Download - 2 3 4 5 Aliasing Cancellation Condition Aliasing Cancellation Condition : Perfect Reconstruction Condition Perfect Reconstruction Condition :

Non-Maximal Decimated Filter Bank (NMDFB) and

Its Application in Wideband Signal Processing

Dec 6, 2012

Xiaofei Chen

http://www.sdsu.edu/

2

Non-Maximal Decimated Filter Bank

3

Non-Maximal Decimated Filter Bank, Cont’d

4


𝑌 (𝑍 )= 1𝐷𝑮1×𝑀

𝑇 ( 𝒁 )𝕂𝑀 ×𝑀 (𝒁 )ℍ𝑀 ×𝐷 ( 𝒁 ) 𝑿 𝐷×1 ( 𝒁 )= 1𝐷𝑻𝕂

1×𝐷 ( 𝒁 ) 𝑿𝐷× 1 ( 𝒁 )

𝑻𝕂1×𝐷 (𝒁 )≝𝑮1×𝑀

𝑇 (𝒁 )𝕂𝑀×𝑀 ( 𝒁 )ℍ𝑀 ×𝐷 (𝒁 )= [𝑇 𝑠𝕂 (𝑍 )𝑻 𝐴

𝕂 (𝑍 ) ]𝑌 (𝑍 )= 1

𝐷𝑇 𝑠

𝕂 (𝑍 )𝑋 (𝑍 )+ 1𝐷𝑻 𝐴

𝕂 (𝑍 ) 𝑿 (𝑍 )

5


𝐻 (𝑍𝑊 𝐷𝑑 )𝐺 (𝑍 )=0 , ∀𝑑=1 ,…,𝐷−1

∑𝑚=0

𝑀−1

𝐻 (𝑍𝑊𝑀𝑚 )𝐺 (𝑍𝑊𝑀

𝑚 )=𝑍−𝑛𝐷

Aliasing Cancellation Condition:

Perfect Reconstruction Condition:

6

Polyphase Implementation of NMDFB (AFB)

M-path Partition of the mth channel filter:

7

Polyphase Implementation of NMDFB (SFB)

M-path Partition of the mth channel synthesis filter:

8

Filtering with NMDFB

NMDFB filter:

𝑌 𝑓 (𝑍 )=𝑇 𝑠𝕂 (𝑍 ) 𝑋 (𝑍 )+𝑻 𝐴

𝕂 (𝑍 ) 𝑿 (𝑍 )≅𝑇 𝑠𝕂 (𝑍 ) 𝑋 (𝑍 )

Time Domain filter:

𝑌 𝑡 (𝑍 )=𝑆 (𝑍 )𝑍−𝑛𝐷 𝑋 (𝑍 )

9

Filtering with NMDFB, Cont’d

Error between two filtering models:

Error Transfer Function:

ℰ (𝑍 )=𝑌 𝑡 (𝑍 )−𝑌 𝑓 (𝑍 )= [𝑇 𝑠𝕂 (𝑍 )𝑍𝑛𝑑−𝑆 (𝑍 ) ] 𝑋 (𝑍 )𝑍−𝑛𝑑

10


Piecewise Constant Approximation:

Linear Interpolation

~h𝑁𝑌𝑄 (𝑛)= 1𝑀𝑆𝐼𝑁𝐶2( 1𝑀 𝑛) , 𝑓𝑜𝑟 −𝑛𝑑≤𝑛≤𝑛𝑑

~𝐻𝑁𝑌𝑄 (𝜔−𝜔𝑚)={1−|𝜔−𝜔𝑚

2𝜋 /𝑀 |,𝜔∈[𝜔𝑚−2𝜋𝑀,𝜔𝑚+ 2𝜋

𝑀]

0 , h𝑂𝑡 𝑒𝑟𝑤𝑖𝑠𝑒

11


Piecewise Constant Approximation Performance:

|𝑇 ℰ ,𝑚𝑅𝑒 (𝜔 )|≤ |�̇�𝑅𝑒 (𝜔 )|

𝜔∈ [𝜔𝑚−𝜋𝑀,𝜔𝑚±

𝜋𝑀 ]Max

∙𝜋𝑀

=𝐵ℰ ,𝑚𝑅𝑒

|𝑇 ℰ ,𝑚𝐼𝑚 (𝜔 )|≤ |�̇�𝐼𝑚 (𝜔 )|

𝜔∈ [𝜔𝑚−𝜋𝑀,𝜔𝑚±

𝜋𝑀 ]Max

∙𝜋𝑀

=𝐵ℰ ,𝑚𝐼𝑚

|𝑇 ℰ ,𝑚 (𝜔 )|≤√ (𝐵ℰ ,𝑚𝑅𝑒 )2+(𝐵ℰ ,𝑚

𝐼𝑚 )2≝𝐵ℰ ,𝑚

𝜙𝑚≤𝑎𝑡𝑎𝑛( 𝐵ℰ ,𝑚

√ (𝛾𝑠 ,𝑚 )2− (𝐵ℰ ,𝑚 )2 ) , 𝑓𝑜𝑟 𝛾𝑠 ,𝑚>𝐵ℰ ,𝑚

𝛾𝑠 ,𝑚≝ |𝑆 (𝜔 )|𝜔∈[𝜔𝑚−

𝜋𝑀,𝜔𝑚 ±

𝜋𝑀 ]𝑀𝑖𝑛

12


Linear Interpolation Approximation Performance:

|𝑇 ℰ ,𝑚𝑅𝑒 (𝜔 )|≤ |�̈�𝑅𝑒 (𝜔 )|𝜔∈ [𝜔𝑚 ,𝜔𝑚+1]

