46
H 0 (z) x(n) o (n ) M G 0 (z) M + v o (n ) y o (n ) H 1 (z) 1 (n ) M G 1 (z) M v 1 (n ) y 1 (n ) f o (n ) f 1 (n ) y(n ) Figure 31 H M-2 (z) M- 2 (n) M G M-2 (z) M v M- 2 (n) H M - 1 (z) M- 1 (n) M G M-1 (z) M v M- 1 (n) f M- 2 (n) f M- 1 (n) y M- 1 (n)

H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

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Citation preview

Page 1: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

H0(z)

x(n)

o(n)

M G0(z)M

+

vo(n) yo(n)

H1(z)1(n)

M G1(z)M

v1(n) y1(n)

fo(n)

f1(n)

y(n)

Figure 31

HM-2(z)

M-2(n)

M GM-2(z)M

vM-2(n)

HM-1(z)

M-1(n)

M GM-1(z)M

vM-1(n)

fM-2(n)

fM-1(n)yM-1(n)

Page 2: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Without the decimator and interpolator, the input/output relationship is straightforward

Y z G z H z X zk kk

M

0

1

(32)

With the decimator and interpolator, we have to take it step-by-step

X z z V z Y zk k

Page 3: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

Page 4: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)

Decimated signal

Images

W er j r M 2 /

Page 5: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Page 6: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Y z G z F zk kk

M

0

1

(36)

Page 7: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Y z G z F zk kk

M

0

1

(36)

Y z G z V zk kM

k

M

0

1

Page 8: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Y z G z F zk kk

M

0

1

(36)

Y z G zM

zWk kr

r

M

k

M

1

0

1

0

1

(37)

Page 9: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Y z G z F zk kk

M

0

1

(36)

(37) Y z G zM

H zW X zWk kr

r

M

k

Mr

1

0

1

0

1

Page 10: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

k kz H z X z k M 0 1 2 1, , ,....., (33)

V zM

z Wk kr

MM r

1

0

1 1

(34)W er j r M 2 /

F z V zk kM (35)

Y z G z F zk kk

M

0

1

(36)

(37) Y zM

X zW H zW G zrk

rk

r

M

k

M

1

0

1

0

1

Page 11: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Y z

MX z X zW

H z H z H z

H zW H zW H zW

H zW H zW H zW

G z

G z

G z

M

o M

o M

oM M

MM

M

1 1

1 1

1 1

11

11

1

0

1

1

,....

.....

....

....

.

zzzzzzzM

zY TTT xAAxgHx 1

(38)

(39)

Alias Component (AC) matrix

Note Y(z) is a 1x1 matrix!

Page 12: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

zG

zG

zG

zWHzWHzWH

zWHzWHzWH

zHzHzH

zWXzXM

zzzY

MM

MMM

o

Mo

Mo

M

T

1

1

0

11

11

1

11

11

11.

....

....

.....

,....

Ax

Aliasing error

0

0

zT

zz gH Aliasing error free condition

(40)

Page 13: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Y z

MX z X zW

H z H z H z

H zW H zW H zW

H zW H zW H zW

G z

G z

G z

M

o M

o M

oM M

MM

M

1 1

1 1

1 1

11

11

1

0

1

1

,....

.....

....

....

.

Aliasing error

Aliasing error free condition

(41)

0

0

onz

zzz tgH

Page 14: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Condition for perfect reconstruction is simple in theory, as

zzzzzz tHgtgH 1

However,

zz

zAdjzzz t

Hdet

HtHg 1

Is complicated to solve and resulted in IIR synthesis filters.

Page 15: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

An effective solution employing Polyphase decomposition

Mkl

M

l

lk zEzzH

1

0

Recalling,

1

1

111101

111110

100100

1

1

0 1

MMMM

MM

MM

MM

MM

MM

MM

M z

z

zEzEzE

zEzEzE

zEzEzE

zH

zH

zH

.

....

....

.....

,,,

,

,

Or simply

i.e.,

zzz M eEh

Page 16: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Similar treatment to the synthesis filter gives

Mlk

M

l

lMk zRzzG

1

0

1

.

....

....

.....

