fundermentals of signal processing
TRANSCRIPT
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F u n d a m e n t a l s o f S i g n a l P r o c e s s i n g
M i n h D o
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F u n d a m e n t a l s o f S i g n a l P r o c e s s i n g
C o u r s e A u t h o r s :
M i n h D o
C o n t r i b u t i n g A u t h o r s :
R i c h a r d B a r a n i u k
H y e o k h o C h o i
M i n h D o
B e n j a m i n F i t e
A n d e r s G j e n d e m s j o
M i c h a e l H a a g
D o n J o h n s o n
D o u g l a s J o n e s
R o b N o w a k
R i c a r d o R a d a e l l i - S a n c h e z
J u s t i n R o m b e r g
C l a y t o n S c o t t
I v a n S e l e s n i c k
M e l i s s a S e l i k
O n l i n e :
h t t p : / / c n x . o r g / c o n t e n t / c o l 1 0 3 6 0 / 1 . 3 /
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P H H T f n k D r o k o g o D w n h o D f n j m n p D e n q j n m j o D
w l r D h o n t o n o n D h o l t o n D o x o k D o l l E n z D t n
o m D g l o n o D s n l n k D w l l k
T h i s w o r k i s 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 :
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 / 2 . 0 /
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T a b l e o f C o n t e n t s
1 p o n o n
1 . 1 n l n s n o m o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q
1 . 2 s n o o n o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R
1 . 3 h E m n l n m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V
1 . 4 v n m E s n n m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I H
1 . 5 h E m g o n o l o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I
1 . 6 o v n e l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I S
1 . 7 r l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F P W
1 . 8 n l i n o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q H
1 . 9 p o e n l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q Q
1 . 1 0 g o n n o E m p o n o m @ g p A F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q R
1 . 1 1 h E m p o n o m @ h p A F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q T
1 . 1 2 h p w y o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q T
1 . 1 3 p p e l o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q V
2 m l n n p n e n l
2 . 1 s n o o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R S
2 . 2 o o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R T
2 . 3 s l l o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R W
2 . 4 m l n n o n o n w l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F S Q
2 . 5 m o m l n n o n o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F S Q
2 . 6 m l n g n l X e p n h o m n F F F F F F F F F F F F F F F F F F F F S S
2 . 7 h p X p n h o m n g o m e n l F F F F F F F F F F F F F F F F F F F S U
2 . 8 h E m o n o g n l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F T S
2 . 9 o m p o n o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F U P
2 . 1 0 o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V Q
2 . 1 1 p l n h p F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V T
2 . 1 2 s m o o n f F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W S
3 h l p l n
3 . 1 h ' n i o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W W
3 . 2 n o m X h ( n o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I H Q
3 . 3 l o g o m m o n z E n o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I H T
3 . 4 n n n o l G o l o o n E l n F F F F F F F F F F F F F F F F F F F F F F F F F F F F I H V
3 . 5 p l n n p n h o m n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I I
3 . 6 v n E p s p l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I T
3 . 7 p l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I W
3 . 8 y o h l p l h n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I I W
3 . 9 n o h n w o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I P I
3 . 1 0 p n m l n h n w o o p s ( l F F F F F F F F F F F F F F F F F F F F F F F I P P
3 . 1 1 k E w g l l l n p s p l h n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I P R
3 . 1 2 p s p l h n n w e v e f F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I Q H
3 . 1 3 w e v e f p s p l h n i F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I Q P
4 l n e n l o n
4 . 1 s n o o n o n o m n l n o F F F F F F F F F F F F F F F F F F F F F F F F F F F F I Q U
4 . 2 o n n x o n o n n o m o F F F F F F F F F F F F F F F F F F F F F F F F F F I R H
4 . 3 n o m o X w n n n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I R P
4 . 4 g o l o n n g o n o n o m n l F F F F F F F F F F F F F F F F F F F F F F F F F F F I R T
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4 . 5 e o o l o n o n o m o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I S H
4 . 6 g o o l o n o n o m o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I S Q
4 . 7 s n o o n o e p l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I S R
4 . 8 h E m D g l n p l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I S S
4 . 9 l s n n p l s m l m n o n F F F F F F F F F F F F F F F F F F F F F F F F F F I S V
4 . 1 0 w n m z o n n q n h n F F F F F F F F F F F F F F F F F F F F F F F F F F F I S W
4 . 1 1 v w e p l e l o m F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I T H
4 . 1 2 p y g o n n e n l o v w e l o m F F F F F F F F F F F F F F F F F I T P
4 . 1 3 e i l z o n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I T S
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I
0 I n t r o d u c t i o n t o F u n d a m e n t a l s o f S i g n a l P r o c e s s i n g
0 . 1 W h a t i s D i g i t a l S i g n a l P r o c e s s i n g ?
o n n D i g i t a l S i g n a l P r o c e s s i n g ( D S P ) l 9 m n o o
o m n F S i g n a l n l n n o m o n F P r o c e s s i n g
o o o o n o l n F s o S i g n a l
P r o c e s s i n g o n o m o n o m n l o o o n n o m o n E
n n l o m o n o m o n o F p o m l D o n n k n
n l D n n o n o n m n o m n n l o F p n l l D
o D i g i t a l n h l n l o n m n o o n o m D
m o o o D o l o F
( l h n n ( n l o l l o
l o m n n o m n o l o n n E o n F g o n n l D h
l n n n l m o n o l n n o l n n n n
n o l o F n l o m n n h n n m n n m n E
n l n l o m m n o n D m l m D m l n n ( m n D n
m n E o m n o n F
o l l l o l o n n m o h D o n l o m n o
l m F o n l ( l m m m n l l o n l o o o l
l n D l n l m D o o n o o m F n
l m ( n o D l n o o m o ( l m m F
f m o o o o m m o o l D m o o h l o m
n l o o l m o o o l n m o ( n l m
l F n k o l o m n D l o o n m E l m
n o l n n ( l m m F e l o m n o l m
o l l o n @ m o n A D o m l l z m n o l n D
m l o n h o o o o o m F i n l l D l m E
n o l o o m o n l o o o o m l l m o n F n ( n m l
n n n m n o l o n o h l o m m n o n D l
m n D l l o n D n o o n F
s n m m D h o n o n o n w m D D n g o m n D n
n o k n l n n o l o n n m o l o n F
0 . 2 O v e r v i e w o f K e y C o n c e p t s i n D i g i t a l S i g n a l P r o c e s s i n g
o m n n h s i g n a l s n s y s t e m s F e s i g n a l ( n n l
n o n o m o n n n l m @ o n E m n o n l
n l A D o @ P E h o Q E h n l A F n l n l F s n l E o l D m o
o n l c o n t i n u o u s - t i m e o a n a l o g s i g n a l s l o n n o l l
o m F o o o m D o n n o E m n l o ( s a m p l e d
n m n o d i s c r e t e - t i m e s i g n a l o l o m n n n
o n o m m m o l o o n F p m o D n o o o l o
D n l l o q u a n t i z e d n o o l D n
