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Frequency Domain Analysis

Spectral Analysis

Spectral analysis involves decomposing a Time Series (TS) into \(\sin\) and \(\cos\) functions of different frequencies. This approach is particularly appropriate when you observe cycle dynamics, such as periodicity or cyclicity.

Examples:
- Infectious disease data: Identifying the periodicity of immunity.
- Business cycles: Identifying the periodicity of economic cycles.

The goal is to decompose a stationary TS \(\{ Y_{t} \}\) into a combination of sinusoids1 with uncorrelated random coefficients:

\[Y_{t} = \phi_{1}\sin(A_{1}) + \phi_{2}\cos(A_{2})\]

where \(A_{1}\) and \(A_{2}\) are frequencies. Our task is to identify those frequencies which are particularly important or "strong" in the data.

Time Domain vs. Frequency Domain

  • Time Domain approach: Focuses on regression on past values of TS (\(y_{t-i}\)) and shocks (\(e_{t}\)).
  • Frequency Domain (Spectral) approach: Considers regression on sinusoids. To do this, a spectral density function is required.

Trigonometry Refresher

To work effectively in the frequency domain, we utilize several key trigonometric identities:

  • \(\sin A \pm \sin B = 2 \sin \dfrac{A\pm B}{2} \cos \dfrac{A\mp B}{2}\)
  • \(\cos A + \cos B = 2 \cos \dfrac{A+ B}{2} \cos \dfrac{A- B}{2}\)
  • \(\cos A - \cos B = -2 \sin \dfrac{A+ B}{2} \sin \dfrac{A- B}{2}\)
  • \(\sin k\pi = 0\) for all \(k = \pm 1,\pm 2,\dots\)
  • \(\cos k\pi = (-1)^{|k-1|}\) for all \(k = \pm 1,\pm 2,\dots\)
  • \(\sin(-A) = -\sin A\)
  • \(\cos (-A) = \cos A\)

Fourier and Inverse Fourier Transforms

The Discrete Fourier Transform (DFT) of a function \(h(t)\) for \(t \in \{ \dots, -1, 0, 1,\dots \}\) is:

\[H(\omega) = \sum_{t=-\infty}^{\infty} h(t)e^{ -i \omega t } \quad (-\pi \leq \omega \leq \pi)\]

The Inverse Fourier Transform of \(H(\omega)\) is given by:

\[h(t) = \dfrac{1}{2\pi}\int_{-\pi}^{\pi} H(\omega) e^{ -\omega t } \, d\omega\]

Properties

If the function is symmetric, \(h(t) = h(-t)\), the transform simplifies:

\[H(\omega) = h(0) + \sum_{t=1}^{\infty} h(t) (e^{ -\omega t } + e^{ i \omega t})$$ or $$H(\omega) = h(0) + 2 \sum_{t=1}^{\infty} h(t) \cos \omega t, \quad (-\pi \leq \omega \leq \pi)\]

Consequently, \(h(t) = \dfrac{1}{\pi} \int_{0}^{\pi}H(\omega) \cos(\omega t)\ d\omega\).

Notation for a Periodic Process

A typical periodic process can be represented as:
$\(y_{t} = R \cos (\omega t + \nu) + u_{t}\)$

  • \(\omega\): Frequency of periodic variation \((0 \leq \omega \leq 2\pi)\).
  • \(R\): Amplitude of variation.
  • \(\nu\): Phase.
  • \(\{ u_{t} \}\): Purely random process.

Stationarity Note:
- If \(R\) and \(\nu\) are constants, \(E(y_{t}) = R \cos (\omega t + \nu)\), which is \(t\)-dependent and thus becomes non-stationary.
- However, if we assume \(R\) has a 0 mean and finite variance, or if the phase \(\nu \sim \text{Uniform}(0,2\pi)\), then \(E(y_{t})=0\). In this specific case, the process becomes stationary.

Practical Applications of Fourier Transform

  1. Signal Processing
    • Audio processing: Noise reduction, equalization (adjusting the balance of frequency components), and compression (representing signals compactly by prioritizing significant frequencies).
    • Image processing: Edge detection (detecting outliers) and image compression (JPEG uses the Discrete Cosine Transform/DCT).
  2. Communications
    • Modulation and Demodulation.
    • Spectrum Analysis.
  3. Physics and Engineering
    • Wave Analysis.
    • Optics.
    • Structural Analysis (e.g., studying vibrations).

  1. Sine and cosine function together are called sinusoids ↩