Bandwidth fn (= Nyquist frequency). This value indicates the theoretical maximum frequency that can be determined by the FFT.
For example at a sampling rate of 48 kHz, frequency components up to 24 kHz can be theoretically determined. In the case of an analog system, the practically achievable value is usually somewhat below this, due to analog filters - e.g. at 20 kHz.
Measurement duration D. The measurement duration is given by the sampling rate fs and the blocklength BL.
D = BL / fs.
At fs = 48 kHz and BL = 1024, this yields 1024/48000 Hz = 21.33 ms
Frequency resolution df. The frequency resolution indicates the frequency spacing between two measurement results.
df = fs / BL
At fs = 48 kHz and BL = 1024, this gives a df of 48000 Hz / 1024 = 46.88 Hz.
In practice, the sampling frequency fs is usually a variable given by the system. However, by selecting the blocklength BL, the measurement duration and frequency resolution can be defined. The following applies:
It can be seen that condition 2. would apply only to very few signals. The sampling of a signal whose frequencies are not an integer multiple of df would begin and end within a block of 2^n samples with different values. This results in a jump in the time signal, and a "smeared" FFT spectrum. (aka Leakage)
In order to prevent this smearing, in practice "windowing" is applied to the signal sample. Using a weighting function, the signal sample is more or less gently turned on and off. The result is that the sampled and subsequent "windowed" signal begins and ends at amplitude zero. The sample can now be repeated periodically without a hard transition.
As explained in the first part, the sampling rate fs of the measuring system and the block length BL are the two central parameters of an FFT. The sampling rate indicates how often the analog signal to be analyzed is scanned. When recording wav files via a commercially-available PC sound card, for example, the audio signal is usually sampled 44,100 times per second.
Harry Nyquist was the discoverer of a fundamental rule in the sampling of analog signals: the sampling frequency must be at least double the highest frequency of the signal. If, for example, a signal containing frequencies up to 24 kHz is to be sampled, a sampling rate of at least 48 kHz is required for this purpose. Half the sampling rate, in this example 24 kHz, is called the "Nyquist frequency".
But what happens if signals above the Nyquist frequency are fed in to the system?
For the most, a signal is sampled with a more-than-sufficient number of samples. With a 48 kHz sampling rate, for example, the 6 kHz frequency is sampled 8 times per cycle, while the 12 kHz frequency is only sampled 4 times per cycle. At the Nyquist frequency, only 2 samples are available per cycle.
With 2 samples or more it is still possible to reconstruct the signal without loss. If, however, less than 2 samples are available, artifacts which do not occur in the sampled (original) signal are generated.
In the FFT, these artifacts appear as mirror frequencies. If the Nyquist frequency is exceeded, the signal is reflected at this imaginary limit and falls back into the useful frequency band. The following video shows an FFT system with 44.1 kHz sampling rate. A sweep signal of 15 kHz to 25 kHz is fed in to this system.
These unwanted mirror frequencies are counteracted with an analog low-pass filter (anti-aliasing filter) before the scanning. The filter ensures that frequencies above the Nyquist frequency are suppressed.
In the case of periodically-continuous signals, the time windowing serves to smooth the undesired transitional jumps at the end of the scanning (see part 1). This prevents smearing in the spectrum. There are numerous types of windows, some of which differ only slightly. When selecting the time window, the following rule applies: Each window requires a compromise between frequency selectivity and amplitude accuracy.
Power vs. Peak detector
Modern high-resolution FFT analyzers offer the possibility to decouple the number of measurement results from the FFT block length. This results in an increase in measurement performance time, especially for high-resolution FFTs. Thus, for example, with a 2MB block length it is no longer necessary to measure and represent more than 1 Million points (bins), but only the number necessary for the display, e.g. 1024.
The value chosen for each FFT bin can be defined in two ways:
Calculations with FFT results
FFTs are mainly used to visualize signals. However, there are also applications where FFT results are used in calculations. For example, very simple levels of defined frequency bands can be calculated by adding them via an RSS (Root Sum Square) algorithm.
Another application is the comparison of spectra. The example below shows an acoustic measurement of a cordless screwdriver. The measured spectrum is subtracted from a defined reference spectrum. This difference is compared against an upper and lower tolerance. The upper spectrum shows a functional cordless screwdriver. In the lower, the acoustic spectrum suggests that the test specimen is defective.