2 Discrete-Time Signals and Systemscontents.kocw.net/KOCW/document/2014/Chungbuk/... ·...
Transcript of 2 Discrete-Time Signals and Systemscontents.kocw.net/KOCW/document/2014/Chungbuk/... ·...
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