"Data" may be a 2-column matrix with the independent
variable (x-values) in the first column and dependent variable (y
values) in the second column, or separate x and y vectors, or a
single y-vector (in which case the data points are plotted against
their index numbers on the x axis). Only the first argument (Data)
is required; all the others are optional. Returns the processed
dependent axis ('pY') vector (and, in the Spectrum Mode, the frequency
spectrum matrix, 'Spectrum') as the output arguments. Plots
the data in the figure window, the lower half of the window showing
the entire signal, and the upper half showing a selected portion
controlled by the pan and zoom keys (the four cursor arrow keys),
with the initial pan and zoom settings optionally controlled by
input arguments 'xcenter'
and 'xrange', respectively.
Other keystrokes allow you to control the smooth type, width, and
ends treatment, the derivative order (0^{th} through 5^{th}),
and peak sharpening. (The initial values of all these parameters can
be passed to the function via the optional input arguments SmoothMode, SmoothWidth, ends, DerivativeMode, Sharpen,
Sharp1, Sharp2, SlewRate, and MedianWidth.
See the examples below). Press K
to see all the keyboard commands. Note:
Make sure you don't click on the “Show Plot Tools” button in
the toolbar above the figure; that will
disable normal program functioning. If you do; close the
Figure window and start again.
Smoothing
The S key (or input
argument "SmoothMode")
cycles through five smoothing modes:
If SmoothMode=0,
the signal is not smoothed.
If SmoothMode=1,
rectangular (sliding-average or boxcar)
If SmoothMode=2,
triangular (2 passes of sliding-average)
If SmoothMode=3,
pseudo-Gaussian (3 passes of sliding-average)
If SmoothMode=4,
Savitzky-Golay
smooth (thanks to Diederick).
The A and Z keys (or optional input
argument SmoothWidth)
control the SmoothWidth, w. Example shown in the figure on the right.
The X key toggles "ends" between 0 and 1. This
determines how the "ends" of the signal (the first w/2 points and the last w/2 points) are handled when
smoothing:
If ends=0,
the ends are zero.
If ends=1,
the ends are smoothed with progressively smaller smooths the closer
to the end. Generally, ends=1
is best, except in some cases using the derivative mode when ends=0 result in better vertical
centering of the signal.
Note: when you are smoothing peaks, you can easily measure the
effect of smoothing on peak height and width by turning on peak
measure mode (press P)
and then press S to
cycle through the smooth modes.
There are two
special functions for removing or reducing sharp spikes in
signals: the M key, which implements a median filter (it asks
you to enter the spike width, e.g. 1,2, 3... points) and
the ~
key, which limits the maximum rate of change.
Reference: http://terpconnect.umd.edu/~toh/spectrum/Smoothing.html
Differentiation
The D
/ Shift-D
keys (or optional input argument “DerivativeMode”)
increase/decrease
the derivative order. The default is 0. Careful optimization
of smoothing of derivatives is critical for acceptable
signal-to-noise ratio. Example
shown in the figure on the right. In SmoothModes 1 through 3, the
derivatives are computed with respect to the independent variable
(x-values), corrected for non-uniform x axis intervals. In
SmoothMode 4 (Savitzky-Golay) the derivatives are computed by the
Savitzky-Golay algorithm.
Reference: http://terpconnect.umd.edu/~toh/spectrum/Differentiation.html
Peak
sharpening (resolution enhancement)
The E key (or optional
input argument "Sharpen")
turns off and on peak sharpening (resolution enhancement). The
sharpening strength is controlled by the F and V
keys (or optional input argument "Sharp1")
and B and G keys (or optional argument "Sharp2"). The optimum values
depend on the peak shape and width. For peaks of Gaussian shape, a reasonable value for Sharp1 is PeakWidth^{2}/25
and for Sharp2 is PeakWidth^{4}/800 (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.
However, you don't need to do the math yourself; iSignal can calculate
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 Yfor
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.
