[Introduction]
[Signal
arithmetic] [Signals
and noise] [Smoothing]
[Differentiation]
[Peak
Sharpening] [Harmonic
analysis] [Fourier
convolution] [Fourier
deconvolution] [Fourier
filter] [Peak
area measurement] [Linear
Least Squares] [Multicomponent
Spectroscopy] [Iterative
Curve Fitting] [Hyperlinear
quantitative absorption spectrophotometry] [Appendix
and Case Studies] [Peak
Finding and Measurement] [iPeak]
[iSignal]
[Peak
Fitters] [iFilter]
[iPower]
[List
of downloadable software] [Interactive
tools]

Video Demonstration

This 15-second, 1.7 MByte video in WMV format (ResEnhance3.wmv ) demonstrates this resolution enhancement technique. The signal consists of four overlapping, poorly-resolved Lorentzian bands. First, the 2nd derivative factor (Factor 1) is adjusted, then the 4th derivative factor (Factor 2) is adjusted, then the smooth width (Smooth) is adjusted, and finally the Factor 2 is tweeked again.

**The "enhance" command-line function**

`function
Enhancedsignal=enhance(signal,factor1,factor2,SmoothWidth)`

Basic function for resolution enhancement by the even-derivative
method. The arguments **factor1** and **factor2** are the
weighting factors for the 2^{nd} and 4^{th}
derivatives. **SmoothWidth** is the width of the smoothing
function applied to the derivatives. Optimum values for **factor1**
and **factor2** depend on the width and the shape of the peaks
in the signal, and also on the desired trade-off between
resolution enhancement (peak width reduction) and baseline
artifacts that are a by-product of the method. As a starting
point, a reasonable value for factor1 is PeakWidth^{2}/25
and for factor 2 is PeakWidth^{4}/833 for peaks of
Gaussian shape (or PeakWidth^{2}/6 and PeakWidth^{4}/700
for
Lorentzian peaks), where PeakWidth is the full-width at half
maximum of the peaks expressed in number of data points. The
easiest way to determine the optimum values for your data is to
use iSignal or InteractiveResEnhance, described below.

Interactive Resolution Enhancement using the iSignal
function

iSignal is a Matlab function
that performs resolution enhancement for time-series signals,
using the above enhance function, with keystrokes that allow you
to adjust the 2^{nd} and
4^{th} derivative
weighting factors and the smoothing continuously while observing
the effect on your signal dynamically. View the code here or dowload the ZIP file with sample data
for testing. Just place isignal.m in
the Matlab path and type

>> isignal(DataMatrix); or

>> isignal(x,y); or

>> isignal(y);

where DataMatrix is a matrix with x
values in the first row or column and y values in the
second. Press K
to see all the keyboard commands. Use the cursor arrow
keys to pan and zoom.

The E key turns
the resolution enhancement function on and off. iSignal is
particularly convenient to use because it calculates the
sharpening and
smoothing settings for Gaussian and for Lorentzian peak
shapes using the Y and U keys,
respectively. Just isolate a single typical peak in the
upper window using the pan and zoom keys, then press Y for Gaussian or U for Lorentzian
peaks. (The
optimum settings depends on the width of the peak, so if
your signal
has peaks of widely different widths, one setting will not
be optimum
for all the peaks). You can fine-tune the sharpening with
the F/V and G/B keys and the
smoothing with the A/Z keys. Press K to see all the keyboard
commands.

The graphic example on
the right is the result of the following commands:

>> x=[0:.005:2];y=humps(x);Data=[x;y];

>>
isignal(Data,0.3,0.5,1,3,1,0,1,220,5400);

Press the 'E' key to toggle sharpening ON/OFF to compare before
and after peak sharpening.

There is also an older version (which works only in Matlab version 6.5, but not in more recent versions) that uses sliders rather than keyboard controls. This version, however, does not have the automatic calculation of the sharpening and smoothing settings for Gaussian and for Lorentzian peaks. Click here to download the ZIP file "InteractiveResEnhance.zip" that also includes supporting functions, self-contained demos to show how it works. You can also download it from the Matlab File Exchange.

Interactive optimization of derivative resolution
enhancement for your own data. Requires Matlab 6.5. To use
this, place the data to be enhanced in the global vector
"signal", then execute this file. It plots the data and
displays sliders for separate real-time control of 2^{nd}
and 4^{th} derivative weighting factors (factor and
factor2) and smooth width. (Larger values of factor1 and
factor2 will reduce the peak widths but will cause artifacts
in the baseline near the peak. Adjust the factors for the
best trade-off). Use the minimum smooth width needed to
reduce excess noise. The resolution-enhanced signal is
placed in the global vector "Enhancedsignal". (If the range
of the sliders is inappropriate for your signal, you can
adjust the slider ranges in lines 27-29). |

Self-contained demo of resolution enhancement for a
simulated signal of four overlapping peaks. Requires Matlab
6.5. Displays sliders for separate real-time control of 2^{nd}
and 4^{th} derivative weighting factors (factor and
factor2) and smooth width. Larger values of factor1 and
factor2 will reduce the peak width but will also cause
artifacts in the baseline near the peak. Adjust these
factors for the the best compromise. Use the minimum smooth
width needed to reduce excess noise (too much smoothing will
reduce the resolution enhancement). |

Similar to DemoResEnhance, but for a single Gaussian peak. This allows you to experiment with the adjustable parameters that work best for a peak of Gaussian shape. You can change the width of the peak in line 26. The estimated width of the resolution-enhanced peak is computed and displayed above the graph, to make it easier to determine the extent of resolution enhancement quantitatively. Try to adjust the parameters until the estimated peak width is as small as possible, while still giving acceptable baseline flatness. |

Similar to DemoResEnhance, but for a single Lorentzian peak. Requires Matlab 6.5. This allows you to experiment with the adjustable parameters that work best for a peak of Lorentzian shape. You can change the width of the peak in line 26. The estimated width of the resolution-enhanced peak is computed and displayed above the graph, to make it easier to determine the extent of resolution enhancement quantitatively. Try to adjust the parameters until the estimated peak width is as small as possible, while still giving acceptable baseline flatness. |

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