Fourier Interpolation

The Fourier Interpolation option in the Process menu or the Process toolbar offers true interpolation based upon the frequency spectrum. When thresholds are set, this option also performs filtering and reconstruction.

This option presents a dual AutoSignal graph with the frequency domain decomposition in the upper graph and the input (unfiltered) and output (filtered) time domain data in the lower graph.

Interpolation

Fourier interpolation is based upon the frequency spectrum. This means that any size reconstruction will generate the same Fourier spectrum. For interpolation only, the thresholds should remain empty.

Filtering

This procedure is similar to the Fourier Filtering and Reconstruction option in that frequency and spectral thresholds can be set which include or exclude specific FFT channels from the reconstruction.

Reconstruction

In this procedure, the reconstructed data are computed directly from the amplitude, frequency, and phase of the continuous sine components. While this can be appreciably slower than an inverse FFT, this does offer a reconstruction where the data count and limits are variable. In addition to reconstructing the basic data, this option also makes it possible to reconstruct the first, second, third, or fourth derivatives. If a derivative is reconstructed, the thresholding should be specified to sufficiently smooth the data in order to produce a the cleanest possible derivative.

Post-Filtering

When this option is used to increase the data count within the original time interval, the Nyquist frequency will increase. A follow up FFT of the reconstructed data will contain frequencies that lie beyond those in the original FFT. This may be fine, representing a valid extrapolation and gentle decay. On the other hand, all frequencies beyond the original Nyquist can be zeroed by checking the Low Pass post-filter option.

Transform

This procedure fixes the FFT method to the Best Exact N algorithm. It is used to produce the Fourier decomposition and for the forward and inverse transforms in the post-filter procedure. The Best Exact N composite algorithm produces the fastest possible exact n FFT.

Plot

The frequency domain information can be plotted in a variety of formats. In the following table, Re is the real component of the FFT at a given frequency, Im is the imaginary component, n is the data set size, dx is the sampling interval, and var is the variance of the data series.

· Magnitude, sqrt(Re*Re+Im*Im)

· Amplitude, 2.0*sqrt(Re*Re+Im*Im)/n

· Phase, sine-based, Pi/2+atan(Im/Re)

· Variance, Power normalized by variance, (Re*Re+Im*Im)/n/var

· dB, decibels, 10.0*log10(Re*Re+Im*Im)

· dB Norm, decibels, normalized to 0 for frequency channel with maximum power

· PSD SumSq, Power as Sum Squared Amplitude, 2.0*(Re*Re+Im*Im)/n

· PSD MeanSq, Power as Mean Squared Amplitude, 2.0*(Re*Re+Im*Im)/n/n

· PSD TimeInt, Power as Time-Integral Squared Amplitude, 2.0*dx*(Re*Re+Im*Im)/n

In an amplitude plot, you see the actual amplitude of sine components. In a normalized decibel plot, the highest peak is at 0dB, a peak at -3dB would have half the power, and a peak at -6dB would have half the amplitude. The PSD TISA (time-integral squared amplitude power) is the actual integral under the curve defined by the square of the raw data.

If the data set size is 512 or lower (257 or lower pts in the frequency spectrum), a bar format is automatically used. Otherwise, the points in the spectrum will be rendered directly with lines connecting the points. You can override either of these point formats using the Modify Point Format option and their respective states will be saved across sessions.

Frequency Thresholds

There are several ways to set the thresholds for Fourier filtering. The first is to define a range of frequency space. This is primarily used to isolate and reconstruct signal components. The Freq1 and Freq2 fields set limits in frequency. The frequency boundaries of this region can either be included (Incl) or excluded (Excl). If a frequency range is included, this means that only the frequencies that fall within this range are used in the data reconstruction. If the region is excluded, only those frequencies that fall outside the region are used for the reconstruction.

The frequencies can be entered numerically. If both fields for frequency are left blank, the full range is assumed for the Incl option, and no range is specified for the Excl option. If only a lower limit is entered, the variable range is assumed to be from this limit and greater for the Incl option, and all values below this limit if the Excl option is set. If only an upper limit is entered, the variable range is assumed to be from this limit and lower for Incl, and all values above this limit for Excl.

It is generally easier to set a frequency range graphically. When the X Sectioning mode is selected, the mouse is used in the upper graph to select the frequency range of interest. Frequencies are toggled off by holding the left mouse button down and sliding right and toggled back on by moving left. Individual frequencies can be toggled on and off by placing the cursor at the point (or top of a bar) and left clicking the mouse.

Spectral Thresholds

Another way to set the thresholds for Fourier filtering involves defining spectral thresholds. This is primarily used to separate signal components by power. It can also be used for noise removal, replicating the functionality in the Fourier Smoothing and Denoising option, or for signal removal, generating a signal-free background for subsequent analysis.

The Spec1 and Spec2 fields set the spectral limits. These can only be entered numerically. If only Spec1 is set, all values at or above this threshold will be used in the reconstruction if Incl is set, and all below this threshold will be used if Excl is checked. If only Spec2 is set, all values at this threshold and below will be used in the reconstruction if Incl is set, and all above this threshold will be used if Excl is set. For this option, the normalized dB values must use the minus sign.

