# Frequency Domain ## Transfer functions The transfer function $W(z)$, being a ratio of two polynomials, acts as a **digital filter**. It is instrumental in transforming $WN$ into an ARMA process and is applied in analyzing stationary processes through digital filter theory. **Shift Operators in Time Domain Analysis** - Backward shift operator $z^{-1}$ shifts a signal back in time - Forward shift operator $z$ shifts a signal forward in time. Properties of $z$ and $z^{-1}$ include linearity, recursive application, and linear composition. Using these properties, we can rewrite an AR model equation in terms of $z$ and $z-1$, revealing the relationship between the operators and the AR model. The resulting equation shows that the output $y_t$ is the **ratio** of two polynomials of $z$: $y(t)=\frac{c_{0}+c_{1}z^{-1}+\cdots c_{n}z^{n}}{1-a_{1}z^{-1}+\cdots a_{m}z^{-m}}e(t)=\frac{C_{(z)}}{A(z)}e(t)=W(z)e^{it}(t)$ ## Asymptotically stability Stability analysis involves examining the poles of the transfer function: $y(t)=\frac{C(z)}{A(z)}e(t)$ - Zeros are values of $z$ where the transfer function equals zero - Poles are the inverse of the transfer function's roots (roots of $A(z)$ in case of no cancellations). The locations of poles and zeros, particularly within the unit circle in the complex plane, are indicative of system stability. - To compute zeros, set the numerator of the transfer function equal to zero. - For poles, set the denominator equal to zero. A system is considered asymptotically **stable** if all **poles** lie strictly inside the unit circle, and it is deemed a minimum phase system if both poles and zeros reside within the unit circle. The asymptotic stability is often necessary to demonstrate if a process $y(t)=W(z)e(t)$ is stochastic process: - the process transfer function is asymptotically stable (its poles are all located within the unitary circle) - the input (e.g. $e(t)$) is a stationary stochastic process ## Canonical representation of $SSP$ Stationary stochastic processes (so including ARMA, MA and AR) can be equivalently represented in 4 distinct ways, each offering unique insights and applications: **Probabilistic representation**: $y(t)= m_y,\gamma_y(\tau)$ the mean provides information about the central tendency of the process, while the covariance function captures how data points at different time lags are related: particularly useful for understanding the process's statistical properties and for generating simulations. **Difference equations** (time domain): $y(t)=a_1y(t-1)+\cdots a_my(t-m)+ c_o e(t)+\cdots c_ne(t-n)$ essential in modeling and forecasting **Transfer function** (operatorial): $y(t)=\frac{C(z)}{A(z)}e(t)$ Transfer functions are commonly used in control theory and engineering applications, where the focus is on manipulating and controlling the process. **Frequency** $y(t)=m_y,\Gamma_y(\omega)$It is especially valuable in applications involving signal processing, where the focus is on understanding the frequency content of the data. They are all equivalent but not **unique**. The uniqueness aspect it will be problematic, and so before introducing prediction theory, we need to better understand the non-uniqueness property and find ways to make the representation unique. Taken the process in operatorial representation: $y(t)=\frac{C(z)}{A(z)}e(t)$ We can adopt this "convention" to be sure about the uniqueness of representation: 1) **monic**: the term with the highest power has coefficient equal to $1$ 2) **coprime**: no common factors to that can be simplified 3) have **same degree** 4) have **roots inside the unit circle** ## The spectral factorization theorem In the analysis of discrete-time signals through the use of the transfer function $W(z)$, if $y(t)=W(z)v(t)$ and: - $v(t)$ is stationary - $W(z)$ is stable $y(t)$ is well defined and stationary. ## Final value