ke L

k

k k

d Cx xA e P

dx x

ω ω

+

=+

=

−= =

−

∑

∑

2

1

1L

ki k ii k

bx x

+

=≠

=−∏

( )jeA e ω

iω

sω ω π≤ ≤δ±

ka δand

( )( )( )

2

112

1

;1

k

Lj

k dk

kLk

k k

b H e

bW

ω

δ

ω

+

=++

=

=−

∑

∑

( )jeA e ω

1 Kδ±

δ± 2L +

0 pω ω≤ ≤





Example: 7 L =

( )jeA e ω

ka δand

1 Kδ±




• Characteristics of optimum FIR filters

– Size of the filter tap( )( )

1 210log 132.324 s p

Mδ δω ω

− −=

−




• Filter design examples by Parks-McClellan algorithm

26M = 27M =

0.4 ; 0.6p sω π ω π= =

2 31 210 ; 10δ δ −= =




• Filter design example (cont.)

– Compensation of zero-order holder

28 14M L= ⇒ =

( ) ( )/ 2 ; 0

sin / 20;

pjd

s

H e ω

ω ω ωω

ω ω π

⎧ ≤ ≤⎪= ⎨⎪ ≤ ≤⎩

2 31 210 ; 10 ; 0.4 ; 0.6p sδ δ ω π ω π−= = = =




• Optimum bandpass filter design

– Local extrema may occur in the transition

regions, which may not be acceptable

– The approximation need not be equi-ripple

in the approximation regions

( )

( )

0; 0 0.31; 0.35 0.30; 0.7

1; 0 0.31; 0.35 0.30.2; 0.7

jd

j

H e

W e

ω

ω



≤ ≤⎧⎪= ≤ ≤⎨⎪ ≤ ≤⎩

≤ ≤⎧⎪= ≤ ≤⎨⎪ ≤ ≤⎩

74 37M L= ⇒ =



6. Discrete Fourier transform

Discrete Fourier transform• Discrete Fourier series representation of periodic sequences

– Let be a periodic sequence with period N such that for any

integer m

Example A periodic complex exponential

– A set of N periodic complex exponentials defines all the

frequency components

– Fourier series representation

[ ]x n

[ ]{ }, 0,1,2,... 1,ke n k N= −

[ ] [ ] k k Ne n e n+=Note that for any integer

[ ] [ ]x n x n mN= +

[ ] [ ]2

; knj

Nk ke n e e n mN k

π

= = + is an interger

[ ] [ ]

[ ] [ ]

21

0

21

0

1

knN jN

k

mnN jN

n

x n X k eN

X k x n e

π

π

−

=

− −

=

⇒ =

=

∑

∑

[ ] [ ]21

0

1 knN jN

k

x n X k eN

π−

=

= ∑




The Fourier series coefficient can be obtained by

The Fourier series coefficients is a periodic sequence

– Discrete Fourier series (DFS)

Define by

[ ] [ ]

[ ]( )

( )

2 2 21 1 1

0 0 0

21 1

0 0

21

0

1

1

1,10, otherwise

rn kn rnN N Nj j jN N N

n n k

k r nN N jN

k k

k r nN jN

n

x n e X k e eN

X k eN

k r Ne

N

π π π

π

π

− − −− −

= = =

−− −

= =

−−

=

=

=

− =⎧= ⎨⎩

∑ ∑ ∑

∑ ∑

∑Since

[ ] [ ]21

0

rnN jN

nx n e X r

π−

=

=∑

[ ] [ ]DFSx n X k←⎯⎯→

[ ] [ ]

[ ] [ ]

21

0

1

0

;

1

N jkn NN N

n

Nkn

Nk

X k x n w w e

x n X k wN

π− −

=

−−

=

= =

=

∑

∑

Analysis equation :

Synthesis equation :

[ ] [ ]

[ ] [ ] [ ] [ ]

21

0

2 ( ) 21 12

0 0

knN jN

n

k N n knN Nj j j nN N

n n

X k x n e

X k N x n e x n e e X k

π

π ππ

− −

=

+− −− − −

− −

=

+ = = =

∑

∑ ∑




Example Periodic pulse train

Example Duality in DFS

Let the DFS coefficient be a periodic impulse train

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

1

0

21 1

0 0

1, 0, otherwise

, 0 1

1

1 1 1

r

Nkn

Nn

N N j knkn kn NN N

k k

n mMx n n mN

x n n n N

X k n w

x n X k w w eN N N

π

δ

δ

δ

∞

=−∞

−−

=

− −− −

= =

=⎧= − = ⎨

⎩= ≤ ≤ −

= =

⇒ = = =

∑

∑

∑ ∑ ∑

Since

[ ] [ ] [ ] [ ] Y k Nx n y n X n= ↔ =

[ ] [ ]1

0

01,

Nkn

N Nk

y n N k w w nδ−

− −

=

⇒ = = = ∀∑

[ ] [ ]r

Y k N k rNδ∞

=−∞

= −∑




Example Periodic rectangular pulse train

Any periodic sequence can be represented as a sum of complex exponential

sequences

[ ]2 454 4

1010 1010

0 0 10

sin1 21 sin

10

kkj kn jnkk

n n

kWX k W e ekW

π ππ

π− −

− =

−= = = =

−∑ ∑




• Properties of DFS

– Linearity

– Shift of a sequence

– Duality

– Symmetry properties

[ ] [ ] [ ] [ ]1 2DFSax n bx n aX k bX k+ ←⎯⎯→ +

[ ] [ ] ( ) [ ] [ ]

[ ] [ ][ ] [ ]

2 2 2 21 1

0 0

N Nj kn j n m j j km kmN N N NN

n n

DFS kmN

DFSnN

x n m e x n m e e X k e X k W

x n m X k W

x n W X k

π π π π− −− − − − −

= =

−

− = − = =

⇒ − ←⎯⎯→

←⎯⎯→ −

∑ ∑

[ ] [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ]

1 1

1

0

0 0

1

,

Nkn

Nk

Nkn

Nk

DFS DF

k

S

Nkn

N Nx n k W

Nx

x n X k

k x n W

x n k X n Nx

XN

k

w

X

−

=

−

−−

=

=

− =

⇒ − =

←⎯⎯→ ←⎯⎯→

= ⇒

−

∑ ∑

∑If

[ ] [ ][ ] [ ]

