Package xal.tools.dsp

Class FourierSineTransform

java.lang.Object

xal.tools.dsp.FourierSineTransform

public class FourierSineTransform
extends Object

This class encapsulates many aspects of the Fourier sine transform. Although this is a discrete transform, it is not a Fast Fourier Transform (FFT). Specifically, we do not fully exploit the dyadic nature of the transform kernel. In fact, we explicitly compute this kernel which, in the discrete version, is a matrix. Thus, this transform can potentially be quite expensive for large data sets. What we have here then, is essentially a convenient wrapper around a real, symmetric matrix which makes this class look like a magical transformer.

The advantages of the sine transform are that it only involves real numbers (i.e., primitives of type double). The transform is somewhat generalized in that we assume a base frequency of π rather than 2π so that we can transform even functions as well as odd ones. (The FFT has "positive" and "negative" frequencies.) The only caveat is that the function f being transformed is assumed to have value zero at the boundaries (as does the sine kernel). There are two approaches here, either we pad the given function by zero (adding two data points), or we construct the transform so that the recovered function (via inverse transform) always has a zero values at the boundaries. Here we choose the later, preferring not to change the dimensions of the data.

The transform performed here is given by

[f^] = [K]·[f]

where [f^] is the vector of transformed data, [K] is the real symmetric matrix kernel, and [f] is data vector of input function values. The elements K_mn of the matrix kernel are given by

K_mn = (2/(N-1))^½ sin πmn/(N-1)

where N is the size of the data vector [f], and indices m, n range over the values 0,…,N. The factor (2/(N-1))^½ is a normalization constant; specifically, the value of the L₂ norm ||sin πn/(N-1)||. It's presence ensures the dual nature of the transform, specifically [K].[K] = [I] where [I] is the identity matrix (sans the initial and final 1 entry).

From the value of K_mn we see that the stride in [f^] is 1/2T, where T is the period of the interval over which f is defined. Thus, the largest frequency we can see is (N-1)/2T. Also, we see that there is no specific DC component. That is, the DC (or zero-frequency) component of the signal has a projection upon all the odd-frequency components in this general Fourier sine transform. Thus, it cannot be identified separately as in the traditional Fourier transform.

Author:

Christopher K. Allen

Constructor Summary

Constructors

Constructor	Description
FourierSineTransform(int szData)	Create a new sine transform object for transforming data vectors of size szData.

Method Summary

Modifier and Type	Method	Description
double	compFreqStrideFromInterval(double dblDelta)	Compute and return the value of the frequency stride for this transform given the time stride (time interval between data points).
double	compFreqStrideFromPeriod(double dblPeriod)	Compute and return the value of the frequency stride for this transform given the total time period over which the data is taken.
int	getDataSize()	Return the expected size of the data, which is also the dimensions of the kernel.
double[]	powerSpectrum(double[] arrFunc)	Compute and return the discrete power spectrum for the given function.
double[]	transform(double[] arrFunc)	Compute and return the Fourier sine transform of the given function.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

FourierSineTransform

public FourierSineTransform(int szData)

Create a new sine transform object for transforming data vectors of size szData. During construction the szData×szData transform matrix kernel is built. Upon completion the returned transform object is able to transform any double[] object of the appropriate size.

Parameters:

szData -

See Also:

• transform(double[])

Method Details

getDataSize

public int getDataSize()

Return the expected size of the data, which is also the dimensions of the kernel.

Returns:

size of the transform vectors (that is, the value N)

compFreqStrideFromPeriod

public double compFreqStrideFromPeriod(double dblPeriod)

Compute and return the value of the frequency stride for this transform given the total time period over which the data is taken.

Parameters:

dblPeriod - total length of the data window

Returns:

frequency interval (stride) between transformed data points

compFreqStrideFromInterval

public double compFreqStrideFromInterval(double dblDelta)

Compute and return the value of the frequency stride for this transform given the time stride (time interval between data points).

Parameters:

dblDelta - time interval between data points

Returns:

frequency interval (stride) between transformed data points

transform

public double[] transform(double[] arrFunc)
                   throws IllegalArgumentException

Compute and return the Fourier sine transform of the given function. Note that this transform is essential dual to itself, thus, the transform and the inverse transform are the same operation. This fact follows from the normalization used so that the transform matrix is essentially a root of unity.

The returned values are ordered so that the lowest frequency components come first. That is, the components are indexed according to their discrete frequency. Note also that the zero-frequency component of a sine transform is identically zero, as is the N^th component. Thus, the first and last values will always be zero.

Parameters:

arrFunc - vector array of function values (zero values on either end)

Returns:

vector array of transformed value

Throws:

IllegalArgumentException - invalid function dimension

powerSpectrum

public double[] powerSpectrum(double[] arrFunc)
                       throws IllegalArgumentException

Compute and return the discrete power spectrum for the given function. The power spectrum is the square of the frequency spectrum and, therefore, is always positive.

The returned values are ordered so that the lowest frequency components come first. That is, the components are indexed according to their discrete frequency. Note also that the zero-frequency component of a sine transform is identically zero, as is the N^th component. Thus, the first and last values will always be zero.

Parameters:

arrFunc - discrete function

Returns:

discrete power spectrum of given function

Throws:

IllegalArgumentException - invalid function dimension