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Additional resources for Earth Soundings Analysis - Processing vs Inversion
If you see only half of an axis, you cannot tell whether the function is even or odd or neither. , the function that vanishes for t < 0. Its even part cancels its odd part on t < 0. The ro transforms to an IO, which, being imaginary, is not shown. The third interpretation of these displays is that the frequency function is one-sided, and the time signal is complex. ” For analytic signals, RE extinguishes RO at negative ω, and the imaginary even part, ie, is not displayed. In summary, plots that show only half the axes can be correctly interpreted in three ways: left side even[ f (t)] causal(t) f (t) right side even[ F(ω)] F(ω) OneSided(ω) How can we compute these invisible imaginary parts?
INVERTIBLE SLOW FT PROGRAM 29 noise in the imaginary part due to numerical roundoff. The size of the imaginary part, theoretically zero, is typically about 10−6 of the real part. This is also about the size of the error on the real part of a signal after inverse transform. It is almost always much smaller than experimental errors and is of little consequence. As a check, I viewed these near-zero imaginary parts, but I do not show them here. A more serious error is a relative one of about 1/N on an N -point signal.
The slow FT code The slowft() routine exhibits features found in many physics and engineering programs. For example, the time-domain signal (which I call “tt()"), has nt values subscripted, from tt(1) to tt(nt). The first value of this signal tt(1) is located in real physical time at t0. The time interval between values is dt. The value of tt(it) is at time t0+(it-1)*dt. I do not use “if” as a pointer on the frequency axis because if is a keyword in most programming languages. Instead, I count along the frequency axis with a variable named ie.