You can expect a decrease in peak width (and corresponding
increase in peak height) of about 20% - 50%, depending on the
shape of the peak (the peak area is largely unaffected by
sharpening). Excessive sharpening leads to baseline
artifacts and increased noise. iSignal allows you to experimentally determine
the values of these parameters that give the best trade-off
between sharpening, noise, and baseline artifacts, for your
purposes. You can easily
measure the effect of sharpening quantitatively by turning on
peak measure mode (press P)
and then press E to
toggle the sharpen mode off and on. Note: only
the Savitzky-Golay smooth mode is used for peak sharpening. Example
shown in the figure on the right.
Reference: http://terpconnect.umd.edu/~toh/spectrum/ResolutionEnhancement.html
Signal
measurement
The cursor keys control the position of the green cursor
(left and right arrow keys) and the distance between the dotted red
cursors (up and down arrow keys) that mark the selected range
displayed in the upper graph window. The label under the top graph
window shows the value of the signal (y) at the green cursor, the
peak-to-peak (min and max) signal range, the area under the signal,
and the standard deviation within the selected range (the dotted
cursors). Pressing the Q
key prints out a table of the signal information in the command
window.
Signal is measured by placing the green
center cursor on top of the peak
Noise is the standard deviation measured
on a flat portion of the baseline
Signal-to-noise ratio (SNR) measurement of a signal with very high
SNR. Left: The peak height of the largest signal peak is
measured by placing the green center cursor on the largest peak;
peak-to-peak signal=66,000. Right: The noise is measured on a flat
portion of the baseline: standard deviation of noise=0.01,
therefore the SNR=66,000/.01 = 6,600,000
If the optional output arguments maxy, miny, area,
stdev are specified, isignal returns the maximum value
of y, the minimum value of y, the total area under the curve,
and the standard deviation of y, in the selected range displayed
in the upper panel. The demo script demoisignal42.m
demonstrates this version.
Frequency
Spectrum mode
The Frequency Spectrum mode,
toggled on and off by the Shift-S
key, computes the Fourier
frequency spectrum of the segment of the signal displayed in
the upper window and displays it in the lower window (temporarily
replacing the full-signal display). Use the pan and zoom keys to
adjust the region of the signal to be viewed. Press Shift-A to cycle through four
plot modes (linear, semilog X, semilog Y, or log-log) and press Shift-X to toggle between a frequency on the x axis
and time on the
x-axis. All signal processing functions remain active in
the frequency spectrum mode (smooth, derivative, etc) so
you can observe the effect of these functions on the frequency
spectrum of the signal, as in the animated figure on the right.
Press Shift-S again to
return to the normal mode. Spectrum mode is a visible mode,
indicated by the label at the top of the figure. To start off
in the spectrum mode, set the 13th input argument, SpectrumMode, to
1. To
save the spectrum as a new variable, call isignal
with the output arguments [pY,Spectrum]:
>> x=0:.1:60; y=sin(x)+sin(10.*x); >>
[pY,Spectrum]=isignal([x;y],30,30,4,3,1,0,0,1,0,0,0,1); >> plot(Spectrum) or isignal(Spectrum);
or ipf(Spectrum); or ipeak(Spectrum);
or whatever.
Shift-Z toggles on and off peak detection and labeling on the
frequency/time spectrum; peaks are labeled with their frequencies.
You can adjust the peak detection parameters in lines 2192-2195 in
version 5; see http://terpconnect.umd.edu/~toh/spectrum/PeakFindingandMeasurement.htm.
The Shift-W command displays the 3D waterfall
spectrum, by dividing up the signal into segments and
computing the power spectrum of each segment. This is
mostly a novelty, but it may be useful for signals whose
frequency spectrum varies over the duration of the signal. You are
asked to choose the number of segments into which to divide the
signal (that is, the number of spectra) and the type of 3D display
(mesh, contour, surface, etc).
Background subtraction
There are two ways to subtract the background from the signal:
automatic and manual. To select an automatic baseline correction
mode, press the T key
repeatedly; it cycles thorough four modes: No baseline
correction, linear baseline subtraction, quadratic baseline
subtraction, flat baseline correction, then back to no
baseline correction. When baseline mode is linear,
a straight-line baseline connecting the two ends of the signal
segment in the upper panel will be automatically
subtracted. When baseline mode is quadratic, a
parabolic baseline connecting the two ends of the signal segment in
the upper panel will be automatically subtracted. The baseline is
calculated by computing a linear (or quadratic) least-squares fit to
the signal in the first 1/10th of the points and the
last 1/10th of the points. Try to adjust the pan and zoom
to include some of the baseline at the beginning and end of the
segment in the upper window, allowing the automatic baseline
subtract gets a good reading of the baseline. The flat baseline
mode is used only for peak fitting. The
calculation of the signal amplitude, peak-to-peak signal, and peak
area are all recalculated based on the baseline-subtracted
signal in the upper window. If you are measuring peaks superimposed
on a background, the use of the autozero mode will have a big effect
on the measured peak height, width, and area, but very little
effect on the peak x-axis position, as demonstrated by these two
figures.
