# Mathematical description of functions

I am solving an optimization problem which is somewhat easy to solve.

The function I want to minimize is the absolute value of the real part of the FFT of a signal at a specific frequency.

But what are the correct mathematical terms?

First question:
What is the correct name for the real part of the FFT of a function at a specific frequency. Is it
the amplitude of this frequency signal? The signed amplitude?

Second question:
I optimize a parameter such that the absolute value signed amplitude is minimized. If I plot the signed amplitude vs the parameter I want to determine the derivative is always positive. Is there a mathematical term for a function that has a derivative that is always positive?

So I could say I am solving a convex optimization problem. But this is not what I am doing, what
I am doing is more like a root finding problem which is not so easy due to the high level of noise
that is present. Which mathematical property of the problem do I exploit if I formulate this problem
as root finding problem?

So to answer 2.: The term you seek is (strictly) monotonically increasing.

For 1. I am not sure whether a dedicated term exists for â€śthe real part of the Fourier componentâ€ť. It is not amplitude though. Amplitude is the magnitude of the component so the square-root of the sum of squares of real and imaginary part |f(\omega)| = \sqrt{Re(f(\omega))^2+Im(f(\omega))^2}. There might exist domain specific names for real/imaginary parts though like the real part of the complex-valued impedance is the resistivity of the circuit. If you give some context to your problem, maybe someone can help you find a proper term.

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Hard to say much without knowing what your optimization parameters are. What are you optimizing with respect to?

In signal processing a signal at a specific frequency can be decomposed into an â€śin-phaseâ€ť and a (not quite intuitively named) â€śquadratureâ€ť component, e.g. here. The DFT of the in-phase component is real, of the quadrature one is imaginary. So you seem to want to optimize for a maximum quadrature and a minimum in-phase component at a specific frequency (this would not necessarily require to compute the entire DFT).

Nit-picks: FFT is a fast algorithm/program to compute the discreter Fourier transform (DFT), but not the only available one. The Fourier transform of a function is normally an analytic integral giving, well, the Fourier transform of the function, and the FFT of this discretized function would be an often relatively poor numerical approximation of this integral transform.

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I think the term â€śin-phase componentâ€ť for the real part of the FFT is better than â€śsigned amplitudeâ€ť, thanks for suggesting that. And â€śmonotonically increasingâ€ť is also a good term.

@abraemer You can find more about the context of my work here: A learning algorithm for the calibration of internal model uncertainties in advanced wind turbine controllers: A wind speed measurement-free approach | IEEE Conference Publication | IEEE Xplore But the general context is control systems, signal processing is also not wrong as context because we are exiting the system and measuring an error signal.