( n l l l l d i g i t a l s i g n a l F n n z o n ' n o D m
E m n l n l n l n n n l F
s n n l o n D s y s t e m ( n o o n n o E
n l F e n m o n l o m l o l i n e a r t i m e - i n v a r i a n t @ o s h i f t - i n v a r i a n t A
s y s t e m s F m m k l o o m n o m l l
z n i m p u l s e r e s p o n s e f u n c t i o n @ o m m l o l l p o i n t s p r e a d f u n c -
t i o n A D n m ( n c o n v o l u t i o n @ l o o l t e r i n g A o o n F
D l n m E n n m l n o @ l n A l t e r F v n m E n n
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P
m l ( n o o D o n i t e - d u r a t i o n i m p u l s e r e s p o n s e ( F I R )
n o n i n n i t e - d u r a t i o n i m p u l s e r e s p o n s e ( I I R ) F
e n l n v e c t o r n v e c t o r s p a c e F D l i n e a r a l g e b r a o
o l m o k o n l n l n m F s n l D n o D
n l n n @ o n A l i n e a r c o m b i n a t i o n o f e l e m e n t a r y s i g n a l s F
m o m o n s i g n a l e x p a n s i o n s o F o u r i e r t r a n s f o r m s F p o
n o m D n l n o m D o n ' l o n o m o l m
o m o n o m n o n o o m n m o o l o n l z F o
o m n o p o n o m l m n n n l l t i m e d o m a i n n
f r e q u e n c y d o m a i n F
S a m p l i n g D o o n o n o c o n t i n u o u s - d o m a i n r e a l - l i f e s i g n a l s o d i s c r e t e n u m b e r s
n o o m D n l n n l o n
l o l F s m o n o n n o n n o n n n l n m
n l o l n n o m F o n n o n o n n n o n l z n
n o m n F w o o D m n n l n m ( f r e q u e n c y
c h a r a c t e r i s t i c s F
f n l i n e a r t i m e - i n v a r i a n t s y s t e m n z l t e r D n o
m o l o n o n o ( l F l l D n l t e r d e s i g n
o D m n o 0 n o n p s o s s ( l l o l o m
f r e q u e n c y r e s p o n s e ( o n F o p o n o m D z - t r a n s f o r m
o n ' o o l o n l z n n l ( l F
s n m n l o n D n l o n n n l s t a t i s t i c a l m o d e l s r a n d o m
s i g n a l s F s m k l o m m l n ( l @ n n o m i n i m u m m e a n - s q u a r e
e r r o r A D o l l W i e n e r l t e r s D n m n n o n l s e c o n d - o r d e r s t a t i s t i c s @ a u -
t o c o r r e l a t i o n n c r o s s c o r r e l a t i o n n o n A o s t a t i o n a r y p r o c e s s F n
n n o ( o n o n o m D n m l o a d a p t i v e l t e r s D
( l o 0 n o n l F m o o l l o m o
l j ( l o 0 n l e a s t - m e a n s q u a r e ( L M S ) l o m F
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C h a p t e r 1
F o u n d a t i o n s
1 . 1 S i g n a l s R e p r e s e n t I n f o r m a t i o n
n l o o l D n o m o n n n m n l n n l E
l n n n X s i g n a l F n m m l m D a s i g n a l i s m e r e l y a f u n c t i o n F
e n l o n l o n n o E l Y l n l E l F n n n
l o n l o l m @ D o m l A D @ m A D o n
@ n o n n n o l n n m n o o l l o A F
1 . 1 . 1 A n a l o g S i g n a l s
A n a l o g s i g n a l s a r e u s u a l l y s i g n a l s d e n e d o v e r c o n t i n u o u s i n d e p e n d e n t v a r i a b l e ( s ) F
1
o o o l o n o o n n n o o l F l
o n n D n n l o o n o n o n
n n n n l o n m n l o o n n o X
s (x, t) @ r o n o o n x o n o l o o n A F n o o o m o n l k n D o l n n l l l l o o n D
x0 F e n m l o l n o m s (x0, t) o n n ( @ p I F I A F
o o D n o n n o E l n l ( n o F f l k E
n E m o n l o n l o n n D m o n o o l
) o n o F s n p I F P D n m o n D m o n n @ n l l o
m l l A n o n o o n n n l l F
g o l o m l o ) n o n o l m F
n l o n o o n m n o o m n o n o o E l l m o l o !
D l l o n l ! n o l o l o m F D m o l l
o o n l o n n D ' n o o l o X
r o m o D g r e e n n l n F w m l l D o l o m l l !
o E l ! n l X
s (x) = (r (x) , g (x) , b (x))T
F
s n n o n n l o n l n n o o n o n n o l D
m D o n l F p o m l D m n k n
o o n n o ! n l o ! l D n l 9 n n n l @ n l l A
n F
1
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 4 9 / l a t e s t /
Q
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R C H A P T E R 1 . F O U N D A T I O N S
S p e e c h E x a m p l e
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
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Amplitude
F i g u r e 1 . 1 : A s p e e c h s i g n a l ' s a m p l i t u d e r e l a t e s t o t i n y a i r p r e s s u r e v a r i a t i o n s . S h o w n
i s a r e c o r d i n g o f t h e v o w e l " e " ( a s i n " s p e e c h " ) .
1 . 1 . 2 D i g i t a l S i g n a l s
o 4 l 4 m n E l n m l n l n n E l n E
n n l F h l n o m o n n l n m n m o l @ o n
k o D o m l A F g o m l o n l n o n o n o m o n o
m n l n n o m n o m o n F m o l o n o n m l D n
n n n m F e g s s o E n l o
D n m D n o n m k D n o o m o l n
n E n F p o m l D e g s s o n l a n m 97 n l A 65 F p I F Q o n n o n l o n n o n o n o n n F
1 . 2 I n t r o d u c t i o n t o S y s t e m s
S i g n a l s a r e m a n i p u l a t e d b y s y s t e m s F w m l l D n m o
n o o n y (t) = S(x (t))D x n n n n l n y o n l F
n o o n m m m m l m o l o o n o n X e m 9 n
n l o o o n n n n l n o n n l F p o
m m l l n l n D m f u n c t i o n a l X n o n o n o n @ n l
n o n A F
m l m n o n n o ! o n m 9 o o m n o 9 n !
o o m l o m o l l n F s n o n n o n o o l o n o m l D
l l o n o o n o n n o n o m F
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S
L e n a
( a )
( b )
F i g u r e 1 . 2 : O n t h e l e f t i s t h e c l a s s i c
L e n a
i m a g e , w h i c h i s u s e d u b i q u i t o u s l y a s a t e s t
i m a g e . I t c o n t a i n s s t r a i g h t a n d c u r v e d l i n e s , c o m p l i c a t e d t e x t u r e , a n d a f a c e . O n t h e
r i g h t i s a p e r s p e c t i v e d i s p l a y o f t h e L e n a i m a g e a s a s i g n a l : a f u n c t i o n o f t w o s p a t i a l
v a r i a b l e s . T h e c o l o r s m e r e l y h e l p s h o w w h a t s i g n a l v a l u e s a r e a b o u t t h e s a m e s i z e . I n
t h i s i m a g e , s i g n a l v a l u e s r a n g e b e t w e e n 0 a n d 2 5 5 ; w h y i s t h a t ?
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T C H A P T E R 1 . F O U N D A T I O N S
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a c -
t e r
H H n l H I o H P H Q H R o H S n H T k H U l
H V H W H e n l H f H g n H h H i o H p
I H l I I I I P P I Q Q I R R I S n k I T n I U
I V I W m I e I f I g I h I i I p
P H P I 3 P P 4 P Q P R 6 P S 7 P T 8 P U 9
P V @ P W A P e B P f C P g D P h E P i F P p G
Q H H Q I I Q P P Q Q Q Q R R Q S S Q T T Q U U
Q V V Q W W Q e X Q f Y Q g
Q p c
R H d R I e R P f R Q g R R h R S i R T p R U q
R V r R W s R e t R f u R g v R h w R i x R p H
S H S I S P S Q S R S S S T S U
S V S W S e S f S g \ S h S i S p T H 9 T I T P T Q T R T S T T T U
T V T W T e j T f k T g l T h m T i n T p o
U H U I U P U Q U R U S U T U U
U V U W U e z U f { U g | U h } U i U p l
F i g u r e 1 . 3 : T h e A S C I I t r a n s l a t i o n t a b l e s h o w s h o w s t a n d a r d k e y b o a r d c h a r a c t e r s
a r e r e p r e s e n t e d b y i n t e g e r s . T h i s t a b l e d i s p l a y s t h e s o - c a l l e d 7 - b i t c o d e ( h o w m a n y
c h a r a c t e r s i n a s e v e n - b i t c o d e ? ) ; e x t e n d e d A S C I I h a s a n 8 - b i t c o d e . T h e n u m e r i c c o d e s
a r e r e p r e s e n t e d i n h e x a d e c i m a l ( b a s e - 1 6 ) n o t a t i o n . T h e m n e m o n i c c h a r a c t e r s c o r r e s p o n d
t o c o n t r o l c h a r a c t e r s , s o m e o f w h i c h m a y b e f a m i l i a r ( l i k e
c r
f o r c a r r i a g e r e t u r n ) a n d
s o m e n o t (
b e l
m e a n s a " b e l l " ) .