In addition to the four autozero baseline subtraction modes for peak
measurement, a piecewise linear baseline can be subtracted manually
from the entire signal
in one operation. The Backspace
key starts background correction operation. In the command
window, type in the number of background points to click and press
the Enter key. The cursor
changes to crosshairs; click along the presumed background in the
figure window, starting to the left of the x axis and placing the
last click to the right of the x axis. When the last point is
clicked, the linearly interpolated baseline between those points is
subtracted from the signal. To restore the original background (i.e.
to start over), press the '\'
key (just below the backspace key).
Peak and valley
measurement
The P key toggles off and
on the "peak parabola" mode, which attempts to measure the one peak
(or valley) that is centered in the upper window under the green
cursor by superimposing a least-squares
best-fit parabola, in red, on the center portion of the signal
displayed in the upper window. (Zoom in so that the red overlays
just the top of the peak or the bottom of the valley as closely as
possible). The peak position, height, and "Gaussian width" are
measured by least-squares curve fitting of a parabola to the
isolated peak. The "RSquared" value is the coefficient of
determination; the closer to 1.000 the better. The peak widths will
most accurate if the peaks are Gaussian; other shapes, and very
noisy peaks of any shape, will give only approximate results.
However, the position, height, and area values are pretty good for
any peak shape as long as the "RSquared" value is at least 0.99. The
"total area" is measured by the trapezoidal method over the entire
selected segment displayed in the upper window. The "SNR" is the signal-to-noise-ratio
of the peak under the green cursor; it's the ratio of the peak
height to the standard deviation of the residuals between the data
and the best-fit line in red. Example shown in the figure on the
right. If the peaks are superimposed on a non-zero background,
subtract the background before measuring peaks, either by using the
autozero mode (T key) or the
multi-point background subtraction (backspace key). Press the R key to print out the peak
parameters in the command window.
Peak area is measured two ways: the "Gaussian area" and the
"Total area". The "Gaussian area" is the area under the
Gaussian that is best fit to the center portion of the signal
displayed in the upper window, marked in red. The "Total area"
is the area by the trapezoidal method over the entire selected
segment displayed in the upper window. (In version 4.3, the percent
of total area is also calculated). If the portion of the signal
displayed in the upper window is a pure Gaussian with no noise and a
zero baseline, then the two measures should agree almost exactly.
If the peak is not Gaussian in shape, then the total area is
likely to be more accurate, as long as the peak is well separated
from other peaks. If the peaks are overlapped, but have a
known shape, then peak fitting (Shift-F) will give more
accurate peak areas.
