wave2lp
Syntax
Description
[___,
also returns the perfect reconstruction condition PRCond,AACond
] = wave2lp(wname
)PRCond
and the
anti-aliasing condition AACond
.
Examples
Laurent Polynomials Associated with Wavelet
Obtain the four Laurent polynomials associated with the orthogonal wavelet db3
. Also obtain the perfect reconstruction and anti-aliasing conditions.
[LoDz,HiDz,LoRz,HiRz,PRC,AAC] = wave2lp("db3")
LoDz = laurentPolynomial with properties: Coefficients: [0.0352 -0.0854 -0.1350 0.4599 0.8069 0.3327] MaxOrder: 5
HiDz = laurentPolynomial with properties: Coefficients: [0.3327 -0.8069 0.4599 0.1350 -0.0854 -0.0352] MaxOrder: 1
LoRz = laurentPolynomial with properties: Coefficients: [0.3327 0.8069 0.4599 -0.1350 -0.0854 0.0352] MaxOrder: 0
HiRz = laurentPolynomial with properties: Coefficients: [-0.0352 -0.0854 0.1350 0.4599 -0.8069 0.3327] MaxOrder: 4
PRC = laurentPolynomial with properties: Coefficients: 2.0000 MaxOrder: 0
AAC = laurentPolynomial with properties: Coefficients: 0 MaxOrder: 0
Verify the perfect reconstruction condition.
eq(LoRz*LoDz + HiRz*HiDz,PRC)
ans = logical
1
Verify the anti-aliasing condition. Use the helper function helperMakeLaurentPoly
to obtain , where is the Laurent polynomial LoDz
. Use the helper function helperMakeLaurentPoly
to obtain , where is the Laurent polynomial HiDz
.
LoDzm = helperMakeLaurentPoly(LoDz); HiDzm = helperMakeLaurentPoly(HiDz); eq(LoRz*LoDzm + HiRz*HiDzm,AAC)
ans = logical
1
Helper Functions
function polyout = helperMakeLaurentPoly(poly) % This function is only intended to support this example. % It may change or be removed in a future release. polyout = poly; cflen = length(polyout.Coefficients); cmo = polyout.MaxOrder; polyneg = (-1).^(mod(cmo,2)+(0:cflen-1)); polyout.Coefficients = polyout.Coefficients.*polyneg; end
Input Arguments
wname
— Wavelet
character vector | string scalar
Wavelet, specified as a character vector or string scalar.
wname
must be one of the wavelets supported by
liftingScheme
. See the Wavelet property of liftingScheme
for the
list of wavelets.
Example: [LoDz,HiDz,LoRz,HiRz] =
wave2lp("db2")
Data Types: char
| string
PmaxHS
— Maximum power
0
(default) | integer
Maximum power of the Laurent polynomial LoRz
,
specified as an integer.
Example: If [~,~,LoRz,HiRz] = wave2lp("db2",3)
, then the
maximum power, or order, of the Laurent polynomial LoRz
is 3.
Data Types: double
AddPOW
— Integer
0
(default) | integer
Integer to set the maximum order of the Laurent polynomial
HiRz
. PmaxHiRz
, the maximum
order of HiRz
, is
PmaxHiRz =
PmaxHS+length(HiRz.Coefficients)-2+AddPow
.
AddPOW
must be an even integer to
preserve the perfect reconstruction condition.
Data Types: double
Output Arguments
LoDz
— Laurent polynomial
laurentPolynomial
object
Laurent polynomial associated with the lowpass analysis filter, returned
as a laurentPolynomial
object.
HiDz
— Laurent polynomial
laurentPolynomial
object
Laurent polynomial associated with the highpass analysis filter, returned
as a laurentPolynomial
object.
LoRz
— Laurent polynomial
laurentPolynomial
object
Laurent polynomial associated with the lowpass synthesis filter, returned
as a laurentPolynomial
object.
HiRz
— Laurent polynomial
laurentPolynomial
object
Laurent polynomial associated with the highpass synthesis filter, returned
as a laurentPolynomial
object.
PRCond,AACond
— Perfect reconstruction and anti-aliasing conditions
laurentPolynomial
objects
Perfect reconstruction and anti-aliasing conditions, returned as
laurentPolynomial
objects. The perfect reconstruction
condition PRCond
and anti-aliasing condition
AACond
are:
PRCond(z) = LoRz(z) LoDz(z) + HiRz(z) HiDz(z)
AACond(z) = LoRz(z) LoDz(-z) + HiRz(z) HiDz(-z)
The pairs (LoRz
, HiRz
) and
(LoDz
, HiDz
) are associated with
perfect reconstructions filters if and only if:
PRCond(z) = 2
, andAACond(z) = 0
If PRCond(z) = 2
zd
,
a delay is introduced in the reconstruction process.
Version History
Introduced in R2021bR2021b: wave2lp
input syntax has changed
The wave2lp input syntax has changed.
You can now set the maximum order of
LoRz
usingPmaxHS
.You can now set the maximum order of
HiRz
usingAddPOW
.
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