fourier-spectral-analysis

name	fourier_spectral_analysis
description	Use when decomposing signals or fields into frequency/momentum components, applying Fourier transforms, or using spectral methods to solve differential equations.

Fourier and Spectral Analysis

Apply this skill when the problem involves transforming between position/time and momentum/frequency domains, analyzing spectral content of signals or fields, or solving differential equations via spectral decomposition.

Goal

Perform Fourier transforms (continuous or discrete), analyze spectral properties, and apply spectral methods to solve physical problems efficiently.

Scope

• Continuous Fourier transform and its inverse (1D and multi-dimensional)
• Discrete Fourier transform (DFT) and Fast Fourier Transform (FFT)
• Power spectral density and Parseval's theorem
• Convolution theorem and filtering
• Spectral methods for solving ODEs and PDEs (Fourier, Chebyshev)
• Sampling theorem (Nyquist) and aliasing
• Windowing and spectral leakage in finite data sets
• Laplace and Z-transforms when appropriate

Inputs

• signal_or_field: The function, data set, or field to be transformed
• domain: Spatial/temporal domain specification (extent, sampling rate, boundary conditions)
• transform_type: Which transform to apply (continuous FT, DFT/FFT, Laplace, etc.)
• analysis_goal: What to extract (frequency content, transfer function, spectral solution of PDE, etc.)

Outputs

• transformed_result: The Fourier (or other spectral) transform of the input
• spectral_analysis: Power spectrum, dominant frequencies, bandwidth
• spectral_solution: Solution to a differential equation obtained via spectral methods
• numerical_code: Python implementation using numpy.fft or scipy.fft

Workflow

1. Choose the appropriate transform.

   • Continuous data on infinite domain: continuous Fourier transform.
   • Discrete sampled data: DFT/FFT.
   • Periodic boundary conditions: Fourier series / FFT.
   • Non-periodic on finite domain: Chebyshev or sine/cosine transforms.

2. Apply the transform.

   • Use consistent sign and normalization conventions; state them explicitly.
   • Common physics convention: f_hat(k) = integral f(x) e^{-ikx} dx.
   • numpy convention: uses e^{-2 pi i k n / N} for DFT.

3. Analyze in the spectral domain.

   • Identify dominant frequencies/modes.
   • For PDEs: algebraic equations in Fourier space replace differential operators (d/dx -> ik).
   • Apply filters or solve algebraic equations as needed.

4. Transform back (if needed).

   • Apply the inverse transform.
   • Verify by comparing with the original (Parseval's theorem as a consistency check).

5. Handle numerical aspects.

   • Respect the Nyquist limit: maximum resolvable frequency = sampling_rate / 2.
   • Apply windowing (Hann, Hamming) to reduce spectral leakage for finite-length data.
   • Zero-pad for finer frequency resolution if needed.

Quality Checks

• Parseval's theorem: total power in time/space domain must equal total power in frequency domain.
• The inverse transform of the forward transform must recover the original signal (within numerical precision).
• Nyquist criterion: the sampling rate must be at least 2x the highest frequency present.
• For spectral PDE solutions, verify against known analytic solutions or convergence under refinement.

Constraints

• Always state the Fourier convention (sign, normalization) being used; mixing conventions is a common source of errors.
• Do not interpret FFT results beyond the Nyquist frequency; those are aliases.
• Spectral methods assume smoothness; for discontinuous solutions (shocks), use appropriate filtering or switch to finite-difference/finite-volume methods.
• When applying DFT, the data is implicitly assumed periodic; non-periodic data requires windowing or alternative transforms.