Peak
fitting (NEW PROCEDURE in version 5) iSignal has an iterative
curve fitting method performed by peakfit.m. This is the most
accurate method for the measurements of the areas of highly
overlapped peaks. Press the Shift-F
key, then type the desired peak shape by number from the menu
displayed in the Command window (shown below), enter the number of
peaks, enter the number of repeat trial fits (usually 1-10), and
finally click the mouse pointer on the top graph where you think
the peaks might be. A graph of the fit is displayed in
Figure window 2 and a table of results is printed out in the command
window. Version 5 of iSignal can fit many different combinations of
peak shapes and constraints:
Gaussians: y=exp(-((x-pos)./(0.6005615.*width)) .^2) Gaussians with independent positions and
widths...................1 (default) Exponentially-broadened Gaussian (equal time
constants)...........5 Exponentially-broadened equal-width
Gaussian......................8 Fixed-width exponentionally-broadened
Gaussian...................36 Exponentially-broadened Gaussian (independent time
constants)....31 Gaussians with the same
widths....................................6 Gaussians with preset fixed
widths...............................11 Fixed-position
Gaussians.........................................16 Asymmetrical Gaussians with unequal half-widths on
both sides....14 Lorentzians:
y=ones(size(x))./(1+((x-pos)./(0.5.*width)).^2) Lorentzians with independent positions and
widths.................2 Exponentially-broadened
Lorentzian...............................18 Equal-width
Lorentzians...........................................7 Fixed-width
Lorentzian...........................................12 Fixed-position
Lorentzian........................................17 Gaussian/Lorentzian blend (equal
blends)...........................13 Fixed-width Gaussian/Lorentzian
blend............................35 Gaussian/Lorentzian blend with independent
blends)...............33 Voigt profile with equal
alphas)...................................20 Fixed-width Voigt profile with equal
alphas......................34 Voigt profile with independent
alphas............................30 Logistic: n=exp(-((x-pos)/(.477.*wid)).^2);
y=(2.*n)./(1+n).........3 Pearson:
y=ones(size(x))./(1+((x-pos)./((0.5.^(2/m)).*wid)).^2).^m..4 Fixed-width
Pearson..............................................37 Pearson with independent shape factors,
m........................32 Breit-Wigner-Fano..................................................15 Exponential pulse:
y=(x-tau2)./tau1.*exp(1-(x-tau2)./tau1)..........9 Alpha function:
y=(x-spoint)./pos.*exp(1-(x-spoint)./pos);.........19 Up Sigmoid (logistic function):
y=.5+.5*erf((x-tau1)/sqrt(2*tau2)).10 Down Sigmoid
y=.5-.5*erf((x-tau1)/sqrt(2*tau2))....................23 Triangular.........................................................21
Polynomial fitting. Shift-o fits a simple polynomial (linear, quadratic, cubic,
etc) to the upper panel segment and displays the coefficients (in
descending powers) and the correlation coefficient R^{2}
Saving the results
To save the processed signal to the disc as a x.y matrix in a.mat
file, press the 'o' key, the type in the desired file name, then
press Enter or click Save.
Other keystroke
controls
The Shift-G key toggles on and off a temporary grid on the
upper and lower panel plots. The L
key toggles off and on the Overlay mode, which shows the original
signal as a dotted line overlaid on the current processed signal,
for the purposes of comparison. The tab key restores the original signal and cursor
settings. The ";" key sets
the selected region to zero (to eliminate artifacts and spikes). The
"-" (minus sign) key is used to negate the signal (flip + for
-). Press H to toggle
display of semilog y plot in the lower window, which is useful for
signals with very wide dynamic range, as in the example in the
figures below (zero and negative points are ignored in the log
plot). Press '+' key
to take the absolute value of the entire signal (and follow this by
a smooth to create an amplitude
modulation "detector").
Linear y-axis mode
Log y mode (H key)
The C key condenses the
signal by the specified factor n, replacing each group of n points
with their average (n must be an integer, such as 2,3, 4, etc). The I key replaces the signal with a
linearly interpolated version containing m data
points. This can be used either to increase or decrease the x-axis
interval of the signal or to convert unevenly spaced values to
evenly spaced values. After pressing C or I, you
must type in the value of n or m respectively.
You can press Shift-C, then click on the graph to print out
the x,y coordinates of that point. This works on both the
upper and lower panels, and on the frequency spectrum as well. Playing data as audio.
Press Spacebar or Shift-P to play the segment of
the signal displayed in the upper window as audio through the
computer's sound output. Press Shift-R to set the sampling
rate - the larger the number the shorter and higher-pitched will be
the sound. The default rate is 44000Hz. Sounds or music files in WAV
format can be loaded into Matlab using the built-in "wavread"
function. The example on the right shows a 1.5825 sec duration audio
recording of the phrase "Testing, one, two, three" recorded at 44000
Hz, saved in WAV format, loaded into iSignal and zoomed in on the
"oo" sound in the word "two". Press Spacebar to play the
selected sound; press Shift-S to display the frequency spectrum of the
selected region, and press Shift-Z to label the peaks in the
frequency spectrum with their frequencies (graphic).
>> v=wavread('TestingOneTwoThree.wav'); >> t=0:1/44001:1.5825; >> isignal(t,v(:,2));
Press Shift-R and type 44000 to set the sampling rate.
