In the previous discussion, we talked about how, in DFT (Discrete Fourier Transform) calculations, the minimum frequency resolution and the length of the time window to be cut (sampling time) are inversely related. So, what determines the maximum frequency that can be analyzed?
In digital signal processing, a signal that changes continuously over time (analog signal) is converted into a discontinuous data sequence (digital signal) by extracting data at regular time intervals (this process is called A/D conversion). This time interval is called the "sampling period." Its reciprocal, fs (= 1/Δt), is called the "sampling frequency," and it represents the speed or number of times data is extracted per second.
The maximum frequency that can be analyzed depends on this sampling frequency.
If we let fm be the frequency of the sine wave signal we want to analyze, it's intuitively clear that we need to sample at a frequency higher than fm to obtain accurate data. But then, at what frequency should we sample to reproduce the data?
Generally, the Nyquist sampling theorem states that "data can be correctly reproduced if it is sampled at a frequency of at least twice the input frequency." For example, if the input sine wave frequency is fm, the sampling frequency fs must be 2fm or higher. This lower limit sampling frequency of 2fm is sometimes called the "Nyquist frequency." Failure to adhere to this can result in the acquisition of false data, a phenomenon known as "aliasing."
Conversely, if we determine the sampling frequency fs, the maximum frequency that can be analyzed will be fs/2.
In digital signal processing, when extracting data over a finite time length T, the "number of sampling points" is also important. If N points are sampled with a sampling period Δt, then T = NΔt (= N/fs). Specifically, the number of sampling points N is usually a power of 2 (for example, 1024, 2048, etc.) for the convenience of FFT calculations.
From N time signals, a meaningful spectrum with up to N/2 points can be obtained. However, due to various factors, it is possible to obtain a spectrum with fewer points, specifically N/2.56 points. For example, when N = 1024 and 2048, the spectra will have 400 and 800 lines, respectively.
If you want to raise the analysis frequency line, you need to set N to a larger value. Similarly, the relationship between the sampling frequency fs and the frequency range fR of the resulting spectrum is also this factor ratio of 2.56 (fR = fs/2.56).
To summarize what we've discussed so far, there are three parameters: sampling frequency fs, sampling points N, and acquisition time window length T. If you decide on two of them, the third one will be automatically determined.
In a typical FFT analyzer, the user can select the frequency range and the number of sampling points.
For specific numerical examples, please refer to the FAQ section on Ono Sokki website below.
FFT Basics FAQ - Relationship between Data Length, Frequency Resolution, and Time Length
(Excerpt from the email newsletter issued on January 24, 2003)