D e n i t i o n o f a s y s t e m
Systemx(t) y(t)
F i g u r e 1 . 4 : T h e s y s t e m d e p i c t e d h a s i n p u t
x (t)a n d o u t p u t
y (t). M a t h e m a t i c a l l y ,
s y s t e m s o p e r a t e o n f u n c t i o n ( s ) t o p r o d u c e o t h e r f u n c t i o n ( s ) . I n m a n y w a y s , s y s t e m s a r e
l i k e f u n c t i o n s , r u l e s t h a t y i e l d a v a l u e f o r t h e d e p e n d e n t v a r i a b l e ( o u r o u t p u t s i g n a l ) f o r
e a c h v a l u e o f i t s i n d e p e n d e n t v a r i a b l e ( i t s i n p u t s i g n a l ) . T h e n o t a t i o n
y (t) = S(x (t))c o r r e s p o n d s t o t h i s b l o c k d i a g r a m . W e t e r m
S()t h e i n p u t - o u t p u t r e l a t i o n f o r t h e s y s t e m .
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U
c a s c a d e
S1[] S2[]
x(t) y(t)w(t)
F i g u r e 1 . 5 : T h e m o s t r u d i m e n t a r y w a y s o f i n t e r c o n n e c t i n g s y s t e m s a r e s h o w n i n t h e
g u r e s i n t h i s s e c t i o n . T h i s i s t h e c a s c a d e c o n g u r a t i o n .
p a r a l l e l
x(t)
x(t)
x(t)
+y(t)
S1[]
S2[]
F i g u r e 1 . 6 : T h e p a r a l l e l c o n g u r a t i o n .
1 . 2 . 1 C a s c a d e I n t e r c o n n e c t i o n
m l o m n o n m 9 o o n n o n l o n o 9 n F w E
m l l D w (t) = S1 (x (t))D n y (t) = S2 (w (t))D n o m o n o n n n x (t) o ( D n o n m F s n o m D o n o m
m D n o o n o F p o m l D n n m n l m o l o o m m n o n
2
o n m o n l m F
1 . 2 . 2 P a r a l l e l I n t e r c o n n e c t i o n
e n l x (t) o o o @ o m o A m D n l n n o l l m m l n o l n l n F f l o k m o n n o n
n l o n o m o n o n m n o l n o l o n F o o
m o m o o n x (t) n o o o o y (t) F D y (t) = S1 (x (t)) +S2 (x (t)) D n n o m o n n x (t) o l o m F
1 . 2 . 3 F e e d b a c k I n t e r c o n n e c t i o n
l n o n n o n o n ( o n m 9 o l o o n n o
n F i n n o l o 4 k 4 o n o m P D n
m n o l o F m m l m n o k n o n n o n @ p I F U A
2
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 0 2 / l a t e s t / # c o m m s y s
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V C H A P T E R 1 . F O U N D A T I O N S
f e e d b a c k
S1
[]x(t) e(t) y(t)
S2[]
+
F i g u r e 1 . 7 : T h e f e e d b a c k c o n g u r a t i o n .
E o m o o X
y (t) = S1 (e (t))F n
e (t) l n n l m n o o o m o m 9 o o y (t) X e (t) =x (t)S2 (y (t))F p k m o m n n n o n o l o l m D o n l o j o o o m o n o n ( n n @ o n o l l n A n l F
p o m l D n 9 o n o l m D x (t) o n n n n o n D n y (t) 9 m o m F s n l o n D m P n m @ o l n A F
1 . 3 D i s c r e t e - T i m e S i g n a l s a n d S y s t e m s
w m l l D n l o n l n o n n n n n l o n n E
o n D n m F h E m n l n o n ( n o n
n Y n F e n l o n l D k o o m o n
E m n l n o m l o m o n n F f o l n o
n n n o n l D n l o o n n o m o n
@ o n n n o m o n n l A n o n o m o n @
n o m o n n A F p o m o l E l n l D o ' n X
l o o m m o n n o n o l l m o l E l n l o n m o
n o m o n o n n n n ( F p o m n n o m o n n o n E
D m o m o n o m D o o l E l n m o l E l n l D
0 n X m o m o n o n o m o n n o m o n o
n l F
1 . 3 . 1 R e a l - a n d C o m p l e x - v a l u e d S i g n a l s
e E m n l n m o l l l
s (n) D n ={
. . . ,
1, 0, 1, . . .}
F
l l E m n l m l o o m z
n o n ( n o n l o n n F n l E m n l n n
j n l o o n F e l n m l o n (n m)D n l o n n n = mF
1 . 3 . 2 C o m p l e x E x p o n e n t i a l s
m o m o n n l D o o D c o m p l e x e x p o n e n t i a l s e q u e n c e F
s (n) = ej2fn @ I F I A
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W
C o s i n e
n
sn
1
F i g u r e 1 . 8 : T h e d i s c r e t e - t i m e c o s i n e s i g n a l i s p l o t t e d a s a s t e m p l o t . C a n y o u n d t h e
f o r m u l a f o r t h i s s i g n a l ?
U n i t s a m p l e
1
n
n
F i g u r e 1 . 9 : T h e u n i t s a m p l e .
1 . 3 . 3 S i n u s o i d s
h E m n o o o o m s (n) = Acos(2f n + ) F e o o o n l o o m l o n n l n n o n n n l l D
n o E m o n l n o m o n l y n
fl n
n l
12 , 12 F o n l n o o n o n n n n o n o E m o m l o n n l n o ' o n n l 9
l F
ej2(f+m)n = ej2fnej2mn
= ej2fn@ I F P A
o n o l l o o m l o n n l l n n m l l o
2 l o n F
1 . 3 . 4 U n i t S a m p l e
o n E m o m o n E m n l u n i t s a m p l e D ( n o
(n) =
1 n = 0
0 o @ I F Q A
i m n o n o E m n l 9 l o D l k o o n n l o n n
p I F V D l l l n l o n o n o l n l n m l F
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I H C H A P T E R 1 . F O U N D A T I O N S
f l o n n m n o s (m) n n m l l o o m n (n m)D n o m o a n y n l m o n m l l o o l o o n n l n l l F
s (n) =
m=(s (m) (n m)) @ I F R A
k n o o m o o n n o E m n l D n l l o l E
n l F
1 . 3 . 5 S y m b o l i c S i g n a l s
e n n n o E m n l l o n o n o l
n m F o l E l E m n l l k n o D l o
n l n o n o o n k o F
n l n 9 l n m D n o l l o n o o l n l l D l l
m m l o n m m o F w o o m l l D
l m n o s y m b o l i c - v a l u e d n l s (n) k o n o n o l {a1, . . . , aK} o m a l p h a b e t
AF n l m n o l o o n o m n m o l
o n m m o i n l o q k l F o l n k o E
D @ V E n A D n o n l m F o n o l l
o o n o D E m m l m l o n o m l D
o n e n t i r e l y o n l o l m n F p m o D n m o n n E
o n o E m n l D l k E m l D o m l n l o n l n m F
n n n o E m n n l o n l n m n n
m n o l o o F
1 . 3 . 6 D i s c r e t e - T i m e S y s t e m s
h E m m n o n E m n l n m l o o o n n
n l o n l n m F f o o l o o n E m m D m n
m o ' n m n n o n n 4 o n 4 o m n n
n l o n l F s n D l l o n l o n l n o n n o E
m n l D o o D n o n k n o n n l o n l D l l o
n o n o o F p o n l D m n l o n o D
l n n l o l z o n 0 l D n o m o l D o n F
1 . 4 L i n e a r T i m e - I n v a r i a n t S y s t e m s
e E m n l
s (n) d e l a y e d n0 m l n s (n n0) D n0 > 0Fg o o n
n0 o n n n l l o n n F e o o o n l o
l
3
D E m l n o n l y n l F s n n o m n D l n
n l o o n o l n o n l 9 E m p o n o m X s (n n0) e(j2fn0)S
ej2f
F
L i n e a r d i s c r e t e - t i m e s y s t e m s o o n o F
S u p e r p o s i t i o n
S(a1x1 (n) + a2x2 (n)) = a1S(x1 (n)) + a2S(x2 (n)) @ I F S A
3
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 0 6 / l a t e s t / # d e l a y
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I I
e E m m l l s h i f t - i n v a r i a n t @ n l o o o m E n n n l o E
m
4
A l n n l o o n n o F
S h i f t - I n v a r i a n t
s S(x (n)) = y (n) D n S(x (n n0)) = y (n n0) @ I F T A m E n n o m z l n o n l n l n
E m D l n n l o n l D l n l l F
n o o n n o n m o l n n E n n F s l l
l l o l l o o n E o m n n l n m l m n o n F
f n o l o n n n 4 o n n 4 m D n o n l
m m l ( o n F s n n l o m D ' n l o n ( n E
o l o n n m E o m n F o o n n E m ( o n
d i e r e n c e e q u a t i o n F
T h e D i e r e n c e E q u a t i o n
y (n) = a1y (n 1) + + apy (n p) + b0x (n) + b1x (n 1) + + bqx (n q) @ I F U A r D o n l y (n) l o p a s t l y (n l) D l = {1, . . . , p} D n o n n l o n n l x (n) F m 9 m n o o n m o o 0 n p n q n o 0 n 9 l
{a1, . . . , ap} n {b0, b1, . . . , bq} F
