Radio Background

Basic Concepts

• Diffraction: Radio frequnce waves bend around an obstacle.
• Scattering: When an radio frequence wave encounters an obstacle, it will scatter into multiple waves.
• Additive Noise: Noise is introduced to the signal of interest via addition.
• Decibel(dB): Decibel is a logarithmic unit that expresses the ratio of two physical quantities of the same unit. It is often used to measure the relative magnitude of sound intensity or electrical signal power.
• Large-scale signal attenuation: Signal loss during long-distance propagation; large scale means it focuses on the macroscopic changes in signal with distance/terrain...
• Small-scale fading: Severe fluctuations in received signal strength over extremely short distances (wavelength levels) or over extremely short periods of time.
  • It originates from the superposition of scattered wavefronts. When a signal is emitted from a base station, it reaches receiver via different paths. Because the length of each path is different, the phase of the arriving signal is also different.
  • Wavefront: TODO
• Second Moment: Usually refer to the average of the squares of a random variable. $E[X^2]$
  • Unit Second Moment: $E[X^2]=1$
• Baseband: It is the range of frequencies occupied by a signal that has not been modulated to higher frequencies.
  • Carry the most original data; typically originate from transducers, converting some other variable into an electrical signal.
  • Frequency is usually very low, starting from zero.
• Power Spectral Density(PSD): A physical quantity that describes how signal power is distributed at different frequencies.
• Bandwidth: The range of frequencies contained within a signal. The unit is Hertz (Hz).
• Delay Spread: The time difference between the first arriving signal and the last arriving signal.
• Spectrum: A spectrum of radio arranged continuously by frequency or wavelength.
  • Spectrum Map: The distribution of signal strength in geographic space within a specific frequency.
  • Spectrum Map Interpolation: Estimate the occupancy and signal strength of a map at a specific frequency using a small number of measurement points.
• Shadowing Effect: When a signal encounters an obstacle, it is blocked or absorbed, resulting in a significant decrease in signal strength behind the obstacle.

Degrees of Freedom (DoF)

• The number of independent parallel data streams that a channel can carry, under the high SNR limit.
  • High SNR: The transmission power approaches infinity, or the ambient noise approaches zero.
  • High SNR means the signal is mainly affected by interference, while a low SNR means the signal is mainly affected by noise.

Shannon Formula

• Channel Capacity: the maximum theoretical rate without errors in a noisy communication channel
  • The unit of channel capacity is bits per second (bps).

$$C = W \log_2 (1 + SNR)$$

Notation:

• $C$: channel capacity
• $W$: bandwidth
• $SNR$: signal-to-noise ratio

• In High SNR environment, Shannon Formula has an approximate form: $C \approx W \log_2(SNR)$
  • Channel capacity is linearly proportional to the signal strength in dB.

Generalized Degrees of Freedom (GDoF)

• Related Paper: On the Optimality of Treating Interference as Noise

Pathloss & Radio Map

• Pathloss is a quantity that measures the loss of signal strength (reduction in power, or attenuation) between a transmitter and receiver.
• Radio Map is a spatial visualization of Pathloss, it is a function TODO: Is my understanding of the relationship between Radio Map and Pathloss accurate?
• Gain: Signal gain; it is usually the inverse of the loss.

Antenna Gain/Directivity

• How well the antenna converts input power into radio waves headed in a specified direction.
• It is the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.

General Pathloss Model

$$P_r(d) = \frac{P_t G_t G_r \lambda^2}{(4\pi)^2d^2L}$$

Notation:

• $d$: distance between the transmitter and receiver (in meters)
• $G_t$/$G_r$: transmit/receiver gain, related link
• $\lambda$: wavelength
• $P_t$/$P_r$: transmission/received power
• $L$: system loss factor, which is independent of propagation environment. The system loss factor represents overall attenuation or loss in the actual system hardware, including transmission line, filter, and antennas. Typically $L>1$ , with $L=1$ as ideal.

Free Space Pathloss (FSPL)

According to General Pathloss Model, we can set $L$ to 1 to get Free Space Pathloss (FSPL)

$$PL_F(d) = 10 \log (P_t/P_r) = -10 \log (\frac{G_t G_r\lambda^2}{(4\pi)^2d^2}) = -20 \log (\frac{\sqrt{G_t G_r} \lambda}{4\pi d})$$

Application Scenarios

• Device-to-device (D2D) link scheduling.
• Station assignment / User-cell site association: Assign a set of wireless devices to a set of cellular base stations. Related link.
• Fingerprint based localization: Locate device by matching signal characteristics.
• Physical-layer security, power control in multi-cell massive MIMO systems, user pairing in MIMO-NOMA systems, precoding in multi-cell large scale antenna systems, path planning, and activity detection.