,,,

,

,

MMM

MM

MM

MM

MM

MM

MMT

M

MT

M zRzRzR

zRzRzR

zRzRzR

z

z

zG

zG

zG

111101

111110

100100

2

1

1

1

0

1

Or simply

i.e.,

MTMT zzzz Reg 1

Page 17: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Diagram illustration:

zH0

zHM 1

zH1

MzE

z-1

z-1

z-1

Page 18: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Diagram illustration:

zG0

zGM 1

zG1

MzR

z-1

z-1

z-1

Can be replaced with their polyphase components, as

Page 19: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

zH0

zHM 1

zH1

zG0

zGM 1

zG1

M

M

M

M

M

M

Can be replaced with their polyphase components, as

nx ny

Page 20: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

M

M

M

M

M

M

MzR

z-1

z-1

z-1

MzE

z-1

z-1

z-1

nx

ny

Page 21: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

M

M

M

M

M

M

zR

z-1

z-1

z-1

zE

z-1

z-1

z-1

nx

ny

Page 22: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

M

M

M

M

M

M

z-1

z-1

z-1

zP

z-1

z-1

z-1

nx

ny

IczzEzRzP m Results in perfect reconstruction

Page 23: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

1

1

0

0

z

Iz M

Implementing FIR filters for M-channel filter banks, to begin with,

10

011

MIzz

MM Iz

z

Izzz 1

11

1

0

0

Noted that

Page 24: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

01 RzzRzRzE JJ .....

Let

111

10

J

RzRzRzR .....

zzRzzRzzRRzzRzR

zEzRzP

JJJ

0111

11

0 ..........

Compute P(z)

zzRzzRzzzRzR J

011

11

0 ..........

I

zzRzzRRzRzR JJ

.......

.......... 0111

11

10

Page 25: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

121

012

001

0R

It can be seen that both analysis and synthesis filters are FIR

100

210

121

01TRzR

123

012

0011

0R

111

111

111

01

2

24122

225

zzz

zzz

zzz

RzRzE

100

210

3211

1R

Page 26: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

All the analysis filters can be generated from

111

111

111

01

2

24122

225

zzz

zzz

zzz

RzRzE

54310 225 zzzzzH

54311 2422 zzzzzH

5432 2 zzzzH

Page 27: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

All the analysis filters can be generated from

111

111

111

11

13183

852

32

10

zzz

zzz

zzz

RzRzR

5430 23 zzzzF

5431 258 zzzzF

5432 38131 zzzzF

Page 28: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure
Page 29: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure
Page 30: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure
Page 31: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Spectral AnalysisSpectral Analysis

sk

N

sf k k s N

0

1

0 1*

Given f(n) = [f0, f1, ..... , fN-1] and an orthonormal basis

0 1 1n n nN, ,....,

rk

N

sk k r s otherwise

0

1

1* = for = and 0 i.e.,

The spectral (generalized Fourier) coefficients of f(n) are defined as

f k k k Nss

N

s

0

1

0 1

(66)

(67)

Eqn. 66 and 67 define the orthonormal transform and its inverse

Page 32: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

Spectral AnalysisSpectral Analysis

If the members of are sinusoidal sequences, the transform is known as the Fourier Transform

The Parseval theorem - Conservation of Energy in orthonormal transform

f kk

N

nn

N

0

1 2

0

1 2

(68)

Page 33: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

An Application - Spectral AnalysisAn Application - Spectral Analysis

0 1* N n N

N

N

1 1* N n

N N n 1 1*

f(n)

0

1

N-1

Orthonormal spectral analyser implemented with multirate filter bank

Figure 32

Page 34: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

An Application - Spectral AnalysisAn Application - Spectral Analysis

Transform efficiency - measured by decorrelation and energy compactness

Correlation - Neighboring samples can be predicted from the current sample : an inefficient representation.

Energy Compactness - The importance of each sample in forming the entire signal. If every sample is equally important, everyone of them has to be included in the representation: again an inefficient representation.

An ideal transform: 1. Samples are totally unrelated to each other.2. Only a few samples are necessary to represent the

entire signal.