* *

* *

DFS

DFS

x n X k

x n X k

←⎯⎯→ −

− ←⎯⎯→




⇒ If is real,

– Periodic convolution

Duality

Difference between aperiodic convolution and periodic convolution

• The sum is over an interval

•

[ ] [ ][ ] [ ]

*X k X k

X k X k

= −

= −

[ ]x n

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ]

1 1

3 1 2 2 10 0

1 1

3 1 20 0

1 1

1 20 0

1

2 1 1 20

N N

m m

N Nkn

Nn m

N Nkn

Nm n

Nkm

Nm

x n x m x n m x m x n m

X k x m x n m W

x m x n m W

X k x m W X k X k

− −

= =

− −

= =

− −

= =

−

=

= − = −

⎛ ⎞= −⎜ ⎟

⎝ ⎠

= −

= =

∑ ∑

∑ ∑

∑ ∑

∑

0 1m N≤ ≤ −

[ ] ( )2 2x n m x m n− = − −⎡ ⎤⎣ ⎦

[ ] [ ] [ ] [ ]1

1 2 1 20

NDFS

mx m x n m X k X k

−

=

⇒ − ←⎯⎯→∑

[ ] [ ] [ ] [ ]1

1 2 1 20

1 NDFS

m

x n x n X m X k mN

−

=

←⎯⎯→ −∑




Example Periodic convolution




– Properties of DFS




– Properties of DFS (cont.)




• Fourier transform (F.T.) of periodic signals

– A sequence should be absolutely summable to guarantee the uniform convergence of

its Fourier transform

– Periodic signals are not absolutely summable

– The F.T. of a periodic signal can be interpreted as an impulse train in the frequency

domain with impulse values proportional to the DFS coefficients for the sequence

– If is periodic with period N, the Fourier transform of is defined by

Although the F.T. of a periodic sequence does not converge in the normal sense,

the introduction of impulses enables to use the F. T.

Note: has a necessary periodicity with period

[ ] ( ) [ ]2 2j

k

kx n X e X kN N

ω π πδ ω∞

=−∞

⎛ ⎞⇔ = −⎜ ⎟⎝ ⎠

∑

[ ]x n[ ]x n

( ) [ ]

[ ]

[ ] [ ]

2 2

0 0

2

0

2

1 1 2 2 ; 0 2 /2 2

1 2

1

j j n j n

k

j n

k

j knN

k

kX e e d X k e d NN N

kX k e dN N

X k e x nN

π ε π εω ω ω

ε ε

π ε ω

ε

π

π πω δ ω ω ε ππ π

πδ ω ω

− −

− −

∞ −

−=−∞

∞

=−∞

⎛ ⎞= − < <⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

= =

∑∫ ∫

∑ ∫

∑

( )jX e ω 2π




Example: Fourier transform of a periodic impulse train

– For a finite-length sequence

Thus, is obtained by sampling the Fourier transform of a finite sequence

[ ] [ ] [ ]

[ ] [ ] [ ]

( ) ( ) ( )

( )2

2 2

2 2

j j j

j

k

kjN

k

m m

x n x n p n

X e X e P e

kX e

mN

N N

kX eN

x n x n mN

N

ω ω ω

ω

π

π πδ

ω

δ

ω

π πδ

∞ ∞ ∞

=−∞ =−∞ =−

∞

=−∞

∞

=−

∞

∞

= ∗

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= −

− −

⎜ ⎟ ⎜ ⎟⎝ ⎠⎝

−

⎠

= =∑ ∑ ∑

∑

∑

[ ] [ ] [ ]

( )

1,

2

2m

j

k

p n n mN P k k

kP e

NNω

δ

πδ ω

π

∞

=−∞

∞

=−∞

= − ⇒ = ∀

⇒ = −⎡ ⎤⎢ ⎥⎣ ⎦

∑

∑[ ] [ ]( ) , 0 for 0 and x n x n n n N= < ≥., i.e

[ ] ( ) [ ]2 21

20

Nj k j kj N Nk nN

X k X e X e x n eπ π

ωπω

− −

= =

⎛ ⎞⇒ = = =⎜ ⎟

⎝ ⎠∑

[ ] x n[ ] X k




Example: Relationship between the DFS coefficients and the FT of one period

[ ] 1; 0 40;

x nn≤ ≤⎧

= ⎨⎩ otherwise

( ) ( )( )

42

0

sin 5 / 2sin / 2

j j n j

nX e e eω ω ωω

ω− −

=

⇒ = =∑




• Sampling the Fourier transform

– Since is periodic with period is also periodic in k with period N.

– Since

The samples of the F.T. of can be thought of as DFS coefficients of a periodic sequence obtained through periodic replication of

( )jX e ω [ ]2 , X kπ

[ ] ( ) [ ]

[ ] [ ]

[ ] ( )

[ ] [ ]

[ ] [ ]

[ ]

2 21

20

2 21

0

21

0

,

1

1

Nj k j kj N Nk nN

N j mk j knN N

k m

N j k m nN

m k

m

X k X e X e x n e

x n x m e eN

x m eN

x m n m N

x n n N

x n N

π πω

πω

π π

π

δ

δ

− −

==

− ∞ −

= =−∞

∞ − −

=−∞ =

∞ ∞

=−∞ =−∞

∞

=−∞

∞

=−∞

⎛ ⎞= = =⎜ ⎟

⎝ ⎠⎡ ⎤

= ⎢ ⎥⎣ ⎦

=

= − −

= ∗ −

= −

∑

∑ ∑

∑ ∑

∑ ∑

∑

∑

[ ]x n[ ]x n[ ]x n




– A periodic sequence can be interpreted as equally spaced samples of the Fourier transform of one period of

Example

[ ] [ ]

( ) [ ] [ ]

[ ] ( )

1 1

0 0

2

, 0 10 , otherwise

2

N Nj j n j n

n n

jk

N

x n n Nx n

X e x n e x n e

X k X e

ω ω ω

ωπω

πω

− −− −

= =

=

⎧ ≤ ≤ −= ⎨⎩

= =

⇒ = ⇐

∑ ∑

Sampling of the Fourier transform at a rate of

[ ]

( ) [ ] ( )( )

