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 $y(t)$:

• $\mathbb{E}[y(t)]=m_y \text{ } \forall t$
• $\gamma_y(\tau)=\mathbb{E}[(y(t)-m_y)(y(t-\tau)-m_y)]=m_y$

Then we define the spectrum as the discrete Fourier transform of the covariance function $\gamma_y(\tau)$ of the stationary stochastic process $y(t)$: $$\Gamma(\omega):=\sum_{\tau=-\infty}^{+\infty}{\gamma_y(\tau)e^{-j\omega\tau}}$$

Example: spectrum of the white noise

Given the white noise $e(t) \sim EN(\mu,\lambda^2)$ which has the covariance function: $$ \gamma_e(\tau)= \begin{cases} \lambda^2 &\text{if } \tau=0 \newline 0 &\text{if } \tau\neq0 \end{cases} $$ The spectrum is: $$\Gamma(\omega)=\sum_{\tau=-\infty}^{+\infty}{\gamma_e(\tau)e^{-j\omega\tau}}=\gamma_e(\tau)e^{-j\omega0}=\lambda^2$$

Example: spectrum of MA(1) process

Consider the process $y(t)=e(t)+ce(t-1)$, his covariance function is: $$ \gamma_y(\tau)= \begin{cases} (1^2+c^2)\lambda^2 &\text{if } \tau=0 \newline (1 \times c)\lambda^2 &\text{if } \tau=\pm1 \newline 0 &\text{if } \tau\neq0 \end{cases} $$ The spectrum of the process is then: $$\Gamma(\omega)=\sum_{\tau=-\infty}^{+\infty}{\gamma_y(\tau)e^{-j\omega\tau}}=(1+c^2)\lambda^2e^{-j\omega0}+c\lambda^2e^{-j\omega1}+c\lambda^2e^{-j\omega(-1)}=\lambda^2[1+c^2+c(e^{-j\omega}+e^{j\omega})]$$ Since we know that: $e^{j\omega}=cos(\omega)+jsin(\omega)$ and then $e^{-j\omega}=cos(\omega)-jsin(\omega)$ We can rewrite the spectrum as: $\Gamma(\omega)=\lambda^2(1+c^2+2cos(\omega))$

Properties of the spectrum

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

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

Spectrum of output of digital filters

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

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

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

Given the $MA(1)$ process $y(t)=e(t)+ce(t-1)=(1+cz^{-1})e(t)$ with $e(t) \sim EN(\mu,\lambda^2)$ the spectrum of the process is: $$\Gamma_y(\omega)=|W(e^{jw})|^2\Gamma_e(\omega)=|1+ce^{-jw}|^2\lambda^2=\lambda^2(1+c^2+2cos(\omega))$$

Questions from past exams

What is the spectrum of a stationary stochastic process and explain the most important properties.

See above.

Give the definition of spectral density of a stationary stochastic process. Then, prove that the spectral density of the sum of two uncorrelated stationary processes is equal to the sum of the spectral densities.

As far as I know we did not see this proof in class. Probably starting from the definition it can be worked out.