a s i d e : n m m n o 0 n X a0 c o 0 n
o l m l l y (n) m n ' n o n @ i o n I F U A F n l l o n D o n o n n E o
l o n F o n n o n a0 l o n F
e o o o ' n l o n D o n l o n i m p l i c i t o n o
m @ m o m o o l ' n l o n A D ' n o n o n
e x p l i c i t o o m n o o n n F m l ' n
o n o m l l o o m o o l D n
n n o n F
1 . 5 D i s c r e t e - T i m e C o n v o l u t i o n
1 . 5 . 1 O v e r v i e w
g o n o l o n o n n o l l m o l i n e a r a n d t i m e -
i n v a r i a n t
5
@ L T I A F o d i s c r e t e - t i m e c o n v o l u t i o n l m o
o n n o E m o n o l o n
6
F p o o n D m l o l o o k o o n o
l o n n n o m l m o n o n F l l o n o l o n
o l o o l n m n n m 9 o o m k n o l o n n
n m 9 m l o n F s l l l o l l o o n o l o n l l
o o n n o l o n o m D o m n l
7
l l
o n n n F o l l o ' ' n o o l o n F
4
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 0 7 / l a t e s t / # t i m e i n v
5
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 8 4 / l a t e s t /
6
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 8 5 / l a t e s t /
7
h t t p : / / w w w . j h u . e d u / s i g n a l s
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I P C H A P T E R 1 . F O U N D A T I O N S
1 . 5 . 2 C o n v o l u t i o n S u m
e m n o n o D o n o l o n m o o n D m m l o
o o n v s m o n n E m n n l n m 9
o n F c o n v o l u t i o n s u m
y [n] =
k=(x [k] h [n k]) @ I F V A
e o n n o E m D o n o l o n n m o l B D n n n
y [n] = x [n] h [n] @ I F W A f m k n m l n o l n o o n o l o n m D k = n k D n l o o n o l o n c o m m u t a t i v e X
x [n]
h [n] = h [n]
x [n] @ I F I H A
p o m o n o m o n o n o o n o l o n D o o o
g o n o l o n
8
F
1 . 5 . 3 D e r i v a t i o n
k n o n E m n l n n m m o n o l n
E m m l F n m n m o l n n m E
n n D o l m o o n n n n l o m o m o l n
m l o l o n o o m o m o l n
m l o n F l o n c o n v o l u t i o n F f l o n m o
o o n m m l l o o k o n X
v n H h v s m D o l l o n o n n o k o
o n o n o l o n m 3
y [n] = H [x [n]]= H
k= (x [k] [n k])
=
k= (H [x [k] [n k]])
=
k= (x [k]H [ [n k]])=
k= (x [k] h [n k])
@ I F I I A
v k k l o o k k n n o o n F e o n l o n D
n h n o
9
o n o n D x [n]D m o n o n m n m l F x D n m o o n H o o n m m o n
H [] l n D h m F f o l n n x [k] o n n D n l l o m n o n o n n o n m l m l l H []Fp n l l D H
[] m n n n o o o ( n l E
o n o l o n m 3
e k l m l m l n m o n n o n o l o n o k F
1 . 5 . 4 C o n v o l u t i o n T h r o u g h T i m e ( A G r a p h i c a l A p p r o a c h )
s n o n l l l o o n l n o n o E m o n o l o n F
l l n n o n o l o n m l l o n x o l D l n E m n l
8
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 8 8 / l a t e s t /
9
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 5 9 / l a t e s t / # s i f t i n g
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I Q
F i g u r e 1 . 1 0 : A s i n g l e i m p u l s e i n p u t y i e l d s t h e s y s t e m ' s i m p u l s e r e s p o n s e .
F i g u r e 1 . 1 1 : A s c a l e d i m p u l s e i n p u t y i e l d s a s c a l e d r e s p o n s e , d u e t o t h e s c a l i n g
p r o p e r t y o f t h e s y s t e m ' s l i n e a r i t y .
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I R C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 1 2 : W e n o w u s e t h e t i m e - i n v a r i a n c e p r o p e r t y o f t h e s y s t e m t o s h o w t h a t a
d e l a y e d i n p u t r e s u l t s i n a n o u t p u t o f t h e s a m e s h a p e , o n l y d e l a y e d b y t h e s a m e a m o u n t
a s t h e i n p u t .
F i g u r e 1 . 1 3 : W e n o w u s e t h e a d d i t i v i t y p o r t i o n o f t h e l i n e a r i t y p r o p e r t y o f t h e s y s t e m
t o c o m p l e t e t h e p i c t u r e . S i n c e a n y d i s c r e t e - t i m e s i g n a l i s j u s t a s u m o f s c a l e d a n d s h i f t e d
d i s c r e t e - t i m e i m p u l s e s , w e c a n n d t h e o u t p u t f r o m k n o w i n g t h e i n p u t a n d t h e i m p u l s e
r e s p o n s e .
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I S
F i g u r e 1 . 1 4 : T h i s i s t h e e n d r e s u l t t h a t w e a r e l o o k i n g t o n d .
n h o l D l n E k D v s m F ( n m m o n D
y [n] =m1
l=0(x [l] h [n l]) @ I F I P A
x o o n n n m o o o xl n m E l hl F o m l l m o x m o m E h n
m F
q o n k o o m l X
o n n o m o n o n n m l o n n m
n 4 l k n o 4 n n l F g l l D l m l l n D
n n m m n m l o n F
o o m E n D n l n o o m m o n o o n n
o n o l o n D n m o n o o n o l o n l n o o m F
1 . 6 R e v i e w o f L i n e a r A l g e b r a
o n l o j o n l n l F e o l
( n o ( l o l F
1 . 6 . 1 F i e l d s
e ( l
F o o o n D o n n m l l o n D n o n n n
o l m m H n I @ 0 = 1A D o l l {a,b,c} FI F @ A
a + b F
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I T C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 1 5 : H e r e w e r e v e r s e t h e i m p u l s e r e s p o n s e , h , a n d b e g i n i t s t r a v e r s e a t t i m e
0.
F i g u r e 1 . 1 6 : W e c o n t i n u e t h e t r a v e r s e . S e e t h a t a t t i m e
1, w e a r e m u l t i p l y i n g t w o
e l e m e n t s o f t h e i n p u t s i g n a l b y t w o e l e m e n t s o f t h e i m p u l s e r e s p o n s e .
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I U
F i g u r e 1 . 1 7
@ A a + b = b + a
@ A (a + b) + c = a + (b + c)
@ A a + 0 = a
@ A a a + (a) = 0P F @ A ab F
@ A ab = ba
@ A (ab) c = a (bc)
@ A a 1 = a@ A
a1
aa1 = 1
Q F a (b + c) = ab + ac
w o o n l
I F
F n l n o n o n
P F F n l n o n m l l o n
Q F m l l o n o o n
1 . 6 . 1 . 1 E x a m p l e s
QD R D C
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I V C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 1 8 : I f w e f o l l o w t h i s t h r o u g h t o o n e m o r e s t e p , n = 4, t h e n w e c a n s e e t h a t w e p r o d u c e t h e s a m e o u t p u t a s w e s a w i n t h e i n i t i a l e x a m p l e .
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I W
1 . 6 . 2 V e c t o r S p a c e s
v F ( l D n V F V i s a v e c t o r s p a c e o v e r F o o o n D
( n o l l a
FD u
V n v
V X
o o n X @ u D v A u + v V l m l l o n X @ aD v A av V
n n l m n n o
0 V D o l l o n o l o l l a FDb FD n u V D v V D n w V
I F @ A
u + (v + w) = (u + v) + w
@ A u + v = v + u
@ A u + 0 = u
@ A u u + (u) = 0P F @ A a (u + v) = au + av
@ A (a + b)u = au + bu
@ A (ab)u = a (bu)
@ A 1 u = uw o o n l D
I F V n l n o n l
P F x l o o l m l l o n
1 . 6 . 2 . 1 E x a m p l e s
RN o o R CN o o C CN o o R RN n o t o o C
l m n o V l l v e c t o r s F
1 . 6 . 3 E u c l i d e a n S p a c e
o o o l l n k o n l o
x =
x1x2
F
F
F
xN
= x1 x2 . . . xN T
m l {xi} o l m l o m ( n o n D o n n o m n l D o E m l F
e n l l l l o n o o n o o o X
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P H C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 1 9 : S i s a n y l i n e t h r o u g h t h e o r i g i n .
