a sub-band filter design approach for sound field reproduction

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12-11-2020

A Sub-band Filter Design Approach for Sound Field Reproduction A Sub-band Filter Design Approach for Sound Field Reproduction

Yongjie Zhuang Purdue University, [email protected]

Guochenhao Song Purdue University, [email protected]

Yangfan Liu Purdue University, [email protected]

Follow this and additional works at: https://docs.lib.purdue.edu/herrick

Zhuang, Yongjie; Song, Guochenhao; and Liu, Yangfan, "A Sub-band Filter Design Approach for Sound Field Reproduction" (2020). Publications of the Ray W. Herrick Laboratories. Paper 225. https://docs.lib.purdue.edu/herrick/225

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A Sub-band Filter Design Approach for Sound Field Reproduction

Yongjie Zhuang Guochenhao Song

Yangfan Liu

Ray W. Herrick Laboratories, school of Mechanical Engineering,

Purdue University [email protected]

mailto:[email protected]

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Content • Introduction

• Methodology

• Results

• Conclusions

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Statement of research problem

Sound field reproduction uses loudspeakers to produce desired sound at locations.

When designing filter for sound that spans a wide frequency range:

Low frequency band longer time span Large number of filter coefficients High frequency band higher sampling frequency

Impulse response

𝑛𝑛

long time span

high sampling frequency



Statement of research problem

An approach is proposed to design filter in a sub-band form:

Design all sub-band filters directly in one optimization problem:

The transition region between two sub-band filters can be designed conveniently

The computational load can be reduced even if sub-band filters structure is not required




• Methodology

• Results

• Conclusions

,Ir



Designing filter directly

𝒌𝒌𝒌𝒌 desired

added delay signal d(n)

Σy(n) +

e(n) -r(n) 𝑾𝑾𝒙𝒙 x(n)

𝑯𝑯 output error filtered original

filter signal system signal signal signal response

Example: Cost function: Constraints: Use loudspeaker to produce Minimizing the power of Filter response 𝑾𝑾𝑥𝑥(f) desired sound at certain locations error signal 𝒆𝒆

-.

;-- - - - - - - - - - - -I

I I

I I I

I I

I ,Ir

I ,. I , . - - -

I - - - - . .

I ' ~ I

I I - I I

I I I I

I , ___________ _



Designing filter when sub-band technique is used

𝒌𝒌𝒌𝒌 desired

added delay signal d(n)

x(n) Σ

𝑾𝑾1

sub-band filter 1

𝑾𝑾𝑁𝑁

sub-band filter N

+... y(n)

+ e(n) -r(n) 𝑯𝑯

output error filtered original signal system signal signal

signal response

𝑾𝑾𝒙𝒙

Example: Cost function: Constraints: Use loudspeaker to produce Minimizing the power of Filter response 𝑾𝑾𝑖𝑖 (f) desired sound at certain locations error signal 𝒆𝒆

( ) ( )~ ( ) [ -]

[



Expressing sub-band filters as one equivalent filter

Conventional method (one single filter)

frequency response of designed filter at frequency 𝑓𝑓𝑘𝑘 :

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘(𝑁𝑁𝑡𝑡−1)= 𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠, 𝑁𝑁𝑡𝑡 𝑤𝑤𝑥𝑥 , 𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠, 𝑁𝑁𝑡𝑡 =𝑊𝑊𝑥𝑥 𝑓𝑓𝑘𝑘 1 𝑓𝑓𝑠𝑠 ⋯ 𝑓𝑓𝑠𝑠

𝑓𝑓𝑠𝑠 is the sampling frequency, 𝑁𝑁𝑡𝑡 is the number of filter coefficients, 𝑤𝑤𝑥𝑥 is the filter coefficients

Sub-band structure

frequency response of designed filter at frequency 𝑓𝑓𝑘𝑘:

𝑁𝑁 𝑁𝑁

−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘 −𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘(𝑁𝑁𝑡𝑡−1) 𝑊𝑊𝑖𝑖 (𝑓𝑓𝑘𝑘) = 𝑓𝑓𝑠𝑠𝑖𝑖 𝑓𝑓𝑠𝑠𝑖𝑖 𝑤𝑤𝑖𝑖1 𝑒𝑒 ⋯ 𝑒𝑒

𝑖𝑖=1 𝑖𝑖=1

( )

( ) [ ( ) --+

( ) [ ( ) ( )]



Expressing sub-band filters as one equivalent filter

So designing sub-band filters can be treated as: designing one filter 𝑤𝑤𝑥𝑥 with modified Fourier matrix 𝐹𝐹 𝑓𝑓𝑘𝑘

𝑁𝑁 𝑁𝑁

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘(𝑁𝑁𝑡𝑡−1)