Pathloss Prediction Methods

There are two main categories of methods for Pathloss Function/Radio Map Prediction: data-driven interpolation and model-based data fitting. We will introduce some representative methods below.

Kriging

• It is a data-driven interpolation method.
• It uses measured values from known points to predict signal strength at unmeasured points through mathematical algorithms.
• Assuming that points with closer geographical locations have more similar signal characteristics.
• It strives to be a BLUE (Best Linear Unbiased Estimator) method.
  • Best: To minimize variance, the results will be more stable, with less fluctuation in deviation from the true value.
  • Linear: Linear combination of the original observation data.
  • Unbaised: The expected value (average) of the estimated value equals the true value.

Radial Basis Function (RBF) Interpolation

• It is a data-driven interpolation method.
• Radial means the value depends only on the distance from the center. Basis Function is a set of simple, predefined mathematical functions.
• Prediction result is a linear combination of the basis functions.

$$f(y) = \sum_{i=1}^{k} w_i \cdot \phi(\|y - x_i\|)$$

Notation: $y$: Location of target point. $x_i$: Location of the $i$-th measurement point. $w_i$: Corresponding weight coefficients. $\phi$: A radial basis function.

Tomograph Methods

• It is a model-based data fitting method.
• It treats the Spatial Loss Field (SLF) as a blocking factor and calculates the degree of signal weakening by measuring the set of blocking factors along the path from transmitter to receiver.
  • Spatial Loss Field (SLF): A function describing the signal blocking capability of each point in space.
  • SLF can be calculated in reverse by measuring the total attenuation between many transmitter-receiver pairs.
• Low-rank structure assumption: Spatial attenuation distribution has some repetition or regularity.
  • Rank represents the number of independent pieces of information contained in the matrix.
  • Low rank implies strong correlation or similarity among many rows or columns of a matrix, allowing for the reconstruction of the entire matrix with very little known data.
• Sparsity assumption: Only a few places (such as buildings) will create significant blockings, most areas are empty (0).
• Piecewise homogeneity assumption: SLF values within the same area should be the same or similar, such as the values within the same building are homogeneous.
• Transmission is taken into account, while (diffraction)[#basic-concepts], reflection and scatter are ignored.

Ray-tracing

• It is a model-based data fitting method.
• Assuming the radio wave frequency is high enough (the wavelength is much smaller than the size of the obstacle), the propagation of radio waves can be treated as the propagation of light.
• Calculate the reflection, diffraction, transmission, and scattering that the rays undergo from the transmitter to the receiver.
• High accuracy, no measurement data or pre-training required.
• Slow calculation

Intelligent Ray Tracing (IRT)

• Use preprocessing to accelerate computation, see details in section B. 3D Ray-optical model based on preprocessing of this article.
  • building will be divided into several units according to some standards.
    • Differences in interaction types: If a unit interacts differently with a ray, it must be further subdivided.
    • Approximately constant: Ensure that all physical parameters remain nearly constant when different parts within a unit interact with the ray.
    • Visibility: Construct a visibility tree based on the visibility relationships between different units.
    • Nodes of a tree store physical parameters rather than actual values.
    • Visibility here is a probability; different rays result in different visibility, therefore only a subset of branches will be activated and used.
• In real-time computation, only the top layer of the visibility tree needs to be recalculated; subsequent layers can use the pre-processed data.
• IRT[X]: Limit the interaction between the ray and the environment to [X] (such as, 2 or 4) times.
  • Assuming that after [X] or more interactions, the intensity of the ray has been weakened to a negligible level.

Dominant Path Model (DPM)

• It is a model-based data fitting method.
• Focus on only one or a few dominant paths that play a decisive role in receiving energy.
• Computational load is light.
• By constructing a tree structure, the dominant path from transmitter to receiver can be found.
  • The root node is the transmitter.
  • Branches of the current node are convex angles or receivers within the current node's line of sight.
    • Convex angles: Angles greater than 180°, e.g. building exterior corner.
    • Concave corner: Angles less than 180°, e.g. building interior corner.
    • When rays strike a concave corner, they are usually further blocked by the wall or reflected multiple times, making it difficult to form a "dominant path" that can bypass the building and carry a large amount of energy.
• High robustness.
• Waveguide effect: When radio waves propagate through environments like streets or pipeline, they gain energy due to continuous reflection from the walls of buildings on either side.

