3 Power Spectrum - WordPress.com · Advanced Digital Signal Processing Asst. Prof. Dr....
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3Power Spectrum
Asst. Prof. Dr. Peerapol Yuvapoositanon, PhD, DIC
Department of Electronic Engineering
and
Graduate School of Electrical Engineering
Mahanakorn University of Technology
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Power Spectrum
• ใชการแปลงของฟูริเยรแบบไมตอเนื่องทางเวลา (Discrete-time Fourier Transform) หรือ DTFT
• แต x(n) เปนสัญญาณสุมที่ไมอยูในรูปที่จะสามารถทําอินทีเกรทกําลังสองได
2( ) ( ) j fn
n
X f x n e
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Z-transform
• โดยทั่วไปเรามักจะพบกับระบบ z-transform มากกวา DTFT เพราะใชงานหลากหลายกวา
( ) ( ) n
n
X z x n z
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Z-transform v.s. DTFT
• เราจะเปลี่ยนการแปลงแซดใหเปนการแปลงฟูริเยรแบบไมตอเนื่องทางเวลาไดอยางสะดวกโดย
2j f
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Z-plane
• โดยทั่วไปเราพิจารณา z-transform
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ความถี่ไนควิสต
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Power Spectrum และ Auto-Correlation
• สเปคตรัมกําลัง คือการแปลงฟูริเยรแบบไมตอเนื่องทางเวลาของสหสัมพันธตัวเองของสัญญาณ x(n)
2( ) ( ) j fm
m
S f r m e
xx xx
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กราฟ Power Spectrum
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การซ้ําคาของสเปคตรัมกําลัง
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• การหาคากําลังในชวงระหวาง f1 และ f2
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สเปคตรัมกําลังและระบบเชิงเสน (Power Spectrum and Linear Systems)• ระบบเชิงเสน (Linear system) แบบไมแปรเปลี่ยนตามการเลื่อน ซึ่งมีคาผลตอบสนองอิมพัลสเปน hk
( ) ( ) ( )k kk
y n h x n h x n k
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• a) คาเฉลี่ยกลาง
• คา DC
2( ) j fkk
k
H f h e
(0) kk
H h
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{ ( )} ( )
( )
(0)
kk
kk
kk
E y n E h x n k
h E x n k
h
H
x
x
X(n) เปน
WSS
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• คาเฉลี่ยกลางของเอาทพุท = คาเฉลี่ยกลางของอินพุท * DC Gain ของระบบ
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• b) สหสัมพันธตัวเอง
( ) ( ) ( )
( ) ( )
( ) ( )
( )
kk
kk
kk
r n E y m x m n
E h x m k x m n
h E x m k x m n
h r n k
yx
xx
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( ) ( )
( )
kk
k
r n h r n k
h r n
yx xx
xx
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• ดวยความเปนสมมาตรของฟงกชันสหสัมพันธตัวเอง ดังนั้นเราแทน n ไดดวย -n
( ) ( )
( )
( )
kk
k
r n r n
h r n k
h r n
yy yy
yx
yx
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( ) ( )
( )
( ) ( )
k
k k
r n h r n k
h h r n
r k r n
yy yx
xx
hh xx
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2
( ) ( ) ( ) ( )
( ) ( )
S f H f H f S f
H f S f
yy xx
xx
( ) ( ) ( )r n r k r n yy hh xx
DTFT
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สเปคตรัมกําลังกับตัวเขารหัสเสียงพูด (Power Spectrum and Speech Coder)
• เปนสัญญาณรบกวนสีขาว (White Noise) ซึ่งมีสเปคตรัมกําลังเปนคาคงที่หรือ k
2( ) ( )S f k H fyy
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• หรือหมายความวา สเปคตรัมกําลังของสัญญาณ เปนรูปแบบใดๆ ก็ขึ้นกับรูปรางของ โดยมีเงื่อนไขก็คือตองให x(n) เปนสัญญาณรบกวนสีขาว
2( )H f
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Randn(.)
