A Semi-Analytic Fourier Transform for Spline Signals

David MerrellAdvised by Dr. Derek Thomas

BYU Department of Physics and Astronomy

Contents

• The Big Picture: Isogeometric Analysis (IGA)

• Zooming In: B-Splines

• My Work:The Semi-Analytic Fourier Transform

• Ideas for Future Work

Big Picture: Isogeometric Analysis

IGA Streamlines the Engineering Workflow

Big Picture: Isogeometric Analysis

IGA is Superior to FEA in Many Situations:

• Mechanics of Thin Shells • Fluids (John Evans colloquium)• Electricity and Magnetism• Acoustics• Rolling/Contact• Mechanical Vibration

Big Picture: Isogeometric Analysis

IGA is Superior to FEA in Many Situations:

• Mechanics of Thin Shells • Fluids (John Evans colloquium)• Electricity and Magnetism• Acoustics• Rolling/Contact• Mechanical Vibration

Big Picture: Isogeometric Analysis

IGA is Superior to FEA in Many Situations:

• Mechanics of Thin Shells • Fluids (John Evans colloquium)• Electricity and Magnetism• Acoustics• Rolling/Contact• Mechanical Vibration

In summary, when physics is highly dependent on geometry, IGA will beat Finite Elements.

Zooming In: B-Splines

Some 1-Dimensional B-Spline Functions

• The secret behind IGA’s accurate geometry is SPLINES: piecewise polynomial parametric functions

• CAD software already uses splines to represent geometry; IGA uses those same splines as a basis to represent the solution of a PDE.

Zooming In: B-Splines

A 2-Dimensional B-Spline Surface

• The secret behind IGA’s accurate geometry is SPLINES: piecewise polynomial parametric functions

• CAD software already uses splines to represent geometry; IGA uses those same splines as a basis to represent the solution of a PDE.

• The secret behind IGA’s accurate geometry is SPLINES: piecewise polynomial parametric functions

• CAD software already uses splines to represent geometry; IGA uses those same splines as a basis to represent the solution of a PDE.

A 3-dimensional T-spline solid,along with the IGA solution of itssixth vibrational mode

Zooming In: B-Splines

Zooming In: B-Splines

• There are many kinds of splines.

• The simplest are B-Splines.

• We’ll quickly explore some relevant B-Spline anatomy…

Relevant B-Spline Anatomy• A B-Spline is a linear combination of

piecewise-polynomial basis functions.

= Σ

= Σλi( )i

Relevant B-Spline Anatomy• We can construct B-Spline basis functions of

any polynomial degree.

0 1 2 3

…

…

Relevant B-Spline Anatomy• B-Spline basis functions of higher degree can be

made by convolving basis functions of lower degrees:

=

*=

=

*

*

Relevant B-Spline Anatomy

= * * *…

Pth degreebasis function

(P + 1) zeroth degree basis functions

• B-Spline basis functions of higher degree can be made by convolving basis functions of lower degrees:

In general,

Relevant B-Spline Anatomy• A common technique that allows a B-Spline to

efficiently resolve localized detail is known as hierarchical refinement.

Without Hierarchical Refinement:

Projection

Relevant B-Spline Anatomy• A common technique that allows a B-Spline to

efficiently resolve localized detail is known as hierarchical refinement.

With Hierarchical Refinement:

Projection

The Semi-Analytic Fourier Transform

• Since a zeroth degree basis function is a rectangular pulse, we know that its Fourier transform is a sinc function:

F[ ] =

The Semi-Analytic Fourier Transform

• It follows from the convolution property of Fourier transforms that the Fourier transform of a Pth degree basis function is simply the (P+1)th power of a sinc function.

F[ ] = ( )P+1

The Semi-Analytic Fourier Transform• Since a spline function is a linear combination of basis

functions, it follows that its Fourier transform is a linear combination of powers of sinc functions (each term shifted by the appropriate phase factor). This gives us the Semi-Analytic Fourier Transform.

F[ ] = Σλke-ic ω( ) P+1

k

The Semi-Analytic Fourier Transform• Generally, the Semi-Analytic Fourier Transform can’t be

reduced. However, under certain assumptions, it takes a special form. Assume that all of the basis functions have the same width. Then we can do the following:

Σλke-ic ω( ) = ( ) Σλke-ic ω

P+1

k

P+1

k

k

k

The Semi-Analytic Fourier Transform• Generally, the Semi-Analytic Fourier Transform can’t be

reduced. However, under certain assumptions, it takes a special form. Assume that all of the basis functions have the same width. Then we can do the following:

Σλke-ic ω( ) = ( ) Σλke-ic ω

P+1

k

P+1

k

k

k

The Semi-Analytic Fourier Transform• Consider this sum. • Generally, it can’t be reduced.

λke-ic ωkΣk=0

“ck” is the location of the center of the kth basis function in the time domain.

“ω” may represent any point in the frequency domain that we wish to know about.

“λk” is the scaling factor of the kth basis function in the time domain

N-1

“N” is the number of basis functions (and, hence,the number of coefficients).

The Semi-Analytic Fourier Transform• If we make some additional assumptions, though, this sum

takes a familiar form:– Assume the signal is 2π-periodic. Then its Fourier transform is nonzero

only at integer values of ω. So we allow ω = n = 0, 1, 2,…, N-1.– Moreover, assume that the basis functions are uniformly spaced. Then

ck = 2πk/N.– These assumptions reduce the sum to:

λke-i2πkn/N =Σk=0

N-1The Discrete FourierTransform of theλk array.

The Semi-Analytic Fourier Transform• In summary:– Generally, the Semi-Analytic Fourier transform is a

sum of sinc functions, multiplied by phase and scale factors.

– Under special circumstances, it can be thought of as a sinc function multiplied by the DFT of the λk

array. The implications of this are not yet fully understood.

Σλke-ic ω( ) P+1

kk

Strengths of the Semi-Analytic Fourier Transform:

• The Semi-Analytic Fourier Transform is a natural way to evaluate the Fourier transform of IGA outputs.

• The Semi-Analytic Fourier Transform may be used to view any desired interval of the frequency domain.

• The Semi-Analytic Fourier Transform may be used with hierarchically refined spline bases.

Weaknesses of the Semi-Analytic Fourier Transform

• Inevitably, it is more computationally expensive than traditional FFT methods.

• Currently, it is not as well-understood as traditional FFT methods.

Demonstration: Enveloped Sine

Demonstration: Enveloped Spline

Demonstration: Fubini Solution for Burger’s Equation;

hierarchical refinement

Demonstration: Fubini Solution for Burger’s Equation;

hierarchical refinement

Future Work• Explore the relationship between the Semi-

Analytic Fourier Transform and the Discrete Fourier Transform of the coefficient array.

• Application: Nonlinear Acoustics– Some frequency-space metrics have been shown to

be highly dependent on sampling frequencies. Since the Semi-Analytic Fourier Transform is not dependent on sampling frequency in the ordinary sense, it could afford a superior way to compute those metrics.

Future Work

• Another Application: Time Series Signal Processing– There exists an efficient way to project time series

data onto B-spline basis functions; SAFT could allow the real-time computation of the Fourier transform, while allowing non-uniform sampling rates

Questions?