L41 Frequency Domain Analysis

Spectral Analysis

• Decomposing a TS into \(\sin\) and \(\cos\) functions of different frequencies.
• Appropriate when you observe cycle dynamics (periodicity, cyclicity)
  • Infectious disease data (periodicity of immunity)
  • Business cycles (periodicity of cycles)
• Decompose stationary TS \(\{ Y_{t} \}\) into 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. We have to identify those frequencies which are particularly important (strong).

• Till now, time domain approach
  • Regression on past values of TS (\(y_{t-i}\)) and shocks (\(e_{t}\))
• Regression approach considers regression on sinusiods
  • Spectral density function is required

Trigonometry refresher

\[ \sin A \pm \sin B = 2 \sin \dfrac{A\pm B}{2} \cos \dfrac{A\mp B}{2} \]

\[ \cos A + \cos B = 2 \cos \dfrac{A+ B}{2} \cos \dfrac{A- B}{2} \]

\[ \cos A - \cos B = -2 \sin \dfrac{A+ B}{2} \sin \dfrac{A- B}{2} \]

• \(\sin k\pi = 0\) 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

• DFT of a function \(h(t)\) for \(t \in \{ \dots, -1, 0, 1,\dots \}\) is

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

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 \(h(t) = h(-t)\)

$$
H(\omega) = h(0) + \sum_{t=1}^{\infty} h(t) (e^{ -\omega t } + e^{ i \omega t})
$$
or

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

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

Notation

\[ y_{t} = R \cos t(\omega t + \nu) + u_{t} \]

• \(\omega:\) frequency of periodic variation \((0 \leq \omega \leq 2\pi)\)
• \(R:\) amplitude of variation
• \(\nu\) phase
• \(\{ u_{t} \}:\) purely random process

If \(R, \nu\) are constants

\[ E(y_{t} ) = R \cos (\omega t + \nu) \]

• Assume, \(R\) has 0 mean and finite variance, or \(\mu \sim Uniform(0,2\pi)\) then \(E(y_{t})=0\). Thus the process is stationary.

Practical applications of Fourier Transform

1. Signal procesing
   • Audio processing
     • Noise Reduction
     • Equalization (adjust the balance of frequency components)
     • Compression (represent audio signals more compactly by prioritizing significant frequency components)
   • Image processing
     • Edge detection (detect outlier)
     • Image compression (JPEG uses DCT)
2. Communications
   • Modulation
   • Demodulation
   • Spectrum Analysis
3. Physics and Engineering
   • Wave Analysis
   • Optics
   • Structural Analysis

which is \(t\)-dependent and thus becomes non-stationary.

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1. Sine and cosine function together are called sinusoids ↩