𝑊𝑊𝑥𝑥 𝑓𝑓𝑘𝑘 = 𝑊𝑊𝑖𝑖 (𝑓𝑓𝑘𝑘) = 𝑓𝑓𝑠𝑠𝑖𝑖 𝑓𝑓𝑠𝑠𝑖𝑖 𝑤𝑤𝑖𝑖 = 𝐹𝐹 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 ,1 ⋯𝑖𝑖=1 𝑖𝑖=1

𝐹𝐹 𝑓𝑓𝑘𝑘 = 𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠1 , 𝑁𝑁𝑡𝑡1

⋯ 𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠𝑁𝑁 , 𝑁𝑁𝑡𝑡𝑁𝑁

,

𝑤𝑤1𝑤𝑤𝑥𝑥 = ⋮

𝑤𝑤𝑁𝑁

So all the sub-band filters can be designed in one optimization problem if designed in the frequency domain. The transition region can be designed more conveniently.

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Overview of proposed design process

Conventional method General Nonlinear algorithm

(e.g., sequential quadratic programming)

Design Problem

Convex Optimization

Proposed method Primal-dual Cone

Programming Interior-point Global minimum Possible to apply • Efficient Cost function: usually guaranteed efficient algorithms • Upper-bound for

iteration times Power of 𝒆𝒆

Constraints: Filter response

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Problem formulation

Design Problem Expressed in Convex Problem Cost function: Total power of e:

𝑘𝑘2 𝑘𝑘2 𝑘𝑘2 𝑘𝑘2𝑗 𝑤𝑤𝑥𝑥

T 𝐴𝐴𝐽𝐽 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 2Re 𝑏𝑏𝐽𝐽T 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 𝑐𝑐𝐽𝐽 𝑓𝑓𝑘𝑘,𝐸𝐸 𝑓𝑓𝑘𝑘 𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1

𝑘𝑘=𝑘𝑘1 Convex • Quadratic

Constraints: • 𝐴𝐴𝐽𝐽 𝑓𝑓𝑘𝑘 p.s.d

Filter response: The magnitude of frequency response:

Vector norm Convex ‖𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠1 , 𝑁𝑁𝑡𝑡1

𝑤𝑤𝑖𝑖‖𝑗 − 𝐶𝐶𝑖𝑖 (𝑓𝑓𝑘𝑘) ≤ 0≤ 𝐶𝐶𝑖𝑖(𝑓𝑓𝑘𝑘)𝑊𝑊𝑖𝑖 𝑓𝑓𝑘𝑘

( ( )

? • ( ) -+ ....



Cone Programming Reformulation

Convex Problem Standard Cone Programming Cost function:

Cost function: 𝑐𝑐T𝑥𝑥 𝑘𝑘2 𝑘𝑘2

𝑤𝑤𝑥𝑥T

𝐴𝐴𝐽𝐽 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 2Re 𝑏𝑏𝐽𝐽T 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 𝑐𝑐𝐽𝐽 𝑓𝑓𝑘𝑘

𝑘𝑘2

𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1 Constraints: 𝑥𝑥 ∈ 𝐾𝐾𝑖𝑖 , 𝑖𝑖 = 1, 2, 3 …

Constraints: 𝐴𝐴𝑥𝑥 = 𝑏𝑏

𝑐𝑐 to be a constant vector ‖𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠1 , 𝑁𝑁𝑡𝑡1

𝑤𝑤𝑖𝑖‖𝑗 − 𝐶𝐶𝑖𝑖(𝑓𝑓𝑘𝑘) ≤ 0 𝐾𝐾𝑖𝑖 to be a convex cone

𝐴𝐴, 𝑏𝑏 to be a constant matrix and vector

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Convex Problem Cone Programming • Reformulate quadratic cost function

Cost function: 𝑥𝑥T 𝐴𝐴 𝑥𝑥 + 𝑏𝑏T𝑥𝑥 + 𝑐𝑐

Cost function:

Constraints: �̃�𝑡0 = 1

Linear cost function

Rotated second-order cone

Linear constraint

• The vector norm constraint

Constraints: ‖𝑥𝑥‖𝑗 − 𝑐𝑐 ≤ 0

‖ 𝐴𝐴 𝑥𝑥‖𝑗 ≤ 𝑡𝑡0 �̃�𝑡0

𝑡𝑡0 + 𝑏𝑏T𝑥𝑥

Constraints: ‖𝑥𝑥‖𝑗 ≤ 𝑡𝑡 Second-order cone

𝑡𝑡 = 𝑐𝑐 Linear constraint

( ( ) ( ) --+

( ) -+ (




Convex Problem Cone Programming

Cost function: Cost function: 𝑘𝑘2𝑘𝑘2 𝑘𝑘2 𝑘𝑘2T

𝑤𝑤𝑥𝑥 𝐴𝐴𝐽𝐽 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 2Re 𝑏𝑏𝐽𝐽T 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥 + 𝑐𝑐𝐽𝐽 𝑓𝑓𝑘𝑘 𝑡𝑡0 + 2Re 𝑏𝑏𝐽𝐽T 𝑓𝑓𝑘𝑘 𝑤𝑤𝑥𝑥𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1 𝑘𝑘=𝑘𝑘1

