Lecture Notes on Wavelet Transform and Signal Processing

Overview of Signals

• Real-world signals are noisy and irregular but have inherent structure.
• Signal processing is a field dedicated to analyzing these signals.
• Example: Electrical activity recordings from mouse brains show oscillation patterns.

Characterization of Signals

• Scientists require objective mathematical methods to analyze and describe signals.
• Task: Quantify structure present in noisy data.
• Introduction of a mathematical tool: Wavelet Transform.

Time-Frequency Duality

• Time-Frequency Duality: Different representations of the same data.
  • Example: Sending two numbers (x1, x2) vs. their sum (y1) and difference (y2).
  • y1 (low frequency) and y2 (high frequency) provide alternative views of the same information.
• Joseph Fourier's insight: Functions can be decomposed into sums of sine and cosine waves using the Fourier Transform.

Limitations of Fourier Transform

• Fourier Transform loses temporal information; it squishes the signal in time for frequency analysis.
• Example: Traffic light signal analysis shows peaks in frequency but lacks temporal context.
• Heisenberg Uncertainty Principle: Trade-off between time and frequency resolution.
  • Knowing one means losing information about the other.

Wavelet Transform as a Compromise

• Wavelet Transform allows for time-frequency analysis by utilizing wavelets instead of sine functions.
• Wavelet Definition: A short-lived oscillation localized in time (family of functions).

Properties of Wavelets

1. Zero Mean: Must have no zero frequency component; the integral over the function should equal zero.
2. Finite Energy: The area under the squared function must be a finite number (localized in time).

Wavelet Transform Mechanics

• Fourier Transform yields a one-dimensional frequency representation.
• Wavelet Transform produces a two-dimensional function (time and frequency).
• Daughter Wavelets: Result from scaling and translating a mother wavelet.
  • Time and frequency adjustments allow for examining the signal at different scales.

Convolution and Similarity Measurement

• The contribution of wavelets to the signal is assessed through a dot product (similarity measure).
• The convolution process slides the wavelet across the signal, measuring where they align.
• Areas of good alignment (green) vs. poor alignment (red) are mathematically evaluated.

Implementation of Wavelet Transform

• Convolution of signal with both real and imaginary components of wavelets.
• The absolute value of the complex result measures the power of frequency components over time.
• Visual representation: Wavelet Scalogram displays frequency contributions dynamically.

Application and Utility

• Wavelet Transform is useful for analyzing signals in various fields: fluid dynamics, engineering, neuroscience, medicine, astronomy.
• Example: Brain signal analysis reveals distinct low and high-frequency rhythms.

Trade-off of Time and Frequency Resolution

• Wavelet Transform balances time and frequency resolution, addressing the uncertainty principle.
• Heisenberg boxes visualize the trade-off; wider boxes for low frequencies, narrower for high frequencies.

Conclusion

• Wavelet Transform serves as an effective tool for revealing the inherent structure of signals over time and frequency.
• Future signals and data interpretations can benefit from this nuanced analysis approach.