Page 35: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

E f n n

Given a signal f(n), define the mean and autocorrelation as

and E f n f n k R n k n ,

Assume f(n) is wide-sense stationary, i.e. its statistical properties are constant with changes in time

cons t and R n k n R ktan ,

Define and

(69)

(70)

1

1

1

1

1

0

Nf

f

f

f

Page 36: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

R

R R R N

R R R N

R N R N R

N

N

N N

0 1 1

1 0 2

1 2 0

1 1 1

1 1 2

1 2 1

....

....

.....

....

....

.....

= 2

where R k k and 2 0 1 (71)

Equation 69 can be rewritten as

C conv f E f fT

The covariance of f is given by

R E f f T (72)

(73) R T

Page 37: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

The signal is transform to its spectral coefficients with eqn 66

sk

N

sf k k s N

0

1

0 1*

Comparing the two sequences:

f n f f f andN N 0 1 1, ,..., , ,...., 0 1 -1

Page 38: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

The signal is transform to its spectral coefficients with eqn 66

sk

N

sf k k s N

0

1

0 1*

Comparing the two sequences:

f n f f f andN N 0 1 1, ,..., , ,...., 0 1 -1

a. Adjacent terms are relatedb. Every term is important

a. Adjacent terms are unrelatedb. Only the first few terms are

important

Page 39: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

The signal is transform to its spectral coefficients with eqn 66

sk

N

sf k k s N

0

1

0 1*

similar to f, we can define the mean, autocorrelation and covariance matrix for

R E T

Page 40: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

f n f f f andN N 0 1 1, ,..., , ,...., 0 1 -1

a. Adjacent terms are related a. Adjacent terms are unrelated

Adjacent terms are uncorrelated if every term is only correlated to itself, i.e., all off-diagonal terms in the autocorrelation function is zero.

Define a measurement on correlation between samples:

f fjj i

N

i

N

jj i

N

i

N

R i j and R i j

, , 1

1

1

1

1

1

1

1

(74)

Page 41: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

We assume that the mean of the signal is zero. This can be achieved simply by subtracting the mean from f if it is non-zero.

The covariance and autocorrelation matrices are the same after the mean is removed.

Page 42: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

f n f f f andN N 0 1 1, ,..., , ,...., 0 1 -1

b. Every term is important b. Only the first few terms are important

0

1

1

0

1

1N

r

r

r

r N

Note:

If only the first L-1 terms are used to reconstruct the signal, we have

f L r rr

L

0

1

(75)

Page 43: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

If only the first L-1 terms are used to reconstruct the signal, the error is

The energy lost is given by e eLT

L rr L

N

21

e f fL L r rr L

N

1

r rk

NT

rf k k f

*

0

1

but,

hence r rT T

rf f2

(76)

(77)

(78)

Page 44: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

Eqn. 78 is valid for describing the approximation error of a single sequence of signal data f. A more generic description for covering a collection of signal sequences is given by:

J E e e E

E f f R

L LT

L rr L

N

rT T

rr L

N

rT

f rr L

N

'

21

1 1

(79)

An optimal transform mininize the error term in eqn. 79. However, the solution space is enormous and constraint is required. Noted that the basis functions are orthonormal, hence the following objective function is adopted.

Page 45: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

J RrT

f r r rT

rr L

N

11

(80)

The term r is known as the Lagrangian multiplier

The optimal solution can be found by setting the gradient of J to 0 for each value of r, i.e.,

rr

JJ

0

Eqn 81 is based on the orthonormal property of the basis functions.

(81)

Page 46: H 0 (z) x(n)x(n) o(n)o(n) M G 0 (z) M + vo(n)vo(n) yo(n)yo(n) H 1 (z) 1(n)1(n) M G 1 (z) M v1(n)v1(n) y1(n)y1(n) fo(n)fo(n) f1(n)f1(n) y(n)y(n) Figure

How to derive the optimal transform?How to derive the optimal transform?

R f r r r

The solution for each basis function is given by

(82)

ris an eigenvector of Rf and r is an eigenvalue

Grouping the N basis functions gives an overall equation

R fT T

N where 0 1,......., (83)

R = Rf= which is a diagonal matrix.The decorrelation criteria is satisfied

(84)