Introduction Recovery Algorithm Mathematical Model Performance Testing Conclusion

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# Introduction Recovery Algorithm Mathematical Model Performance Testing Conclusion - PowerPoint PPT Presentation

A Frequency Domain Technique for Correction of Audio Distortion Caused by Variable Recording Speed: Authors Robert Hembree Henry Skiba Alex Smith Advisor Dr. Marcus Pendergrass. Outline. Introduction Recovery Algorithm Mathematical Model Performance Testing Conclusion. Introduction.

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A Frequency Domain Technique for Correction of Audio Distortion Caused by Variable Recording Speed:AuthorsRobert HembreeHenry SkibaAlex SmithAdvisorDr. Marcus Pendergrass

Outline

• Introduction
• Recovery Algorithm
• Mathematical Model
• Performance Testing
• Conclusion

Introduction

• Wow and Flutter Distortion
• Audio distortion caused by variations in the speed at which data was recorded
• Wow refers to low frequency variations in the recording speed
• Flutter refers to high frequency variations in the recording speed
• We will use the term “wobble” to refer to either wow or flutter distortion

Introduction

Input signal

Signal to be recorded

Position of record head at time t

Position function for record head (record function)

The recording

Data value recorded at position p

Recording

Introduction

Because

and the record function Ψr is assumed invertible, we have

or

Recording

Introduction

Position function for playback head (playback function)

Position of playback head at time t

Played-back signal

Signal played back from the recording

Because

we have

Combining this with our previous expression for r gives

Introduction

Reciprocity Theorem

for all s if and only if

for all s if and only if

proof

So a mismatch between Ψr and Ψpb causes distortion:

Introduction

• The wobble w(p)at position p in the recording is defined as the timing error at position p during the record process
• The timing error is the difference between the actual time that the record head was at position p, and the nominal time it would have been at position p had the record function been ideal (i.e. equal to playback function).

= actual time during record process when the record head was at position p

= nominal time the record head would have been at position phad the record function been ideal

So

w(p) is the timing error at position p

Introduction

• Knowing the wobble function w(p)would enable us to correct for distortion caused by a mismatch between Ψr and Ψpb .

Nominal time - incorrect

0.8156

Original recording

p

Introduction

• Knowing the wobble function w(p)would enable us to correct for distortion caused by a mismatch between Ψr and Ψpb .

actual time

0.8156

Original recording

p

Introduction

• Knowing the wobble function w(p)would enable us to correct for distortion caused by a mismatch between Ψr and Ψpb .

actual time

0.8156

0.7942

Original recording

p

p

Resample

Corrected recording

Introduction

Basic Assumptions

• The playback function is ideal, constant velocity:
• The record function is unknown, but invertible.
• The record and playback functions are continuous and differentiable.
• The recording contains an isolated sinusoid of known frequency. This will be used as a reference by the recovery algorithm.

Wobble Recovery Algorithm

• To correct wobble distortion, we need information about the wobble
• We might know something about the original signal that was recorded.
• We will focus on the case when the recording contains a sine wave of known frequency

Wobble Recovery Algorithm

• We can use that information, along with the distorted recording, to deduce the wobble function.
• This wobble function can be used to resample the corrupt file in order to recover the original file

Wobble Recovery Algorithm

BandPass Filter

f

f

Wobble Recovery Algorithm

Inverse Fourier Transform

Shift Baseband

f

0

Wobble Recovery Algorithm

p

w(p)+ϑ

Complex ln

(timing error)

t

q is a phase shift in the model, which only introduces a delay

Interpolate

Modeling the Wobble

• In order to test our recovery algorithm, we need a variety of distorted recordings to work with.
• A model for the record function Ψr was developed.
• Model encompasses a variety of distortion scenarios
• Weak to strong distortion
• Slowly-varying to quickly-varying distortion

Modeling the Wobble

• Begin by modeling the velocity function of the record head.
• Without loss of generality the average velocity is 1.
• User specifies
• standard deviation of the velocity fluctuations
• maximum frequency of the velocity fluctuations in time.

Modeling the Wobble

• Velocity is modeled as
• n(t) is a band-limited Gaussian noise process
• lognormal velocity process
• maximum frequency in n(t) is fmax (user specified)
• a and b are chosen so that

user specified

Performance Testing

How well does the Algorithm actually work?

Henry Skiba

Introduces White Gaussian noise to the recording at varying power

Robert Hembree

Steps a structured interferer through a sine wave

Conclusion

• Developed wobble recovery algorithm

-Uses a reference sinusoid to recover wobble, and then resample the recording

• Performance Tests were performed on the algorithm
• Random Noise

10^-2 relative error at 10 dB SNR

- Structured Interferer

10^-3 relative error at 10dB SNR

Interpolation raises the noise floor