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Frequency domain and spectrum

The frequency domain is another way to obtain the weak (wide sense) characterization of a stationary stochastic process.

Consider a stationary stochastic process y(t)y(t):

  • E[y(t)]=my ∀t\mathbb{E}[y(t)]=m_y \text{ } \forall t
  • γy(τ)=E[(y(t)−my)(y(t−τ)−my)]=my\gamma_y(\tau)=\mathbb{E}[(y(t)-m_y)(y(t-\tau)-m_y)]=m_y

Then we define the spectrum as the discrete Fourier transform of the covariance function γy(τ)\gamma_y(\tau) of the stationary stochastic process y(t)y(t): Γ(ω):=∑τ=−∞+∞γy(τ)e−jωτ\Gamma(\omega):=\sum_{\tau=-\infty}^{+\infty}{\gamma_y(\tau)e^{-j\omega\tau}}

Example: spectrum of the white noise

Given the white noise e(t)∼EN(μ,λ2)e(t) \sim EN(\mu,\lambda^2) which has the covariance function: γe(τ)={λ2if τ=00if τ≠0 \gamma_e(\tau)= \begin{cases} \lambda^2 &\text{if } \tau=0 \newline 0 &\text{if } \tau\neq0 \end{cases} The spectrum is: Γ(ω)=∑τ=−∞+∞γe(τ)e−jωτ=γe(τ)e−jω0=λ2\Gamma(\omega)=\sum_{\tau=-\infty}^{+\infty}{\gamma_e(\tau)e^{-j\omega\tau}}=\gamma_e(\tau)e^{-j\omega0}=\lambda^2

Example: spectrum of MA(1) process

Consider the process y(t)=e(t)+ce(t−1)y(t)=e(t)+ce(t-1), his covariance function is: γy(τ)={(12+c2)λ2if τ=0(1×c)λ2if τ=±10if τ≠0 \gamma_y(\tau)= \begin{cases} (1^2+c^2)\lambda^2 &\text{if } \tau=0 \newline (1 \times c)\lambda^2 &\text{if } \tau=\pm1 \newline 0 &\text{if } \tau\neq0 \end{cases} The spectrum of the process is then: Γ(ω)=∑τ=−∞+∞γy(τ)e−jωτ=(1+c2)λ2e−jω0+cλ2e−jω1+cλ2e−jω(−1)=λ2[1+c2+c(e−jω+ejω)]\Gamma(\omega)=\sum_{\tau=-\infty}^{+\infty}{\gamma_y(\tau)e^{-j\omega\tau}}=(1+c^2)\lambda^2e^{-j\omega0}+c\lambda^2e^{-j\omega1}+c\lambda^2e^{-j\omega(-1)}=\lambda^2[1+c^2+c(e^{-j\omega}+e^{j\omega})] Since we know that: ejω=cos(ω)+jsin(ω)e^{j\omega}=cos(\omega)+jsin(\omega) and then e−jω=cos(ω)−jsin(ω)e^{-j\omega}=cos(\omega)-jsin(\omega) We can rewrite the spectrum as: Γ(ω)=λ2(1+c2+2cos(ω))\Gamma(\omega)=\lambda^2(1+c^2+2cos(\omega))

Properties of the spectrum

The properties of the spectrum are inherited from the properties of γy(τ)\gamma_y(\tau).

  1. Γ(ω)∈R ∀ω\Gamma(\omega) \in \mathbb{R} \text{ } \forall\omega the spectrum is real valued. This happens, intuitivelly, because since γy(τ)\gamma_y(\tau) = γy(−τ)\gamma_y(-\tau) then the immaginary parts cancel out.
  2. Γ(ω)≥0 ∀ω\Gamma(\omega) \ge 0 \text{ } \forall\omega the spectrum is positive.
  3. Γ(ω)=Γ(−ω) ∀ω\Gamma(\omega) = \Gamma(-\omega) \text{ } \forall\omega the spectrum is an even function.
  4. Γ(ω)=Γ(ω+2kπ) ∀ω,k=0,±1,...\Gamma(\omega) = \Gamma(\omega +2k\pi) \text{ } \forall\omega,k=0,\pm1,... the spectrum is periodic with period 2π2\pi.

Spectrum of output of digital filters

Given the well defined stationary stochastic process y(t)=W(z)v(t)=A(z)B(z)v(t)y(t)=W(z)v(t)=\frac{A(z)}{B(z)}v(t) we can compute the covarianca function and then the spectrum.

There exists an easier way... an important theorem states that: Γy(ω)=∣W(ejw)∣2Γv(ω)\Gamma_y(\omega)=|W(e^{jw})|^2\Gamma_v(\omega) Which means that the spectrum of the output is the spectrum of the input multiplied by the transfer function evaluated for z=ejwz=e^{jw} absolute value square.

Example: spectrum of MA(1) using digital filter th.

Given the MA(1)MA(1) process y(t)=e(t)+ce(t−1)=(1+cz−1)e(t)y(t)=e(t)+ce(t-1)=(1+cz^{-1})e(t) with e(t)∼EN(μ,λ2)e(t) \sim EN(\mu,\lambda^2) the spectrum of the process is: Γy(ω)=∣W(ejw)∣2Γe(ω)=∣1+ce−jw∣2λ2=λ2(1+c2+2cos(ω))\Gamma_y(\omega)=|W(e^{jw})|^2\Gamma_e(\omega)=|1+ce^{-jw}|^2\lambda^2=\lambda^2(1+c^2+2cos(\omega))

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