Specifications

EAW Smaart 6 Operation Manual Concepts, Glossary, and Bibliography

2.1.5 Frequency Resolution

The frequency resolution (Q) of an FFT is equal to the sampling rate divided by the FFT

size. The frequency data points in an FFT are distributed linearly along the frequency

axis at intervals of Q Hz, from 0 to the Nyquist Frequency (1/2 the sampling rate). For

example, with a sampling rate of 44.1 kHz, an FFT size of 4096 (4 k) yields a frequency

resolution of 10.77 Hz. The resulting FFT has a data point every 10.77 Hz, 0-22.05 kHz.

Fixed-Point Per Octave (FPPO) Frequency Response Display

One problem associated with the linear distribution of FFT data points arises from the

fact that we hear frequencies logarithmically. Human hearing perceives each doubling

of frequency as an equal interval so each higher octave contains twice as many frequencies

as the one below. Using the example discussed on the previous page, in an FFT with a

frequency resolution of 10.77 Hz, there will be only three data points in the range 31.5-

63 Hz (the center frequencies of the two lowest octaves), providing very poor resolu-

tion. In the two highest octaves, the span between the center frequencies (8-16 kHz) is

8 kHz, yielding more than 700 data points. When viewed using a logarithmic frequency

scale, these data points are densely packed, creating a very difficult display to interpret.

Smaart 6 addresses this problem by using multiple FFTs, at different sampling rates and

FFT sizes, then combining the results to provide equal resolution in every octave, except

the two lowest. The resolution of the Real-time mode display is 24 points per octave

above 44 Hz, with 24 points distributed over the two lowest octaves. Note that using

multiple FFTs results in a longer time window at lower frequencies and a shorter time

window at higher frequencies.

Frequency Resolution and Octave/Fractional Octave Band Displays

For Spectrum measurements, the multiple-FFT technique used to measure Frequency

Response is not an option due to a mathematical limitation and so all RTA displays are

created from single FFTs. Since the linear distribution of FFT points in a single FFT

yields lower resolution in the lower than higher octaves, there may be bands at the low end

that contain only 0 or 1 data point, depending on the display and FFT input parameters.

The wider spacing between FFT data points in the lower octaves accounts for the missing

teeth seen at the low end in banded displays on some FFT-based analyzers. Smaart uses

an advanced algorithm to properly distribute energy into bands at low frequencies but

very sparse FFT data limits its effectiveness. Therefore, it is still advisable to select FFT

parameters that provide good frequency resolution at the lowest frequencies required.