2장 Discrete-Time Signals and Systems

2.1 Discrete-Time Signals ( sequences )

• ¥<<¥-= n,))nT(x)(n(x c

• x(n) is defined only for integer values of n

2.1.1 Basic Signals

• 00

n1n0

)n( ççè

æ=¹

=d

• å¥

-¥=

-d=k

)kn()k(X)n(X

(1) Unit sample sequence ( impulse )

1

(2) Unit step sequence

0n0n

01

)n(u<³

ççè

æ=·

å

å¥

=

-¥=

-d=·

d=·

--=d·

0k

n

k

)kn()n(u

sumrunning:)k()n(u

differencefirst:)1n(u)n(u)n(

running sum running sum 2

2.1.2 Complex exponential

fw =a=aa= jjn eAA,e,A)n(x 0

(1) Real exponential• A and a are Real

real exponential signal3

(2) Sinusoidal signal

)ncos(A)n(xeA)n(x

1

0

)n(j 0

f+w=×

=×

=a×f+w

(3) General complex exponential signals)n(jn

0eA)n(x f+wa=×

discrete-time sinusoidal signal4

(4) Periodicity properties of discrete-time complex exponential

• 주파수 w0 와 (w0 ±2pr) 은 같은주파수이다.

njn)2(j 00 ee wp+w =

• 고려해야할 주파수 범위 : –p< w0 £ p or 0 < w0 £ 2 p

• Periodic Sequence

)(Nk

2,k2N

1e,ee

)n(x)Nn(x

00

Njnj)Nn(j 000

비유리수=pw

p=w

==

=+ww+w

ççè

æWWW

함수주기값에서도어떤의

신호다른서로다르면가

:e0

0tj 0•

5

• 주기가 N인신호의 기본주파수

개주파수다른서로고조파관련된와 N :1N,.......,1,0k,N

k2 N2

k0 -=p

=wp

=w

• w0 = 2pk 근처에서는낮은 주파수, w0 = (p + 2pk) 근처에서는 높은 주파수

sinusoidal sequence for different frequencies6

2.2 Discrete-Time Systems

)]n(x[T)n(y =

• Ideal Delay System

)nn(x)n(y d-=

• Moving Average

å-=

-++

=2

1

M

Mk21

)kn(x1MM

1)n(y

입력신호 n번째샘플근처의 (M1+M2+1)개의입력샘플평균을출력의 n번째

샘플로계산한다 . : 일종의저역통과필터 ( LPF )특성

2.2.1 Basic system properties

(1) Memoryless

• The output y(n) at every value of n depends only on the input x(n) at the

same value of n7

(2) Linear• 중첩의정리성립 ( The principle of superposition )

propertyscaling:)n(ay)]n(x[aT)]n(ax[Tpropertyadditivity:)n(y)n(y)]n(x[T)]n(x[T)]n(x)n(x[T 212121

==+=+=+

• 일반적으로 입력 x(n)=Sakxk(n) 에 대한 linear system 의출력은

y(n)=Sakyk(n)

• Accumulator

)n(u responseimpulseaccumlator:)n(u)n(ythen),n()n(xif

)k(x)n(yn

k

는의=d=

= å-¥=

• A System for which a time shift of input sequence causes a corresponding

shift in the output sequence.

)nn(y)nn(x)n(y)n(x

00 -®-®

(3) Time - Invariant

8

• x(n) = d(n) ® y(n) = h(n) : impulse response

(4) Causality

• A system is causal if the output sequence value at the index n=n0 depends

only on the input sequence values for n £ n0

(5) Stability

• A system is stable if and only if every bounded input sequence produces

a bounded output sequence.

• BIBO ( Bounded Input, Bounded Output )

nallforB)n(y,B)n(x yx ££×

2.2.2 Linear Time - Invariant Systems

9

• Linear and time invariant

(1) Commutative and distributing properties

)n(h)n(x)n(h)n(x))n(h)n(h()n(x)n(x)n(h)n(h)n(x)n(y

2121 *+*=+*×*=*=×

ò

å

å

¥

¥-

¥

-¥=

¥

-¥=

*=tt-t=

*=-=

-d=

egralintnconvolutio:)t(h)t(xd)t(h)(x)t(y

sumnconvolutio:)n(h)n(x)kn(h)k(x)n(y

)kn()k(x)n(x

k

k

10

(2) Cascade and parallel connection

• h(n)=h1(n) * h2(n)

• h(n)=h1(n) + h2(n)