42 4 /10

0

1, 0 40, otherwise

5sin sin / 22sin /10sin

2

j j n j j k

n

nx n

kX e e e X k e

kω ω ω π

ωπ

ω π− − −

=

≤ ≤⎧= ⎨⎩

= = ⇒ =∑

[ ]x n[ ]X k




• Fourier representation of finite-duration sequences: Discrete Fourier transform (DFT)

– can be visualized as warping the finite sequence around the cylinder

– Since the Fourier series coefficients of is also a periodic sequence with

a period of N, the DFT of can be obtained by

– Since

[ ] [ ] [ ] [ ] ( )( )

( )( ) ( )

, 0 1

0 , otherwise

modulo

N

N

x n n Nx n x n x n N x n

x n x n N

∞

=−∞

⎧ ≤ ≤ −⎡ ⎤= ⇔ = − =⎨ ⎣ ⎦

⎩⎡ ⎤ ⎡ ⎤⎣ ⎦⎣ ⎦

∑

where

[ ]x n

[ ] [ ]

[ ] ( )( ) [ ] [ ]

, 0 10 , otherwise

Nm

X k k NX k

X k X k X k X k mN∞

=−∞

⎧ ≤ ≤ −⎪= ⎨⎪⎩

⎡ ⎤= ⇔ = −⎣ ⎦ ∑

[ ]x n[ ]X k[ ]x n

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

1 1

0 0

1 1

0 0

1 ,

1, 0 1 , 0 1

0 , otherwise 0 , otherwise

N Nnk nk

N Nn k

N Nnk nk

N Nk k

X k x n W x n X k WN

x n W n N X k W n NX k x n N

− −−

= =

− −−

= =

= =

⎧ ⎧≤ ≤ − ≤ ≤ −⎪ ⎪= =⎨ ⎨

⎪ ⎪⎩ ⎩

∑ ∑

∑ ∑

and

and

[ ] [ ]DFTx n X k←⎯⎯→




– Example

N=5;

[ ]2 21

20

, 0, , 2 ,10, otherwise

1

j kN jN

kjn N

N k N NeX k ee

π π

π

−− −

−=

= ± ± ⋅ ⋅ ⋅⎧−= = = ⎨

⎩−∑

[ ]X k [ ]X k

[ ]x n

[ ]x n




– Example: N=10

x n⎡ ⎤⎣ ⎦

x n⎡ ⎤⎣ ⎦

[ ]X k




• Properties of the DFS

– Linearity

– Circular shift

Let

Since

[ ] [ ] [ ] [ ] [ ] [ ]1 2 1 2

1 2 1, 2

a

m x

DFT DFTx n x n X k X k

N N N

x n X k

N N

α β α β⇒ + +

⎡ ⎤≥ ⎣ ⎦

←⎯⎯→ ←⎯⎯→

with zero padding

[ ] [ ] ( )1DFT j j mx n x n m X e eω ω−←⎯⎯→= −

[ ] [ ] [ ]2

1 1

j kmFT NDx n X k X k eπ

−←⎯⎯→ =

[ ] [ ][ ]

1 1 0 0 -1, x n n N x n

x n

= ≤ ≤outside cannot be obtained by simply time shifting

[ ] ( )( ) [ ] ( )( )

[ ] ( )( )( )( )

( )( ) [ ]

[ ] [ ] ( )( )

1 1 1 1

2 2 2

1

11

, 0 1

0 , otherwise

N

N N

m k km kmj j jN N N

N

DFT

N

N

x n x n X k X k

X k X k e X k e X k e

x n x n m n Nx n

π π π− − −

⎡ ⎤ ⎡ ⎤⇒ = =⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⇒ = = =⎣ ⎦ ⎣ ⎦⎧ ⎡ ⎤= − ≤ ≤ −⎪ ⎣ ⎦⇒ =

←⎯⎯→

⎨⎪⎩

( )( ) [ ]2

, 0 1: kmj

NN

DFTx n m X k e n Nπ

−←⎡ ⎤⇒ − ≤ ≤⎣ ⎯ −⎯→⎦ circular shift




– Example: circular shift




– Duality

Let

– Symmetry

[ ] ( )( ) [ ] ( )( )[ ] [ ]

DFTN

D

N

FT

x n X n X k X k

X k Nx k

⎡ ⎤ ⎡ ⎤= ←⎯ =⎣ ⎦ ⎯→

←⎣

⇒ −⎯⎯→⎦

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

( )( )

[ ] ( )( )

1 1

1

1

, 0 10, otherwise

, 0 1

0, otherwise

, 0 1

N

DFT

DFN

T

Nx k k NX k

N

x n X n x n

x k k N

X n N x k k

X k Nx

N

k←⎯⎯→= ⇒

←⎯⎯→

=

⎧ − ≤ ≤ −= ⎨⎩⎧ ⎡ ⎤− ≤ ≤ −⎪ ⎣ ⎦= ⎨⎪⎩

⎡ ⎤⇒ − ≤⎣ ⎦

−

≤ −

[ ] [ ] ( )( )[ ] ( )( )( )( ) [ ]

,

0 1DFT

N

DFT

N

N

x n

x n

X k n N

x n X k

x n x n∗ ∗

∗ ∗

⎡ ⎤=

⎡ ⎤⇒ ←⎯⎯→ − ≤ ≤ −⎣ ⎦⎡ ⎤− ←⎯⎯→⎣

= ⎣

⎦

⎦DFT of DFS coefficients of

[ ] [ ] [ ]{ } [ ] ( )( )

[ ] [ ] [ ]{ }0

1 ; 212

e Nx n x n x n x n x n

x n x n x n

∗

∗

⎡ ⎤= + − = ⎣ ⎦

= − −




Similarly,

– N-point Circular convolution

[ ] [ ]

( )( ) ( )( ){ }[ ] [ ]{ } ( )( ) ( ) ( )( )

[ ] [ ] [ ] [ ]{ } [ ]{ }

, 0 11 21 2

1 Re2

; ; 0 1

DFT

e e

N

e e

N N

N

x n x n n N

x n x n

x n x n N n nN n

x n X k X k

n n

X k

N

k X

∗

∗

∗

= ≤ ≤ −

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣

⇐ − = −

⎦

=

←⎯⎯

= ≤+ −

= + =

−

→

≤

[ ] [ ][ ] [ ]{ }0

, 0 1

Imo

DF

o

T

x n x n n N

x n j X k

= ≤ ≤ −

⇒ ←⎯⎯→

[ ] [ ] [ ]

