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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 1
T Y P E S O F D I S C R E T E - T I M E S Y S T E M S
∗
N g u y e n H u u P h u o n g
T h i s w o r k i s p r o d u c e d b y T h e C o n n e x i o n s P r o j e c t a n d l i c e n s e d u n d e r t h e
C r e a t i v e C o m m o n s A t t r i b u t i o n L i c e n s e
†
D i c r e t e - t i m e ( d i g i t a l ) s y s t e m s c o m p r i s e o f s e r v e r a l b a s i c t y p e s w i t h d i e r e n t c h a r a c t e r i s t i c s . T h e c a t e g o -
r i z a t i o n g i v e s u s a d e e p e r u n d e r s t a n d i n g o f s y s t e m s a n d t h e c h o i c e o f a p p r o p r i a t e a n a l y s i s m e t h o d .
1 M e m o r y l e s s s y s t e m s , a n d s y s t e m s w i t h m e m o r y
A m e m o r y l e s s ( o r s t a t i c ) s y s t e m d o e s n o t n e e d m e m o r y . I t p r o c e s s e s t h e i n p u t a n d o u t p u t s i g n a l s t a k i n g
p l a c e a t t h e s a m e i n s t a n t . F o r e x a m p l e
y (n) = 2x (n)
y (n) = 2x (n)− x2 (n)
A c t u a l l y t h e r e i s a s m a l l d e l a y b e t w e e n i n p u t a n d o u t p u t d u e t o t h e p r o p a g a t i o n d e l a y o f t h e s y s t e m .
A s y s t e m w i t h m e m o r y ( o r d y n a m i c ) n e e d s m e m o r y t o s t o r e p a s t a n d f u t u r e v a l u e s n e e d e d f o r t h e
p r o c e s s i n g . F o r e x a m p l e
y (n) = x (n) + 0.8x (n− 1) : one memory cell
y (n) = 1
3[x (n − 1) + x (n) + x (n + 1)] : two memory cell
y (n) =+∞
k=−∞ x (n− k) : infinite memory
2 C a u s a l a n d n o n c a u s a l s y s t e m s
I n c a u s a l s y s t e m t h e r e s u l t c o m e s a f t e r t h e c a u s e , o r , a t t h e s a m e t i m e ( s i m u l t a n e o u s l y ) . T h i s i s t o s a y
t h a t t h e o u t p u t a t i n d e x n o n l y d e p e n d s o n t h e i n p u t a t n , n 1 , n 2 , . . ., a n d n o t o n n + 1 , n + 2 , . . . I n
n o n c a u s a l s y s t e m s , o n t h e o t h e r h a n d t h e o u t p u t a l s o d e p e n d s o n f u t u r e i n p u t s . F o l l o w i n g i s a f e w e x a m p l e s .
(a) y (n) = 2x (n)− 3x2 (n) : n o n c a u s a l
(b) y (n) = 13 [x (n− 1) + x (n) + x (n + 1)] : n o n c a u s a l d u e t o t h e l a s t t e r m
(c) y (n) =∞
k=0x (n− k) : c a u s a l
(d) y (n) =∞
n=−∞ x (n) : n o n c a u s a l
(e) y (n) =∞
k=−∞ x (n− k) : n o n c a u s a l
(f ) y (n) = x (−n) : n o n c a u s a l
∗V e r s i o n 1 . 1 : J u l 9 , 2 0 0 9 5 : 0 1 a m G M T - 5
†h t t p : / / c r e a t i v e c o m m o n s . o r g / l i c e n s e s / b y / 3 . 0 /
h t t p : / / c n x . o r g / c o n t e n t / m 2 8 7 4 1 / 1 . 1 /
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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 2
(g) y (n) = x2 (n) : n o n c a u s a l
(h) y (n) = x
n2
: n o n c a u s a l
I n r e a l - t i m e p r o c e s s i n g ( o r o n - l i n e p r o c e s s i n g ) , s y s t e m s m u s t b e c a u s a l , o - l i n e p r o c e s s i n g ( o r
b a t c h p r o c e s s i n g o r b l o c k p r o c e s s i n g ) s y s t e m s c a n b e n o n c a u s a l s i n c e a l l s a m p l e s h a v e b e e n s t o r e d i n
m e m o r y , m a n y o f t h o s e w i l l b e f u t u r e v a l u e s w i t h r e s p e c t t o t h e c h o s e n t i m e o r i g i n .
