scalar quantization with memory...scalar quantization with memory speech and images have redundancy...
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Scalar quantization with memory
Speech and images have redundancy or memory that scalar
quantization can not exploit.
Scalar quantization for source with memory provides
which is rather far from
VQ could attain better but usually at the cost of
significant increasing of computational complexity.
Another approach leading to better and preserving
rather low computational complexity combines linear
processing with scalar quantization.
)(DR).(DH
)(DR
)(DR
Scalar quantization with memory
Preprocessing
block Scalar
quantizer
Input
signal Quantized
data
Scalar quantization with memory
Predictive coding Transform coding
Speech coding Audio, images, video coding
Discrete-time filters Analog or continuous system is something that transforms an
input signal into an output signal , where both
signals are continuous functions of time.
A discrete-time system transforms an input sequence
into output sequence
Any discrete-time system can be considered as a discrete filter.
Thus predictive coding and transform coding are kinds of
discrete filtering.
A discrete-time system (filter) is said to be linear if the
following condition holds
)(tx )(ty
)( snTx
).( snTy
L
,)()()()( 2121 ssss nTxLnTxLnTxnTxL
where and are arbitrary real or complex constants.
Discrete-time filters
Description by means of recurrent equations
A discrete filter of th order is described by the
following linear recurrent equation N
N
j
ssj
M
i
ssis jTnTybiTnTxanTy10
),()()(
where
),( snTx
)( snTy are samples of input and output signals, ,ia
jb are constants which do not depend on ).( snTx
The discrete-time counterpart to Dirac’s delta function is
called Kronecker’s delta function )( snT
1, 0( ) .
0, 0s
nnT
n
If )()( ss nTnTx then ),()( ss nThnTy where )( snTh
is the discrete-time pulse response.
(4.1)
An N-th order time-invariant discrete filter
Discrete-time filters
The output can be expressed via the input
)( snTy
)( snTx
and
)( snTh as a discrete-time convolution
.)()()()()(0 0
n
m
n
m
sssssss mTnTxmThmTnThmTxnTy
If )( snTh has nonzero samples only for
Mn ,...,0 we
call the filter a finite response (FIR) filter.
If in (4.1) we set ,0jb Nj ,...,1 then we obtain a FIR
filter of th order. M
Since FIR filters do not contain feedback paths they are also
called nonrecursive filters.
(4.2)
An M-th order FIR filter
Discrete-time filters
It follows from (4.1) that the output of the FIR filter is
).(...)()()( 10 ssMssss MTnTxaTnTxanTxanTy
Formula (4.2) in the case of FIR filter has the form
.)()()()()(0 0
M
m
M
m
sssssss mTnTxmThmTnThmTxnTy
(4.3)
(4.4)
Comparing (4.3) and (4.4) we obtain that ,)( is aiTh .,...1,0 Mi
A filter with an infinite (im)pulse response is called IIR
filter. In general case the recursive filter described by (4.1)
represents IIR filter.
Discrete-time filters
In order to compute the pulse response of the discrete IIR
filter it is necessary to solve the corresponding recurrent
equation (4.1).
Example. The first order IIR filter is described by the
equation: ).()()( ssss TnTbynTaxnTy
It is evident that ),()0()0( sTbyaxy
).()0()()( 2
sss TaxabxTybTy
Continuing in such a manner we obtain
.)()()(0
1
n
i
ss
i
s
n
s iTnTaxbTybnTy
Setting and we get
0)( sTy )()( ss nTnTx .)( n
s abnTh
Discrete-time filters
Fig.4.3
Discrete-time filters
Description by means of z-transform
The z-transform of a given discrete-time signal is
defined as
)( snTx
,)()(
n
n
s znTxzX ,xEz
where
z is a complex variable,
)(zX denotes a function
of ,zxE is called the existence region.
If is causal then one-sided z-transform is defined as )( snTx
,)()(0
n
n
s znTxzX .xEz
z-transform allows to replace recurrent equations over
sequences by algebraic equations over z-transforms like the
Laplace transform allows to replace differential equations by
algebraic equations.
