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Converted document FAST FOURIER TRANSFORMS

FAST FOURIER TRANSFORMS

Periodic data, such as vibration, periodic loading of rotor or wind turbine blades, and active control, are highly relevant in aerospace engineering. These data can be modelled in the temporal or time domain, using trigonometric or sinusoidal functions (sin, cosine, tangent, ramp, pulse, square waves). There are several characteristics that can be used to describe these data:

amplitude: the height (maximum minus minimum) of the oscillatory motion during a single period;
phase: the oscillatory response in an engineering application is a result to an oscillatory forcing function (for example, airfoil lift responding to angle of attack changes). The response is typically not instantaneous, but occurs some time after (lags) the forcing function (which leads the response);
frequency: how often the cyclic motion repeats. Frequency, f, is defined as cycles per time (usually seconds), and its counterpart, angular frequency, ω, provides the information in more useful engineering units, radians/time. Angular frequency and frequency are related via the equation ω = 2πf. The period, T, is the time per cycle, and is the inverse of the frequency (T = 1 ⁄ f).

A periodic waveform can be described via a series of trigonometric functions, known as a Fourier series:

F(t) ≈ C₀ + A₁sin(ωt) + B₁cos(ωt) + … + A_isin(iωt) + B_icos(iωt) + … ≈ ^∞⎲⎳_i = 0C_ie^ijωt

where j is the imaginary symbol (j = √( − 1)). When initial conditions are provided, the series can be resolved. The response will be defined as a complex function. The amplitude of the response is defined as the square root of the square of the real and imaginary components of the function (A = √(R² + I²)) and the phase angle as the inverse tangent of the fraction of the real over the imaginary components, θ = tan^− 1(R ⁄ I).

The unknown coefficients of the series can be computed using the the following expressions

C₀ = (1)/(T)^T⌠⌡₀F(t)dt A_i = (2)/(T)^T⌠⌡₀F(t)sin(iωt)dt B_i = (2)/(T)^T⌠⌡₀F(t)cos(iωt)dt

While these are continuous functions in time, their periodicity means that, using the characteristics described above, these values can be used to describe the motion at each frequency (the fundamental frequency, ω and its higher harmonics). This is a very efficient way to define the motion. These scalar variables constitute the behavior of the motion in the frequency domain rather than the time domain.

To take advantage of this, the motion needs to be converted from the time to the frequency domain. This is accomplished via a Fourier integral. For a generic Fourier series

F(t) = ^∞⎲⎳_{i = − ∞}C_ie^jiωt

where the coefficients, C_i, are defined as

C_i = (1)/(T)^T ⁄ 2⌠⌡_{− T ⁄ 2}F(t)e^jiωtdt

The Fourier integral converts the periodic function to an aperiodic function. Given the function, F(t),

F(t) = (1)/(2π)^∞⌠⌡_− ∞ℱ(jω)e^jωtdω

where the frequency-based coefficient, ℱ(jω), can be defined as a Fourier integral as

ℱ(jω) = ^∞⌠⌡_− ∞F(t)e^− jωtdt

This pair of integral equations is known as the Fourier Transform (pair) of F(t). This pair of equations permits the switching of the function between the temporal and frequency domains.