T h e c o n c e p t o f c a u s a l i t y i s a l s o a p p l i e d t o s i g n a l s b u t t h e d e n i t i o n i s m o d i e d . A s i g n a l x ( n ) c a n b e
c l a s s i e d a s
• C a u s a l ( o r r i g h t - s i d e d ) i f x ( n ) = 0 f o r n < 0
• A n t i c a u s a l ( o r l e f t - s i d e d ) i f x ( n ) = 0 f o r n ≥ 0
• T w o - s i d e d ( o r b i l a t e r a l ) i f x ( n ) e x i s t s f o r a l l n ( < 0 a n d ≥ 0 )
F o r e x a m p l e , t h e u n i t s t e p u ( n ) i s c a u s a l , u ( - n - 1 ) i s a n t i c a u s a l ,
a|n|
i s t w o - s i d e d . W e c a n p l o t o u t t h e s e
s i g n a l s t o r e a l l y s e e t h e d i e r e n c e .
3 T i m e - i n v a r i a n t a n d t i m e - v a r i a n t s y s t e m s
T h e c h a r a c t e r i s t i c s o f a s y s t e m m a y c h a n g e w i t h t i m e s o t h a t t h e o u t p u t d e p e n d s o n t h e i n p u t a s w e l l a s
t h e i n s t a n t t h e i n p u t i s a p p l i e d . T h i s i s a t i m e - v a r i a n t s y s t e m . O n t h e o t h e r h a n d , m a n y s y s t e m s c a n b e
a s s u m e d t o b e t i m e - i n v a r i a n t , i . e . t h e o u t p u t d o e s n o t d e p e n d o n t h e t i m e t h e i n p u t i s a p p l i e d . T h e t e r m s
s h i f t - v a r i a n t a n d s h i f t - i n v a r i a n t c a n b e u s e d i n s t e a d o f t i m e - v a r i a n t a n d t i m e - i n v a r i a n t r e s p e c t i v e l y .
T h e t i m e ( s h i f t ) i n v a r i a n c e i s j u d g e d a s f o l l o w s .
I f x (n) → y (n)
t h e n x (n− k) → y (n− k)
T h i s c r i t e r i o n i s i l l u s t r a t e d i n F i g u r e 1
F i g u r e 1 : T i m e ( s h i f t ) i n v a r i a n t s y s t e m
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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 3
E x a m p l e 1
A r e t h e f o l l o w i n g s y s t e m s t i m e - i n v a r i a n t ?
(a) y (n) = 1
3[x (n − 1) + x (n) + x (n + 1)]
(b) y (n) = n x (n)
(c) y (n) = x (−n)
S o l u t i o n
( a ) F o r t h e s y s t e m
y (n) = 1
3[x (n− 1) + x (n) + x (n + 1)]
I f t h e p r e s e n t i n p u t i s d e l a y e d b y k ( i . e . b y r e p l a c i n g x ( n ) b y x ( n k ) . . . ) t h e n t h e o u t p u t i s
y (n− k) = 1
3[x (n− 1− k) + x (n− k) + x (n + 1 + k)]
a n d i f t h e p r e s e n t o u t p u t i s d e l a y e d b y k ( i . e . b y r e p l a c i n g n b y n k )
y ' (n − k) = 1
3[x (n − 1− k) + x (n− k) + x (n + 1 + k)]
S i n c e
y'
(n − k) = y (n − k)t h e s y s t e m i s t i m e - i n v a r i a n t .
( b ) F o r t h e s y s t e m
y (n) = n x (n)
i f t h e p r e s e n t i n p u t i s d e l a y e d b y k t h e n t h e o u t p u t i s
y (n− k) = n x (n− k)
a n d i f t h e p r e s e n t o u t p u t i s d e l a y e d b y k t h e n t h e o u t p u t i s
y ' (n − k) = (n− k) x (n − k)
S i n c e
y ' (n − k) = y (n − k)
t h e s y s t e m i s t i m e - v a r i a n t .
( c ) F o r t h e s y s t e m
y (n) = x (−n)
w e h a v e
y (n− k) = x (−n− k)
y ' (n − k) = x [− (n− k)] = x (−n + k) = y (n− k)
S o t h e s y s t e m i s t i m e - v a r i a n t .
4 L i n e a r a n d n o n l i n e a r s y s t e m s
T h e s i g n i c a n c e o f l i n e a r i t y a n d n o n l i n e a r i t y f o r d i s c r e t e - t i m e s y s t e m s i s a b o u t t h e s a m e a s f o r a n a l o g
s y s t e m s . S u p p o s e t w o i n p u t s i g n a l s x1 (n) a n d x2 (n) w h e n a p p l i e d s e p a r a t e l y t o a s y s t e m g i v e c o r r e s p o n d i n g
o u t p u t s
y1 (n)a n d
y2 (n). N o w i f a l i n e a r c o m b i n a t i o n o f t h e t w o i n p u t s g i v e t h e s a m e l i n e a r c o m b i n a t i o n
o f t h e o u t p u t s t h e n t h e s y s t e m i s l i n e a r , o t h e r w i s e t h e s y s t e m i s n o n l i n e a r . T h u s l i n e a r i t y i m p l i e s b o t h
s c a l a b i l i t y ( p r o p o r t i o n a l i t y ) a n d s u p e r p o s i t i o n . T h e d e n i t i o n o f l i n e a r i t y i s i l l u s t r a t e d i n F i g u r e 2 .