Constraints: Constraints:

‖𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠1 , 𝑁𝑁𝑡𝑡1

𝑤𝑤𝑖𝑖‖𝑗 − 𝐶𝐶𝑖𝑖(𝑓𝑓𝑘𝑘) ≤ 0 ‖𝐹𝐹 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠1 , 𝑁𝑁𝑡𝑡1 𝑤𝑤𝑖𝑖‖𝑗 ≤ 𝑡𝑡3,𝑘𝑘 ,

𝑡𝑡3,𝑘𝑘 = 𝐶𝐶(𝑓𝑓𝑘𝑘)

( )

( )

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A reduced order technique

Sometimes, the designed filter has high frequency response concentrated in small time span: 𝒘𝒘𝟏𝟏 𝒏𝒏 Number of

coefficients: 𝒘𝒘𝒙𝒙 𝒏𝒏

𝑛𝑛 𝑇𝑇 × 0.1𝑓𝑓𝑠𝑠

𝑛𝑛 0.2𝑇𝑇 × 𝑓𝑓𝑠𝑠𝒘𝒘𝟐𝟐 𝒏𝒏

𝑛𝑛 0.1𝑇𝑇 × 𝑓𝑓𝑠𝑠Number of coefficients: 𝑇𝑇 × 𝑓𝑓𝑠𝑠 𝑀𝑀

In this case, 𝑤𝑤𝑖𝑖 (with higher sampling frequency) can be chosen to start with 𝑡𝑡 = 𝑀𝑀Δ , where 𝑀𝑀 > 0 , then we have:

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘𝑀𝑀

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘(𝑀𝑀+1)

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑓𝑓𝑘𝑘(𝑁𝑁𝑡𝑡−1)𝐹𝐹𝑟𝑟 𝑓𝑓𝑘𝑘 , 𝑓𝑓𝑠𝑠, 𝑁𝑁𝑡𝑡 = 𝑓𝑓𝑠𝑠 𝑓𝑓𝑠𝑠 ⋯ 𝑓𝑓𝑠𝑠




• Methodology

• Results

• Conclusions

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Experimental setup • An experimental setup for psychoacoustic listening test • Speaker should produce desired sound at listening location

Listening location Speaker



Experimental setup Required sampling frequency: 48 kHz (𝚫𝚫 =20.83 us)

Desired delay: 19200 𝚫𝚫

Two sub-band filters:

Sampling frequency Filter coefficients Starting time

Filter 1 2.4 kHz 1920 0 Filter 2 48 kHz 3000 17700 𝚫𝚫

SeDuMi is used to solve the reformulated cone programming problem



Result

-40

-20

0

20

ampl

itude

(dB)

w1 w2

The frequency response of both filter around 1200 Hz

0 500 1000 1500 2000

frequency (HZ)

( ) ( ) I I I I I

I I I I

I I I I I

X

X

X



0

Result

The frequency response of H(f)𝑊𝑊𝑥𝑥 𝑓𝑓 H(f) 𝑊𝑊𝑥𝑥 𝑓𝑓 around 1200 Hz 5

ampl

itude

(dB) 0

ampl

itude

(dB)

-20

-5-40

0 0.5 1 1.5 2 -10 600 800 1000 1200 1400 1600 1800

10 4frequency (HZ) frequency (HZ)

10 4

0 -2000

-2

phas

e (ra

d)

phas

e (ra

d)

-4

-6

-3000

-40000 0.5 1 1.5 2

frequency (HZ) 10 4 600 800 1000 1200 1400 1600 1800

frequency (HZ)



Result Combining two sub-band filters together in time domain

1500 The combination is done by:

1000

• Upsampling the sub-band filter 1 500

0 0.2 0.4 0.6

time (s)

with lower sampling frequency

• Adds the upsampled filter 1 with ampl

itude

0

filter 2 -500

-1000

-1500

The designed filter coefficients are 1920+3000 = 4920, which is much smaller than 48000 × 19𝑗0

𝑗400 = 38400




• Methodology

• Results

• Conclusions



Conclusions

The proposed method can design sub-band filters for sound field reconstruction in one optimization problem, so designing transition region is more convenient.

The optimization problem can be reformulated to a convex problem, then further reformulated to a cone programming problem. These guarantees the global optimal solution can be found in an efficient way.

A reduced-order technique can be used to reduce the variables in filter design problem if different frequency bands of required filter have impulse response concentrated in different time intervals.

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a sub-band filter design approach for sound field reproduction

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