This recorded sound example allows you to experiment with the effect
of smoothing, differentiation, and interpolation on the sound of
recorded speech. Interestingly, different degrees of smoothing and
differentiation will change the timbre of the
voice but has little effect on the intelligibility. This is
because the sequence of frequency components in the signal is not
shifted in pitch or in time but merely changed in amplitude by
smoothing and differentiation. Even computing the absolute value
(+ key), which effectively doubles the fundamental frequency, does
not make the sound unintelligible.
Shift-Ctrl-F transfers the current signal to
Interactive Peak Fitter (ipf.m) and Shift-Ctrl-P
transfers the current signal to Interactive Peak Detector
(iPeak.m), if those functions are installed in your Matlab path.
Press K to see all the
keyboard commands.
EXAMPLE 1:
Single input argument; data in a two columns of a matrix [x;y] or in
a single y vector
>> isignal(y);
>> isignal([x;y]);
EXAMPLE 2: Two input
arguments. Data in separate x and y vectors.
>> isignal(x,y);
EXAMPLE 3: Three or four
input arguments. The last two arguments specify the initial values of pan (xcenter) and
zoom (xrange) in the last two input arguments. Using data in the ZIP
file:
>> load data.mat
>> isignal(DataMatrix,180,40); or
>> isignal(x,y,180,40);
EXAMPLE 4: As above, but
additionally specifies initial values of SmoothMode, SmoothWidth,
ends, and DerivativeMode in the last four input arguments.
>> isignal(DataMatrix,180,40,2,9,0,1); EXAMPLE 5: As above, but
additionally specifies initial values of the peak sharpening
parameters Sharpen, Sharp1, and Sharp2 in the last three input
arguments. Press the E
key to toggle sharpening on and off for comparison.
>>
isignal(DataMatrix,180,40,4,19,0,0,1,51,6000);
EXAMPLE 6:
Using the built-in "humps" function: >>
x=[0:.005:2];y=humps(x);Data=[x;y];
4th derivative of the peak at x=0.9: >>
isignal(Data,0.9,0.5,1,3,1,4); (shown on the right)
Peak sharpening applied to the peak at x=0.3: >>
isignal(Data,0.3,0.5,1,3,1,0,1,220,5400);
(Press 'E' key to toggle sharpening ON/OFF to compare)
EXAMPLE 7: Measurement of
peak area. This example generates four Gaussian peaks, all
with the exact same peak height (1.00) and area (1.77). The first
peak (at x=4) is isolated, the second peak (x=9) is slightly
overlapped with the third one, and the last two peaks (at x= 13 and
15) are strongly overlapped. To measure the area under a peak
using the perpendicular drop method, position the dotted red marker
lines at the minimum between the overlapped peaks.
Greater accuracy in peak area
measurement using iSignal can be obtained by using the peak sharpening function to
reduce
the overlap between the peaks. This reduces the peak widths,
increases the peak heights, but has no effect on the peak
areas.
EXAMPLE 8: Single peak with random spikes (shown in the figure on the right). Compare smoothing vs spike filter (M key) and slew rate
limit (~ key) to remove spikes.
x=-5:.01:5; y=exp(-(x).^2); for n=1:1000, if randn()>2,y(n)=rand()+y(n), end, end; isignal(x,y);
EXAMPLE 9:
The demo function isignaldemo2
creates a test signal containing three peaks with heights 1,
2, and 3, but with equal widths, superimposed on a very
strong curved baseline, plus added random white noise. The
objective is to extract a measure that is proportional to
the peak height but independent of the baseline strength.
Suggested approaches: (a) Use automatic or manual baseline
subtraction to remove the baseline; or (b) use
differentiation (with smoothing) to suppress the baseline.
EXAMPLE 10:
Direct entry into frequency spectrum mode, plotting returned
frequency spectrum.
>> x=0:.1:60; y=sin(x)+sin(10.*x); >>
[pY,SpectrumOut]=isignal([x;y],30,30,4,3,1,0,0,1,0,0,0,1); >> plot(SpectrumOut)
EXAMPLE 11:
The demo script demoisignal.m
is a self-running demo that requires iSignal 4.2 or later and the
latest version of plotit.m to be
installed.