h t t p : / / c n x . o r g / c o n t e n t / m 2 8 7 4 1 / 1 . 1 /
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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 4
F i g u r e 2 : L i n e a r s y s t e m s
E x a m p l e 2
C o n s i d e r t h e l i n e a r i t y o f t h e f o l l o w i n g s y s t e m s :
(a) y (n) = n2x (n)
(b) y (n) = n x
n2
(c) y (n) = n x
2 (n)
(d) y (n) =A x
(n) + B, A, Bc o n s t a n t s
S o l u t i o n
( a ) T h e s y s t e m i s
y (n) = n2x (n)
T h e t w o s e p a r a t e i n p u t s a n d c o r r e s p o n d i n g o u t p u t s a r e
y1 (n) = n2x1 (n)
y2 (n) = n2x2 (n)
N o w f o r t h e c o m b i n e d i n p u t
x (n) = a1x1 (n) + a2x2 (n)
t h e o u t p u t i s
y (n) = n2 [a1x1 (n) + a2x2 (n)] = a1
n21 (n)
+ a2
n22 (n)
= a1y1 (n) + a2y2 (n)
S o t h e s y s t e m i s l i n e a r .
( b ) T h e s y s t e m i s
h t t p : / / c n x . o r g / c o n t e n t / m 2 8 7 4 1 / 1 . 1 /
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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 5
y (n) = n x
n2
T h e p r o c e d u r e i s s u m m a r i z e d a s f o l l o w s
x1 (n) → y1 (n) = n x 1
n2
x2 (n) → y2 (n) = n x 2
n2
x (n) = a1x1 (n) + a2x2 (n)
t h e n
y (n) = a1 n x 1
n2
+ a2 n x 2
n2
= a1y1 (n) + a2y2 (n)
S o t h e s y s t e m i s l i n e a r .
( c ) T h e s y s t e m i s
y (n) = x2 (n)
T h e r e a s o n i n g i s
x1 (n) → y1 (n) = x21 (n)
x2 (n) → y2 (n) = x22 (n)
x (n) = a1x1 (n) + a2x2 (n)
t h e n
y (n) = [a1x1 (n) + a2x2 (n) ]2
= a21y21 (n) + a22y22 (n) + 2a1a2x1 (n) x2 (n)
S o t h e s y s t e m i s n o n l i n e a r
( d ) T h e s y s t e m i s
y (n) = A x (n) + B, A, B
T h e r e a s o n i n g i s
x1 (n) → y1 (n) = A x 1 (n) + B
x2 (n) → y2 (n) = A x 2 (n) + B
x (n) = a1x1 (n) + a2x2 (n)
y (n) = A [a1x1 (n) + a2x2 (n)] + B
= a 1 A x 1 (n) + a2 A x 2 (n) + B
= a 1 [A x 1 (n) + B] + a2 [A x 2 (n) + B] + B − a1B − a2B
= a 1y1 (n) + a2y2 (n) + (1 − a1 − a2) B
h t t p : / / c n x . o r g / c o n t e n t / m 2 8 7 4 1 / 1 . 1 /
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C o n n e x i o n s m o d u l e : m 2 8 7 4 1 6
D u e t o t h e p r e s e n c e o f t h e l a s t t e r m t h e s y s t e m i s n o n l i n e a r . W h e n B = 0 ( t h e s y s t e m i s s a i d r e l a x e d )
t h e s y s t e m b e c o m e s l i n e a r .
H e r e a f t e r , a l l s y s t e m s a r e a s s u m e d t o b e l i n e a r a n d t i m e ( s h i f t ) i n v a r i a n t ( L T I o r L S I ) , o t h e r w i s e
s t a t e d .
S t i l l , t h e r e i s a n i m p o r t a n t c h a r a c t e r i s t i c o f s y s t e m s , i . e . t h e s t a b i l i t y . S y s t e m s a r e e i t h e r s t a b l e o r
n o n s t a b l e ( a s t a b l e ) . S t a b i l i t y w i l l b e d i s c u s s e d i n n e x t c h a p t e r ( S e c t i o n 2 . 4 ) .
h t t p : / / c n x . o r g / c o n t e n t / m 2 8 7 4 1 / 1 . 1 /