Is there a proper way to determine when and when not to add quantities in quadrature (ie. adding the squares)? Or is it one of those black arts?
In what follows, is what I actually teach to my students. I say it is partly a black art. But am I really correct, or is there a firmer underpinning I can give it?
a) First off, I tell my students that if quantities are vectors at right angles you add in quadrature. That's easy and there is no problem seeing that is the case, simply by looking at the geometry of the situation. No black art yet, so far so good.
b) Second, imagine two Gaussian white-noise signals. Do we add these in quadrature or not? Initially this is hard for the student to see the actual geometry of the situation, but then I write down the formula for the correlation between the two noise signals. I then tell the students that they can imagine these two noise signals as vectors, where the magnitudes represent the rms amplitudes and the angle between the vectors represents the degree of correlation. Now they can see the geometry! So they understand now why you linearly add when the two signals are correlated and why you add the noise in quadrature when it is uncorrelated. Because they are at 90deg when they are uncorrelated. No problem, we are all happy. Perhaps it was a little bit of a stretch with the vector analogy, so this part is only a grey art so far.
c) Third, now for the black art. Note that there are many situations in physics and engineering similar to what I am about to describe. I am giving just one illustrative example here. But I want to make the point that this type of question is ubiquitous. The example is this: imagine a photodiode. The diode has a response time determined not only by the transit time of carriers across the depletion region, but also by its circuit RC time constant. Recall that C is the junction capacitance, and R is any load resistance hanging off the diode. Now my students can easily calculate both the transit time and the RC time, with no problem. But when it comes to calculating the total response time, should they linearly add those two times or should they add the times in quadrature?
Unfortunately, as a lecturer I totally fail here because I have no nice geometric picture to give my students, like I did in cases (a) and (b). When we look in the literature, we are told those two terms must be added in *quadrature*. Does the literature tell us why? No.
So this is what I tell my students. I say that we cannot necessarily know, a priori, if these two quantities should be added in quadrature or not. We can *suspect* that we should add those times in quadrature because they come from two physically different origins that are apparently independent. But we do not really know at first glance if there is any degree of correlation or not. Therefore, we must go away and make empirical measurements to be really sure. I tell them it is a black art. We have no firm theoretical underpinning to decide the correct way, other than just going and doing the experiment to see if the quadrature or linear model fits the measurements better.
Am I right to say this, or can we claim some firmer principle?
Remember, I am looking for a general principle and not simply an answer to the photodiode case, which was only one of many examples.
Is it possible to theoretically predict when and when not to add in quadrature, or am I right that in many cases it can only be finally decided by what a real experiment tells you to do?
All I can offer, my own explanation (as true, literature does not say) would be time in reality (v. Newtonian time) is curved as in curved spacetime, especially in dealing with light (photodiodes) with definition as speedXspeed, or acceleration, c-SQUAREd (QUADrature-curved parabolically, which also note has the squared formula, y=x^2 + x...; also as you know electromagnetism, or light, being orthogonal, electrical field perpendicular to its magnetic field, 90 degrees). Furthermore, also note 'squared' relates geometrically to area, of course, which in calculus represents the integral, or sum of triangles under a curve or change or motion (tangent, derivative, speed), area of which for the triangle is 1/2 of a square (or 1/2 bXh) (attributable again in my mind to the 1/2 sometimes placed in front of integrals, either the definite or probably indefinite, forget exactly when that 1/2 was introduced in calculus)... Not in lecture form, but for what it's worth, I'll leave that to you!
The "quadrature" calculation arises from the same fundamental principle in both the probability calculation and the time delay calculation, because both problems depend on the calculation of a convolution.
1. For uncorrelated random variables (not only gaussian, but the independence implies uncorrelatedness) the density function of a sum is found from the convolution of the separate densities. For functions that are positive the second moments add under convolution. The power of a variable is of course the second moment. Compare the parallel axis theorem of mechanics.
Reference: classical texts on probability etc eg. Probability, Random Variables and Stochastic Processes by Athanasios Papoulis
The case of two sinusoids in quadrature should fit in this context, because they are uncorrelated.
2. For time delay systems e.g. the delay of a pulse observed via a photodiode time-spread then passed through a first-order linear filter, the process is convolution. If the original function and the delay impulse response are positive only, then the second moments add as in 1 above. IF the delays are estimated as "moment delays" then the additivity of the second moments results in the "quadrature " calculation.
Reference: classical texts on linear systems, Fourier etc eg. The Fourier Integral and Its Applications by A. Papoulis
Also: Bogner, R.E. “Estimation of Time Delay”.Invited paper for special issue of IEEE Trans. on Sonics and Ultrasonics, Vol. SU-31, No. 4, July 1984, pp.373-390. 68.
The texts say that when you when you cascade two RC circuits, you add their rise times in quadrature. But this seems to be an approximation.
You can precisely calculate the correct answer, in this case, and the correct answer is not the same as linearly adding nor adding in quadrature. However, adding in quadrature gives the closest value to the correct answer.
It seems to me that adding in quadrature in these types of examples is a fudge that happens to approximately work by luck, or am I missing something?
And why is adding rise times in the earlier photodetector example, any different to the cascaded RC example?
When the impulse responses are positive only, then the second-moment widths add in quadrature precisely - just as the variances add with indep. random variables. See the references above; also the Wikipedia article at http://en.wikipedia.org/wiki/Moment_%28mathematics%29 re random variables.
The only possibly-unsatisfying aspect is that the moment widths of pulses are not the same as other measures of pulse width or rise time. In practice rise times are measured from 10% to 90%, which may be quite different from the moment (or square-root variance) rise time. But the quadrature relationship is precise and rigorous.
The combination of durations of interacting processes to a duration of the combined process is obviously more general (and more interesting) than the combination of signals going over the same line.
Mountaineers have rules for estimating the time needed for reaching the top of a mountain. For the horizontal distance they calculate a time tH based on a speed of, say 3 km/h (depends on the weight of the rucksack and the character of the terrain). For the vertical distance they calculate a time tV based on a vertical speed of, say 200 m/h (depends heavily on the difficulty of the ascend). For the total time they take Max(tH,tV) + 0.5*Min(tH,tV), which is a simple computational approximation to adding the squares and taking the square-root of the result. The rule works surprisingly well, although the underlying (already idealized) law is quite different: you have to follow the actual trail with a actual velocity (i.e. tangential to the surface) depending in a monotone nonlinear manner on the local steepness.
So, we seem to be in the dark-grey regime. This turns into a bright one if we apply the modern art of computer simulation to the problem. Then we easily random-generate thousands of pseudo-realistic samples, compute the times with various models and do means and variances. Very likely one will find good statistics for the rule we started from.
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