A Frequency Domain

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



•
Introduction

•
Recovery Algorithm

•
Mathematical Model

•
Performance Testing

•
Conclusion


Outline

•
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


s
t


Signal to be recorded

Input signal


p


r
t


Position of record head at time
t

Position function for record head (record function)


r
p


Data value recorded at position
p

The recording

)
(
t
p
r


Recording

Record Head

8156
.
0
)
(

p
r
Introduction


s
t



r
p



r

r
t




Because

and the record function
Ψ
r

is assumed invertible, we have


s

r

r
or


r

s

r

1
)
(
t
p
r


Recording

Record Head

8156
.
0
)
(

p
r
Introduction


ˆ
s
t



r
p



r

pb
t




we have

pb
r
s


ˆ
Combining this with our previous expression for
r

gives

pb
r
s
s


1
ˆ


Position function for playback head (playback function)


p


pb
t


Position of playback head at time
t

Played
-
back signal


ˆ
s
t


Signal played back from the recording

Because

Introduction

proof

Reciprocity Theorem


ˆ
s

s
for all
s
if and only if



pb


r


r

1

pb

identity

ˆ
s

s
for all
s
if and only if

So a mismatch between
Ψ
r

and
Ψ
pb

causes distortion:



pb


r

ˆ
s

s
Introduction


t


r

1
p


•
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


p
v
0
= nominal time the record head would have been at
position
p

had the record function been ideal

So


w
p




r

1
p



p
/
v
0

t

p
/
v
0
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
Ψ
r

and
Ψ
pb

.

0.8156

p


t

p
/
v
0
Nominal time
-

incorrect

Original recording

Introduction

•
Knowing the wobble function
w
(
p
)

would enable us to correct for
distortion caused by a mismatch between
Ψ
r

and
Ψ
pb

.

0.8156

p

actual time


t

p
/
v
0

w
p


Original recording

Introduction

•
Knowing the wobble function
w
(
p
)

would enable us to correct for
distortion caused by a mismatch between
Ψ
r

and
Ψ
pb

.

Resample

0.8156

p


t

p
/
v
0

w
p


Original recording

actual time

0.7942

p


t

p
/
v
0
Ready for
Ψ
pb

Corrected recording

Introduction

Basic Assumptions



pb
t



v
0
t
•
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.

Introduction

Wobble

f

f

Wobble
-
Induced Distortion in the
Frequency Domain

•
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

Baseband Shift

f

f

Correcting for Wobble in a Sinusoid

f

Inverse FT



i
p
w
if
e

)
(
2
0
Correcting for Wobble in a Sinusoid

Wobble

f

f

Wobble
-
Induced Distortion in the
Frequency Domain

f

BandPass Filter

f

Wobble Recovery Algorithm

Shift Baseband

Inverse Fourier Transform

f

0



i
p
w
if
e

)
(
2
0
Wobble Recovery Algorithm

Complex ln

t

p

w(p)+
ϑ


Interpolate

(timing error)



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

Wobble Recovery Algorithm

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
Ψ
r

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.



v
t



dp
dt



r
t


•
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



v
t



exp
a

bn
t




•
n
(
t
)

is a band
-
limited Gaussian noise process

•
lognormal

velocity process

•
maximum frequency in
n
(
t
)

is
f
max

(user specified)

•
a

and
b

are chosen so that


E
v
t





1

Var
v
t






2
user specified

Simulated Wobble Example




0
.
1
,
f
max

1
Hz
Simulated Wobble Example




0
.
1
,
f
max

10
Hz
Simulated Wobble Example




0
.
5
f
max

10
Hz
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

Performance Testing

•
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




Conclusion