(4.5)
Discrete-time filters
Properties of z-transform
Linearity: If
)(zX is the z-transform of and
)( snTx
)(zY
is z-transform of
)( snTy then )()( ss nTbynTax has
z-transform ).()( zbYzaX
Discrete convolution: If )(zX is z-transform of )( snTx and
)(zY is z-transform of )( snTy and )()()( zYzXzR
then )(zR is z-transform of )( snTr
n
m
n
m
sssssss mTymTnTxmTnTymTxnTr0 0
)()()()()(
Delay: If )(zX is z-transform of )( snTx then z-transform of
)()( sss mTnTxnTy is
.
.)()( mzzXzY
1, 0( )
0, 0s
nx nT
n
1)( zX
1, 0( )
0, 0s
nx nT
n
11
1)(
zzX
sanT
s AenTx
)( 11)(
ze
AzX
saT
)sin()( s
anT
s nTenTx s 221
1
cos21
sin)(
zeTez
TezzX
ss
s
aT
s
aT
s
aT
)cos()( s
anT
s nTenTx s 221
1
cos21
cos1)(
zeTez
TzezX
ss
s
aT
s
aT
s
aT
ba
a
nTx nn
s
42/
),()(
2
2,1
1
2
1
1
))((
1
1
1)(
21
21
zzbzazzX
, 0( )
0, 0
sj nT
s
e nx nT
n
11
1)(
zezX
sTj
Name of
the
sequence
Sequence
z-transform
Unit
impulse
Unit step
Exponent
Damped
sine
Damped
cosine
Complex
exponent
Response of
the 2nd
order filter
Example
Let .
12
3
2
1
1)(
12
zz
zX
Decompose )(zX into sum of simple fractions. We obtain
11
11
2
11
1
2
11)1(
1)(
z
B
z
A
zz
zX(4.6)
Multiplying both parts of (4.6) by
11
2
11)1( zz we get
).1(2
111 11
zBzA
Equating coefficients of like powers of z we obtain
1
02
A B
AB
.
2
11
1
1
2
2
11)1(
1)(
11
11
zz
zz
zXand
Example
Comparing the obtained results with examples in the
table we obtain that the first summand corresponds to
2, 0( )
0, 0s
nx nT
n
and the second summand corresponds to .2)( n
snTx
Thus we obtain
.
2 2 , 0( )
0, 0
n
s
nx nT
n
.
Inverse z-transform
There is one-to-one correspondence between a discrete
sequence and its z-transform.
According to definition )( snTx represents the inverse
z-transform of ).(zX
It can be found from (4.5). First multiply both parts by 1kzand then take the contour integral of both parts of equation.
The contour integral is taken along a counterclockwise
arbitrary closed path that encloses all the finite poles of 1)( kzzX and lies entirely in .xE
dzznTxdzzzX nk
n
s
k 1
0
1 )()( (4.7)
Inverse z-transform
It follows from the Cauchy theorem that if the path of
integration encloses the origin of coordinates then
01 dzz nk
for all k except .nk For nk it is equal to
.21 jdzz
Thus (4.7) reduces to
dzzzX
jkTx k
s
1)(2
1)(
which describes the inverse z-transform.
(4.8)
Inverse z-transform
The contour integral (4.8) can be evaluated via the
theorem of residues.
Theorem. Let function )(zf be unique and analytic
(differentiable) everywhere along the contour
L and inside it
except the so-called singular points ,kz ),,...,1( Nk lying
inside .L Then
,),(Res2)()( 1
L
N
k
kzzfjdzzf
that is, the contour integral is equal to the sum of residues of
integrand function taken in the singular points lying inside the
path of integration, multiplied by .2 j
Inverse z-transform
A residue in the th order pole can be calculated as m0z
0
1
0 01
1Res ( ), lim ( ) ( )
( 1)!
mm
mz z
df z z z z f z
m dz
If function represents a ratio of two finite functions
)(zf
( ) ( ) / ( )f z g z h z and it being known that )(zh has
zero of the 1st order in az and that 0)( ag then
the residue in az can be computed as
( )
Res ( ),'( )
g af z a
h a
If has finite poles at 1)( nzzX K ,iaz Ki ,...2,1 then
.