KEYBOARD CONTROLS (Version 5.1): Pan left and
right..........Coarse pan: <
and >
Fine
pan: left and right cursor arrows
Nudge:
[ and ]
Zoom in and out.............Coarse zoom: / and "
Fine
zoom: up and down cursor arrows
Resets pan and zoom.........ESC
Select entire signal........Ctrl-A Display Grid (on/off).......Shift-G
Temporarily displays grid on
the plots Adjust smooth width.........A,Z (A=>more, Z=>less) Adjust smooth
type..........S (cycles
through None, Rectangular, Triangle, Gaussian,and Savitzky-Golay)
Toggle smooth ends..........X (0=ends zeroed 1=ends smoothed (slower)
Adjust derivative order.....D/Shift-D Increase/Decrease
derivative
order
Toggle peak sharpening......E (0=OFF 1=ON)
Sharpening for Gaussian.....Y Set sharpen settings for Gaussian
Sharpening for Lorentzian...U Set sharpen settings for Lorentzian
Adjust sharp1...............F,V
F=>sharper, V=>less sharpening
Adjust sharp2 ............G,B G=>sharper, B=>less sharpening
Slew rate limit (0=OFF).....~ Largest allowed y change between points
Spike filter width (0=OFF)..Mmedian
filter eliminates sharp spikes
Toggle peak parabola........P fits parabola to center, labels vertex
Fits peak in upper window...Shift-F (Asks for shape, number of peaks,
Number of trials, etc)
Fit polynomial..............Shift-o Fits
polynomial to data in
upper panel
Spectrum mode on/off........Shift-S (Shift-A
and Shift-X to change
axes)
Peak labels on spectrum.....Shift-Z in
spectrum mode
Display Waterfall spectrum..Shift-W
Allows choice of mesh, surf,
contour, or pcolor
Click graph to print x,y....Shift-C
Click graph to print coordinates
Print peak report...........R prints position, height, width, area
Toggle overlay mode.........L Overlays original signal as dotted line
Toggle log y mode...........H semilog plot in lower window Cycle baseline mode.........T none,
linear, quadratic, or flat
baseline
mode
Restores original signal....Tab key resets to original signal
and modes
Baseline subtraction........Backspace, then click baseline at
multiple points
Restore background..........\ to cancel previous background
subtraction
Invert signal...............- Invert (negate) the signal (flip + / -)
Remove offset...............0 (zero) set minimum signal to zero
Sets region to zero.........; sets selected region to zero.
Absolute value..............+ Computes absolute
value of entire
signal
Condense signal.............C Condense oversampled signal by factor N
Interpolate signal..........i Interpolate (re-sample) to N points
Print keyboard commands.....K prints this list
Print signal report.........Q prints signal info and current settings
Print isignal arguments.....W prints isignal (current arguments)
Save output to disk.........O as .mat file with processed signal
matrix and (in spectrum mode) the
frequency spectrum.
Play signal as sound........Spacebar or
Shift-P Play upper panel segment through computer sound system
Set sound sample rate.......Shift-R for the
Shift-P command.
Switch to ipf.m.............Shift-Ctrl-FTransfer current signal
to
Interactive Peak Fitter Switch to
iPeak.............Shift-Ctrl-PTransfer current signal
to
Interactive Peak Detector Earlier versions
Earlier versions of iSignal as also available:1.5, 1.6,
1.7, 1.8,
1.9,2.0, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 4.5.
ProcessSignal, a Matlab/Octave
command-line function that performs smoothing and
differentiation on the time-series data set x,y (column or row
vectors). Type "help ProcessSignal". Returns the processed
signal as a vector that has the same shape as x, regardless of
the shape of y. The syntax is
Processed=ProcessSignal(x, y, DerivativeMode, w, type, ends,
Sharpen, factor1, factor2, SlewRate, MedianWidth)
January, 2016. This page is part of "A
Pragmatic Introduction to Signal Processing", created
and maintained by Prof.
Tom O'Haver , Department of Chemistry and Biochemistry, The
University of Maryland at College Park. Comments, suggestions and
questions should be directed to Prof. O'Haver at toh@umd.edu.