I. The Core Distinction: Decoupling

The fundamental difference between Wavelet Convolution and the Filter-Hilbert method lies in how they handle the two primary tasks of time-frequency analysis: Filtering (isolating frequencies) and Analytic Representation (extracting power/phase).

  • Wavelet Convolution: Performs both tasks simultaneously. A complex Morlet wavelet is, by definition, a bandpass filter and an analytic signal combined.
  • Filter-Hilbert Method: Decouples these steps.
    1. Step 1 (Filtering): Apply a bandpass filter to isolate the frequency of interest.
    2. Step 2 (Hilbert): Apply the Hilbert transform to extract the analytic signal (complex power/phase information).

The Main Advantage: By decoupling these steps, you gain significantly more control over the frequency characteristics of the filter.

II. The Hilbert Transform

The Hilbert transform is a mathematical operation used to extract complex information from a real-valued signal.

1. Purpose

It converts a real-valued signal (which only has cosine components, $M\cos(2\pi ft)$) into an analytic (complex) signal by generating and adding the imaginary part.

  • Real Part: Original signal ($M\cos(2\pi ft)$).
  • Imaginary Part: Phase quadrature component ($iM\sin(2\pi ft)$).

2. Implementation (The Frequency Domain Trick)

In practice (like in MATLAB’s hilbert function), we do not compute this by integrating in the time domain. Instead, we manipulate the Fourier spectrum:

  1. FFT: Convert signal to the frequency domain.
  2. Rotate: Double the positive-frequency coefficients and set the negative-frequency coefficients to zero.
  3. IFFT: Convert back to the time domain.

III. The Advantage: Filter Control (Gaussian vs. Plateau)

The primary limitation of Morlet wavelets is that their frequency shape is always Gaussian. You cannot change this shape; you can only change its width (via the number of cycles).

The Filter-Hilbert method allows you to design filters with custom shapes:

  • Plateau Shapes: Unlike the Gaussian peak of a wavelet, a bandpass filter can have a flat top (“plateau”).
  • Benefit: This offers better frequency specificity. You can define a hard boundary for the frequencies you want to keep versus those you want to remove, resulting in time-frequency plots that are smoother along the frequency axis.

IV. Filter Types

Using the Filter-Hilbert method, you can apply four standard types of frequency filters before extracting the analytic signal:

  • Bandpass: Keep activity between two frequencies (most useful for time-frequency decomposition).
  • Band-stop: Remove activity between two frequencies (notch filter).
  • High-pass: Retain frequencies above a cutoff.
  • Low-pass: Retain frequencies below a cutoff.

Summary: While wavelet convolution is mathematically elegant (one step), the Filter-Hilbert method is practically powerful because it allows you to shape your frequency isolation (e.g., using a plateau filter) before extracting the power and phase.

V. Comparison with Wavelet Convolution

1. Smoothness in the Frequency Domain

While the Filter-Hilbert method allows for “plateau” shaped filters (which have sharp boundaries), this sharpness can sometimes be a disadvantage.

  • The Wavelet Advantage: Because Morlet wavelets always have a Gaussian shape in the frequency domain, they produce time-frequency plots that appear smoother along the frequency axis compared to the Filter-Hilbert method.

2. Artifact Minimization (Time Domain)

The shape of the window matters significantly for preventing artifacts in your data.

  • No Sharp Edges: Morlet wavelets use a Gaussian taper. Unlike “box-car” (square) filters, Gaussian windows have no sharp edges.
  • Consequence: Sharp edges in a filter kernel can produce “edge artifacts” or ripples in the time domain. The smooth Gaussian taper of the Morlet wavelet prevents these artifacts, ensuring that the influence of surrounding time points is dampened smoothly.

3. The Optimal Precision Trade-off

The Gaussian shape of the Morlet wavelet is not arbitrary; it is mathematically optimized.

  • Control: The Gaussian window allows you to precisely control the trade-off between temporal precision and frequency precision.
  • Stationarity: The stationarity assumption for wavelets is limited to the brief window where the wavelet is non-zero. This is a much safer assumption for neural data (which is often stationary for only hundreds of milliseconds) compared to methods that require longer windows.

4. Summary: When to Use Which?

  • Use Filter-Hilbert: When you need exact control over the filter shape (e.g., a flat “plateau” to treat all frequencies in a band exactly equally).
  • Use Morlet Wavelets: When you prioritize smoothness in your results, artifact minimization, and a mathematically robust trade-off between time and frequency precision.