• สราง x(n) ที่มี mean =0 และ ความแปรปรวน=1
0 10 20 30 40 50 60 70 80 90 100-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
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IID
• x(n) นั้นเปน เปนอิสระตอกัน และกระจายอยางเทาเทียมกัน (Independent and IdenticallyDistributed) หรือ เรียกยอๆ วา IID
( ) { ( ) ( )}
( )
r n m E x m x n
n m
xx
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เมตริกซสหสัมพันธตัวเอง
• ของ IID จะเปนไดแอกอนัล (Diagonal matrix)2
2
2
2
0 0
0 0
0 0
xxR
I
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ไดแอกอนัล กับ เมตริกซผกผัน
• จาก เรื่องการหาการผกผันของเมตริกซ (Inverse Matrix) เราจึงไดโดยงายวา
12
xx
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• และจากเรื่อง System identification เราจึงได
1
2
xx dx
dx
h R r
r
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สัญญาณสุมแบบเกาส
• สัญญาณสุมแบบเกาส
2
2
1 ( ) ( )( ( )) exp
2 2i i
x i if x i
( ) ~ ( ( ), )ix i iN
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1 1 0 0
1 1 0 0
1 0 1 0 0 1 1 0
1 0 1 0
( ) ( ) ( ) ( )
( ) ( ) } { ( ) ( )
{ ( )} { ( )} ( ) { ( )} ( ) { ( )} ( ) ( )
( ) ( ) ( ) ( )
0
E x n n x n n
E x n n E x n n
E x n E x n n E x n n E x n n n
n n n n
x(n) เปน IID
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การสรางแบบจําลองของสัญญาณสุมModelling of Random Signals
• Transfer function
10 1
11
( )( )
( )1
NN
MN
b b z b z B zH z
A za z a z
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ระบบแบบเฟสนอยสุดMinimum Phase Systems
• ตําแหนงของซีโรที่อยูภายในวงกลมหนึ่งหนวยจะเปนตัวกําหนดความเปนระบบเฟสนอยสุด
• ซีโรของระบบจะตองอยูภายในวงกลมหนึ่งหนวย
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• ตัวอยาง
• ซีโรเพียงตัวเดียวซึ่งอยู ณ ตําแหนง z=-1
1( ) 1H z z
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ผลตอบสนองความถี่
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• ตัวอยาง 1
1( ) 1 0.5H z z
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Part
Imag
inar
y P
art
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• ตัวอยาง
-2 -1.5 -1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Real Part
Imag
inar
y P
art
12( ) 1 2H z z
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30
-20
-10
0
Normalized Frequency ( rad/sample)
Pha
se (
degr
ees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Normalized Frequency ( rad/sample)
Mag
nitu
de (
dB)
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200
-150
-100
-50
0
Normalized Frequency ( rad/sample)
Pha
se (
degr
ees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
Normalized Frequency ( rad/sample)
Mag
nitu
de (
dB)
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ซีโรอยูในวงกลมหนึ่งหนวย
• ตําแหนงซีโรที่อยูในวงกลมหนึ่งหนวย ถูกแปลงเปนโพลในวงกลมหนึ่งหนวยและมีสเถียรภาพ
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ซีโรอยูนอกวงกลมหนึ่งหนวย
• ตําแหนงซีโรที่อยูนอกวงกลมหนึ่งหนวยนี้จะกอใหเกิดความไมมีเสถียรภาพ (Instability)
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ตัวอยางการใชงานสเปคตรัมกําลัง
• ตองการสงสัญญาณเสียงพูดผานสายโทรศัพทที่ม ีแบนดวิทต่ํา ๆ ( 4 kHz)
• เพื่อลดปริมาณการสงขอมูล เราใชการเขารหัสที่เรียกวา Linear Prediction Coding (LPC)
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สเปคตรัมกําลังของเสียงพูด
• เราใชการสงขอมูลสําคัญคือ ความถี่ฟอรแมนท (Formant frequencies)
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การเขารหัสทํานายคาเชิงเสน (Linear Predictive Coding LPC)
• ประมาณคา a ของระบบ MA แลวสงเฉพาะ a
2 3 1(̂ ) ( 1) ( 2) .... ( )px n a x n a x n a x n p
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การสรางคืนสัญญาณเสียงพูด
• สรางสัญญาณพูดจากระบบ AR โดยใชเฉพาะ a
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ตัวอยาง
• ตัวอยาง
1( ) 1 0.8MAH z z
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สหสัมพันธตัวเองของ x(n)
• ให x(n) เปนสัญญาณรบกวนสีขาว ( IID )
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2(0) { ( )}yyr E y n
1
0 10
( ) ( ) ( ) ( 1)kk
y n h x n k h x n h x n
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• เพราะ x(n) เปนสัญญาณรบกวนสีขาว
212 2 2 20 0 1 1
0
2 2 2 20 0 1 1
0
2 2 2 20 1
( ) ( ) 2 ( ) ( 1) ( 1)
( ) 2 { ( ) ( 1)} ( 1)
kk
x x
E h x n k E h x n h x n h x n h x n
h E x n h h E x n x n h E x n
h h
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• ที ่n =0
12 2 2 2
0
(0) 1 0.8yy k xk
r h
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• ที ่n =-1, +1
0 1(1) ( 1) 1 0.8yy yyr r h h
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