( )( ) ( )( )

[ ] ( )( )[ ] [ ]

( )( ) [ ]

[ ] [ ]

1

3 1 20

1

1 20

1

1 20

1 2

1

1 20

2 1

, 0 1

N

m

N

N Nm

N

Nm

N

Nm

x n x m x n m n N

x m x n m

x m x n m

x n x n

x n m x m

x n x n

−

=

−

=

−

=

−

=

= − ≤ ≤ −

⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦

⎡ ⎤= −⎣ ⎦

Δ

⎡ ⎤= −⎣ ⎦

=

∑

∑

∑

∑

N

N




Example

[ ] [ ]1 1x n nδ= −




Example: N-point circular convolution

N=L:

N =2 L: Linear convolution ?

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] ( )( )

[ ] [ ]

1 2 1 2

1

1 2 3 1 20

1 2

1

1

DFS

NDFS

N

x n x n X k X k

x n x n X k X X kN

X k X kN

−

=

←⎯⎯→

⎡ ⎤⇔ ←⎯⎯→ = −⎣ ⎦∑

N

N

[ ] [ ]

[ ] [ ] [ ]

[ ]

1

1 20

2

3 1 2

3

, 01

0, otherwise

, 00, otherwise

, 0 1

Nkn

Nn

N kX k X k W

N kX k X k X k

x n N n N

−

=

=⎧= = ⋅ = ⎨

⎩=⎧

⇒ = = ⎨⎩

⇒ = ≤ ≤ −

∑

[ ] [ ]1 2

1 0 10 otherwise

n Lx n x n

≤ ≤ −⎧= = ⎨

⎩




Example: N(=2L)-point circular convolution

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] ( )( )

[ ] [ ]

1 2 1 2

1

1 2 3 1 20

1 2

1

1

DFT

NDFT

N

x n x n X k X k

x n x n X k X X kN

X k X kN

−

=

←⎯⎯→

⎡ ⎤←⎯⎯→ = −⎣ ⎦

=

∑

N

N

[ ] [ ]1 2

1 0 10 otherwise

n Lx n x n

≤ ≤ −⎧= = ⎨

⎩




• Linear convolution using the DFT

– -point circular convolution with zero padding

0 L

0 p0 L p+ −1

[ ]2x n

[ ] [ ] [ ]3 1 2m

x n x m x n m∞

=−∞

= −∑

( )1N L pΔ + −

[ ] [ ] [ ] [ ], 0 1;

0, otherwise mp

x n n Nx n x n x n mN

≤ ≤ −⎧= = −⎨⎩

∑

[ ]1x n

( ) ( ) ( ) [ ] ( )

( ) ( )

3 1 3

1 2

2 3 2

2

,

0

1

j j

j j j j

k

k

N

N

X

X e X e X e X k X e k N

e X e

ω ω ω

ω

ω

ω

π

π

ω

ω

=

=

= ⇒ = ≤ ≤

=

−

[ ] [ ][ ] [ ] [ ]

1 2

3 1 2

p

X k X k

x n x n x n

=

⇒ = N




– The circular convolution of two finite-length sequences can be equivalent to linear

convolution of the two sequences.

– Calculation of the output of an LTI using the DFT

– How to resolve the issue when is infinite?

Assume

The sequence can be decomposed in to a sum of shifted finite-length

segments of length L

[ ][ ]3

3

, 0 1

0, otherwisem

p

x n mN n Nx n

− ≤ ≤ −⎧⎪= ⎨⎪⎩

∑

0 p

h n

zero padding

0

L −1

L L

x n [ ] [ ] [ ]y n h n x n= ∗

[ ]x n

[ ] 0 0x n n= <for

[ ]x n

[ ] [ ] [ ] [ ]0

, 0 1;

0 , otherwisemm m

x n mL n Nx n x n mL x n

∞

=

+ ≤ ≤ −⎧= − = ⎨

⎩∑




– Illustration of warping around due to improper circular convolution

N L= 1N L p= + −




• Linear convolution using the discrete Fourier transform

– How to resolve the issue when is infinite?

Assume

The sequence can be decomposed in to a sum of shifted finite-length

segments of length L

[ ]x n

[ ] 0 0x n n= <for

[ ]x n

[ ] [ ]

[ ] [ ]0

;

, 0 1

0 , otherwise

mm

m

x n x n mL

x n mL n Nx n

∞

=

= −

+ ≤ ≤ −⎧= ⎨⎩

∑

where




– Overlap-add method [ ] [ ] [ ]y n h n x n= ∗

[ ] [ ]

[ ] [ ] [ ]0m

m

m m

y n y n mL

y n h n x n

∞

== −

= ∗

∑

[ ] [ ], 0 1mx n x n mL n L= + ≤ ≤ −

-N point circular convolution




– Overlap-save method [ ] [ ] [ ]y n h n x n= ∗

-N point circular convolution

[ ] ( ) 1 1

0 1mx n x n m L p p

n L

= + − + − +⎡ ⎤⎣ ⎦≤ ≤ −

[ ] ( )

[ ] [ ] [ ]0

1 1m

m

m m

y n y n m L p p

y n h n x n

∞

== − − + + −⎡ ⎤⎣ ⎦

= ∗

∑




• The discrete Cosine transform (DCT)

– General class of finite-length transform

where are referred to the basis sequences s.t.

e.g. DFT:

– DCT definition

The basis sequences are cosine functions

Assume that are periodic and even symmetric

– Type-I periodic sequence: even periodic symmetry about n=0, N-1, 2(N-1),

[ ] [ ] [ ]

[ ] [ ] [ ]

1

0

1

0

1

N

kn

N

kk

k x n n

x n k nN

φ

φ

−∗

=

−

=

Α =

= Α

∑

∑[ ]{ }k nφ [ ] [ ]

1

0

1,10,

N

k mn

m kn n

m kNφ φ

−∗

=

=⎧= ⎨ ≠⎩

∑[ ]