Max∙12 ( 𝜋𝑀 )

2

=𝐵ℰ ,𝑚𝑅𝑒

|𝑇 ℰ ,𝑚𝐼𝑚 (𝜔 )|≤ |�̈�𝐼𝑚 (𝜔 )|𝜔∈ [𝜔𝑚 ,𝜔𝑚+1]

Max∙12 ( 𝜋𝑀 )

2

=𝐵ℰ ,𝑚𝐼𝑚

|𝑇 ℰ ,𝑚 (𝜔 )|≤√ (𝐵ℰ ,𝑚𝑅𝑒 )2+(𝐵ℰ ,𝑚

𝐼𝑚 )2≝𝐵ℰ ,𝑚

𝜙𝑚≤𝑎𝑡𝑎𝑛( 𝐵ℰ ,𝑚

√ (𝛾𝑠 ,𝑚 )2− (𝐵ℰ ,𝑚 )2 ) , 𝑓𝑜𝑟 𝛾𝑠 ,𝑚>𝐵ℰ ,𝑚

𝛾𝑠 ,𝑚≝ |𝑆 (𝜔 )|𝜔∈ [𝜔𝑚,𝜔𝑚+1 ]𝑀𝑖𝑛

13

NMDFB Design Example

M = 64, D = 32Rectangular

02

M

2

M

4

M

4

M

S ynthe s is F ilte rG (Z)

M o d u la ted Im a g eo f H (Z)

02

M

2

M

4

M

4

M

S ynthe s is F ilte rG (Z )


0 4

M

4

M

S ynthe s is F ilte rG (Z )


8

M

8

M

M = 64, D = 32Triangular

M = 64, D = 16Triangular

14

NMDFB Simulation

M = 64, D = 32 RectangularImpulse Response and Filter Spectra

690 700 710 720 730 740 750 760 770 7800

0.5

1

1.5

Analysis / Synthesis Impulse Response

Samples / n

Am

plit

ud

e

200 400 600 800 1000 1200-2

-1

0

1

2x 10

-5

X: 1121Y: -1.343e-005

Details of the Impulse Response Artifacts

Samples / n

Am

plit

ud

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-100

-80

-60

-40

-20

0

Frequency (fs = 64)

Log

Mag

(dB

)

Analysis Filter Spectra

H

0()

H1()

-4 -3 -2 -1 0 1 2 3 4

-100

-50

0

Frequency (fs = 64)

Log

Mag

(dB

)

Analysis Filter and Synthesis Filter Spectra

H

0()

G0()

H2()

H62

()

15

NMDFB Simulation, Cont’d

M = 64, D = 16 TriangularImpulse Response and Filter Spectra

450 460 470 480 490 500 510 520 530 5400

0.5

1

1.5


Samples / n

Am

plit

ud

e

0 200 400 600 800 1000-2

-1

0

1

2x 10

-5

X: 626Y: -4.975e-006


Samples / n

Am

plit

ud

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-100

-50

0

Frequency (fs = 64)

Log

Mag

(dB)


H

0()

H1()

-8 -6 -4 -2 0 2 4 6 8

-100

-50

0

Frequency (fs = 64)

Log

Mag

(dB)


H

0()

G0()

H4()

H60

()

16

NMDFB Simulation, Cont’d

M = 64, D = 16 TriangularImpulse Response and Filter Spectra

450 460 470 480 490 500 510 520 530 5400

0.5

1

1.5


Samples / n

Am

plit

ud

e

0 200 400 600 800 1000-2

-1

0

1

2x 10

-5

X: 626Y: -4.975e-006


Samples / n

Am

plit

ud

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-100

-50

0

Frequency (fs = 64)

Log

Mag

(dB)


H

0()

H1()

-8 -6 -4 -2 0 2 4 6 8

-100

-50

0

Frequency (fs = 64)

Log

Mag

(dB)


H

0()

G0()

H4()

H60

()

17

NMDFB Filtering Simulation 1

M = 64, D = 32 Rectangular / M = 256 Triangular Linear Phase Filtering

0 0.1 0.2 0.3 0.4 0.5

-100

-50

0

Normalized Frequency

Log

Mag

(dB

)