theorem The final value theorem or theorem of the gain is: $E[y(t)]=W(1) \cdot E[u(t)]$ **NB**: The final value theorem can be applied to $W(z)$ that are not in canonical form. Final value theorem is useful when we want to define de-biased process: 1. Find mean with gain theorem: $ E[y(t)] = W(1) \cdot \mu = \bar{y} $ 2. Define de-biased processes: $ \begin{aligned} \tilde{y}(t) &= y(t) - \bar{y} \\ \tilde{e}(t) &= e(t) - \mu \end{aligned} $ 3. Do whatever you want, like computing the covariance $\gamma_{\tilde{y}}(\tau)=\gamma_{\gamma}(\tau)\quad\forall\tau$ ## All Pass Filter In the context of transfer functions $W(z)$ and digital filters, an all pass filter is a particular process which transform a white noise in a white noise with a different variance.  An APF is useful for converting a system to canonical form: we can do this by aggregating coefficients and multiplying by a fraction of identical polynomials (so that we essentially multiply by one) and then we redefine a new white noise process. Apart the variance everything remains the same; for example if input $u(t)$ and output $y(t)$: - If $u(t)$ is constant, $y(t)$ is constant. - If $u(t)$ is sinusoidal, $y(t)$ is sinusoidal. ## Spectral representation Spectral density of a stationary stochastic process  $\Gamma_{y}(w)=\sum_{\tau=-\infty}^{\infty}\gamma_{y}(\tau)\cdot e^{-jw\tau}$ For a stationary process $y(t)$, $\Gamma_y(\omega)$: - positive - real - even $\Gamma(-\omega) = \Gamma(\omega)$ - periodic with period $T = 2\pi$. Its graph can be represented in the upper half plane of a 2D plot, as the imaginary part is always zero.  The "general rule" regarding the effect of the poles and zeros of the transfer function is that if the generic point $e^{j\omega}$ is: - Near **zeroes**, the $\Gamma _y (\omega)$ exhibits attenuation. For a zero on the unit circle there is the so called **blocking property** of the transfer function, which refers to the phenomenon where a zero on the unit circle in the z-domain causes the system to completely attenuate that specific frequency component of the input signal. As the frequency moves away from this zero, the attenuation decreases. - Near **poles**, the $\Gamma _y (\omega)$ is high. The frequency response is amplified. If a pole is near the unit circle, it signifies a strong frequency response at the angle $\omega$ corresponding to the pole's position. ## Spectrum Properties For stationary processes, the following properties hold for the power spectral density $\Gamma(\omega)$: 1. Scalar Multiple If $y(t) = ax(t)$, then: $ \Gamma_y(\omega) = a^2 \Gamma_x(\omega) $ 2. Sum of Uncorrelated Processes: If $z(t) = x(t) + y(t)$ and $x(t)$ and $y(t)$ are uncorrelated, then: $ \Gamma_z(\omega) = \Gamma_x(\omega) + \Gamma_y(\omega) $ 3. Inverse transform returns the covariance function based on the spectral density, noted as the inverse discrete Fourier transform. $\gamma_{y}(z)=F^{-1}(\Gamma_{y}(\omega))=\frac{1}{2\pi}\int_{-\pi}^{\pi}\Gamma_{y}(\omega) d \omega$ 4. The spectrum of a $WN$ is constant and equal to its variance: $\Gamma_{e}(\omega)==Var[e(t)]=\lambda^{2}$ ### Example If $v(t) = ax(t) - by(t)$ and $x(t)$ and $y(t)$ are uncorrelated, then: $\Gamma_v(\omega) = a^2\Gamma_x(\omega) + b^2\Gamma_y(\omega)$ ## Fundamental theorem of spectral analysis Very useful theorem to compute the spectrum: $\Gamma_y(\omega)=W(z=e^{j\omega})W(z=e^{-j\omega})\Gamma_u(\omega)$ It's possible to apply the fundamental theorem of spectrum analysis also if $W(z)$ is not in the canonical form :) **Recall of useful formulas** for spectrum qualitative study are: - $e^{j\omega } =\cos \omega +j \sin \omega$ - $e^{-j\omega }=\cos \omega -j\sin \omega$ - $e^{-j\omega}+e^{+j\omega}=2\cos(\omega)$ Remember that is always possible to write the spectral density function as a real-valued function composed by cosine terms easier to interpret.