2 knjN

k n eπ

φ =

[ ]{ }k nφ cos knNπ

[ ]x n

[ ] ( )( ) ( )( )1 2 2 2 2N Nx n x n x nα α− −

⎡ ⎤ ⎡ ⎤= + −⎣ ⎦ ⎣ ⎦

[ ] [ ] [ ]

[ ]1 0 and -121 1 2

x n n x n

n Nn

n N

α α

α

=

⎧ =⎪= ⎨⎪ ≤ ≤ −⎩

:8 - 29, - 31, - 34, - 36, - 39, - 49, & - 64Due

Homework #

Ju y

7

b ne 9




DCT-1 is defined by

– Type-II periodic sequence: periodic with period 2N

DCT-2 is defined by

[ ] [ ] [ ]

[ ] [ ] [ ]

1

0

1

0

1 1

1 1

2 cos , 0 11

1 cos , 0 11 1

N

n

N

k

c

c

knX k n x n k NN

knx n k X k n NN N

πα

πα

−

=

−

=

= ≤ ≤ −−

= ≤ ≤ −− −

∑

∑

[ ] ( )( ) ( )( )2 22 1N N

x n x n x n⎡ ⎤ ⎡ ⎤= + − −⎣ ⎦ ⎣ ⎦

[ ]1 , 021 , 1 1

kk

k Nβ

⎧ =⎪= ⎨⎪ ≤ ≤ −⎩

[ ] [ ] ( )

[ ] [ ] [ ] ( )

1

0

1

0

2 2

2 2

2 12 cos , 0 1

22 11 cos , 0 12

N

n

N

k

c

c

k nX k x n k N

Nk n

x n k X k n NN N

π

πβ

−

=

−

=

+= ≤ ≤ −

+= ≤ ≤ −

∑

∑




– Relationship between DFT and DCT-1

2(N-1)-point DFT of

The DCT-1 of N-point sequence is identical to 2(N-1)-point DFT of

[ ] ( )( ) ( )( ) [ ]1 12 2 2 2, 0 2 3

N Nx n x n x n x n n Nα α− −

⎡ ⎤ ⎡ ⎤= + − = ≤ ≤ −⎣ ⎦ ⎣ ⎦

[ ]1 .x n

[ ] ( ) [ ] ( )

( ) [ ] ( ) [ ] ( )

( ) [ ] [ ]

( ) [ ] [ ]

22 32 1

10

1 2 31 1

0

1 21 1

0 1

21 1

1

1

1 1

1 1

1 1 1

12 1

1 2( 1)2 1

1 2 1

1 02 1

knN jN

k

kn knN Nj jN N

k k N

kn knN Nj jN N

k k

kn knN j jN N

k

c c

c c

c c c

x n X k eN

X k e X N k eN

X k e X k eN

X k e e X X NN

π

π π

π π

π π

−−

=

− −− −

= =

− −− −

= =

− −− −

=

=−

⎡ ⎤= + − −⎢ ⎥− ⎣ ⎦

⎡ ⎤= +⎢ ⎥− ⎣ ⎦

⎛ ⎞= + + +⎜ ⎟− ⎝ ⎠

∑

∑ ∑

∑ ∑

∑ [ ]( )

[ ] [ ] [ ]

11

1

01

1

1 , 0 and 11 cos , 0 1; 21 1

N nj

N

N

k

c

e

k Nnkk X k n N kN N k N

π

πα α

−−

−

=

⎡ ⎤−⎢ ⎥

⎣ ⎦

⎧

1 , 1 2

= −⎪= ≤ ≤ − = ⎨− − ⎪ ≤ ≤ −∑

⎩

[ ] [ ] [ ] [ ]{ }1 2Re , 0 2 3X k X k X k X k k Nα α α∗= + = ≤ ≤ −

[ ] [ ] [ ] ( )[ ]

1

0

1

22 cos2 1

N

n

c

knX k n x nN

X k

απα

−

=

=−

=

∑

[ ]1 :x n

[ ] ( )

[ ]( )

[ ]

2 31

2 221

1

21

1

1

1

1

2 2

knN jN

k N

N m nN jN

m

nmN jN

m

c

c

c

X N k e

X m e

X m e

π

π

π

−−

=

− −−−

=

−−−

=

− −

=

=

∑

∑

∑




Note : DCT-1 involves only real-valued coefficients

⇒ reduce the implementation complexity.

– Relationship between DFT and DCT-2

2N-point DFT of 2N-point sequence

[ ] ( )( ) ( )( ) [ ]2 22 21 , 0 2 1

N Nx n x n x n x n n N⎡ ⎤ ⎡ ⎤= + − − = ≤ ≤ −⎣ ⎦ ⎣ ⎦

[ ] [ ] [ ][ ] [ ]( )

[ ]{ }

* 2 / 2

/ 2 / 2 * / 2

/ 2 / 2

2

2 Re , 0 2 1

j k N

j k N j k N j k N

j k N j k N

X k X k X k e

e X k e X k e

e X k e k N

π

π π π

π π

−

−

= +

= +

= ≤ ≤ −

[ ]{ } [ ]

[ ] [ ]{ }[ ] [ ]

1/ 2

0

/ 2

/ 2

2

2 2

(2 1)Re cos2

2Re ; 0 1

; 0 1

Nj k N

k

j k N

j k N

c

c

k nX k e x nN

X k X k e k N

X k X k e k N

π

π

π

π−−

=

−

−

+=

⇒ = ≤ ≤ −

⇒ = ≤ ≤ −

∑

[ ] [ ] [ ]1

02 2

(2 )(2 1)2 2 cos ; 0 2 12

N

k

c cN k nX N k x n X k k NN

π−

=

− +− = = − ≤ ≤ −∑

[ ]

[ ][ ]

[ ]

/ 2

/ 2

2

22

2

0 , 0

, 1, , 10,

2 , 1, ,2 1

j k N

j k N

c

c

c

X k

X k e k NX k

k NX N k e k N N

π

π

=⎧⎪

= ⋅ ⋅ ⋅ −⎪⇒ = ⎨=⎪

⎪− − = + ⋅ ⋅ ⋅ −⎩

[ ]2x n

[ ] [ ]2 1

2 / 2

02 2

1 , 2

0,1, ,2 1

Nj k N

kx n X k e

Nn N

π−

=

⇔ =

= ⋅ ⋅ ⋅ −

∑




– Energy compaction property of DCT-2

Since DCT-2 coefficients of a finite-length sequence are concentrated in the low

indices than DFT, DCT-2 is often used for data compression applications in

preference to DFT

Example

[ ] [ ] [ ]1 12 2

0 02

1N N

n k

cx n k X kN

β− −

= =

=∑ ∑

[ ] ( )0cosnx n a nω φ= +

32⇐ −point DCT

32 - point DFT




Energy concentration can be examined by measuring the truncation error of the

transformations.