Magnitude Responses of Original Filter and Synthesized Filter

OriSyn

0 0.1 0.2 0.3 0.4 0.5

-100

-50

0


Log

Mag

(dB

)

Magnitude Error between Original Filter and Synthesized Filter

Mag ErrErr Bound

0 0.1 0.2 0.3 0.4 0.5-1

0

1x 10

-4 Phase Error between Original Filter and Synthesized Filter


Nor

mal

ized

Ang

le

Phase ErrErr Bound

0 0.1 0.2 0.3 0.4 0.5

-100

-50

0


Log

Mag

(dB

)


OriSyn

0 0.1 0.2 0.3 0.4 0.5

-100

-50


Log

Mag

(dB

)


Mag ErrMinimax BoundSub-Opt Bound

0 0.1 0.2 0.3 0.4 0.5

-101

x 10-5 Phase Error between Original Filter and Synthesized Filter


Nor

mal

ized

Ang

le

Phase ErrorMinimax BoundSub-Opt Bound

18


M = 64, D = 32 Rectangular / TriangularNon Linear Phase Filtering

-0.5 0 0.5

-20

0

20


Log

Ma

g (

dB

) Magnitude Responses of Original Filter and Synthesized Filter

OriSyn

-0.5 0 0.5

-20

0

20


Log

Ma

g (

dB

) Magnitude Error between Original Filter and Synthesized Filter

Mag ErrErr Bound

-0.5 0 0.5-0.5

0

0.5Phase Error between Original Filter and Synthesized Filter


Norm

alize

d A

ng

le

Phase ErrErr Bound

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-20

0

20


Log

Mag

(dB)


OriSyn

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-60

-40

-20


Log

Mag

(dB)



-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.01

0

0.01

Phase Error between Original Filter and Synthesized Filter

Normalized FrequencyNo

rmali

zed

Angl

e


19


M = 256, D = 64 TriangularNon Linear Phase Filtering

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-20

0

20


Log

Mag

(dB

)


OriSyn

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-70

-60

-50

-40


Log

Mag

(dB

)



-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-1

0

1

x 10-3 Phase Error between Original Filter and Synthesized Filter


Nor

mal

ized

Ang

le


20

NMDFB Filtering: Fractional Delay

M = 64, D = 16 TriangularFractional Delay Filtering

-0.5 0 0.5-5

0

5x 10

-3


dB

Mag Res, Dly = 0 SMP

-0.5 0 0.5-5

0

5x 10

-3


dB

Mag Res, Dly = 0.5 SMP

-0.5 0 0.5-5

0

5x 10

-3


dB

Mag Res, Dly = -0.5 SMP

-10 -5 0 5 10

0

0.5

1Impz Dly = 0

SMPs / n

Am

p

ReIm

-10 -5 0 5 10

0

0.5

1Impz Dly = 0.5

SMPs / n

Am

p

ReIm

-10 -5 0 5 10

0

0.5

1Impz Dly = -0.5

SMPs / n

Am

p

ReIm

21

NMDFB Filtering Workload

=

: Analysis Filter Bank Length: Synthesis Filter Bank LengthM : Number of PathsN: Number of Intermediate Processing Elements

22

NMDFB APPLICATIONS

1. Wideband Signal Processing: Effectively reducing the hardware processing rate via NMDFB.

2. Filtering: Linear phase, Non-linear phase, Fractional delay, Masking, Cascade Filtering.

3. Support block timing varying filtering. 4. Support wideband power allocation.

23

Communication Example

Time Domain Timing Recovery & Matched Filtering

NMDFB Domain Timing Recovery & Matched Filtering

256 Real Multiplies per Output

240 Real Multiplies per Output

24

Communication Example

NMDFB Timing Recovery Simulation (Submitted to ICASSP 2013)20 dB SNR / AWGN channel / 0.25 Ts Timing Error

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-40

-20

0

Normalized Freq

Mag /

dB

Spectrums: Time Domain MF, NMDFB MF

TD MF

NMDFB MF

-0.5 0 0.5-120

-100

-80

-60

-40

-20

X: 0.3125Y: -32.22

Normalized Freq

Mag /

dB

Mag Error

simulated err

err bound

-0.5 0 0.5-0.02

-0.01

0

0.01

0.02Phase Error

Normalized Freq

Radiu

s

simulated err

err bound

-2 -1 0 1 2-2

-1

0

1

2

Received Constellation = 0.25Ts

-2 -1 0 1 2-2

-1

0

1

2Timing Recovered Constellation

0 0.5 1 1.5 2

x 104

0

0.1

0.2

0.3

0.4PHASE ACCUMULATOR TIME PROFILE

Sample Index (n)

Tim

ing O

ffset

Phase Acc

Defined Timing Offset

0 0.5 1 1.5 2

x 104

-2

0

2

4

6x 10

-3 Timing Error

Sample Index (n)A

mplit

ude

10x Loop Filter Input

Loop Filter Output

25

Thanks! Open for questions Now!