Linear filters -- applications

Why is the assumption that $R_{0}(x)$ is zero outside the interval $\left|x\right| \leq d < f_{0}$ important in (8.21)?

Context restated from the book's Section 8.3.3 (Theorem 8.4): let $\left\{ X(t)\right\}$ be a narrow-banded stationary Gaussian process with mean 0 and a spectral density symmetric around $\pm f_{0}$,

where $R_{0}(x)$ is an even function that is zero outside the interval $\left|x\right| \leq d$, with $d < f_{0}$. Let $\left\{ Y(t)\right\}$ be the Hilbert transform of $\left\{ X(t)\right\}$, and let $\left\{ X_{1}(t)\right\}$, $\left\{ X_{2}(t)\right\}$ be the independent Gaussian processes, each with spectral density $R_{0}(f)$, obtained from the complex representation $Z(t) = X(t)+iY(t) = e^{i2\pi f_{0} t}\left(X_{1}(t)+iX_{2}(t)\right)$. Equation (8.21) is the resulting cross-covariance function,

which is used to show that $\left\{ X_{1}(t)\right\}$ and $\left\{ X_{2}(t)\right\}$ are uncorrelated (hence independent, being jointly Gaussian).

A stationary sequence $\left\{ S(t), t=0,\pm 1,\pm 2 \ldots \right\}$ has expectation zero and covariance function,

The process is disturbed by colored noise $\left\{ N(t), t=0,\pm 1,\pm 2 \ldots \right\}$ with covariance function,

The process and the disturbance are independent. Determine the frequency function $H(f)$, $-1/2 < f \leq 1/2$, and the impulse response $h(t)$, $t = 0,\pm 1,\pm 2 \ldots$, for the optimal Wiener filter that minimizes $\mathbb {E}\left[\left(Y(t)-S(t)\right)^{2}\right]$ when $Y(t)$ is the output signal from the filter with $X(t) = S(t)+N(t)$ as input signal.

Suppose that we want to measure the temperature, $X(t)$, in a manufacturing process. Since the measurements are corrupted by additive noise, $N(t)$, independent of the temperature, we reconstruct the true temperature with an optimal filter. Assuming $r_{X}(\tau ) = e^{-2\left|\tau \right|}$ and $r_{N}(\tau ) = e^{-20\left|\tau \right|}$, find the covariance function of the output signal, $\left\{ Y(t)\right\}$.

Suppose we want to transmit a bandlimited stochastic signal, $S(t)$, on a channel disturbed by flicker noise. The spectral density of the signal is

and that of the flicker noise is $R_{N}(f) = 100/\left|f\right|$ up to high frequencies.

An AR(2)-process $\left\{ X_{t}\right\}$ has a covariance function, estimated to $\widehat{r}_{X}(0) = 4$, $\widehat{r}_{X}(\pm 1) = 2$, and $\widehat{r}_{X}(\pm 2) = -0.5$. Determine from these values, the parameters for the AR-process, and find the frequency function (in real-valued form) for a Wiener filter which in the best way reconstructs $X_{t}$ from the measured signal $Y_{t} = X_{t}+N_{t}$, when $N_{t}$ is white noise with expected value 0 and variance 1. Assume that $\left\{ X_{t}\right\}$ and $\left\{ N_{t}\right\}$ are independent.

A deterministic signal, $s(t)$, is corrupted by additive white noise with spectral density $N_{0}$. The signal is defined by

The signal and the noise is processed by a linear optimal filter. We read the output of this filter at time $t = T > 0$. The filter has the property that the SNR is maximized at $t = T$.

A sender transmits binary characters to a receiver in a digital communication system. The signal, $s(t)$, see Figure 8.11, is used by the sender to represent a binary one and the absence of this signal is used to represent zero. We model the received signal, $X(t)$, by

where $\left\{ N(t), t \in \mathbb {R}\right\}$ is Gaussian noise with spectral density $R_{0}$. Determine the impulse response of a matched causal filter and compute the SNR (signal-to-noise ratio) of the output signal.

The signal, s(t) in Exercise 8:7.

A deterministic signal, $s_{k}$, in discrete time, is given by

We would like to determine whether this signal has been sent or not, when one receives the signal $s_{k}+N_{k}$ if the signal was sent and $N_{k}$ if it was not. We assume that $N_{k}$ is standardized white Gaussian noise.

A weakly stationary process $\left\{ S_{t}, t=0,\pm 1,\pm 2 \ldots \right\}$ has mean zero and covariance function,

The process is disturbed by colored noise $\left\{ N_{t}, t=0,\pm 1,\pm 2 \ldots \right\}$ with expected value zero and covariance function

The process and the disturbance are independent. In communication systems it is quite usual with echo, i.e., you have a signal according to

where $0.5 S_{t-1}$ is the damped and delayed original process realization. Determine the coefficients $a,b$ for the non-causal impulse response

for an optimal filter that minimizes $\mathbb {E}\left[\left(Y_{t}-S_{t}\right)^{2}\right]$, when $Y_{t}$ is the output signal and $X_{t}$ is the input signal to the optimal filter.

Find explicit expressions for the Kalman filter for an AR(1)-process observed with measurement errors,

when $\left\{ e_{t}\right\}$ and $\left\{ f_{t}\right\}$ are independent normal variables with mean zero and $\operatorname {Var}\left[e_{t}\right] = \sigma_{e}^{2}$, $\operatorname {Var}\left[f_{t}\right] = \sigma_{f}^{2}$.