Page 28 - 2016Fall

Smart People Behaving Foolishly
algorithms is only as good as the quality of the input data. Of course, there is rarely enough funding to do all of this important work.
Note that in a linear system, zero input must yield zero output. This is often overlooked in practice. For example, consider a system with the following system model. Let y(t)=4x(t)+5. If we look at this equation strictly in a mathematical sense, we see that it is a linear equation. However, viewing it as a system model, we see that if the input is zero, then the output is 5. The system, by definition, is not linear. In the real world, the constant 5 implies the existence of an energy source of some kind, so the system is active, not passive. Such a system is known as “incrementally linear” (McGillem and Cooper, 1974). If one is working with an incrementally linear system and using algorithms that assume system linearity, then it is important to preprocess the signals to remove any biases (means) or trends.
Linearity Horror Story About Computing
the Autocorrelation of a Signal with Bias
Dr. James V. Candy of the LLNL contributed this story. Early in his career, he computed the autocorrelation of a signal measured at the output of an apparently linear system and obtained results that made no sense. After some time, he re- alized that the measured signal contained a bias (a nonzero mean). Clearly, the system was actually incrementally linear. When he removed the mean of the signal before computing the autocorrelation, the result made sense and was useful. Note that the autocovariance includes mean removal as part of the calculation (Papoulis and Pillai, 2002).
System Time Invariance
Most signal-processing algorithms assume that the system is linear and time invariant. A system is time invariant if a temporal shift of t0 seconds in the input signal causes a temporal shift of t0 seconds in the output signal. Thus, a lin- ear system input x(t – t0) will produce output y(t – t0). Note that a time-invariant system has a constant gain, but a time- varying system has a time-varying gain. This means that a time-varying system is also nonlinear over time. If a system is time varying, then the measured signals are generally non- stationary. In this case, adaptive algorithms are appropriate (Candy, 2006).
Observability
In signal processing, we are often concerned with estimating models of systems. Observability is a measure of how well the internal states of a system can be inferred from knowl- edge of its external outputs (Kailath, 1980). Generally, the measurements must be at least as numerous as the internal states. All too often, scientists and engineers do not take this
Clark’s Law of Garbage In, Garbage Out
Signal processing algorithms are garbage-in, garbage-out devices (Clark, 2014, 2016).
Clark’s Law of Measurement Quality
The best “signal processing” is a good experiment with good measurements (Clark, 2014, 2016).
Clark’s Law of Signal Preprocessing
You’ll spend about 80% to 90% of your project effort acquiring, modeling, and preparing your data for your main signal processing algorithm (if you do it right) (Clark, 2014, 2016).
Important Algorithm Assumptions
Are Often Overlooked
Every signal processing algorithm is derived under a set of assumptions. If these assumptions are not met by the real- world system under analysis, something must be done to bring the algorithms in line with physical reality. Nonethe- less, people often pay little attention to the algorithm as- sumptions when processing signals.
System Linearity
Linear systems obey the superposition principle (McGillem and Cooper, 1974). Consider a system with system response h(t). Let t denote the continuous time variable and a and b denote real constants. For a system to be linear, if input x1(t) produces output y1(t) and input x2(t) produces output y2(t), then input ax1(t)+bx2(t) must produce ay1(t)+by2(t). Thus, the scaling and additive properties must hold. If they do not, then the system is nonlinear.
In practical systems, one can often construct experiments to test system linearity using this definition. However, in my experience, almost nobody bothers to conduct such an ex- periment. People tend to assume that their system is linear whether it is or not. Then they apply signal-processing algo- rithms that assume system linearity, obtain bad results, and wonder what went wrong. Smart people behaving foolishly.
26 | Acoustics Today | Fall 2016