The number of cycles in a Morlet wavelet is a crucial parameter that significantly influences the time-frequency resolution of the analysis. Here's a breakdown of how to select an appropriate number of cycles:

1. Understanding the Trade-off:

• Time Resolution: A higher number of cycles leads to a narrower frequency band, improving frequency resolution. However, this comes at the cost of reduced time resolution.
• Frequency Resolution: A lower number of cycles broadens the frequency band, improving time resolution but decreasing frequency resolution.

2. Guiding Principles:

• Target Signal Characteristics:If your signal has sharp transients or rapid changes, you'll generally want better time resolution. Opt for a lower number of cycles. If you're interested in analyzing narrow-band signals or fine-grained frequency information, choose a higher number of cycles.
• Desired Analysis Goals:If your primary goal is to detect the presence of specific frequencies, use a higher number of cycles. If you need to pinpoint the exact timing of events, use a lower number of cycles.

3. Practical Approach:

• Start with a Reasonable Range: Begin with a range of cycle values (e.g., 3-8 cycles).
• Visual Inspection: Perform the wavelet transform with different cycle values and visually inspect the results. Look for a balance between time and frequency resolution that best suits your analysis needs.
• Cross-Validation: If you have ground truth or known features in your signal, use those to evaluate the performance of different cycle values.

4. Iterative Refinement:

• Adjust and Repeat: Based on the initial analysis, adjust the number of cycles and repeat the wavelet transform. Iterate this process until you find the optimal value for your specific application.

5. Considerations:

• Signal Length: For shorter signals, using too many cycles might lead to edge effects and inaccurate results.
• Sampling Frequency: The number of cycles should be chosen in relation to the sampling frequency of your signal.

Example:

• If you're analyzing EEG data to detect epileptic spikes (which are characterized by rapid changes), you might choose a lower number of cycles (e.g., 3-5) to capture the transient nature of the spikes.

Software Tools:

• Many wavelet analysis libraries (like MATLAB, Python's SciPy) provide functions for calculating the Morlet wavelet transform. These libraries often allow you to specify the number of cycles as a parameter.

https://sapienlabs.org/time-frequency-analysis-and-wavelets/

https://www.mathworks.com/discovery/wavelet-transforms.html#:~:text=Wavelet%20Toolbox%E2%84%A2%20for%20use,wavelets%20used%20in%20wavelet%20analysis.