Audio Processing

AUDIO PROCESSING
6.3 Quantisation and Transmission of Sound
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INTRODUCTION
 Follows the process of sampling - Nyquist
rate
which is explicitly UNIFORM
 Once quantized –transmitted or stored
 Basically 1 . Linear or Uniform
2. Non Linear or Non Uniform
• Protection of weak passages over loud
• Uniform precision – entire range
• Fewer steps
 Baseband signal over compressor
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•OK Reduction in the amount of Memory
required …….. Standard Answer !!!!
WHAT ELSE????
Whenever there’s non linearity.......
3 stages of compression scheme
1. Transformation
2. Loss
3. Coding

6.3.1 CODING OF AUDIO
 Quantization and Transformation coding of data
 Mu law technique + Simple algorithm further
compression
 Difference in signal
1. size of signal values
2. concentrate histogram pixel values
variance reduction therefore
lossless compression into
shorter bit stream
 Pulse Code Modulation – formal term for sampling and
Quantisation
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Bit rate = bits per sample x the
sampling rate
Compressed- 128kbps 192 kbps
256 kbps
Goal – quantized
sampled output

 “Producing Quantised Sampled output for audio”
 Decission Boundaries for quantizer input intervals –coder mapping
 Representative values(reconstruction levels) output from a quantizer
– decoder mapping
 3 stages of compression scheme
1. Transformation
2. Loss
3. Coding
 PCM in Speech Compression
50 Hz to about 10kHzbandwidth
1. uniform quantization without companding 12 bits bit-rate 240 kbps
2. With companding 8 bits bit-rate to 160 kbps.
3. standard approach to telephony 4 kHz
4. - sampling rate 8kHz companded bit rate reduced to 64
kbps.
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6.3.2 PULSE CODE MODULATION

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2 Wrinkeles
Sounds up to 4 kHz
band limiting filter
Reconstructed after low pass
filtering
Original
analog
Decoded staircase
Reconstructed signal
after low pass
filtering

 Audio often not in PCM –difference –fewer bits
 Peaked histogram maximum at zero
 For example, histogram for a linear ramp signal -
flat, histogram for the derivative of the signal
(i.e., the differences, from sampling point to
sampling point) consists of a spike at the slope
value.
 assign short codes to prevalent values and long
code words to rarely occurring ones.
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DIFFERENTIAL CODING OF AUDIO

 predict the next sample as being equal to the
current sample
 transmitting these using a PCM system.
 Linear prediction
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6.3.4 LOSSLESS PREDICTIVE CODING


1n n
n n n
f f
e f f

2 to 4
1
n n k n k
k
f a f
Linear Predictor Function

CONTD…
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Digital speech signal. Histogram of digital speech signal values
Histogram of digital speech signal
differences.
What if Exceptionally Large Difference?
SU ,SD ,-32 EG 100 –
SU SU SU 4

PREDICTOR EXAMPLE
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

1 2
1
( )
2
n n n
n n n
f f f
e f f ENCODER
DECODER
Calculate f1, f2, f3, f4, f5 = 21, 22, 27, 25, 22.

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6.3.5 DPCM
 Predictive Coding, except that it incorporates a
quantizer step.
   



 
1 2 3_ ( , , ,...) ,
,
[ ],
transmit ( ) ,
ˆreconstruct: .
n n n n
n n n
n n
n
n n n
f function of f f f
e f f
e Q e
codeword e
f f e

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Distortion - Average Squared Error
Lloyd-Max quantizer, which is based on a least-squares
minimization of the error term.
For speech, we could modify quantization steps adaptively by
estimating the mean and variance of a patch of signal values,
and shifting quantization steps accordingly, for every block of
signal values. That is, starting at time i we could take a block of
N values fn and try to minimize the quantization error:
2
1
[ ( ) ] /
N
n n
n
f f N
1
2
[ ]
i N
n n
n i
min f Q f

LLOYD MAX QUANTISER
1. Get Pdf
2. Guess M representation Levels
3. Apply Threshold Condition
4. Apply Mean Square Error Estimation
5. Iteration process (of steps 3 n 4)
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 signal differences peaked, could model them
using a Laplacian probability distribution
function, which is strongly peaked at zero
 for variance σ2.
 one assigns quantization steps for a quantizer with
nonuniform steps by assuming signal differences, dn
are drawn from such a distribution and then choosing
steps to minimize....... ?