[ ] [ ] [ ]

[ ] [ ] [ ]

1 2

0

1 2

0

1

1

N

DFTn

N

DCTn

m

m

c

E m x n x nN

E m x n x nN

−

=

−

=

= −

= −

∑

∑

MS truncation error :

[ ] [ ] [ ]1

02

1 (2 1)cos ; 0 12

N m

k

c cm

k nx n k X k n NN N

πβ− −

=

+= ≤ ≤ −∑

[ ]( )

( ) ( )( )

1, 0 1 / 2

0, 1 / 2 1 / 2

1, 1 / 2 1m

k N m

T k N m k N m

N m k N

≤ ≤ − −⎧⎪

= + − ≤ ≤ − +⎨⎪ + + ≤ ≤ −⎩

[ ] [ ] [ ]1

2 / 2

0

1 , 0,1, ,2 1; 1,3,5, , 1N

j k N

km mx n T k X k e n N m N

Nπ

−

=

= = ⋅ ⋅ ⋅ − = ⋅ ⋅ ⋅ −∑




Discrete Fourier transform• Discrete Fourier series representation of periodic sequences

– Let be a periodic sequence with period N such that for any

integer m

Example A periodic complex exponential

– A set of N periodic complex exponentials defines all the

frequency components

– Fourier series representation

[ ]x n

[ ]{ }, 0,1,2,... 1,ke n k N= −

[ ] [ ] k k Ne n e n+=Note that for any integer

[ ] [ ]x n x n mN= +

[ ] [ ]2

; knj

Nk ke n e e n mN k

π

= = + is an interger

[ ] [ ]

[ ] [ ]

21

0

21

0

1

N j knN

k

mnN jN

n

x n X k eN

X k x n e

π

π

−

=

− −

=

⇒ =

=

∑

∑

[ ] [ ]21

0

1 N j knN

k

x n X k eN

π−

=

= ∑




The Fourier series coefficient can be obtained by

The Fourier series coefficients is a periodic sequence

– Discrete Fourier series (DFS)

Define by

[ ] [ ]

[ ] ( )

( )

2 2 21 1 1

0 0 0

21 1

0 0

21

0

1

1

1,10, otherwise

knN N Nj rn j j rnN N N

n n k

N N j k r nN

k k

N j k r nN

n

x n e X k e eN

X k eN

k r Ne

N

π π π

π

π

− − −− −

= = =

− − −

= =

− −

=

=

=

− =⎧= ⎨⎩

∑ ∑ ∑

∑ ∑

∑Since

[ ] [ ]21

0

N j rn

N

n

x n e X rπ−

=

=∑

[ ] [ ]DFSx n X k←⎯⎯→

[ ] [ ]

[ ] [ ]

21

0

1

0

;

1

N jkn NN N

n

Nkn

Nk

X k x n w w e

x n X k wN

π− −

=

−−

=

= =

=

∑

∑

Analysis equation :

Synthesis equation :

[ ] [ ]

[ ] [ ] [ ] [ ]

21

0

2 21 1( ) 2

0 0

N j knN

n

N Nj k N n j kn j nN N

n n

X k x n e

X k N x n e x n e e X k

π

π ππ

− −

=

− −− + − −

− −

=

+ = = =

∑

∑ ∑




Example Periodic pulse train

Example Duality in DFS

Let the DFS coefficient be a periodic impulse train

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

1

0

21 1

0 0

1, 0, otherwise

, 0 1

1

1 1 1

r

Nkn

Nn

N N j knkn kn NN N

k k

n mMx n n mN

x n n n N

X k n w

x n X k w w eN N N

π

δ

δ

δ

∞

=−∞

−−

=

− −− −

= =

=⎧= − = ⎨

⎩= ≤ ≤ −

= =

⇒ = = =

∑

∑

∑ ∑ ∑

Since

[ ] [ ] [ ] [ ] Y k Nx n y n X n= ↔ =

[ ] [ ]1

0

01,

Nkn

N Nk

y n N k w w nδ−

− −

=

⇒ = = = ∀∑

[ ] [ ]r

Y k N k rNδ∞

=−∞

= −∑




Example Periodic rectangular pulse train

Any periodic sequence can be represented as a sum of complex exponential

sequences

[ ]2 454 4

1010 1010

0 0 10

sin1 21 sin

10

kkj kn jnkk

n n

kWX k W e ekW

π ππ

π− −

− =

−= = = =

−∑ ∑




• Properties of DFS

– Linearity

– Shift of a sequence

– Duality

– Symmetry properties

[ ] [ ] [ ] [ ]1 2DFSax n bx n aX k bX k+ ←⎯⎯→ +

[ ] [ ] ( ) [ ] [ ]

[ ] [ ][ ] [ ]

2 2 2 21 1

0 0

N Nj kn j n m j j km kmN N N NN

n n

DFS kmN

DFSnN

x n m e x n m e e X k e X k W

x n m X k W

x n W X k

π π π π− −− − − − −

= =

−

− = − = =

⇒ − ←⎯⎯→

←⎯⎯→ −

∑ ∑

[ ] [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ]

1 1

1

0

0 0

1

,

Nkn

Nk

Nkn

Nk

DFS DF

k

S

Nkn

N Nx n k W

Nx

x n X k

k x n W

x n k X n Nx

XN

k

w

X

−

=

−

−−

=

=

− =

⇒ − =

←⎯⎯→ ←⎯⎯→

= ⇒

−

∑ ∑

∑If

[ ] [ ][ ] [ ]

* *

* *

DFS

DFS

x n X k

x n X k

←⎯⎯→ −

− ←⎯⎯→




⇒ If is real,

– Periodic convolution

Duality

Difference between aperiodic convolution and periodic convolution

• The sum is over an interval

•

[ ] [ ][ ] [ ]

