Frequency Information Displayed

Two of the principal challenges in scientific analysis are mathematically describing the Fourier transfer and understanding its properties. The most common applications of the Fourier transform are the analysis of linear time-invariant systems and spectral analysis, but this transform is most important because it gives the scientist a way to examine a relationship from the frequency domain point of view.

Most introductory textbooks in linear systems, digital signal processing, image processing, and other related applications discuss the two-sided mathematical description of the Fourier transform

and its inverse

Two-sided information means that all the negative and positive frequencies and time are considered in the mathematical implementation of the forward and inverse Fourier transform. Single-sided or one-sided information considers only the positive frequencies and time history of the signal.

A Fourier transform pair consists of the signal represented in both the time and frequency domain. The notation

x(t) X(f)

is commonly used to represent a Fourier transform pair, for example,

tri(t) sinc^2.

Recent advances in the computation of FFTs, Discrete Fourier Transforms (DFTs), and their inverse operations led to the adoption of the frequency information presentation format used in the Complex FFT VI. This format is used principally for speed and processing convenience.

When dealing with test and measurement applications, frequency information in the format used in Digital Signal Processing VIs often appears awkward to scientists. There are two other common ways to present this information: displaying the DC component in the center and displaying one-sided spectrums.

To convert to a DC-centered Fourier transform, the quick and obvious solution is to copy buffers from one place to another. In the case of converting single-side band information, the solution involves extracting the frequency information and multiplying each value by 2, excluding DC and Nyquist components. A special case occurs when the length of the sequence is even, which is also a requirement for the computation of the FFT because its length has to be a power of 2.

The following examples illustrate how to preprocess time sequences to obtain the appropriate frequency display format. Two Sided DC Centered Fast Fourier Transforms briefly discusses alternate methods for obtaining DC-centered and single-sideband Fourier transforms without manipulating buffers. Frequency Information Obtained from Transforms discusses how to obtain and display correct frequency information of the FFT for a given sampling interval. You can extend this information to incorporate it into the DC-centered and single-sideband FFT cases.