Sparsity in the Delay-Doppler Domain for Measured 60 GHz Vehicle-to-Infrastructure Communication Channels

TL;DR

This study presents millimeter wave V2I channel measurements in Vienna, revealing high sparsity in the delay-Doppler domain, influenced by beam elevation, with implications for urban 60 GHz communication systems.

Contribution

First measurement-based analysis of 60 GHz V2I channels in an urban environment highlighting delay-Doppler sparsity and beam elevation effects.

Findings

High sparsity observed in delay-Doppler domain

Beam elevation impacts propagation characteristics

Urban street environment influences channel sparsity

Abstract

We report results from millimeter wave vehicle-to-infrastructure (V2I) channel measurements conducted on Sept. 25, 2018 in an urban street environment, down-town Vienna, Austria. Measurements of a frequency-division multiplexed multiple-input single-output channel have been acquired with a time-domain channel sounder at 60 GHz with a bandwidth of 100 MHz and a frequency resolution of 5 MHz. Two horn antennas were used on a moving transmitter vehicle: one horn emitted a beam towards the horizon and the second horn emitted an elevated beam at 15-degrees up-tilt. This configuration was chosen to assess the impact of beam elevation on V2I communication channel characteristics: propagation loss and sparsity of the local scattering function in the delay-Doppler domain. The measurement results within urban speed limits show high sparsity in the delay-Doppler domain.

Tables

Table 1. TABLE I: Channel sounder parameters

Parameter	Value
center frequency	$f_c=60.15\mathrm{GHz}$
tone spacing	$\Delta f=4.76\mathrm{MHz}$
number of tones	$K=21$
number of TX antennas	$2$
tone spacing between antennas	$\Delta f_{\text{Ant}}=1.19\mathrm{MHz}$
bandwidth	$B=100\mathrm{MHz}$
maximum alias free delay	${\tau }_{\text{max}}=105\mathrm{ns}$
transmit antennas	$20\mathrm{dBi}$ conical horn
receive antenna	$\lambda /4$ monopole
snapshot time	$T_{\text{snap}}=178.07\mu s$
maximum car speed	$v_{\text{max}}=14m{s}^{-1}$
recording time	$T_{\text{rec}}=3.6s$

Equations

$$\mathbf{H}={\mathbf{F}}_K\mathbf{S}{\mathbf{F}}_M^H,$$

$$\mathbf{h}=\left({\mathbf{F}}_M^H\otimes {\mathbf{F}}_K\right)\mathbf{s},$$

$$\mathbf{h}=\mathbf{As}+\mathbf{w},$$

$$\mathbf{h}=\tilde{\mathbf{A}}\tilde{\mathbf{s}}+\mathbf{w},$$

Full text

I Introduction

Vehicular communications was intensely investigated in the last decade [1, 2, 3, 4]. MmWave have been a research topic in vehicular communications for already several decades[5, 6]. The path loss for train-to-infrastructure scenarios in the millimeter wave (mmWave) frequency band is measured in [5], performance for vehicle-to-vehicle (V2V) communication is investigated in [6].

The effect of an overtaking vehicle on a V2V channel at $60\text{\,}\mathrm{GHz}$ of a communicating car platoon has been investigated in [7, 8]. A more detailed statistical analysis of delay and Doppler spread is provided in [9, 10]. V2V channel measurements with antennas placed in the bumpers of cars at $38\text{\,}\mathrm{GHz}$ and $60\text{\,}\mathrm{GHz}$, using a channel sounder with $1\text{\,}\mathrm{GHz}$ bandwidth, have been conducted in [11]. Further mmWave V2V measurements with approaching cars are shown in [12]. Doppler spectra of vehicle-to-infrastructure (V2I) measurements at $28\text{\,}\mathrm{GHz}$ in an expressway environment are shown in[13]. Doppler spectra of vibrations appearing while the vehicle is in operation are shown in [14] and [15].