*X k X k

X k X k

= −

= −

[ ]x n

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ]

1 1

3 1 2 2 10 0

1 1

3 1 20 0

1 1

1 20 0

1

2 1 1 20

N N

m m

N Nkn

Nn m

N Nkn

Nm n

Nkm

Nm

x n x m x n m x m x n m

X k x m x n m W

x m x n m W

X k x m W X k X k

− −

= =

− −

= =

− −

= =

−

=

= − = −

⎛ ⎞= −⎜ ⎟

⎝ ⎠

= −

= =

∑ ∑

∑ ∑

∑ ∑

∑

0 1m N≤ ≤ −

[ ] ( )2 2x n m x m n− = − −⎡ ⎤⎣ ⎦

[ ] [ ] [ ] [ ]1

1 2 1 20

NDFS

mx m x n m X k X k

−

=

⇒ − ←⎯⎯→∑

[ ] [ ] [ ] [ ]1

1 2 1 20

1 NDFS

m

x n x n X m X k mN

−

=

←⎯⎯→ −∑




Example Periodic convolution




– Properties of DFS




– Properties of DFS (cont.)




• Fourier transform of periodic signals

– A sequence should be absolutely summable to guarantee the uniform convergence of

its Fourier transform

– Periodic signals are not absolutely summable

– The F.T. of a periodic signal can be interpreted to be an impulse train in the frequency

domain with the impulse values proportional to the DFS coefficients for the sequence

– If is periodic with period N, the Fourier transform of is defined as

Although the F.T. of a periodic sequence does not converge in the normal sense,

the introduction of impulses enables to use the F. T.

Note: has the necessary periodicity with period

[ ] ( ) [ ]2 2j

k

kx n X e X kN N

ω π πδ ω∞

=−∞

⎛ ⎞⇔ = −⎜ ⎟⎝ ⎠

∑

[ ]x n[ ]x n

( ) [ ]

[ ]

[ ] [ ]

2 2

0 0

2

0

2

1 1 2 2 ; 0 2 /2 2

1 2

1

j j n j n

k

j n

k

j knN

k

kX e e d X k e d NN N

kX k e dN N

X k e x nN

π ε π εω ω ω

ε ε

π ε ω

ε

π

π πω δ ω ω ε ππ π

πδ ω ω

− −

− −

∞ −

−=−∞

∞

=−∞

⎛ ⎞= − < <⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

= =

∑∫ ∫

∑ ∫

∑

( )jX e ω 2π




Example: Fourier transform of a periodic impulse train

– For a finite-length sequence

Thus, is obtained by sampling the Fourier transform of the finite sequence

[ ] [ ] [ ]

[ ] [ ] [ ]

( ) ( ) ( )

( )2

2 2

2 2

j j j

j

k

kjN

k

m m

x n x n p n

X e X e P e

kX e

mN

N N

kX eN

x n x n mN

N

ω ω ω

ω

π

π πδ

ω

δ

ω

π πδ

∞ ∞ ∞

=−∞ =−∞ =−

∞

=−∞

∞

=−

∞

∞

= ∗

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= −

− −

⎜ ⎟ ⎜ ⎟⎝ ⎠⎝

−

⎠

= =∑ ∑ ∑

∑

∑

[ ] [ ] [ ]

( )

1,

2

2m

j

k

p n n mN P k k

kP e

NNω

δ

πδ ω

π

∞

=−∞

∞

=−∞

= − ⇒ = ∀

⇒ = −⎡ ⎤⎢ ⎥⎣ ⎦

∑

∑[ ] [ ]( ) , 0 for 0 and x n x n n n N= < ≥., i.e

[ ] ( ) [ ]2 21

20

Nj k j kj N Nk nN

X k X e X e x n eπ π

ωπω

− −

= =

⎛ ⎞⇒ = = =⎜ ⎟

⎝ ⎠∑

[ ] x n[ ] X k




Example: Relationship between the DFS coefficients and the FT of one period

[ ] 1; 0 40;

x nn≤ ≤⎧

= ⎨⎩ otherwise

( ) ( )( )

42

0

sin 5 / 2sin / 2

j j n j

nX e e eω ω ωω

ω− −

=

⇒ = =∑



7. Fourier analysis of signals using DFT

• Discrete Fourier transform (DFT) can analyze the frequency content of continuous-time

signal

Fourier Analysis of Signals Using DFT

( ) ( ) ( )( )12

j j jV e X e W e dπ

π

ω θ ω θ θπ

−

−= ∫

( ) 1 2jc

k

kX e XT T T

ω ω π∞

=−∞

⎛ ⎞= +⎜ ⎟⎝ ⎠

∑

[ ] [ ] [ ]

[ ] [ ] ( )21

20

; knN j jN

kn N

v n x n w n

V k v n V e Teπ

ω πω ω

− −

= =

=

= = = Ω∑




– Example




– Length of the DFT

Example: T=1/5000 and 10Hz resolution

– Relationship between DFT values

[ ] [ ] [ ] [ ][ ] [ ] [ ] ( )

( ) [ ] ( )

* *

*

11

11

(( ))

512 11 501 11 2000 12 11 5000/512 2 107.4

11 0.4 1

N

c

V k V k V N k V k

V V V j

X T V jπ π

= − ⇒ − =

⇒ − = = = −

Ω = ⋅ ⋅ = ⋅

⇒ Ω ≈ ⋅ = +

[ ] [ ] ( )21

20

;

2 2

knN jN

kn N

jV eV T

NNT T

k v n e ω

ω

π

π ω

π π

=

− −

=

= Ω

ΔΩ ≤ ⇒ ≥

= =

ΔΩ

∑

9

1/ 5000

2 500010 2 500 512 210T

N NNTππ

=

⋅ ≤ ⇒ ≥ = ⇒ = =

[ ] ( )512; 1/5000

If 11 2000 1 ;N T

V j= =

= +




• DFT analysis of sinusoidal signals

– Windowing and spectral sampling have an important effect on the analysis of

sinusoidal signals using the DFT

– Effect of windowing

After the windowing

it can be suffered from reduced resolution and leakage of spectrum

[ ] [ ] [ ][ ] [ ]

[ ] [ ]( ) [ ] [ ]( )0 0 0 0 1 1 1 1

0 0 0 1 1 1

( ) (0 1

cos( ) cos( )

