Note this statement will not change even if we did measure for $10^ Given a 12 second measurement, applying our knowledge of reality (physics), we can state confidently: "this pendulum will swing at a rate of 3 cycles per second given same environment (wind, gravity, etc) and driver (motor)". How much must we measure? That's a question of statistics, and statistical significance. Must we really measure a process from big bang to heat death to know its frequency? No. We take the Fourier Transform of A and B, and get different results despite the physical process being exactly the same how so?īecause by taking the Fourier Transform of $x$ we're answering the question, "what is the continuum of frequencies of continuous complex sinusoids spanning all time, $-\infty$ to $\infty$, that would infinitesimally sum (integrate) to $x$?" This continuum changes simply because in B we sum to non-zero over a greater interval than in A. We record pendulum positions for 5 seconds, call it "data A". Now suppose despite knowing the swing rate, we seek to measure it and describe it mathematically from those measurements. The 'mathematical construct' here is a continuous-time function that does not assume a predefined domain but rather permits it ( $t_0$ to $t_1$) to be selected on demand. How do we describe this swinging mathematically? We can say,Īnd this would accurately describe the swinging over any arbitrary duration $t_0$ to $t_1$. Suppose we know it swings 3 times per sec because we designed a motor to drive it such. ![]() ![]() The goal is to accurately describe reality. The full question probes as far as "what is science?", so I'll try simplifying.įourier Transform is a tool.
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