1 . 6 . 3 . 1 R e a l E u c l i d e a n s p a c e
x RN @ o R A
1 . 6 . 3 . 2 C o m p l e x E u c l i d e a n s p a c e
x CN @ o CA
1 . 6 . 4 S u b s p a c e s
v V o o FF
e S V l l s u b s p a c e o V S o o F n o n F
E x a m p l e 1 . 1 :
V = R2 D F = RD S = n l n o o n F
e o c
T h e o r e m 1 . 1 :
S V n o n l o l l a F n b F n o l l s S n t SD as + bt S
1 . 6 . 5 L i n e a r I n d e p e n d e n c e
v
u1, . . . ,uk V F o l i n e a r l y d e p e n d e n t l
a1, . . . , ak F
ki=1
(aiui) = 0 @ I F I Q A
n l o n ai = 0 Fs i o n I F I Q o n l o l o a1 = = ak = 0 D o
l i n e a r l y i n d e p e n d e n t F
E x a m p l e 1 . 2 :
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P I
F i g u r e 1 . 2 0 : < S > i s t h e x y - p l a n e .
1
112
2 23
0
+ 1 57
2
= 0 o o l n l n n n R3 F
1 . 6 . 6 S p a n n i n g S e t s
g o n S = {v1, v2, . . . , vk}F h ( n s p a n o S
< S > span(S)
k
i=1(aivi) |ai F
F a c t : < S > o V F
E x a m p l e 1 . 3 :
V = R3 D F = RD S = {v1, v2}D v1 = 10
0
D v2 =
01
0
< S >= E l n F
1 . 6 . 6 . 1 A s i d e
s S n ( n D n o o n o l n n n n n n l n l z X
S l n l n n n D o ( n o l l o n u1, . . . , uk SD @ k A
ki=1
(aiui) = 0 ai = 0 ,
n o S
< S >=
ki=1
(aiui) |ai F ui S k <
N o t e : s n o ( n o n D o n l o n n i t e m F
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P P C H A P T E R 1 . F O U N D A T I O N S
1 . 6 . 7 B a s e s
e B V l l b a s i s o V o F n o n l I F B l n l n n n
P F < B >= V
f o n m n l m o n n n l o n F l l o o o m o
n l n o l n l o k @ o A o n m o l n o o F
E x a m p l e 1 . 4 :
V a @ l o o m l A i l n D RN o CNF
B = {e1, . . . , eN} n
ei =
0F
F
F
1F
F
F
0
I n ith o o n F
E x a m p l e 1 . 5 :
V = CN o CFB = {u1, . . . , uN}
h p F
uk =
1
e(j2kN )
F
F
F
e(j2kN
(N1))
j =
1 F
1 . 6 . 7 . 1 K e y F a c t
s B o V D n v V n n n l @ o o o m A n o m
v =
Ni=1
(aivi)
ai F n vi B F
1 . 6 . 7 . 2 O t h e r F a c t s
s S l n l n n n D n S n n o F s < S >= V D n S o n n F
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P Q
1 . 6 . 8 D i m e n s i o n
v V o B F m n o n o V D n o dim(V) D E n l o
BF
T h e o r e m 1 . 2 :
i o F
T h e o r e m 1 . 3 :
i o o m n l F
dim(V) w e l l - d e n e d Fs dim(V) < D V n i t e d i m e n s i o n a l F
1 . 6 . 8 . 1 E x a m p l e s
v e c t o r s p a c e e l d o f s c a l a r s d i m e n s i o n
RN R
CN C
CN R
i o D n o o n m n o n F
E x a m p l e 1 . 6 :
o S = {u1, . . . , uk} V l n l n n n F n dim(< S >) =
F a c t s
s S o V D n dim(S) dim(V)F s dim(S) = dim(V) < D n S = V F
1 . 6 . 9 D i r e c t S u m s
v V o D n l S V n T V F V d i r e c t s u m o S n TD n V = (S T) D n o n l o
v V D n s S n t T v = s + t Fs V = (S T)D n T l l c o m p l e m e n t o SF
E x a m p l e 1 . 7 :
V = C = left{f : R R|f o n n o right}S = n n o n n C
T = o n o n n C
f(t) =1
2(f(t) + f(t)) + 1
2(f(t) f(t))
s
f = g + h = g + h D g S n g SD h T n h TD n g g = h h o n n D m l
g = g n h = h F
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P R C H A P T E R 1 . F O U N D A T I O N S
1 . 6 . 9 . 1 F a c t s
I F i o m l m n
P F
V = (S T) n o n l @ A S
T = {0}
@ A < S, T >= V
Q F s V = (S T) D n dim(V) < D n dim(V) = dim(S) + dim (T)
1 . 6 . 9 . 2 P r o o f s
s n o k F
1 . 6 . 1 0 N o r m s
v
V o o
FF e n o m m n (V
F)D n o
D
o l l
u V D v V D n FI F u > 0 u = 0P F u = || u Q F u + v u + v
1 . 6 . 1 0 . 1 E x a m p l e s
i l n n o m X
x RNX
x
=
N
i=1xi2
12
x CNX
x =
Ni=1
(|xi|)2
12
1 . 6 . 1 0 . 2 I n d u c e d M e t r i c
i n o m n m o n V
d (u,v) u v l o n o o n o 4 n 4 n o F
1 . 6 . 1 1 I n n e r p r o d u c t s
v V o o FD F = R o CF e n n n o m n V V FD n o D
I F v v 0 D n (v v = 0 v = 0)P F
u v = v u
Q F au + bv w = au w + bv w
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P S
1 . 6 . 1 1 . 1 E x a m p l e s
RN o R X
x y = xTy =Ni=1
(xiyi)
CN o CX
x y = xHy =Ni=1
(xiyi)
s x = (x1, . . . , xN)T CD n
xH
x1
F
F
F
xN
T
l l 4 r m n D 4 o 4 o n j n o 4 o xF
1 . 6 . 1 2 T r i a n g l e I n e q u a l i t y
s ( n u = u u D n
u + v u + v
r n D n n o n n o m F
1 . 6 . 1 3 C a u c h y - S c h w a r z I n e q u a l i t y
p o l l u V D v V D|u v| u v
s n n n o D n o o n o n l n o o X
(u,v) = arccos
u v
u v
[0, 2)
1 . 6 . 1 4 O r t h o g o n a l i t y
u n v o r t h o g o n a l
u v = 0x o o n X (u v)F
s n o n u = v = 1D u n v o r t h o n o r m a l Fs n n o o o n l @ o o n o m l A s e t D o o o o o n l @ o o n o m l A F
1 . 6 . 1 5 O r t h o n o r m a l B a s e s
e n y o n o m l {vi}
vi vi = ij =
1 i = j0 i = j
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P T C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 2 1 : O r t h o g o n a l v e c t o r s i n
R2
.
