Freq Resp and exp^π - Part A

Three interesting numbers in Maths are π, the ratio of a circle's circumference to its diameter, Euler's number e, approx 2.71818, and i, √-1. It is well known that e^iπ = -1. Here we discuss a use of e^π in frequency response, which uses i itself!

In frequency response, we input a sinusoid into a system (here first order linear), and consider its output as we change the frequency f of the input. The output is a sinusoid of the same frequency, but its amplitude changes, as does the delay or phase shift between input and output. Strictly we examine changes in gain (ratio of output and input amplitudes) and phase, with angular frequency (2πf).

In Part A we explain the basic concepts.

In Part B we plot gain and phase vs angular frequency (ω = 2 π * frequency).

In Part C we look at asymptotic approximations to the graphs - and get e^π.

In Part D we show the maths.

See also Original Paper at Control 2012.

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