What are the limitations of the fast fourier transformation on signal processing? Renovation?

Fast Fourier Transform (FFT) has several key limitations in signal processing, but I think you may have accidentally included "renovation" - let me address the FFT question first, then check if you have a renovation-related question.

The primary limitations of FFT in signal processing include frequency resolution constraints - the frequency resolution is limited by the window length, so longer signals are needed for better frequency resolution. Time-frequency trade-offs are another major limitation, as FFT provides excellent frequency information but loses all time localization - you can't determine when specific frequencies occurred in the signal.

Spectral leakage occurs when signal frequencies don't align perfectly with FFT bin frequencies, causing energy to spread across multiple bins and creating artifacts. The FFT also assumes the signal is periodic and stationary, which rarely matches real-world signals that change over time. Window effects introduce additional distortions, as the rectangular windowing inherent in FFT can create side lobes and affect amplitude accuracy.

Computational limitations include the requirement for power-of-2 sample sizes for optimal efficiency, and dynamic range issues where weak signals can be masked by stronger ones. The FFT also struggles with transient signals and non-linear systems, as it's fundamentally designed for linear, time-invariant analysis.

For time-varying signals, techniques like Short-Time Fourier Transform (STFT) or wavelet transforms often provide better analysis. Modern alternatives include adaptive transforms and compressed sensing methods that can overcome some FFT limitations.

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