### 1. Introduction

Fourier transforms have played a significant role in the way engineers and scientists measure and analyze signals. Engineers and scientists have used these methods for a long time, even before computers became as accessible as they are today. Though computers have significantly improved our ability to analyze and process signals, given that a Fourier transform calculation requires a tremendous number of multiplication calculations, it is still a daunting and laborious task.

In the early 1960s, a new method known as the fast Fourier transform (FFT) was introduced, significantly reducing the number of individual calculations required. The FFT is actually an umbrella term for a class of fast algorithms for computation of the discrete Fourier transform (DFT). Although the FFT is traditionally credited to J.W. Cooley and J.W. Tukey, the mathematical trick employed by FFT could be attributed to Gauss in about 1805, which predates even the Fourier series. Cooley and Tukey proposed an efficient method of computing the DFT that reduced computation cost from Order N2 to Order N log(N). As an example, a 1024-point FFT can be 2 orders of magnitude faster to compute than a 1024-point DFT.

The FFT is the basis for computer-based spectrum analysis, which relies on the representation of a signal in the frequency domain and is one of the most-commonly used methods to analyze signals and waveforms. The FFT works on the principle that you can express any waveform as a sum of sinusoids. However, it makes one important assumption -- that the waveform is periodic. In other words, the signal is composed of the same waveforms repeated. When you apply the FFT to the signal of interest, it is broken up into the sum of these sines and cosines.

FFT Array of Numbers Converted into Magnitude and Phase Information about Each Sinusoidal Wave Form

The mathematical representation of this function in effect assumes that the waveform repeats infinitely both before and after the acquired time record. Sometimes however, measuring a signal and determining its periodicity is not a straightforward procedure. Only a part of the actual waveform is extracted and you perform the FFT on this snapshot of the waveform on the assumption that the waveform pattern is infinitely repeated. The result of an FFT is typically an array of real and imaginary numbers, which you can convert into magnitude and phase information about each of the sinusoidal waveforms that make up the signal. If the phase information is not required, you can typically use the FFT-based power spectrum.

There are many frequency analysis tools that are either based on the FFT or attempt to provide similar results. The particular use of the FFT depends on the characteristics of the signal of interest and the desired results.

### 2. Zoom FFT

Zoom FFT analysis is simply an efficient computation of a subset of the FFT. You use this kind of tool when you are mainly interested in a certain frequency band of 10 kHz to 11 kHz. Rather than computing the FFT for the entire frequency range, you only perform computations on a subset of frequencies. Thus, you can save a significant amount of processing power and time using this method.

There are many ways of implementing the zoom FFT. The techniques the National Instruments LabVIEW Sound and Vibration Toolset software uses come from the use of a digital filter-based method and a partial transform. After modulation by the center frequency, the digital filter-based method applies a bandpass filter, eliminating frequency elements outside your range of interest. Once completed, you perform down sampling to reduce the number of data samples. Finally, you compute the regular FFT of the reduced data set. This method is suitable for inline processing.

Spectrum analyzers originally provided the zoom FFT to offer higher frequency resolution over a specific bandwidth, given the limitation of a small amount of on-board memory. With the zoom FFT, you can obtain a very fine frequency resolution (narrow band analysis) without computing the entire spectrum.

Applications of the zoom FFT include ultrasonic blood flow analysis, RF communications, mechanical stress analysis, Doppler radar, side band analysis, and modulation analysis.

### 3. Joint Time-Frequency Analysis (JTFA)

The frequency content of most signals encountered in everyday life changes over time. These signals include biomedical signals, speech signals, stock indexes, and vibrations. These are commonly known as transient signals. Because the basis functions used in classical Fourier analysis do not associate with any particular time instant, the resulting measurements, Fourier transforms, do not explicitly reflect a signal time-varying nature. One way to overcome the deficiencies of the regular Fourier transform is to compare the signal with elementary functions that simultaneously localize in time and frequency domains. JTFA provides this capability. Typical applications where JTFA is the best tool include frequency-domain display of audio signals, time-varying linear and nonlinear filtering, musical sound synthesis via spectral modeling, speech synthesis, time scaling, pitch shifting and detection, noise reduction, audio compression, and engine run-up and coast down.

The most common method used in the domain of JTFA is known as the short-time Fourier transform (STFT). This method involves taking sliding snapshots of your waveform, and then performing an FFT on that subset. By performing this action a number of times, you obtain a sliding window, which emphasizes "local" frequency properties.

FFT Array of Numbers Converted into Magnitude and Phase Information about Each Sinusoidal Wave Form

### 4. Super Resolution Spectral Analysis (SRSA)

SRSA does not incorporate FFT, but rather is a model-based alternative to the FFT for spectral analysis that you can use to estimate frequency information despite a limited number of sample points. With these tools, you can estimate phase, amplitude, and damping factor of the sinusoidal components of a signal. One disadvantage is that it requires prior knowledge of the input signal. In other words, the tools of the SRSA component require an estimate of the number of sinusoidal components in your input signal. Another weakness of this method is that it is more computationally intensive than an FFT – therefore it is suitable for smaller datasets of fewer than 100 points. Typical applications for the SRSA include biomedical research, economics, geophysics, noise, vibration, and speech analysis.

There are several methods available to you for spectral analysis. Choosing the right one is essential to effectively analyze the signals that are of interest to you. NI LabVIEW contains all these and other powerful measurement analysis capabilities, and makes it simple for you to acquire signals, analyze them, extract information, and present the results. Learn more about __LabVIEW-based analysis____.__