Many of our readers are moving into the realm of advanced audio
and acoustic measurement. Even with the diversity in types of available
equipment, most share the same basic principles and terminology.
This article addresses a parameter common to most all computer-based
measurement systems - the FFT size.
FFTs in Audio
The FFT (Fast Fourier Transform) is a mathematical technique in
which time domain signals can be transformed into the frequency
domain. Simply stated, the FFT allows you to “see” the
frequency content of a given time do main signal.
A simple case would be a single-channel FFT-based spectrum analyzer.
Typically an audio signal is picked up by a microphone and the “frequency-content”
of the signal displayed on a frequency vs. magnitude display. The
speed and resolution of the analyzer would be determined by two
parameters (in addition to the speed of the computer hardware required);
the FFT size and the sampling rate. A typical set of values for
these parameters might be a sampling rate of 44.1kHz, and a 4096
point FFT. These two values allow us to calculate two important
factors in FFT based analysis; “the frequency resolution”
of our results and the “FFT time constant” of our measurement.
The FFT time constant is just the amount of time it takes to collect
our data points, in our example 4096 points divided by our sampling
rate 44100 points/sec, yielding a time constant of 0.093 seconds.
The frequency resolution is calculated noting that an FFT of N data
points in the time domain yields N/2 frequency domain points spaced
linearly between DC and the Nyquist frequency (half he sampling
rate). In our example, a 44.1 kHz sampling rate and a 4096 point
FFT provide one frequency domain data point every 10.8 Hz. While
better frequency resolution can be obtained by using larger FFTs,
the nature of audio signals is that they change over time, and larger
FFTs require more time, limiting their responsiveness.
Simple spectrum analysis is only one measurement which uses FFTs.
While looking at the frequency content of a single audio channel
can be useful (particularly for ear training) it is common to compare
two signals in the frequency domain. This comparison is often made
between the input and output of a “system”, and is called
the “transfer function” of the system. Stated simply,
the transfer function of a system describes everything that happens
to the signal from input to output. One reason for making this measurement
in the frequency domain is that it is often easier to “read
and interpret” frequency domain data than time domain data.
The transfer function of a system can contain several important
pieces of information about the system, such as the delay time through
or across the system, the frequency response of the system or even
the reverberation time and distribution of reflections in a room.
Here a system can be a wire, an equalizer, a loudspeaker and even
a room, or any combination of these.
In order to measure the COMPLETE transfer function of a system,
there is a requirement that the FFT size selected be long compared
to the decay through and/or of the decay of the system under test.
This requirement of ten means that extremely long FFTs must be used.
How ever once this measurement is made, the result can be transformed
back into the time domain and the systems impulse response is the
result.
There is one limitation to this process. The process assumes that
the system is linear, meaning that the system does NOT change its
response during the measurement. This is not a problem when measuring
equalizers or room acoustics, however it does mean that devices
such as corn pressor/limiters cannot be measured using this technique.
Getting acquainted with the terminology used in audio-acoustic
measurement tools can be quite a challenge. The best way to. get
familiar with the basics is to relate them to something that you
are already familiar with. To understand FFTs and the way they are
commonly used in audio measurements, it is important to first understand
the basics.
A key parameter when making an FFT measurement is the selection
of the proper FFT size. An analogy will help to convey its meaning
and help the user to make the proper selection.
Getting the Points
In a nutshell, the FFT size determines the length of time of the
measurement. One important rule of FFT measurements is that the
gathered data must represent the entire time record of the event
that is being measured. For instance, if a room takes 2 seconds
to decay, than an FFT size must be chosen that is at least 2 seconds
in length. FFT size is selected by the number of “points”
that will be gathered. If we were sampling at 48 kHz, then 48,000
points would gather one second of data.
This process is very similar to digital recording, where a sampling
rate must be selected. The higher the sampling rate the bigger the
file size. Also, the longer the recording the bigger the file size.
Digital recordings require large amounts of storage space. Fortunately,
hard disks are cheap these days so it is feasible to record fairly
long events onto available disk sizes. Whether recording onto a
hard disk, or making an FFT measurement, the key thing to remember
is that you are acquiring discrete time samples of an event. If
the samples are spaced closely enough, the event can be completely
reconstructed from the samples, with no apparent loss of data.
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