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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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slide1

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

slide2

Outline

  • Introduction
  • Recovery Algorithm
  • Mathematical Model
  • Performance Testing
  • Conclusion
slide3

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
slide4

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

Record Head

Recording

slide5

Introduction

Because

and the record function Ψr is assumed invertible, we have

or

Record Head

Recording

slide6

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

slide7

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:

slide8

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

slide9

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

slide10

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

slide11

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

Ready for Ψpb

Corrected recording

slide12

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.
slide14

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
slide15

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
slide19

Wobble Recovery Algorithm

BandPass Filter

f

f

slide20

Wobble Recovery Algorithm

Inverse Fourier Transform

Shift Baseband

f

0

slide21

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

slide22

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
slide23

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.
slide24

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

slide28

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

slide29

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