Pathloss function:

$$L=20\cdot log(\frac{4\pi}{\lambda}) + 20\cdot p\cdot log(d) + \sum_{i=0}^{n}\alpha(\varphi,i) - \frac{1}{c}\sum_{k=0}^{c}w_k$$

Notation:

• Free space pathloss $log(\frac{4\pi}{\lambda})$
  • $\lambda$: wavelength
• Distance attenuation $p\cdot log(d)$
  • $p$: distance attenuation coefficient; dynamically adjust based on whether the current area is at line of sight or not at line of sight
  • $d$: total length of the path
• Interaction loss $\sum_{i=0}^{n}\alpha(\varphi,i)$
  • $\alpha$: signal loss caused by a change in direction (such as diffraction)
  • $\varphi$: turning angle
  • $n$: total number of interactions
• Waveguide effect $\frac{1}{c}\sum_{k=0}^{c}w_k$
  • $w_k$: waveguiding factor; enhancement effect of waveguide effect
  • $c$: total number of waveguide effect segments involved in the path

Gaussian Noise

• When an electrical variation obeys a Gaussian distribution, it is called Gaussian noise, or Random Noise.
• Its mean is 0 and its variance is constant.
• It can be used to simulate real-word random interference

Gaussian Interference Channel

• Theoretical interference channel model for communication .
• Two point-to-point communications occur over a shared medium in the same time.
• Interference exists during communication.
• Currently, no optimal encoding method has been found to achieve its maximum transmission limit (capacity region)
  • Link scheduling is currently the best practice.
• It yields very good performance in comparison with classical interference avoidance schemes such as CSMA
  • Carrier Sense Multiple Access(CSMA): Node will verifie the absence of other traffic before transmitting on a shared transmission medium

Link Scheduling

• Scheduling subsets of links: From all links that want to communicate, select a group of links that have less interference to make them work at the same time and frequency.
• Treat this sufficiently weak interference as Gaussian Noise.
• In a particular regime of weak interference, Treating Interference as Noise (TIN) is information-theoretic approximately optimal.
• FlashLinQ: A practical such link scheduling algorithm developed by Qualcomm
  • Some information-theoretic inspired D2D link scheduling algorithms, which significantly improve upon FlashLinQ: ITLinQ and ITLinQ+

Cellular Radio

In a cellular radio system, the area to be covered is divided up into a number of small areas called cells, with one radio base station positioned to give radio coverage of each cell

• Frequency Reuse: two cells that are far apart can use the same frequency without interfering with each other.
• Handoff / Handover: when moving across cells, the base station will switch seamlessly.
• Cell Splitting: when the number of users in a cell is too high, causing the base station to be unable to provide sufficient service, this cell can be divided into several smaller cells with more base stations.

Signal-to-Noise Ration (SNR)

It is a measure used in science and engineering that compares the level of a desired signal to the level of background noise.

It is the ratio of signal power to noise power:

$$SNR = \frac{G(x,y)P_{T_{x}}}{N_0 W}$$

Notation:

• $G(x,y)$: Pathloss function / Radio map, it represents large-scale signal attenuation
• $P_{T_x}$: Transmit power, the total energy of the signal when it is emitted.
• $N_0$: Power spectral density of additive noise.
• $W$: Signal bandwidth
• $frac{P_{T_x}}{W}$: Power spectral density of singal.

Baseband Equivalent Channel Model

This model ignores extremely high-frequency carrier waves (consider it would result in an extremely large amount of computation), and directly analyze signal loss and interference at the low-frequency digital signal level (baseband).

It compress the complex radio frequency propagation process into a single mathematical coefficient that directly affects the original/baseband data. It can also be called "Fading Channel Model", because it includes $G(x,y)$ (large-scale decay) and $H$ (small-scale decay). It can also be called "Fading Channel with AWGN(Additive White Gaussian Noise)", because it finally added $Z$ (additive noise).

$$Y = \sqrt{G(x, y)} H X + Z$$

Notation:

• $Y$: Received signal sample
• $G(x, y)$: Pathloss function / Radio map, it represents large-scale signal attenuation
• $H$: Normalized small-scale fading
  • Normalized: $E[|H|^2]=1$
• $X$: Transmitted signal sample
• $Z$: Additive noise

Received Energy per Sample

The following formula describes the average energy per sample of the received signal. It is a physical quantity derived from the baseband equivalent channel model. It is used to measure the total average strength of the signal and noise.

$$E[|Y|^2] = G(x, y)P_{T_x}/W +N_0$$

Notation:

• $G(x, y)P_{T_x}/W$: singal part
  • $G(x, y)$: pathloss function / radio map, it represents large-scale signal attenuation
  • $P_{T_x}$: transmit power, the total energy of the signal when it is emitted.
• $N_0$: noise part; power spectral density of additive noise.

It is derived based on this formula:

• Since $E[|H|^2]=1$, $H$ is eliminated during the averaging process.
• Since $X$ and $Z$ are independent, the cross term is 0.