11

(3) Stability of LTI system

• LTI system are stable if and only if the impulse response is

absolutely summable

å¥

-¥=

¥<k

)k(h

(4) Causality of LTI system

0n,0)n(h])1n(x)1(h)n(x)0(h[])2n(x)2(h)1n(x)1(h[

)kn(x)k(h)kn(x)k(h)kn(x)k(h)n(y

0n,0)n(h

)kn(x)k(h)kn(h)k(x)kn(h)k(x)n(y

00

00

00

1

0k

00

k 0k

<=®+-++++-++-=

-+-=-=×

<=®

-=-=-=×

ååå

å å å

¥-

¥-

¥

-¥=

¥

¥-

¥

-¥=

¥

=

L

L

12

(5) Impulse response of LTI system

• Ideal delay

causalandstable0n),nn()n(h

)nn(x)n(y

dd

d

>-d=-=

• Moving average

)2M,0M(causalandstableOtherwise0

MnM1MM

1)kn(

12M1M1)n(h

)kn(x1MM

1)n(y

21

M

M

2121

M

Mk21

2

1

2

1

³³-

ççç

è

æ ££-++=-d

++=

-++

=

å

å

-

-=

13

• Accumulator

causalandunstable0n0

)n(u;0n1)k()n(h

)k(x)n(y

n

k

n

k

å

å

-¥=

-¥=

ççè

æ<=³

=d=

=

• Forward difference

noncausalandstable)n()1n()n(h)n(x)1n(x)n(y

d-+d=-+=

• Backward difference

causalandstable)1n()n()n(h)1n(x)n(x)n(y

-d-d=--=

14

• FIR (Finite-duration Impulse Response) system

Ideal delay, Moving Average, Forward and Backward difference

• IIR (Infinite-duration Impulse Response) system

Accumulator(6) Interconnections of LTI system

• Forward difference — one-sample delay

differencebackward:)1n()n())n()1n(()1n()1n())n()1n(()n(h

-d-d=d-+d*-d=-d*d-+d=

• The backward difference system is the inverse system for the accumulator

)n()n(h)n(h)n(h)n(h ii d=*=*

• Accumulator-backward difference

systemidentity;)n()1n(u)n(u

))1n()n(()n(u)n(h

d=--=

-d-d*=

15

2.2.3 Linear Constant-Coefficient Difference Equations• N차 LTI 시스템의 일반적인 N차 선형상계수차동방정식은

å å= =

-=-N

0k

N

0mmk )mn(xb)kn(ya

• 이 식을 다시 표현하면

åå==

-+--=M

0k 0

kN

1k 0

k )kn(xab)kn(y

aa)n(y

• 이 식에서 y(0) 를구하려면, 입력 x(n)과 초기조건 y(-1),y(-2),….,y(-N) 이

필요하며 y(1) 을구하려면 , 입력x(n)과초기값y(0),y(-1),….,y(-N+1)이필요

하므로, 즉 y(n)은입력과 그이전의출력값으로부터반복적 ( Recursively )

계산된다.

• 만약 차동방정식 (N ³ 1) 으로표현되는시스템이 Linearity, Time Invariance,

Causality를만족하면초기조건은모두영이되어야한다.

• 그리고, N ³ 1 인시스템은 IIR 시스템이며, N=0인 경우는 FIR시스템이다. 16

2.2.4 Frequency-Domain Representation of Discrete-Time Signals

and Systems

(1) Eigenfunction for LTI systems

• LTI 시스템의입력 x(n)=ejwn이면 출력 y(n)은

å

å

å å

¥

-¥=

w-w

ww¥

¥-

ww-

¥

¥-

¥

¥-

-w

=

==

=-=

n

njj

njjnjkj

)kn(j

e)n(h)e(H

e)e(He)e)k(h(

e)k(h)kn(x)k(h)n(y

• LTI 시스템의입력이정현파나 complex exponential 인경우 출력은

입력과같은형태를가지며크기와위상이시스템에의해결정된다.

• 이러한경우 ejwn을 LTI system의 eigenfunction 이라하며 H(ejw)을

eigenvalue라한다. H(ejw)은주파수응답(frequency response) 이며

일반적으로 복소수이다. 17

• 즉신호의 Fourier 표현은 LTI 시스템해석에서매우유용하게쓰인다.

• 일반적으로 입력신호를 complex exponential의선형조합으로표현하면

åå

ww

w

=

=njkj

k

njk

k

k

e)e(Ha)n(y

ea)n(x

• H(ejw)에서고려해야할 주파수범위는 0 £ w <2p, p < w £ p

• Ideal frequency-selective filter

(2) 주파수응답의주기성

주기함수인주기가 2

)e(He)n(h)e(H jn)2(j)2(j

p

==å¥

¥-

wp+w-p+w

18

ideal lowpass filter

19