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Waveform Coding
Compression: factors to be considered –
BW / storage efficiency
Signal distortion
Computational complexity (for real-time usage)
Two categories: Lossless and Lossy
Redundancies
Statistical; e.g. Huffman coding
Inter-sample; e.g. DPCM
Psycho-acoustic / psycho-visual; e.g. sub-band coding
Speech coding
Classes of coders:
Waveform coders
Time-domain waveform coders
Frequency-domain waveform coders
Source coders – vocoders: Follow a speech
production model, encoding excitation parameters.
Perceptual coders – Make use of hearing properties.
Coding methods
Non-uniform quantization (companding) – A-law, µ-law.
Adaptive-quantizer PCM – vary the quantizer step in
proportion to short-time average speech amplitude.
DPCM
Sub-band coding
Transform coding
Vector quantization
Noise-shaping – hiding quantization noise at times /
frequencies of high speech energy
Noise shaping in frequency
Quantization
All digital speech coders use a form of PCM to convert an
analog signal to a digital representation.
No. of quant. levels = L; generally taken in the form of 2N
why?
Sampling rate Fs – min. sampling rate is Nyquist rate
Bits per sample = R (for fixed-rate coding)
In general,
Bit-rate = Fs R
LRL R
2log2
LR 2log
Quantization
Quantization provides first level of compression with
some loss – discuss.
Input samples:
Quantizer output:
Choice of R (or L) is a compromise between coding rate
(compression) and quality – discuss.
The y’s are called reconstruction level why?
Boundary value between two intervals is called decision
level why?
maxmin xxxxX
LyyyYYXQ ,.....,,;: 21
Quantizer types
Mid-rise – even no. of levels; so L can be of the form
Mid-tread – odd no. of levels; at best L can be of the
form
preferred in case of speech why
Uniform quantizer –
equal length quantization intervals
Reconstruction level = mid-point of the interval
Non-uniform quantizer
Quantization noise
Assumed to be
stationary white noise,
uncorrelated with the input signal,
each error sample uniformly distributed in the range
[- ∆/2, ∆/2] where ∆ is the quantization step.
Components of quantization noise:
Granular noise
Overload noise
Quantization noise
Quantization noise calculation
For uniform quantizer:
Quantizer designed for −xm to + xm
Step size
dqqpqdxpyx Q
L
k
XkQ )(2
1
22
R
mm
QQ xL
xdqqpq 22
2
2222 2
3
1
312)(
Lxm2
Non-uniform quantizer
Lloyd-Max quatizer: pdf optimized
Reconstruction level = centroid in an interval
Decision level = mid-point between two reconstruction
level.
Designing by iterative solution.
1
1
)(
)(
k
k
k
k
x
x
X
x
x
X
k
dxxp
dxxxp
y
21 kkk yyx
Log-companding
Companding = Compressing + expanding.
Basically we need to determine a suitable compressing function c(x).
For constant SNR quantization with large L, we need log-companding.
But, such function undefined for x = 0 so quasi-log companding used.
Compressor
C(x)
Uniform
Quantizer
Q[C(x)]
Expander
C−1(x) Q[x]x
A-law quantizer
Mid-rise type.
Chosen value of K
Compression function (logarithmic for large values of x):
Hence, SNR
Standard value of A = 87.56
mxAln1
11
for )sgn(ln1ln1
)(
mm
m
x
x
Ax
x
xA
A
xxc
22 ln13 AL
A-law quantizer
Compression function (linear for small values of x):
Companding gain (slope):
Step-size:
Step-size ratio = max. step-size / min. step size = A
Ax
xxx
A
Axc
m
10for )sgn(
ln1)(
Axc
dx
d
A
Axc
dx
d
mxxx ln1
1)(
ln1)(
0
AL
x
A
A
L
x mm ln12
maxln12
min
μ-law quantizer
Mid-tread type.
Chosen value of K
Compression function (logarithmic for large values of x):
Hence, SNR
Standard value of μ = 255
mx 1ln
1
1for )sgn(ln
1ln)(
mm
m
x
xx
x
xxxc
22 1ln3 L
μ-law quantizer
Compression function (linear for small values of x):
Companding gain (slope):
Step-size:
Step-size ratio = max. step-size / min. step size = μ
10for )sgn(1ln
1ln)(
mm
m
x
xx
x
xxxc
1ln
1)(
1ln)(
0 mxxx
xcdx
dxc
dx
d
1ln
2max
1ln2min
L
x
L
x mm
Adaptive quantization
Quantizer parameters (all decision levels and reconstruction levels) need to be adjusted with change in signal statistics:
Change in pdf – form of quantizer to be changed, e.g. non-uniform to uniform or vice-versa; needs computationally expensive redesigning.
Change in mean value – decision / reconstruction levels to be shifted by the same amount; so computationally simple.
Change in dynamic range – step sizes to be changed proportionately; decision / reconstruction levels also changes proportionately.
Adaptive quantization
We discuss adaptation strategy for change in dynamic range.
Std. deviation changes proportionately with dynamic range.
So, kth step-size at nth sample instant is taken in proportion to the signal standard deviation at that instant of time.
Std. dev. at nth sample instant is estimated from knowledge of N future samples (AQF – adaptive quantization forward) or N past samples (AQB – adaptive quantization backward).
)()( nn Xkk
Adaptive quantization forward
Estimated std. dev.:
(assuming zero mean):
Problems:
Delay – needs to wait for next N−1 samples to arrive.
Overhead – needs to transmit estimated std. dev. value for
adjustment of reconstruction levels in the receiver.
1
0
22 1)(ˆ
N
i
X inxN
n
Buffer Coder Decoder
Level
Estimator
x(n) y(n)
Adaptive quantization backward
Estimated std. dev.:
(assuming zero mean):
Quantized samples used so that same level estimation
possible also in receiver – so, no overhead.
Problem – estimation based on quantized samples, not
actual samples.
N
i
X inyN
n1
22 1)(̂
Buffer
Coder Decoder
Level
Estimator
x(n)y(n)
Level
EstimatorBuffer
Predictive coding
Also called differential coding DPCM.
Predicted sample:
Input to quantizer is the prediction error.
Quantized prediction error:
Reconstructed sample:
N
k
k knxhnx1
)(ˆ)(~
)(~)()( nxnxnd
)()()( nqndnu
)()()()()(~)()(~)(ˆ nqnxnqndnxnunxnx
Predictive coding system
)(~ nx
Quantizer
Linear
Predictor
Linear
Predictor
x(n)
u(n)
)(ˆ nx
d(n)+
+
+
+
+
)(ˆ nx
)(~ nx
−
Optimum prediction
Prediction gain:
Needs to maximize pred. gain minimize pred. error.
Optimum choice of predictor coefficients hk needed.
Method:
Take approximate prediction:
This assumption holds good for large bit/sample (less quantization error) or small N (less accumulated error)
Calculate prediction error variance.
Choose pred. coefficients so as to minimize this pred. error variance.
22DXpG
N
k
k knxhnx1
)()(~
Optimum prediction coefficients
For N = 1,
For any N:
This relation is known as (1) normal equation, or (2) Yule-Walker prediction, or (3) Weiner-Hopf equation
12,1
)1(
)0(
)1(
X
XX
XX
XXopt
R
R
Rh
optN
opt
opt
XXXXXX
XXXXXX
XXXXXX
XX
XX
XX
h
h
h
RNRNR
NRRR
NRRR
NR
R
R
,
,2
,1
......
)0(....)2()1(
................
)2(....)0()1(
)1(....)1()0(
)(
.......
)2(
)1(