2 2j n j n j n j n

v n x n w n

A w n n A w n nA Aw n e w n e w n e w n eω θ ω θ ω θ ω θ

ω θ ω θ

+ − + + − +

=

= + + +

= + + +

[ ]0 0 0 1 1 1

0 0 0 1 1 1

( ) cos( ) cos( )cos( ) cos( )

cs t A t A tx n A n A n

θ θω θ ω θ

= Ω + + Ω +

⇒ = + + +

( ) ( )( ) ( )( ) ( )( ) ( )( )0 0 1 10 0 1 10 1

2 2j j j jj j j jj A AV e W e e W e e W e e W e eω ω ω ω ω ω ω ωθ θ θ θω − + − +− −⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦




Example Leakage due to the use of a rectangular window

0

1

26

23

πω

πω

=

=




The resolution is primarily affected by the width of the main lobe of

The degree of leakage depends on the relative amplitude of the main and side

lobe of

These are associated with the window length L and the shape of the window( )jW e ω

( )jW e ω




Example: Kaiser window

[ ]( )

2

0

0

11,0 1,

20 ,otherwise

nILw n n L

I

αβα

αβ

⎧ ⎡ ⎤−⎡ ⎤⎪ ⎢ ⎥− ⎢ ⎥⎣ ⎦⎪ ⎢ ⎥ −⎣ ⎦= ⎨ ≤ ≤ − =⎪⎪⎩

sl

A

( )24 121

155sl

ml

AL

π ++

Δ




– Effect of spectral sampling

Example

[ ] ( ) ( )2 ; 2 /jk k

N

V k V e k NTωπω

π=

= Ω =

[ ] 2 4cos 0.75cos ;14 15

0 63

n nv n

n

π π= +

≤ ≤

64N =




– Spectral sampling with frequencies matching DFT frequencies

Example [ ] 2 2cos 0.75cos ; 0 6316 8

n nv n nπ π= + ≤ ≤

64N =




– DFT analysis by Kaiser window

Example [ ] [ ] [ ]2 4cos 0.75 cos ; 0 6314 15K K

n nv n w n w n nπ π= + ≤ ≤

64L = 32L =




– DFT analysis with 32-point Kaiser window and zero padding

Example [ ] 2 4cos 0.75cos ; 5.48, 32 Kaiser window14 15

n nv n Lπ π β= + = =

32N = 128N =

1024N =64N =




– DFT analysis with Kaiser window and zero padding

Example [ ] 2 4cos 0.75cos ; 5.48 Kaiser window14 15

n nv n π π β= + =

1024, 32N L= = 1024, 54N L= =

1024, 64N L= =1024, 42N L= =




• Fourier analysis of stationary random signals: Periodogram

– The sample mean and sample variance respectively defined by

are unbiased and asymptotically unbiased estimators, respectively

– Periodogram analysis: estimation of the power spectrum

Periodogram is the F. T. of aperiodic correlation of windowed data sequence

Modified periodogram: is not a rectangular window

( ) [ ] [ ]1

0

Lj j n

n

V e x n w n eω ω−

−

=

=∑

[ ] [ ]( )21 1

2

0 0

1 1ˆ ˆ ˆ; L L

X X Xn n

m x n x n mL L

σ− −

= =

= = −∑ ∑

[ ]w n

( ) ( ) [ ]

[ ] [ ] [ ] [ ] [ ]

12

1

1

0

1 1 Lj j m

vvm L

L

vvn

I V e c m eLU LU

c m x n w n x n m w n m

ω ωω−

−

=− +

−

=

= =

= + +

∑

∑

( ) 1XX SSP P

T Tωω ⎛ ⎞= ⎜ ⎟⎝ ⎠




– Periodogram can be calculated by means of the DFT

– Properties of the periodogram:

To make the periodogram unbiased,

[ ] [ ] [ ]{ } [ ] [ ] [ ]1

0

; L

xx wwn

m E x n x n m c m w n w n mφ−

=

= + = +∑

( ){ } [ ]{ }1

1

1 Lj m

vvm L

E I E c m eLU

ωω−

−

=− +

= ∑

( ) [ ] 21kI V k

LUω =

[ ]{ } [ ] [ ] [ ] [ ]{ }

[ ] [ ] [ ]

[ ] [ ]

1

0

1

0

L

vvn

L

xxn

ww xx

E c m w n w n m E x n x n m

w n w n m m

c m m

φ

φ

−

=

−

=

= + +

= +

=

∑

∑

( ){ } ( ) ( )( ) ( ) ( ) 21 ; j j jXX WW WWE I P C e d C e W e

LUπ ω θ ω ω

πω θ θ−

−= =∫

( ) [ ] [ ]1 12 2 2

0 0

1 1 112

L Lj

n nW e d w n U w n

LU LU Lπ ω

πω

π

− −

−= =

= = =⇒∑ ∑∫




– As the window size increases, the peridogram becomes a consistent estimate

Example: white noise generation with rectangular windowing

256L =

[ ] ( ) [ ] [ ] [ ]212 2 /

1

1 1 Lj kn N

km L

I k I V k w n x n eL L

πω−

−

=− +

= = = ∑

( ){ } ( )2var k XXI Pω ω64L =

16L =

1024L =

256L =




– Edge effect due to covolution of finite sequences

As m becomes close to L, the

calculation becomes inconsistent

This problem can be alleviated by

averaging multiple independent

periodogram estimates

[ ] [ ] [ ] [ ] [ ]1

0

L

vvn

c m w n w n m x n x n m−

=

= + +∑

( ) ( )1 1 2

0 0

1 1K Lj

rr n

I X eK LU

ωω− −

= =

= ∑ ∑

[ ] [ ] [ ]; 0 1rx n w n x rR n n Q= + ≤ ≤ −




– Example

( ) ( ) ( )2

20 02XX e

AP πω δ ω ω δ ω ω σ⎡ ⎤= − + + +⎣ ⎦

[ ] ( ) [ ] [ ] ( ) 20.5cos 2 / 21 ; 3, 3 1ex n n e n e n Uπ θ σ= + + − ⇒ =∼

1024; 1Q U= =

127, 16K L= =

31, 64K L= =

1, 1024K L= =

seoul national university 1. discrete-time signals and systems

Documents