21. Feb. 2017
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.