Computing Frequency Components

Each frequency component is the result of a dot product of the time-domain signal with the complex exponential at that frequency and is given by the following equation.

The DC component is the dot product of x(n) with [cos(0) - jsin(0)], or with 1.0.

The first bin, or frequency component, is the dot product of x(n) with cos(2πn/N) - jsin(2πn/N). Here, cos(2πn/N) is a single cycle of the cosine wave, and sin(2πn/N) is a single cycle of a sine wave.

In general, bin k is the dot product of x(n) with k cycles of the cosine wave for the real part of X(k) and the sine wave for the imaginary part of X(k).

The use of the FFT for frequency analysis implies two important relationships.

The first relationship links the highest frequency that can be analyzed to the sampling frequency and is given by the following equation.

where F_max is the highest frequency that can be analyzed and f_s is the sampling frequency.

The second relationship links the frequency resolution to the total acquisition time, which is related to the sampling frequency and the block size of the FFT and is given by the following equation.

where	Δf is the frequency resolution
	T is the acquisition time
	fs is the sampling frequency
	N is the block size of the FFT