The mmWave wireless channel has been observed to have a sparse multipath structure in multiple domains [16, 17, 18]. For channel estimation using sparse reconstruction, several techniques exist [19]. Sparse channel fits via the least absolute shrinkage and selection operator (LASSO) applied in the delay domain are provided in [20, 21, 22, 23]. In [22] and [23], Akaike’s information criterion was used to select the appropriate sparsity level, where approximately 4 to 6 clusters in the delay domain have been found. Sparse signal reconstruction using the sparse Bayesian learning (SBL) framework with low-complexity is used in [24, 25]. A fixed-point implementation of SBL [25] is shown in [26].

How to exploit the delay-Doppler sparsity of wireless channels is shown in [27]. The orthogonal time frequency space (OTFS) modulation [28] performs especially well for wireless channels with sparsity in the delay-Doppler domain, such as in mmWave communication systems[29].

Our contribution

In this contribution, we show that the sparsity in the delay-Doppler domain holds true for measured $60\text{\,}\mathrm{GHz}$ V2I communication channels in a street crossing scenario in an urban environment. We estimate the channel and find a representation in the delay-Doppler domain with a multitaper estimator. Furthermore, a sparse estimation with sub-Nyquist sampling in the delay domain is carried out using SBL.

II Measured Scenario

The measurement scenario and placement of the transmitters and the receiver (RX) is shown in Fig. 1a. Our TXs are mounted on the roof rack of a car. The RX is positioned at the crossroads of a street canyon, at $5\text{\,}\mathrm{m}$ height. We are observing multipath components of the wireless channel during the approach and passage of the moving transmitter vehicle at the crossroads. All MPCs are Doppler shifted proportional to their respective radial velocity.

For our setup, the $60\text{\,}\mathrm{GHz}$ free space path loss calculates to $100\text{\,}\mathrm{dB}$ at the maximum distance of interest. Both TXs use horn antennas with a gain of $20\text{\,}\mathrm{d}\mathrm{B}\mathrm{i}$, which point towards driving direction. One horn (TX [math]) emits a beam towards the horizon and the second horn (TX [math]) emits a beam at [math] up-tilt as shown in Fig. 1b. At the receiver side, an omni-directional $\nicefrac{{\lambda}}{{4}}$ monopole antenna is used. The receive antenna gain is approximately $-4\text{\,}\mathrm{d}\mathrm{B}\mathrm{i}$ including cable losses.

III RF Setup

The hardware setup for the measurement is shown in Fig. 2. Both TXs consist of external in-phase quadrature (I/Q) mixer modules, an arbitrary waveform generator (AWG), and a frequency reference. The AWG continuously repeats a baseband sounding sequence as I/Q signal for each external I/Q-mixer module. Each I/Q-mixer up-converts the input signal to passband at $60.15\text{\,}\mathrm{GHz}$ center frequency with a local oscillator, synthesized by a phase-locked loop (PLL). The radio frequency outputs of the I/Q-mixers emit mmWaves through conical horn antennas. The frequency reference feeds each PLL with a $285.714\text{\,}\mathrm{MHz}$ reference and the AWG with a $10\text{\,}\mathrm{MHz}$ reference. Our receiver is a signal analyzer (SA) connected to the omni-directional antenna and a light barrier. We measure the center frequency offset (CFO) between AWG and SA at standstill before each measurement. The oscillator drift during one measurement was found to be less than $1\text{\,}\mathrm{Hz}$. A measurement is started as soon as the moving transmitter vehicle passes the light barrier, which triggers the recording at the RX. We directly access the I/Q samples from the SA with a sampling rate of $125\text{\,}\mathrm{M}\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$.

