# FFT Fundamentals (Sound and Vibration Measurement Suite)

The Fast Fourier Transform (FFT) resolves a time waveform into its sinusoidal components. The FFT takes a block of time-domain data and returns the frequency spectrum of the data. The FFT is a digital implementation of the Fourier transform. Thus, the FFT does not yield a continuous spectrum. Instead, the FFT returns a discrete spectrum, in which the frequency content of the waveform is resolved into a finite number of frequency lines, or bins.

## Number of Samples

The sampled time waveform input to an FFT determines the computed spectrum. If an arbitrary signal is sampled at a rate equal to fs over an acquisition time T, N samples are acquired. Compute T with the following equation:

where

T is the acquisition time

N is the number of samples acquired

fs is the sampling frequency

Compute N with the following equation:

N = T · fs

## Frequency Resolution

For FFT, the spectrum computed from the sampled signal has a frequency resolution df. Calculate the frequency resolution with the following equation:

where

 Note  The frequency resolution is determined solely by the acquisition time. The frequency resolution improves as the acquisition time increases.

### Maximum Resolvable Frequency

The sampling rate of a time waveform determines the maximum resolvable frequency. According to the Shannon Sampling Theorem, the maximum resolvable frequency must be half the sampling frequency. To calculate the maximum resolvable frequency, use the following equation:

where

fmax is the maximum resolvable frequency

fNyquist is the Nyquist frequency

fs is the sampling frequency

### Minimum Resolvable Frequency

The minimum resolvable frequency is 0 (DC). The analysis from 0 to fNyquist is the baseband analysis.

### Number of Spectral Lines

The number of lines in the spectrum is half the number of samples, N, in the waveform.

Directly specify the number of lines in the spectrum when using the Zoom FFT VIs. Specify the number of data samples to control the number of spectral lines when using the Baseband FFT VIs and the Baseband Subset VIs.

## Relationship between Time-Domain and Frequency-Domain Specifications and Parameters

The following table summarizes the relationship of time-domain specifications to frequency-domain parameters.

Frequency Domain
fNyquist fmax # lines df
Time Domain fs, N, T

fs · Eb Eb · N

where

fs is the sampling frequency

Eb is the effective bandwidth

N is the number of samples acquired

T is the acquisition time

The previous table shows that the sampling frequency and the block size acquired during each cycle of a continuous acquisition completely determine the frequency-domain parameters in baseband FFT analysis. However, many stand-alone instruments are operated by specifying the frequency range of interest and the number of lines in the FFT.

Use the information in the following table if you prefer to specify the spectrum parameters and determine the required data-acquisition parameters from these specifications.

Time Domain
fs N T
Frequency Domain fmax, # lines, df

where

fmax is the maximum resolvable frequency

Eb is the effective bandwidth

# lines is the number of lines in the spectrum

df is the frequency resolution

The previous table shows how a stand-alone instrument uses the frequency range of interest and the number of lines in the FFT to determine an appropriate sampling frequency and block size.

In the previous tables, Eb is the ratio of the alias-free bandwidth to the sampling frequency. In traditional instruments, Eb is typically . However, the alias-free bandwidth depends on the hardware used to acquire the dynamic signal. Eb can have a maximum value of 0.5. This maximum value corresponds to a perfect anti-aliasing filter. For example, if 1,024 samples are input to the FFT algorithm, the computed spectrum has 512 non-DC spectral lines. The computed spectrum has a total of 513 lines including the DC component. Acquire these same 1,024 samples on an instrument with a standard effective bandwidth, then use the equation in the first table above to find the expected number of alias-free lines in the computed spectrum. Complete the necessary calculations with the following equations:

# lines = (Eb)(N)

Eb is entirely a hardware property. However, mathematically, you can use the FFT to compute the frequency spectrum up to the Nyquist frequency. Remember to account for the presence or absence of an anti-aliasing filter when performing frequency analysis. The Frequency Analysis VIs compute every spectral line, alias-free or not. Use the frequency range to limit the analysis to the alias-free region of the spectrum with FFT subset and zoom FFT measurements. Use the SVFA Get Spectrum Subset VI to limit the analysis to the alias-free region of the spectrum with baseband FFT measurements.

## Increasing Frequency Resolution

Increasing the frequency resolution helps you distinguish two individual tones that are close together. For example, if you analyze a signal that contains two tones at 1,000 Hz and 1,100 Hz, use a sampling frequency of 10,000 Hz. Acquire data for 10 ms with a frequency resolution of 100 Hz. The following front panel shows the results of this analysis.

Notice that you cannot distinguish one tone from the other.

Increase the acquisition time to 1 s to achieve a frequency resolution of 1 Hz. The following front panel shows the results obtained with a 1 s acquisition time.

You can distinguish the individual tones with the increased acquisition time.

The following strategies achieve a finer frequency resolution:

• Decrease the sampling frequency, fs. Decreasing fs usually is not practical because decreasing fs reduces the frequency range.
• Increase the number of samples, N. Increasing N yields an increased number of lines over the original frequency range.

Implement the decreased fs strategy with zoom FFT analysis. Use baseband FFT and FFT-subset analyses to implement the increased N strategy. Baseband FFT analysis and FFT-subset analysis both achieve the same frequency resolution. However, FFT-subset analysis computes only a narrow subset of the spectrum.

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