K-D model of wave propagation

Planar wave model

Engineers model wave propagation as a finite sum of discrete, locally planar waves instead of a ray tracing model. This reduces computation and lowers requirements for optics knowledge. The waves are considered planar between the transmitters and the receivers. Two other important assumptions are:

• Relative bandwidth is small enough so that the time delay can be simply transformed to a phase shift among the antennas.
• The array aperture is small enough that there is no observable magnitude variation.

Based on such assumptions, the basic signal model is described as:

${\displaystyle h(\alpha ,\tau ,\psi _{R},\upsilon _{R},\psi _{T},\upsilon _{T})=\sum _{p=1}^{P}\gamma _{p}\delta (\alpha -\alpha _{p})\delta (\tau -\tau _{p})\cdot \delta (\psi _{R}-\psi _{R_{p}})\delta (\upsilon _{R}-\upsilon _{R_{p}})\cdot \delta (\psi _{T}-\psi _{T_{p}})\delta (\upsilon _{T}-\upsilon _{T_{p}})}$

where ${\displaystyle \tau _{p}}$ is the TDOA (Time Difference of Arrival) of the wave-front ${\displaystyle p}$. ${\displaystyle \psi _{R_{p}},\upsilon _{R_{p}}}$ are DOA at the receiver and ${\displaystyle \psi _{T_{p}},\upsilon _{T_{p}}}$ are DOD at the transmitter,${\displaystyle \alpha _{p}}$ is the Doppler shift.^[1]

Excitation signal

A multitoned signal is chosen as the excitation signal.

${\displaystyle x(t)=\sum _{k=-N_{c}/2}^{N_{c}/2-1}\sin(2\pi (f_{c}+k\cdot \Delta f)\cdot t+\theta _{k})}$

where ${\displaystyle f_{c}}$ is the center frequency, ${\displaystyle \Delta f=B/N_{c}}$ (${\displaystyle B}$ is Bandwidth, ${\displaystyle N_{c}}$ is Number of multitones) is the tone spacing, and ${\displaystyle \theta _{k}}$ is the phase of the ${\displaystyle k^{th}}$ tone. we can obtain ${\displaystyle \theta _{k}}$ by

${\displaystyle \theta _{k}={{(\pi \cdot k)}^{2} \over N_{c}/2}}$

Data post-processing

RUSK SOUNDING. ${\displaystyle \alpha _{max}}$ is the maximum Doppler frequency.${\displaystyle \tau _{max}}$ is the maximum duration of the impulse response and S is the channel's spread (the red rectangle in the figure).^[4]

1. A DFT over K-1 (one waveform lost due to array switching) waveforms that measured in each channel is performed (K: waveforms per channel).
2. The frequency domain samples at the multitone frequencies are picked at every ${\displaystyle (K-1)^{th}}$ sample.
3. An estimated channel transfer function ${\displaystyle {\hat {H}}(f)}$ is obtained by:

${\displaystyle {\hat {H}}(f)={X_{ref}(f)^{*}\cdot Y(f) \over {\left|X_{ref}(f)\right|}^{2}+c\cdot {\hat {\sigma }}_{N}^{2}(f)}}$

where ${\displaystyle {\hat {\sigma }}_{N}^{2}(f)}$ is the noise power, ${\displaystyle X_{ref}(f)}$ is a reference signal and ${\displaystyle Y(f)}$ is the samples. The scaling factor c is defined as

${\displaystyle c={{\bar {\sigma }}_{ref}^{2} \over max{({\bar {\sigma }}_{Y}^{2}}-{\bar {\sigma }}_{N}^{2}),{\bar {\sigma }}_{N}^{2}}}$