2
( ) (1/ 2 ) ( 2 | | / )x exp x
1
2
[ ] ( ).
i N
n n n
n i
min d Q d l d
QUANTISATION ERROR

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   



 
1 2 3_ ( , , ,...) ,
,
[ ] ,
transmit ( ) ,
ˆreconstruct: .
n n n n
n n n
n n
n
n n n
f function of f f f
e f f
e Q e
codeword e
f f e

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• Notice that the quantization noise, , is equal to the
quantization effect on the error term, .
• Suppose we adopt the particular predictor below:
(1)
so that is an integer.
• As well, use the quantization scheme:
(2)
n nf f
n ne e
1 2
ˆ
n n nf trunc f f 
[ ] 16*trunc 255 /16 256 8
ˆ
n n n
n n n
e Q e e
f f e

 
ˆ
n n ne f f

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en in range Quantized to value
-255 .. -240
-239 .. -224
.
.
.
-31 .. -16
-15 .. 0
1 .. 16
17 .. 32
.
.
.
225 .. 240
241 .. 255
-248
-232
.
.
.
-24
-8
8
24
.
.
.
232
248

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DELTA MODULATION
 This scheme sends only the difference between
pulses, if the pulse at time tn+1 is higher in
amplitude value than the pulse at time tn, then a
single bit, say a “1”, is used to indicate the positive
value.
 If the pulse is lower in value, resulting in a negative
value, a “0” is used.
 This scheme works well for small changes in signal
values between samples.
 If changes in amplitude are large, this will result in
large errors.

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1
1
ˆ ,
ˆ ,
if 0,
ˆ .
n n
n n n n n
n
n
n n n
f f
e f f f f
k e where k is a constant
e
k otherwise
f f e



 
Solution Sampling at many times
greater the Nyquist rate
If the slope of the actual signal curve
is high, the staircase approximation
cannot keep up. For a steep curve,
should change the step size k
adaptively
Adaptive DM

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f1 f2 f3 f4
10 11 13 15

11 10f f
1
1
ˆ ,
ˆ ,
if 0,
ˆ .
n n
n n n n n
n
n
n n n
f f
e f f f f
k e where k is a constant
e
k otherwise
f f e



 
e2 = 11 − 10 =
1,
e3 = 13 − 14 =
−1,
e4 = 15 − 10 =
5,

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Step Size + Decision
Boundaries !

6.3.7 ADPCM
• ADPCM (Adaptive DPCM) takes the idea of adapting the
coder to suit the input much farther.
quantizer and the predictor.
1. In Adaptive DM, adapt the quantizer step size to suit the input.
In DPCM, we can change the step size as well as decision
boundaries, using a non-uniform quantizer.
We can carry this out in two ways:
(a) Forward adaptive quantization: use the properties of the input
signal.
(b) Backward adaptive quantization: use the properties of the
quantized output. If quantized errors become too large, we should
change the non-uniform quantizer.
Multimedia Systems (eadeli@iust.ac.ir)
23

2. We can also adapt the predictor, again using
forward or backward adaptation. Making the
predictor coefficients adaptive is called Adaptive
Predictive Coding (APC):
(a) Recall that the predictor is usually taken to be a linear
function of previous reconstructed quantized values,
.
(b) The number of previous values used is called the
“order” of the predictor. For example, if we use M
previous values, we need M coefficients ai, i = 1..M in a
predictor
(6.22)
24
Multimedia Systems (eadeli@iust.ac.ir)
1
ˆ
M
n i n i
i
f a f

nf

• However we can get into a difficult situation if we try to change
the prediction coefficients, that multiply previous quantized
values, because that makes a complicated set of equations to
solve for these coefficients:
(a) Suppose we decide to use a least-squares approach to solving a
minimization trying to find the best values of the ai:
(6.23)
(b) Here we would sum over a large number of samples fn, for the
current patch of speech, say. But because depends on the
quantization we have a difficult problem to solve. As well, we
should really be changing the fineness of the quantization at the
same time, to suit the signal’s changing nature; this makes
things problematical.
25
2
1
ˆ( )
N
n n
n
min f f

nf

(c) Instead, one usually resorts to solving the
simpler problem that results from using not in
the prediction, but instead simply the signal fn
itself. Explicitly writing in terms of the coefficients
ai, we wish to solve:
(6.24)
Differentiation with respect to each of the ai, and
setting to zero, produces a linear system of M
equations that is easy to solve. (The set of
equations is called the Wiener-Hopf equations.)
26
2
1 1
( )
N M
n i n i
n i
min f a f

nf

Schematic diagram for ADPCM encoder and
decoder
27

Audio Processing

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