DFT & FFT Complexity in MATLAB: Digital Frequency Spectrum Analysis

The complexity issue in Digital Frequency Spectrum Analysis based on DFT and the FFT function in MATLAB refers to the amount of computation required to perform the analysis. The DFT algorithm has a computational complexity of O(N^2), where N is the number of samples in the time domain signal. This means that as the number of samples increases, the computation time required to perform the DFT analysis also increases exponentially.

The FFT algorithm, on the other hand, has a computational complexity of O(N log N), which is much faster than the DFT algorithm. This is because the FFT algorithm breaks down the DFT computation into smaller sub-problems using a divide-and-conquer approach.

In MATLAB, the FFT function is a built-in function that uses the FFT algorithm to compute the frequency spectrum of a signal. This function is much faster than the DFT algorithm and is often used in signal processing applications.

However, it's important to note that the FFT algorithm has some limitations. It requires that the number of samples in the time domain signal is a power of two, and it assumes that the signal is periodic. If these conditions are not met, then the FFT algorithm may not provide accurate results. Additionally, the FFT algorithm may introduce spectral leakage and other artifacts in the frequency spectrum, which can affect the accuracy of the analysis.