E x a m p l e 1 . 8 :
n o RN o CN
E x a m p l e 1 . 9 :
n o m l z h p
uk =1N
1
e(j2kN )
F
F
F
e(j2kN
(N1))
1 . 6 . 1 6 E x p a n s i o n C o e c i e n t s
s n o n o
v o {vi}
v =
(aivi)
n
ai = vi v
1 . 6 . 1 7 G r a m - S c h m i d t
i n n o n o o n o m l F e n @ o n l A n m
o o o n l q m E m o o o n l z o n o F
1 . 6 . 1 8 O r t h o g o n a l C o m p l i m e n t s
v
S V F o r t h o g o n a l c o m p l i m e n t S S = {u |u V u v = 0 v S , }
S l n o F s dim(v) < D n V = S SF
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P U
A s i d e : s dim(v) = D n n o o V = S S V o H i l b e r t S p a c e F
1 . 6 . 1 9 L i n e a r T r a n s f o r m a t i o n s
v o o l k n D l n n o m o n m n o m o n o o n o
p r e s e r v e s o o o n F
w o l D l V D W o o m ( l FF e l i n e a r t r a n s f o r m a -
t i o n m n T : V W
T (au + bv) = aT (u) + bT (v)
o l l a FD b F n u V D v V Fs n l l l o n n l n n o m o n n @ l o o m l A
E u c l i d e a n s p a c e s D o o F
1 . 6 . 2 0 I m a g e
image(T) = {w |w W T (v) = w o o m v}
1 . 6 . 2 1 N u l l s p a c e
e l o k n o n k n l X
ker(T) = {v |v V T (v) = 0}f o m n n l l l n o F
1 . 6 . 2 2 R a n k
rank(T) = dim (image(T))
1 . 6 . 2 3 N u l l i t y
null (T) = dim(ker (T))
1 . 6 . 2 4 R a n k p l u s n u l l i t y t h e o r e m
rank(T) + null (T) = dim(V)
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P V C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 2 2
1 . 6 . 2 5 M a t r i c e s
i l n n o m o n T m a t r i x r e p r e s e n t a t i o n F s T : EN EMD E = R o CD n T n n M N m
A =
a11 . . . a1NF
F
F
F
F
F
F
F
F
aM1 . . . aMN
(a1i, . . . , aMi)T
= T (ei) n ei = (0, . . . , 1, . . . , 0)T
ith s t a n d a r d b a s i s o F
A s i d e : e l n n o m o n n n o n o
EN n EMD l n o ' n AF l l l n l n n o E
m o n n n F
1 . 6 . 2 6 C o l u m n s p a n
colspan (A) =< A >= image (A)
1 . 6 . 2 7 D u a l i t y
s
A : RN RMD n ker (A) = image
AT
s A : CN CM D n ker (A) = image
AH
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P W
1 . 6 . 2 8 I n v e r s e s
l n n o m o n G m A i n v e r t i b l e n o n l m B
AB = BA = I @ n A F y n l s q u a r e m n n l F
T h e o r e m 1 . 4 :
v A : FN FN l n D F = R o CF o l l o n l n X I F
A n l @ n o n n l A
P F rank(A) = N
Q F null(A) = 0
R F detA = 0S F o l m n o A o m F
s A1 = AT @ o AH n o m l A D A o r t h o g o n a l @ o u n i t a r y A F
1 . 7 H i l b e r t S p a c e s
1 . 7 . 1 H i l b e r t S p a c e s
e o S l n n o 1 0 ( n o n l l n i n n e r p r o d u c t
s p a c e D l o n o r m e d l i n e a r s p a c e F e H i l b e r t s p a c e n n n o
o m l o n o m ( n n n n o F r l
n m h r l
1 1
D o l o o o n l
o n F ( n o l n o m n n n o X
x = x x @ I F I R A r l l n n n n l z n o n o p o n o n D
p o n o m D n m o n o o n m m n F r l
n n o n l n l n o m m F
1 . 7 . 1 . 1 E x a m p l e s o f H i l b e r t S p a c e s
f l o l l l m l o r l
1 2
F o n l
n n o o m F
p o Cn D
x
y = yTx = y0 y1 . . . yn1
x0x1
F
F
F
xn1
=n1
i=0
(xiyi)
o ( n n o m l n o n X L2 (R)
f g =
f(t) g (t)
dt
1 0
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 7 5 5 / l a t e s t /
1 1
h t t p : / / w w w - h i s t o r y . m c s . s t - a n d r e w s . a c . u k / h i s t o r y / M a t h e m a t i c i a n s / H i l b e r t . h t m l
1 2
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 4 3 4 / l a t e s t /
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Q H C H A P T E R 1 . F O U N D A T I O N S
o E m m l n X 2 (Z)
x
y =
i=
x [i] y [i]1 . 8 S i g n a l E x p a n s i o n s
1 . 8 . 1 M a i n I d e a
n o k n n l m n m l l o k n l n o m l l D m o
m n l F r o l l n o o n o o o n o n o
1 3
n n o m o n n l n o o n o o l F f o n l
o m l o l l o n o n l n m o n n o n o v s m
1 4
F
x o o l l k o l o o k n l n o n n l D o
o o r t h o n o r m a l b a s i s F n n k o o o n o m l o l n l o k
o o n n o n F l l l n l G o m o
l m n F
E x a m p l e 1 . 1 0 :
o m l n o
1T
ej0nt o l l < n < o m n o o n o m l o L2 ([0, T]) F
s n o p o
1 5
o n D f(t) =
n=
cnej0nt
D {cn} j
n o n o n o
f(t) F
n o t e : p o n l G o n r l
1 6
D n o n o 0 n
o ( n F
1 . 8 . 2 A l t e r n a t e R e p r e s e n t a t i o n
l l o ( n o n o b a s i s X e o o {bi} n o S I F bi l n l n n n F
P F bi n 1 7 SF D n ( n {i}D i C @ l A
x =i
(ibi) , x S @ I F I S A
x o n SD l n CD n b o n SF
g o n o n P n o ( n o n n d e c o m p o s e n o n m o
{bi} F g o n o n I n o m o o n u n i q u e @ n k o o m A F n o t e : {i} o n l n n o n o xF
1 3
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 7 3 6 / l a t e s t /
1 4
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 5 0 0 / l a t e s t /
1 5
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 4 9 6 / l a t e s t /
1 6
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 7 5 5 / l a t e s t / # s e c 2
1 7
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 7 3 4 / l a t e s t / # s p a n _ s e c
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Q I
E x a m p l e 1 . 1 1 :
v l o o k m l m l n R2 D o l l o n o X
x =
12
n f X {e0, e1} =
(1, 0)T
, (0, 1)T
x = e0 + 2e1
e l n f X {h0, h1} =
(1, 1)T
, (1, 1)T
x =3
2h0 +
12
h1
s n n l D n {b0, b1} n o x R2 D o o ( n 0 n 1
x = 0b0 + 1b1 @ I F I T A
1 . 8 . 3 F i n d i n g t h e A l p h a s
x o l o n o o o ( n n i 9 n n l o R2
F
n i o n I F I T o n k o bi 9 o l m n n P P m F x
= 0
b0
+ 1
b1
@ I F I U A
x =
F
F
F
F
F
F
b0 b1F
F
F
F
F
F
0
1 @ I F I V A
E x a m p l e 1 . 1 2 :
r m l m l D o l l m o l o o E
o n F x [0]x [1]
= 0
b0 [0]b0 [1]
+ 1
b1 [0]b1 [1]
=
0b0 [0] + 1b1 [0]0b0 [1] + 1b1 [1]
@ I F I W A
x [0]x [1]
=
b0 [0] b1 [0]b0 [1] b1 [1]
01
@ I F P H A
1 . 8 . 3 . 1 S i m p l i f y i n g o u r E q u a t i o n
o m k n o o n m l D ( n o l l o n o m o m o o n X
B a s i s M a t r i x X
B =
F
F
F
F
F
F
b0 b1F
F
F
F
F
F
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Q P C H A P T E R 1 . F O U N D A T I O N S
C o e c i e n t V e c t o r X =
01
o l l o n D o n o n X
x = B @ I F P I A
l n o x =1
i=0 (ibi) F
E x a m p l e 1 . 1 3 :
q n n D
10
,
01
D n o l l o n m E
X
B =
0 11 0
o i 9 D o l o o 0 n o n i o n I F P I
= B1x @ I F P P A
B1 n m
1 8
o
BF
1 . 8 . 3 . 2 E x a m p l e s
E x a m p l e 1 . 1 4 :
v l o o k n ( n o l l o m F
B = 1 00 1 = I
I i d e n t i t y m a t r i x F s n o o o l o l ( n n o
B ( @ o o l l n A X
B1 =
1 00 1
o D
= B1x = x
E x a m p l e 1 . 1 5 :
v l o o k E o E l l m o o m l o
11
, 11 =
{h0, h1} n o m n n m o m X
B =
1 11 1
B1 =
12
12
12
12
1 8
h t t p : / / c n x . o r g / c o n t e n t / m 2 1 1 3 / l a t e s t /