1
1
( ) Res ( ) ,K
n
s i
i
x nT X z z z a
(4.9)
Example
Continuing our example we obtain that
.
2
11
1
2)(
1
1
1
1
z
dzz
z
dzznTx
nn
s
Using (4.9) we compute
2 2Res ,1
1 1
nz
z
2 2(1/ 2)Res ,1/ 2 2
2 1 2
n nnz
z
Thus .22)( snT
snTx
Example Let us solve the equation
).()()( ssss TnTbynTaxnTy
using z-transform. Using properties (linearity and delay) we obtain
),()()( 1 zaXzzbYzY
,1)(
)()(
1
bz
a
zX
zYzH
where
)(zH is a transfer function of the filter,
)(zX and
)(zY are z-transforms of the input and output. )(zH is
z-transform of
)( snTh .)()(0
n
n
s znThzH
)()()( zHzXzY Since then if
),()( ss nTnTx
.1)( zX
)()( zHzY11 bz
a.
bz
az
This yields
Example
By applying the inverse z-transform to if the )(zH
integration path encloses the pole
bz we obtain
.)( n
s abnTy
Frequency function of discrete filter
The complex frequency function of the discrete-time linear
filter can be obtained by inserting into sTjez
).(zH
Let snTj
s enTx
)( then the corresponding output of the filter
with pulse response )( snTh is .
n
m
n
m
Tmnj
ssssssemThmTnTxmThnTy
0 0
)()()()()(
.
.
n
m
mTj
s
nTj ss emThe0
)(
When n ),()()(0
ssss TjnTj
m
mTj
s
nTj
s eHeemThenTy
where 0
( ) ( )s sj T j mT
smH e h mT e
is the complex
frequency function of the discrete-time linear filter.
Frequency function of discrete filter
The complex frequency function can be represented in the form
)),(Im())(Re()()( )( sss TjTjjTjeHjeHeAeH
where 22
))(Im())(Re()( ss TjTjeHeHA
is the amplitude function and
))(Re(
))(Im(arctan)(
s
s
Tj
Tj
eH
eH
is the phase function.
If
)sin()( ss TnnTx then
.
)).(sin()()( ss TnAnTy
The amplitude and the phase functions described the change in
amplitude and phase of the discrete sinusoid introduced by the
discrete-time linear filter.
Frequency function of discrete filter Properties of the frequency function:
•The frequency function is a continuous function of
frequency
•The frequency function is a periodic function of frequency
with period equal to the sampling frequency.
•For discrete-time linear filters with coefficients which
are real numbers the amplitude function is an even function
of frequency and the phase function is an odd function of
frequency.
•The frequency function can be represented via the pulse
response as
ii ba ,.
.)2sin()2cos()()()()(0 0 0
2
n n n
sss
nfTj
s
nTj
s
TjnfTjnfTnThenThenTheH sss
Frequency function of discrete filter
For convenience of comparison of different frequency
functions instead of we consider the normalized frequency
,/ s where
ss T/2 is the sampling frequency
in radians. Then we obtain
0 0
2 .)2sin()2cos()()()(n n
s
nj
s
TjnjnnThenTheH s
Example
The transfer function of the discrete-time linear filter of
the 1st order has the form
.1
)(1
bz
azH
By inserting sTjez
we obtain the frequency function
.)sin()cos(11
)(ss
Tj
Tj
TjbTb
a
be
aeH
s
s
The amplitude function is
.)cos(21)(sin))cos(1(
)(2222 bTb
a
TbTb
aA
sss
The phase function .
)cos(1
)sin(arctan)(
s
s
Tb
Tb
Example
By normalizing frequency we obtain
,)2cos(21
)(2bb
aA
.)2cos(1
)2sin(arctan)(
b
b
The amplitude function for )(A
,1a
5.0b
is shown in Fig.4.4.
Example
Fig.4.4