Suppose we are writing a seismogram modeling program
and we wish to model an impulsive plane wave of unit amplitude.
Say the signal seen at *x* is
.At the plane wave is shifted in time
so that the impulse lies half way between two points,
say it is
.The question is, ``what should be the value of *a*?''
There are three contradictory points of view:

- 1.
- The amplitude
*a*should be 1 so that the peak amplitude is constant with*x*. - 2.
- The amplitude
*a*should be so that both seismic signals have the same energy. - 3.
- The amplitude
*a*should be 1/2 so that both seismic signals have the same area.

What is important
in the signal
is not the high frequencies
especially those near the Nyquist.
We cannot model the continuous universe with sampled data
at frequencies above the Nyquist frequency
nor can we do it well or easily
at frequencies approaching the Nyquist.
For example, at half the Nyquist frequency,
a derivative is quite different from a finite difference.
What we must try to handle correctly is the low frequencies
(the adequately sampled signals).
The above three points of view are contradictory at low frequencies.
Examine only the zero frequency of each. Sum over time.
Only by choosing equal areas *a*=1/2 do the two signals have equal strength.
The appropriate definition of amplitude on
a sampled representation of the continuum is the *area per unit time.*
Think of each signal value as representing the integral of
the continuous amplitude from
to
.Amplitude defined in this way cannot be confounded by
functions oscillating between the sampled values.

Consider the task of abandoning data: We must reduce data sampled at a two millisecond rate to data sampled at a four millisecond rate. A method with aliasing is to abandon alternate points. A method with reasonably effective antialiasing is to convolve with the rectangle (1,1) (add two neighboring values) and then abandon alternate values. Without the antialiasing, you could lose the impulse on the signal. A method with no aliasing is to multiply in the frequency domain by a rectangle function between Nyquist/2 (equivalent to convolving with a sinc function) and then abandoning alternate data points. This method perfectly preserves all frequencies up to the new Nyquist frequency (which is half the original).

12/26/2000