How do you estimate the frequency of a sinusoidal signal with spectral leakage?

1 What is spectral leakage?

Spectral leakage occurs when you apply a discrete Fourier transform (DFT) to a finite segment of a sinusoidal signal that does not exactly match an integer number of cycles. In this case, the DFT assumes that the signal is periodic and extends it infinitely by repeating the segment. This creates discontinuities at the segment boundaries, which introduce spurious frequency components in the spectrum. These components spread the energy of the signal across multiple frequency bins, reducing the peak amplitude and distorting the true frequency.

3 How to reduce spectral leakage?

One way to reduce spectral leakage is to choose a window function that has a narrow main lobe and low sidelobes, where the sidelobes are the smaller peaks around the main lobe. Different window functions have different trade-offs between main lobe width and sidelobe level. For example, a rectangular window has the narrowest main lobe, but the highest sidelobes, while a Hann window has a wider main lobe, but lower sidelobes. You can compare different window functions and their effects on the spectrum using MATLAB or Python.

4 How to improve frequency resolution?

Another way to improve the accuracy of your frequency estimation is to increase the frequency resolution of your spectrum. The frequency resolution is the smallest difference between two frequencies that can be distinguished in the spectrum, and it depends on the length of the signal segment and the sampling rate. The longer the signal segment, the higher the frequency resolution. However, increasing the signal segment may not be possible or desirable, as it may introduce more noise or non-stationarity in the signal. A better alternative is to use zero-padding, which is adding zeros to the end of the signal segment before applying the DFT. This does not change the frequency content of the signal, but it increases the number of frequency bins in the spectrum, making it easier to identify the peak frequency.

6 How to verify frequency estimation?

Finally, you can verify your frequency estimation by comparing it with the true frequency of your signal, if you know it, or by using another method of frequency estimation, such as zero-crossing, autocorrelation, or phase-locked loop. You can also evaluate the performance of your frequency estimation by calculating the mean squared error (MSE), which is the average of the squared difference between your estimated frequency and the true frequency. The lower the MSE, the better your frequency estimation.