The DFT (Discrete Fourier Transform) size refers to the number of samples, N, that the transform operates on.
In essence, the DFT takes a finite sequence of N complex numbers (samples) and decomposes it into N complex exponentials. This 'size' N is a critical parameter, directly influencing the frequency resolution and the computational complexity of the DFT. The input to the DFT is presumed to be evenly spaced samples over a finite interval, often referred to as a frame. This frame may represent time, distance, or another dimension.
Here's a breakdown:
- N = DFT Size: This is the number of input samples.
- Frequency Resolution: The frequency resolution of the DFT is inversely proportional to the DFT size (approximately
fs/N, wherefsis the sampling frequency). A larger N means finer frequency resolution. - Computational Complexity: The standard DFT has a computational complexity of O(N^2). However, using Fast Fourier Transform (FFT) algorithms, this can be reduced to O(N log N). Therefore, the size of the DFT significantly affects the processing time.
- Example: If you have a signal sampled at 1000 Hz and you perform a DFT of size 1024 (N=1024), your frequency resolution will be approximately 1000/1024 ≈ 0.98 Hz. This means you can distinguish between frequencies that are at least 0.98 Hz apart.
In summary, the DFT size (N) determines the number of discrete points used in the transform, influencing the frequency resolution, computational cost, and the overall characteristics of the analysis.