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Q Q
n o m l n
x = 32
x o o l o
= B1x =
12
12
12
12
32
=
2.50.5
n
x = 2.5h0 + 0.5h1
E x e r c i s e 1 . 1 :
x o n o l l o n m n xX
{b0, b1} =
12
,
30
x = 3
2
p o o l m D m k k o n n n
x n m o b0
n b1 F
n o t e : e n o m l l o o k
x o m 4 ' n F 4 B1
t r a n s f o r m s x o m n o o n D {b0, b1}F x o o l l m n l o F
1 . 8 . 4 E x t e n d i n g t h e D i m e n s i o n a n d S p a c e
n l o n l l j R2 n l o o k m n Rn n Cn F
o n n l l o @ >P A m n o n F q n
{b0, b1, . . . , bn1
} o
RnD n o ( n {0, 1, . . . , n1}
x = 0b0 + 1b1 + + n1bn1 @ I F P Q A e n D l l m
B =
b0 b1 b2 . . . bn1
o l m n l o n l l l n n n m @ l o o m o n o o n l m n o n o o n A F
n n o o i o n I F P I
x =
b0 b1 . . . bn1
0F
F
F
n1
= B
n
= B1x
1 . 9 F o u r i e r A n a l y s i s
p o n l n m n l o n n n o o n l n m F
l o n o i n n o n
1 9
o l n D m E n n @ v s A
2 0
1 9
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 5 0 0 / l a t e s t /
2 0
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 8 4 / l a t e s t /
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Q R C H A P T E R 1 . F O U N D A T I O N S
m F o n l n o o v s m D
l o n o m n o o n o F n D n p o n l l l o o
( n n l n m o n o D l l n o o m n o n n m
' l l o l n o @ n n o n
2 1
A n o m l n n n
o m F p m o D n l o ( n o n o o
m m l l o n o n o n n o n m n D
n o n o n n l o m o m o n o l o n
2 2
@ n m A o
m l l o n @ n n A F p o n l o F
x o D o l l n o l o o n l o m o o n l D m E
m n l m n p o n l F o p o n o m o m
n l p o
2 3
D g o n n o E m p o n o m @ o n P F I H A D
h E m p o n o m @ o n P F I I A n h p o n o m
2 4
F p o
o m n D l l v l n o m
2 5
n E n o m @ o n R F Q A m l
n o n o g p n h p l F e l l o n o m n l l
m D o n n n l n m o n l n n l n n @ n E
o A F r o D n n o n n o ( n l i . e . ( n E o
n ( n E l n n E o o n n o E m A n o n E
o m o o n n l n o n o m n F f l o l o o p o
n o m n n o F s l o n l l n o n o l o n o
( F
T a b l e o f F o u r i e r R e p r e s e n t a t i o n s
T r a n s f o r m T i m e D o m a i n F r e q u e n c y D o m a i n C o n v o l u t i o n
g o n n o E m p o L2 ([0, T)) l2 (Z) g o n n o E m g g o n n o E m p o n o m
L2 (R) L2 (R) g o n n o E m vh E m p o n o m
l2 (Z) L2 ([0, 2)) h E m v nh p o n o m
l2 ([0, N 1]) l2 ([0, N 1]) h E m g
1 . 1 0 C o n t i n u o u s - T i m e F o u r i e r T r a n s f o r m ( C T F T )
1 . 1 0 . 1 I n t r o d u c t i o n
h o l n m o o n n o E m n l n D p o
2 6
o ( l m o o m m n o m o n l n n l
m n n X o o n o n m o n o F x o D n l o o k o
n o n n o E m n o n o n l n m o o o n F f l o
l l n C o n t i n u o u s - T i m e F o u r i e r T r a n s f o r m @ g p A D l o o
j p o n o m @ p A F f g p n o l n o n o n l D
m n o ( n o n l a l l n n n l o n F
1 . 1 0 . 1 . 1 E q u a t i o n s
C o n t i n u o u s - T i m e F o u r i e r T r a n s f o r m
2 1
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 2 8 / l a t e s t /
2 2
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 8 8 / l a t e s t /
2 3
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 9 7 / l a t e s t /
2 4
h t t p : / / c n x . o r g / c o n t e n t / m 0 5 0 2 / l a t e s t /
2 5
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 1 1 0 / l a t e s t /
2 6
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 3 9 / l a t e s t /
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Q S
F i g u r e 1 . 2 3 : M a p p i n g
L2 (R)i n t h e t i m e d o m a i n t o
L2 (R)i n t h e f r e q u e n c y d o m a i n .
F() =
f(t) e(jt)dt @ I F P R A
I n v e r s e C T F T
f(t) =1
2
F() ejtd @ I F P S A
w a r n i n g : h o n o o n n o o n E n o n o m m o n o o
o m l n l l ' n F y n o m o o m m o n ' n m o n
m n o o o n n l n F e o
l n l n o n n l D = 2fD o n l l o n o o n l m o l o n D
j2f tD n o n n l F g l k
2 7
o n o o n o o n n g o n n o n 9 h m o l F
o o n o g p n n o m l o m p o
n o n n n o o 0 n F p o g p m l l z n o n
n m m o n o l o o n l F o l m k n n
o g p m l n n o p o o n l n o n o
n l D n n n m F v o o k h o n o p o
n o m
2 8
o m o n l o o k F
1 . 1 0 . 2 R e l e v a n t S p a c e s
g o n n o E m p o n o m m n ( n E l n D o n n o E m n l n
L2 o n ( n E l n D o n n o E n n l n L2 F p o e n l @ E
o n P F W A o n o o l l n p o n l F
p o m o n o m o n o n o g p D l l o o k m o l o n
o o p o n o m
2 9
F
1 . 1 0 . 3 E x a m p l e P r o b l e m s
E x e r c i s e 1 . 2 :
p n p o n o m @ g p A o n o n
f(t) =
e(t)
t 00 o
@ I F P T A
2 7
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 1 6 1 / l a t e s t /
2 8
h t t p : / / c n x . o r g / c o n t e n t / m 0 0 4 6 / l a t e s t /
2 9
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 1 0 0 / l a t e s t /
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Q T C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 2 4 : M a p p i n g l2 (Z) i n t h e t i m e d o m a i n t o L2 ([0, 2)) i n t h e f r e q u e n c y d o m a i n .
E x e r c i s e 1 . 3 :
p n n p o n o m o ( n
X() = 1 || M0 o @ I F P U A
1 . 1 1 D i s c r e t e - T i m e F o u r i e r T r a n s f o r m ( D T F T )
D i s c r e t e - T i m e F o u r i e r T r a n s f o r m
X() =
n=
x (n) e(jn)
@ I F P V A
I n v e r s e D i s c r e t e - T i m e F o u r i e r T r a n s f o r m
x (n) = 12
2
0
X() ejnd @ I F P W A
1 . 1 1 . 1 R e l e v a n t S p a c e s
h E m p o n o m
3 0
m n ( n E l n D E m n l n l2 o
( n E l n @ o 2 E o A D o n n o E n n l n L2 F
1 . 1 2 D F T a s a M a t r i x O p e r a t i o n
1 . 1 2 . 1 M a t r i x R e v i e w
l l X
o n RNX
x =
x0x1. . .
xN1
, xi R
3 0
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 2 4 7 / l a t e s t /
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Q U
o n CNX
x =
x0x1
. . .xN1
, xi C
n o o n X I F n o X
xT =
x0 x1 . . . xN1
P F o n j X
xH =
x0 x1 . . . xN1
s n n o 3 1 X
I F l X
xTy =
N1i=0
(xiyi)
P F o m l X
xHy =
N1i=0
(xnyn)
w w l l o n X
Ax =
a00 a01 . . . a0,N1a10 a11 . . . a1,N1
F
F
F
F
F
F . . .F
F
F
aN
1,0 aN
1,1 . . . aN
1,N
1
x0x1. . .
xN1
=
y0y1. . .
yN1
yk =N1n=0
(aknxn)
w n o o n X
AT =
a00 a10 . . . aN1,0a01 a11 . . . aN1,1
F
F
F
F
F
F . . .F
F
F
a0,N1 a1,N1 . . . aN1,N1
w n o o n n o l m l n o o l m n F
AH = AT
o o n r m n n o F ATkn
= AnkAHkn
= [A]nk
3 1
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 7 5 5 / l a t e s t /
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Q V C H A P T E R 1 . F O U N D A T I O N S
1 . 1 2 . 2 R e p r e s e n t i n g D F T a s M a t r i x O p e r a t i o n
x o l 9 n h p
3 2
n o E m n o o n F
x =
x [0]x [1]. . .
x [N 1]
X =
X[0]X[1]
. . .
