## 数学代写|信息论作业代写information theory代考|STEM2004

2022年7月21日

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## 数学代写|信息论作业代写information theory代考|The Low-Energy Regime

To explore the low-energy regime, we consider bandlimited signals with $N_{0}$ degrees of freedom subject to the fixed energy constraint (1.22), and assume the addition of random Gaussian noise independently to each degree of freedom, subject to (1.68). In this case, the energy of the signal is bounded, while the total amount of noise is proportional to $N_{0}$, and we have
$$\frac{1}{N_{0}} \sum_{n=1}^{N_{0}} x_{n}^{2} \leq \frac{E}{N_{0}}$$
which tends to zero as $N_{0} \rightarrow \infty$. Substituting $E / N_{0}$ for $P$ into (1.73) and using a first-order Taylor expansion of the logarithmic function, we have
\begin{aligned} C &=\frac{1}{2} \log \left(1+\frac{E}{N_{0} \epsilon^{2}}\right) \ & \simeq \frac{E}{2 N_{0} \epsilon^{2}} \log e \text { bits per degree of freedom. } \end{aligned}
It follows that in a regime where the energy of the signal is negligible compared to the energy of the noise, the total amount of information carried by any one signal in the space and expressed in bits is proportional to the energy of the signal, and remains bounded even if the number of degrees of freedom tends to infinity. On the other hand, the capacity per degree of freedom in (1.77) vanishes as $N_{0} \rightarrow \infty$. This is due to the signal being spread over a large number of degrees of freedom, while a constant amount of noise is added to each degree of freedom.

## 数学代写|信息论作业代写information theory代考|The High-Energy Regime

In both the deterministic model of Kolmogorov and the stochastic model of Shannon, we can increase the amount of information associated with the waveforms in the signals’ space by increasing the signal-to-noise ratio. By (1.24), (1.26), (1.46), and (1.73), this increases entropy and capacity by a logarithmic factor. We now ask whether we can also spend energy to obtain a linear increase of the amount of information, keeping a fixed signal-to-noise ratio. A possible strategy seems to be to increase the number of degrees of freedom, since this increases entropy and capacity linearly, and by (1.15) and (1.18) it can be accomplished by increasing the frequency of radiation. It turns out, however, that high-frequency signals are also observed at a coarser resolution, so that increasing the frequency while keeping the signal-to-noise ratio constant requires a corresponding increase of the energy per degree of freedom of the radiated signal, and an ultimate limit to the amount of information is imposed by the laws of high-energy physics.

To view these effects in more detail, let us have a closer look at the quantities determining the number of degrees of freedom. By (1.15), in a two-dimensional setting the number of space-wavenumber degrees of freedom at every frequency $\omega$ depends on size of the cut-set boundary and on the frequency of radiation. For any arbitrary configuration of sources and scatterers, we can increase the number of space-wavenumber degrees of freedom by transmitting at higher and higher frequencies. This improves the spatial resolution of the received waveform on the cut-set boundary. Similarly, in a three-dimensional setting (1.18) shows that the number of spatial degrees of freedom at each frequency $\omega$ increases with the frequency of radiation.
When radiation occurs over a range of frequencies of support $2 \Omega$ centered around the origin, the total number of degrees of freedom is given by (1.16) and (1.19), in two and three dimensions respectively. These equations show that the number of degrees of freedom grows with the largest frequency $\Omega$ of the radiated signal.

Finally, when radiation occurs over a bandwidth $\Omega$ centered around a carrier frequency $\omega_{\mathrm{c}} \gg \Omega$, as depicted in Figure 1.25, a computation analogous to (1.16) gives the following total number of degrees of freedom in the two-dimensional setting:
$$N_{0}=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \int_{\omega_{1}}^{\omega_{2}} \omega d \omega$$ $=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \frac{\left(\omega_{2}^{2}-\omega_{1}^{2}\right)}{2}$
$=\frac{\Omega T}{\pi} \frac{2 \pi r \omega_{c}}{c \pi} .$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|The Low-Energy Regime

$$\frac{1}{N_{0}} \sum_{n=1}^{N_{0}} x_{n}^{2} \leq \frac{E}{N_{0}}$$

## 数学代写|信息论作业代写information theory代考|The High-Energy Regime

\begin{aligned} &=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \frac{\left(\omega_{2}^{2}-\omega_{1}^{2}\right)}{2} \ &=\frac{\Omega T T}{\pi} \frac{2 \pi r \omega_{c}}{c \pi} . \end{aligned}
$$N_{0}=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \int_{\omega_{1}}^{\omega_{2}} \omega d \omega$$

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## MATLAB代写

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