IV Sounding Sequence Design

We extract an estimate of the channel transfer function with a multitone waveform as excitation signal for channel sounding similar to the approach in [30]. A Zadoff-Chu sequence [31] is used for its constant amplitude zero autocorrelation property to keep the crest factor low and to phase-align both TXs during calibration. For our scenario, we consider a maximum excess delay $\tau_{\text{max}}$ to the assumed propagation delay between the line of sight (LOS) component and MPCs due to single reflections. Paths due to multiple reflections and paths due to single reflections, due to back- and side-lobes at the TXs, are below the receiver sensitivity. Possible single reflections are limited by the surrounding street canyon. The width of the street canyon is $20\text{\,}\mathrm{m}$. Therefore, at a maximum distance of $44\text{\,}\mathrm{m}$ between TXs and RX, we account for path differences between LOS and MPCs, which will not be larger than $16\text{\,}\mathrm{m}$ in length or $54\text{\,}\mathrm{ns}$ in time. Reflections at the crossing and beyond the crossing might lead to delays larger than $100\text{\,}\mathrm{ns}$. The tone spacing $\Delta f$ of our multitone waveform has to fulfill the sampling theorem. Therefore, $\Delta f\leq{1/(2\tau_{\text{max}})}$ needs to hold true for the maximum excess delay $\tau_{\text{max}}$ of the channel. We use a measurement bandwidth $B=$100\text{,}\mathrm{MHz} and $K=$21 tones to get a tone spacing of $\Delta f=$4.76\text{,}\mathrm{MHz}, which leads to a maximum alias free delay of $\tau_{\text{max}}=$105\text{\,}\mathrm{ns}.

To measure the channel simultaneously for our two TXs, we use a frequency division multiplexing (FDM) approach as described in [32]. We derive the multitone waveform for the second TX from the first TX with a frequency offset $\Delta f_{\text{Ant}}=$1.19\text{,}\mathrm{MHz} for every single tone. The superposition of both waveforms leads to a multitone-comb, so that [TXs](#S7.16.16.16) are orthogonally interleaved in frequency. The resulting multitone sounding sequence for each [TX](#S7.16.16.16) has a period of $T=$840\text{\,}\mathrm{ns}, leaving possible additional tones unoccupied.

The snapshot time $T_{\text{snap}}$ for the channel has to fulfill the sampling theorem for a maximum Doppler frequency $\nu_{\text{max}}$, so that $T_{\text{snap}}\leq{1/(2\nu_{\text{max}})}$. For a maximum assumed speed of approaching cars $v_{\text{max}}=$14\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1}, $\nu_{\text{max}}=$2800\text{\,}\mathrm{Hz} and thus $T_{\text{snap}}\leq{1/(2\nu_{\text{max}})}=$178.6\text{,}\mathrm{\SIUnitSymbolMicro s}$$.

The final signal-power to noise ratio (SNR) is enhanced to combat thermal noise by coherently averaging over $N$ channel measurements up-to the maximum snapshot time. By using $N=212$, a processing gain of $23.26\text{\,}\mathrm{dB}$ is achieved and $T_{\text{snap}}=$178.07\text{,}\mathrm{\SIUnitSymbolMicro s}$$ is low enough for $\nu_{\text{max}}$.

Our RX is able to record up to $450\text{\,}\mathrm{M}\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}$ at one shot, which influences the choice of measurement bandwidth. For a sampling rate of $125\text{\,}\mathrm{M}\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$, the recording time is $T_{\text{rec}}=$3.6\text{,}\mathrm{s}$$ which translates to a travel distance of $50.4\text{\,}\mathrm{m}$ at $v_{\text{max}}$. An example of the measurement run is shown in Section VI. The parameters of the channel sounder are summarized in Table I.

V Measurement Evaluation

Frequency and time synchronization is performed utilizing the LOS component.

Frequency synchronization

The CFO between the oscillator of RX and TX is measured at standstill with a correlator before each measurement.

Coherent averaging

I/Q samples of the actual measurement with the car in motion are recorded. A frequency offset of the LOS due to Doppler plus CFO is estimated using correlators for each TX. We calculate the average received sounding sequence by applying a frequency offset correction and coherent averaging. The frequency correction, without the CFO, is reversed afterwards. For averaging, we use a rectangular window without overlap.