X[N 1]
CN
r
x o o m m l n
X o o h p o 0 n F r o
x
n
X l X
X[k] =N1
n=0
x [n] e(j 2N kn)
akn =
e(j2N )kn
= WNkn
o
X = Wx
X h p o D
W m n
x m o m n o F
Wkn =
e(j2N )kn
X = W
x [0]x [1]. . .
x [N 1]
s h p X
x [n] =1
N
N1k=0
X[k]
ej
2N
nk
ej
2N
nk= WN
nk
WNnk
m r m n n o F o D
x =1
NWHX
x m o D
1
NW
H n h p m D n
X h p o F
1 . 1 3 T h e F F T A l g o r i t h m
D e n i t i o n 1 : F F T
@ p p o n o m A e n 0 n c o m p u t a t i o n a l a l g o r i t h m o o m n
h p
3 3
F
3 2
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 2 4 9 / l a t e s t /
3 3
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 2 4 9 / l a t e s t /
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Q W
F i g u r e 1 . 2 5
F i g u r e 1 . 2 6
1 . 1 3 . 1 T h e F a s t F o u r i e r T r a n s f o r m F F T
h p n e x p e n s i v e o o m d i r e c t l y
X[k] =
N1n=0
x [n] e(j2
kNn)
, 0 k N 1
p o
kD m X
N o m l m l l N 1 o m l
o l o o o m o n o n
NE o n h p
N2 o m l m l l N(N 1) o m l
r o m n n m l o r e a l n m c
4 O
N2
4 o m o n l o o n D N l X
N I I H I H H I H H H 106
N2 I I H H I H D H H H 106 1012
p p o m m o e c i e n t o o m n h p F p p
o n l 4 O (NlogN) 4 o m o n o o m NE o n h p F
N I H I H H I H H H 106
N2 I H H I H D H H H 106 1012
Nlog10N I H P H H Q H H H 6 106
r o l o n 1012sec c w o n I H 3 r o l o n 6 106secc p p n l o m o l o n z h @ I W T H E I W V H A F
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R H C H A P T E R 1 . F O U N D A T I O N S
1 . 1 3 . 2 H o w d o e s t h e F F T w o r k ?
p p l o s y m m e t r i e s o o m l o n n l WNkn = e(j 2N kn)
WNkn l l 4 t w i d d l e f a c t o r s 4
S y m m e t r y 1 . 1 : g o m l g o n j m m
WNk(Nn) = WN
(kn) = WNkn
e(j2kN
(Nn)) = ej2kNn = e(j2
kNn)
S y m m e t r y 1 . 2 : o n n n k
WNkn = WN
k(N+n) = WN(k+N)n
e(j2Nkn) = e(j
2Nk(N+n)) = e(j
2N
(k+N)n)
WN = e(j 2N )
1 . 1 3 . 3 D e c i m a t i o n i n T i m e F F T
t o n o m a n y ' n p p l o m i d e a o l h p o o m l l n m l l h p o m o n x [n] n o
m l l n m l l n F
e m N = 2m @ o o P A
1 . 1 3 . 3 . 1 D e r i v a t i o n
N e v e n D o n o m l X[k] n x [n] n o t w o n o l n N2 F
x [n]
N2
n = evenN2 n = odd
X[k] =
N1n=0
x [n] WN
kn
, 0 k N 1
X[k] =
x [n] WNkn
+
x [n] WNkn
0 r N2 1F o
X[k] =N
2 1r=0
x [2r] WN
2kr
+N
2 1r=0
x [2r + 1] WN
(2r+1)k
=N
2 1r=0
x [2r]
WN
2kr
+ WNkN
2 1r=0
x [2r + 1]
WN
2kr @ I F Q H A
WN2 = e(j
2N
2) = ej 2N2
= WN
2F o
X[k] =
N2 1r=0
x [2r] WN
2
kr
+ WNk
N2 1r=0
x [2r + 1] WN
2
kr
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R I
N2 1r=0
x [2r] WN
2
kr
N2
E o n h p o n m l @ G [k]A n N
2 1r=0
x [2r + 1] WN
2
kr
N2
E o n h p o o m l @ H[k]A F
X[k] = G [k] + WNkH[k] , 0 k N 1h o m o o n o n NE o n h p m o P N2 E o n h p F
o l n o o c B e c a u s e i t i s m o r e e c i e n t !
R e c a l l : g o o o m n NE o n h p o m l N2 o m l m l
n F
f o m o o n n o P
N2 E o n h p C o m n o n o n l
N
2
2+
N
2
2+ N =
N2
2+ N
( n m o o m l m l n o
N2 E o n h p D G [k] F
o n n m o o m l m l n o
N2 E o n h p D H[k]F
n m o o m l m l n o o m n o n F e n o l
N2
2 + N o m l m l n F
E x a m p l e 1 . 1 6 : S a v i n g s
p o
N = 1000 DN2 = 106
N2
2+ N =
106
2+ 1000
f I H H H m l l o m o S H H D H H H D
N2
2+ N 10
6
2
o o c 3 u o m o n F f k o N2 E o n h p n o o
N4 E o n h p D e t c . D F F F F
n k o m o n X
N
21=
N
2,
N
4,
N
8, . . . ,
N
2m1,
N
2m
= 1
m = log2N = m
g o m o n l o X
NE h p [U+F577] o N2 E h p F o N
2 2N2 2+NF o l n
N2 E h p o
N4 E h p l l o o
22N4
2
+N
2+ N = 4N
4
2
+ 2N =N2
22+ 2N =
N2
2p+pN
e k o n
p = {3, 4, . . . , m} D m = log2NF o N2
2log2N+ Nlog2N =
N2
N+ Nlog2N = N + Nlog2N
N + Nlog2N o l n m o o m l n m l F p o l ND cost Nlog2N o 4 O (Nlog2N) 4 D n (Nlog2N N) o l NF
N o t e : o o m m l
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R P C H A P T E R 1 . F O U N D A T I O N S
F i g u r e 1 . 2 7 : N = 8 p o i n t F F T . S u m m i n g n o d e s Wnk
t w i d d l e m u l t i p l i c a t i o n f a c t o r s .
F i g u r e 1 . 2 8 : T h i s i s c a l l e d " b u t t e r i e s . "
S o l u t i o n s t o E x e r c i s e s i n C h a p t e r 1
S o l u t i o n t o E x e r c i s e 1 . 1 :
s n o o n
x n m o
b0 n b1 l l o l l o m n
o m l F
B =
1 23 0
B1 =
0 1213
16
= B1x =
123
e n n o n x n m o b0 n b1 F
x = b0 +2
3b1
e n n l n o k n o n l o b0 n b1 o o l F
S o l u t i o n t o E x e r c i s e 1 . 2 :
s n o o l l p o n o m D l l n o i o n I F P R D o m l
o n n l
3 4
D n l l F
F() = f(t) e(jt)dt=
0
e(t)e(jt)dt=
0
e(t)(+j)dt= 0 1
+j
@ I F Q I A
F() = 1 +j
@ I F Q P A
3 4
h t t p : / / c n x . o r g / c o n t e n t / m 1 0 0 6 0 / l a t e s t /
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R Q
S o l u t i o n t o E x e r c i s e 1 . 3 :
r l l i o n I F P S o ( n n p n t = 0 F
x (t) = 12MM ejtd
= 12 ejt|,=ejw
= 1t
sin(M t)
@ I F Q Q A
x (t) =M
sinc
M t
@ I F Q R A
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R R C H A P T E R 1 . F O U N D A T I O N S
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C h a p t e r 2
S a m p l i n g a n d F r e q u e n c y A n a l y s i s
2 . 1 I n t r o d u c t i o n
C o n t e n t s o f S a m p l i n g c h a p t e r
s n o o n @ g n m o l A o o @ o n Q F P A s l l o n @ o n Q F Q A w l i m l @ o n Q F R A r o l o o n 1
m @ o n Q F S A e l n l 2
i 3
l o o m l 4
2 . 1 . 1 W h y s a m p l e ?
o n n o m l n F m l n n n m n o l l l n l
o n n o m m n o n F m l n n ( n o o m n n
n l o n l n o n F
h l n o n o n l o n l o ' n n m o
o n o n o D m n n n n m o G o ) l o n m n D n l o m m o l l o n m n o o m l l o m
1
h t t p : / / c n x . o r g / c o n t e n t / m 1 1 4 5 8 / l a t e s t /
2
h t t p :