Frequency-Spectral Thresholds

The thresholds for Fourier filtering can also consist of a frequency-spectrum region. This allows signal components to be isolated by both frequency and spectral power. This is the most effective way to isolate and reconstruct a single signal component when some number of components are present.

The Freq1 and Freq2 fields set limits in frequency while the Spec1 and Spec2 fields set the spectral limits. The values can be entered numerically. If only a lower limit field is set, all values at or above this threshold will be used in the reconstruction if Incl is set, and all below this threshold will be used if Excl is checked. If only the upper limit is set, all values at this threshold and below will be used in the reconstruction if Incl is set, and all above this threshold will be used if Excl is set. Again, the normalized dB values must use the minus sign.

It is generally easier to set a frequency-spectrum region graphically. When the XY Sectioning mode is selected, the mouse is used in the upper graph to enclose the frequency-spectral region of interest. To include the region that is to be graphically defined, click and hold down the left mouse button at one corner of the frequency-spectral rectangle in the upper graph and drag the rectangle so that it encloses the region desired. This will automatically enter both the frequency and spectral limits and set the process state to include the FFT channels located within the region.

To exclude the region that is to be graphically defined, the same graphical procedure is followed except that the right rather than the left mouse button is used. In this case, the process state will be set to exclude the FFT channels within the region. This is useful for reconstructing data with a single component removed, such as line noise.

Note that it is possible to reconstruct only one frequency-spectral region at a time.

The Clear button is used to clear all of the threshold fields.

The Apply button can be used to immediately update the processing rather than wait for the automatic update.

Popup Information

The Toggle Popup Information Window option is used to report estimated noise and power reductions as well as a correlation coefficient between the filtered and unfiltered data.

AutoSignal offers a robust noise estimation procedure that may be of some value for low-frequency signals. A cubic polynomial interpolation is made for each point using the two points to the left and the two to the right (excluding the current point). The difference between the interpolated and signal values is used to generate a measure of the white noise present in the signal. This assumes that the signal can be locally characterized by a smooth cubic interpolant. Also, the signal component(s) should exist only in the lower quarter of the Nyquist range. If a high frequency signal component is present, these estimates of noise will be invalid. The Noise In value reports the estimated white noise in the incoming data, the Noise Out value the estimated white noise for the filtered signal. The Noise % is given as the amount of estimated noise remaining after filtration.

The TISA In value reports the TISA power in the incoming data. The TISA Out value is the TISA power for the filtered signal. The TISA % is given as the amount of power remaining after filtration.

The r-squared correlation coefficient is also reported. An rē of 1 is a perfect correlation while a value of 0 means the filtered and unfiltered signals are completely uncorrelated.

Output Data

Since the Fourier decomposition can be expressed as continuous infinite-length phase-bearing sinusoids, it is possible to generate interpolated output for any number of values (n) and any range of times (Time1 to Time2). Since the Fourier reconstruction reproduces the same data series outside the range of original data, there is no value in generating extrapolations. This procedure cannot be used for prediction.

In addition to reconstructing the sinusoidal basis functions (Function), it is also possible to reconstruct the first through fourth derivatives (1st Deriv, 2nd Deriv, 3rd Deriv, 4th Deriv). If this option is used to generate derivative information, it is important to remove as much noise as possible by setting the appropriate thresholds.

The output data stream will be limited to 16384 values.

Post-Filter

If the Low Pass item is checked, an additional Fourier-based low pass filtration is performed on the output data. All frequencies greater than the Nyquist limit of the original data series will be zeroed and an inverse FFT will generate the final output. Note that the additional frequency information arising from interpolating a higher data density is often a valid extrapolation. Unlike the Fourier Upsampling procedure, this post filter is optional.

List

The List Data option lists the index, time, and output signal in a three column table. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the time and output signal values to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors).

Save

The Save Data to Disk option writes the time and output values to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat.

Production Facility

The AutoSignal Automation facility allows unattended processing of large numbers of data sets. The data sets can be consolidated in an Excel file or acquired using a DLL. The graphs can be exported to an MS Word RTF file, while the processed data can be exported to an Excel 95 or Excel 97 file.

Evaluation

The Evaluation option offers the means to more extensively evaluate the sum of sinusoids model generated by the Fourier decomposition. The evaluation can be be direct, or the computation can consist of first or second derivatives, roots, or integrated areas. This option can be used to generate a table or file of any size using a generated X grid or by importing X data from supported file formats.

Local Options

A local option changes the data set for the duration of the current procedure only. The main data table is not altered. AutoSignal offers the following local options for this procedure:

Section the data to isolate specific regions for processing.

Detrend for removing mean or subtracting a least-squares trend model.

Eigendecomposition Filtration for isolating spectral components by signal strength.

The Reset button restores the data to its state when first entering the procedure. Note that if you implement sequential local procedures, all of the revisions are discarded upon reset. If an Automation Session is in progress, the Reset button can be used to terminate the automated processing.

When exiting this procedure with the OK button, an option will be presented to update AutoSignal's main data table with the filtered/interpolated data.