Agilent Technologies 89410A Operator's Manual page 227

FFT Background
FFT Basics
The Fourier transform integral converts data from
the time domain into the frequency domain.
However, this integral assumes the possibility of
deriving a mathematical description of the
waveform to be transformed—but real-world
signals are complex and defy description by a
simple equation. The Fast Fourier Transform (FFT)
algorithm operates on sampled data, and provides
time-to-frequency domain transformations without
the need to derive the waveform equation.
The Fast Fourier Transform (FFT) is an
implementation of the Discrete Fourier Transform,
the math algorithm used for transforming data from
the time domain to the frequency domain. Before
an analyzer uses the FFT algorithm, it samples the
input signal with an analog-to-digital converter (the
Nyquist sampling theorem states that if samples
are taken twice as fast as the highest frequency
component in the signal, the signal can be
reconstructed exactly). This transforms the
continuous (analog) signal into a discrete (digital)
signal.
Because the input signal is sampled, an exact
representation of this signal is not available in either
the time domain or the frequency domain.
However, by spacing the samples closely, the
analyzer provides an excellent approximation of the
input signal.
FFT Properties
As with the swept-tuned analyzer, the input to the
analyzer is a continuous analog voltage. The
voltage might come directly from an electronic
circuit (for example, a local oscillator) or through a
transducer (for example, when measuring
vibration). Whatever the source of the input signal,
the FFT algorithm requires digital data. Therefore,
the analyzer must convert the analog voltage in to a
digital representation. So the first steps in building
an FFT analyzer are to build a sampler and an
analog-to-digital converter (ADC) in order to create
the digitized stream of samples that feeds the FFT
processor.
What Makes this Analyzer Different?
The FFT algorithm works on sampled data in a
special way. Rather than acting on each data
sample as it is converted by the ADC, the FFT waits
until a number of samples (N) have been taken and
transforms the complete block of data. The
sampled data representing the time-domain
waveform is typically called a time record of size-N
samples.
But the FFT analyzer cannot compute a valid
frequency-domain result until at least one time
record is acquired—this is analogous to the initial
settling time in a parallel-filter analyzer. After this
initial time record is filled, the FFT analyzer is able
to determine very rapid changes in the frequency
domain. A typical size for N might be 1024
samples in one time record.
During the FFT process, the FFT algorithm
transforms the N time domain samples into N/2
equally-spaced lines in the frequency domain. Each
line contains both amplitude and phase
information—this is why half as many lines are
available in the frequency domain (actually, slightly
less than half the number of lines are used, since
some data is corrupted by anti-aliasing filters).
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