Time synchronization through LOS alignment

We estimate the propagation delay of the LOS component from the known distance between TXs at the light barrier position to the RX. Therefore, we align the delay of the LOS component accordingly.

We calculate estimates of the time-variant transfer function $H_{i}[t,f_{i}]$ similar to [7], where $t$ is the time index $t=\{t_{0},\dots,t_{N_{\text{snap}}-1}\}$ of a snapshot and $f_{i}$ are the frequency indices $f_{i}=\{f_{i,0},\dots,f_{i,K-1}\}$ selected for the given TX $i$ due to the interleaved multitone waveform. For a total recording time $T_{\text{rec}}=$3.6\text{,}\mathrm{s}, the number of snapshots is $N_{\text{snap}}=$20\,275. As common for example in OTFS modulation [28], we characterize the channel in delay-Doppler domain using a symplectic finite Fourier transform (SFFT) on $H_{i}[t,f_{i}]$. We use a multitaper based estimator within an evaluation region of $M$ samples in time and $K$ samples in frequency. The employed tapers are discrete prolate spheroidal sequences[33] to allow averaging of multiple independent spectral estimates of the same measurement with confinement in time and frequency. Furthermore, this type of estimator leads to delay and Doppler confined estimates. This approach is known as local scattering function (LSF)[34, 35, 36]. We use three tapers in time domain and three tapers in frequency domain.

From the LSF, we derive the Doppler power spectral density (DSD) as explained in[34]. For evaluation of the LSF, we set the number of snapshots to $M=360$ and number of frequencies to $K=21$ which corresponds to an evaluation time of $T_{\text{eval}}=$64.3\text{,}\mathrm{ms} and bandwidth of $B_{\text{eval}}=$100\text{\,}\mathrm{MHz}. In Fig. 5 and Fig. 6 we show the DSD and the LSF over the recording time.

Formalism for a sparse representation of the channel

We use the SFFT to transform consecutive time-variant frequency responses $H_{i}[k,l]$ into a Doppler-variant impulse response $S_{i}[n,m]$ and vice versa with the inverse SFFT (ISFFT). We write $H_{i}[k,l]$ and $S_{i}[n,m]$ as matrix $\mathbf{H}\in\mathbb{C}^{K\times M}$ and $\mathbf{S}\in\mathbb{C}^{K\times M}$ respectively for each $i$. $K$ is the number of frequency indices $k$ and delay indices $n$, and $M$ is the number of time indices $l$ and Doppler indices $m$. Splitting up the ISFFT into a discrete Fourier transform (DFT) and an inverse DFT as in [37] leads to

[TABLE]

where the index of the DFT matrix $\mathbf{F}$ denotes the size of the DFT. Using the Kronecker product $\otimes$, we find the vectorization $\text{vec}(\cdot)$ of $\mathbf{H}$ as

[TABLE]

with $\mathbf{h}=\text{vec}\left(\mathbf{H}\right)$ and $\mathbf{s}=\text{vec}\left(\mathbf{S}\right)$. With the transfer matrix $\mathbf{A}=\left(\mathbf{F}_{M}^{H}\otimes\mathbf{F}_{K}\right)$ and additive noise $\mathbf{w}$, $\mathbf{h}$ is expressed as

[TABLE]

where we want to find a vector $\mathbf{\hat{s}}$ which is a sparse representation of the measured channel. In this contribution a sparse fit is obtained with two different methods.

Largest peaks of LSF

We find a sparse fit through trivial two-dimensional peak search of the LSF and selection of the ten largest peaks.

