wave2lp

Laurent polynomials associated with wavelet

Since R2021b

Syntax

[LoDz,HiDz,LoRz,HiRz] = wave2lp(wname)

[___,PRCond,AACond] = wave2lp(wname)

[___] = wave2lp(wname,PmaxHS)

[___] = wave2lp(wname,PmaxHS,AddPOW)

Description

[LoDz,HiDz,LoRz,HiRz] = wave2lp(wname) returns the four Laurent polynomials associated with the wavelet wname. The pairs (LoRz,HiRz) and (LoDz,HiDz) are associated with the synthesis and analysis filters, respectively.

[___,PRCond,AACond] = wave2lp(wname) also returns the perfect reconstruction condition PRCond and the anti-aliasing condition AACond.

[___] = wave2lp(wname,PmaxHS) sets the maximum order of LoRz.

[___] = wave2lp(wname,PmaxHS,AddPOW) sets the maximum order of the Laurent polynomial HiRz.

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 $$LoD(-z)$$, where $$LoD(z)$$ is the Laurent polynomial LoDz. Use the helper function helperMakeLaurentPoly to obtain $$HiD(-z)$$, where $$HiD(z)$$ 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, and

• AACond(z) = 0

If PRCond(z) = 2 zd, a delay is introduced in the reconstruction process.

Version History

Introduced in R2021b

R2021b: wave2lp input syntax has changed

The wave2lp input syntax has changed.

• You can now set the maximum order of LoRz using PmaxHS.

• You can now set the maximum order of HiRz using AddPOW.