Sparse fit using SBL

Moreover, we find a sparse representation of the channel $\mathbf{h}$ with the SBL algorithm in [25] as a solver. Based on $\mathbf{F}_{K}$, we define a DFT matrix $\mathbf{\tilde{F}}_{K}\in\mathbb{C}^{K\times 4K}$ for super-resolution in the delay domain by a factor of $4$. This enhances the resolution of the estimation, which is given by the channel sounder’s sampling in frequency. The modified transfer matrix for the solver is then $\mathbf{\tilde{A}}=\mathbf{F}_{M}^{H}\otimes\mathbf{\tilde{F}}_{K}\in\mathbb{C}^{KM\times 4KM}$, which leads to the model

[TABLE]

with $\mathbf{\tilde{s}}\in\mathbb{C}^{4KM}$, the resolution enhanced version of $\mathbf{s}$. Instead of $\mathbf{\tilde{s}}$, SBL iteratively estimates the channel powers through the variances $\boldsymbol{\gamma}=[\gamma_{1},\dots,\gamma_{4KM}]^{T}$. Furthermore, each iteration in SBL estimates the noise variance based on active set selection using the $P$ largest peaks in $\boldsymbol{\gamma}$. We modify the one-dimensional selection in [25] for a two-dimensional peak selection in $\text{vec}^{-1}\left(\boldsymbol{\gamma}\right)\in\mathbb{R}^{4K\times M}$ according to our delay-Doppler plane. For the SBL algorithm, we use the SBL1 variant in [25] and $P=10$. Initial channel variances are $\boldsymbol{\gamma}=[1,\dots,1]^{T}$ and initial noise variance is $0.1$. Ten iterations are done.

VI Results

As an example, we show a measurement in Fig. 3 with the transmitter car passing through the whole measurement scenario at constant speed. The recording is triggered when the car passes the light barrier $41\text{\,}\mathrm{m}$ before the RX position. The car continues towards the crossroads without stopping at the green traffic light and then passes the RX. The SNR at the RX is shown in Fig. 4. Fading is different for both TX antennas. Fading through MPCs in the first third of the recording is due to a parked truck. In the second third, fading is equal for both TX antennas because only two dominant signal paths exist. As the car passes the RX, SNR decreases first for the horizontal beam and later for the elevated beam due to spatial filtering. The Doppler power spectral density is shown in Fig. 5 with a dominant LOS component. The elevated beam shows fewer MPCs due to spatial filtering of the street level. For both antenna elevations, a LOS component with Doppler shift of more than $2.5\text{\,}\mathrm{kHz}$ is visible at $0\text{\,}\mathrm{s}$. As the car approaches the RX, Doppler shift of the LOS component decreases towards negative frequencies.

The channel power spreading in delay-Doppler domain, over the course of the measurement, is shown in Fig. 6. The spreading is mostly limited to a cluster around the LOS component and is thus described as a sparse channel in the delay-Doppler domain. Components arriving before the LOS are due to the previously mentioned reflections beyond the crossing (aliasing), and DFT leakage. Results from the sparse channel estimation with SBL are shown in Fig. 6. They are in good agreement with the peaks of the LSF. Furthermore, finer sampling in the delay domain reveals additional MPCs.

VII Conclusion

The measured V2I wireless channels in an urban environment are dominated by one cluster in the delay-Doppler domain. In a communication system, this allows for easy compensation for delay and Doppler shifts of the whole cluster, and enables the design of low-complexity receivers. This type of channel is especially suitable for OTFS modulation. Both considered beam elevations of the antennas at the car’s roof have only a weak impact on the V2I channel in delay-Doppler domain. However, the elevated beam is less subject to fading due to weaker illumination of the street level. SBL is able to find a sparse representation of the measured channel, which agrees with results from the LSF, and reveals additional MPCs.

AWG arbitrary waveform generator SBL sparse Bayesian learning DFT discrete Fourier transform LSF local scattering function LOS line of sight DPSS discrete prolate spheroidal sequences OTFS orthogonal time frequency space DSD Doppler power spectral density SA signal analyzer MPC multipath component mmWave millimeter wave V2V vehicle-to-vehicle V2I vehicle-to-infrastructure SFFT symplectic finite Fourier transform ISFFT inverse SFFT TX transmitter RX receiver LO local oscillator RF radio frequency PLL phase-locked loop CFO center frequency offset SNR signal-power to noise ratio ZC Zadoff-Chu CAZAC constant amplitude zero autocorrelation