Conclusions

• Geometry-based stochastic model for V2V channel

• V2V channel structure in the delay Doppler

(Three Key Regions)

• Leakage effect, independent in delay Doppler directions

• Low complexity, structured joint Sparse

• Nested estimation of the channel vector based on

the group and element wise penalty functions

Acknowledgments: This research was funded in part by one or all of these grants: ONR N00014-09-1-0700, AFOSR FA9550-12-1-0215, DOT CA-26-7084-00, NSF CCF-1117896, NSF CNS-1213128, NSF CCF-1410009, NSF CPS-1446901, Barbro Osher Pro Suecia Foundation, Ericsson's Research Foundation FOSTIFT-13:038, and Adlerbert Research Foundation.

Simulation Results Comparison of NMSE v.s SNR Channel estimation: group sparsity

Channel estimation: leakage effect • Non-convex vs. convex regularizers: 3 dB at low SNR and 5 dB at high SNR values with the same computational complexity.

• In MSE = −20 dB, the performance curve related to the the structured estimator shows a 6dB improvement in SNR compared to HSD estimator, and 15 dB improvement in SNR compared to the Wiener Filter.

• Different group sizes for key regions up to 4 dB for high SNR. • Uncompensated leakage effect reduces performance severely, more than 7 dB,

particularly at higher SNR, due to the channel mismatch introduced by the channel leakage.

• Significant improvement (6 dB to 10 dB) of joint-sparsity.

.

Regularized LS Estimator & Leakage

Compensation

Pilots

Proposed Algorithm to Estimate the V2V Channel

Region Finder

Support Detector

Refinement

Nest Sparse Approximation

• Convert 2D Channel to 1D Channel vector: Vector x has both element-wise and

group-wise sparsity.

• Proper Regularization:

Leakage Effect Template Computation

Leaked Channel:

True Channel:

Leakage Template:

(Delay direction) Filters correlation function, Sampling Time

(Doppler direction) Dirichlet function, Pilot sequence length

• Independent in delay and Doppler direction • Given the system parameters, Leakage template is known

depends only on Leakage template

Compensation Approach

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DopplerDelay

Am

plitu

de

Leaked Channel:

Pulse Shape/Leakage Effect

: Interpolating Filter

: Anti-aliasing Filter

Input-Output relationship:

Finite transmit Bandwidth Finite pilot sequence length

Delay-Doppler Representation

• Diffuse contribution (only) in U-shape area

• Huge area with zero/small value components

• Symmetry of Diffuse scatterers contribution

• Diffuse components follow exponential profile (Delay wise)

• Sparse Region = Mobile Discrete Scatterers

• Sparse Components = All Discrete components

• Sparse Components exist in all Three Regions

16

Delay

Do

pple

r

LOSStatic Discrete (SD)Mobile Discrete (MD)Diffuse (DI)

νmax

νp

νS

-νS

-νmax

νS-Δν

Δν-νS R1

R3

R2

R3

R3

Fig. 4. Delay-Doppler domain representation of V2V channel. Delay-Doppler spreading function for diffuse components is

confined in a U-shaped area.

(2K + 1) ≥ N r , is

H l ,i [k, m] =1

2K + 1

N r − 1X

n= 0

hi [n, m]! − nk k 2 K

= ⌘i e− j 2⇡⌫i (mTs −⌧i )p(mTs − ⌧i )w(k,⌫i ),

where K = { 0, ± 1, ± 2, . . . , ± K } and w(k, x) is the (2K + 1)-point DFT of a discrete-time complex

exponential with frequency x and truncated to N r samples

w(k, x) =

8>><

>>:

N r

2K + 1, x = k/ (2K + 1)

e− j ⇡ ( k

2K + 1− x ) ( N r − 1)

2K + 1

sin(⇡( k

2K + 1− x)N r )

sin(⇡( k

2K + 1− x))

, otherwise

We note that the leakage in the delay and Doppler plane is due to the non-zero support of p and w. The

leakage with respect to Doppler decreases with the observation length, N r , and the leakage with respect

to delay decreases with the bandwidth of the transmitted signal. For simplicity, and for the purpose of

deriving a relative low-complexity method to compensate for the leakage, we assume that we can write

⌫i Ts = ki / (2K + 1) and ⌧i = m i Ts, where ki and mi are integers for all i . We can then write

H l ,i [k, m] = ⌘i g[k, m, ki , mi ], k 2 K, m 2 M

October 10, 2014 DRAFT

Geometry-Based Stochastic Model

• Doppler Shift (Specific ensemble):

• Delay (Specific ensemble):

GBSM: For any ensemble of point scatters with V2V channel statistical properties, compute its contribution at the receiver

13

RX

TX

Point scatterery

x

d

D

d

d1

d2

y1,SD

y2,SD

Fig. 2. Geometric representation of the V2V channel. The shaded areas on each side of the road contain static discrete (SD)

and diffuse (DI) scatters, while the road area contains both SD and moving discrete (MD) scatterers.

scatterers will result in multipath components with delay-Doppler pairs inside this region. The maximum

and minimum Doppler of the region is easily found from Eq. (11). In fact, it follows from Eq. (11) that

the Doppler parameter of an MPC due to a static scatterer will be confined to the symmetric interval

[−⌫S,⌫S], where⌫S = 1λ⌫

(vT + vR ).

Static Discrete Scatterers: The static discrete (SD) scatterers can appear outside the shadowed

regions in Fig. 2. In fact, the y-coordinates of the SD scatterers are drawn from a Gaussian mixture

consisting of two Gaussian pdfs with the same standard deviation σy,SD and means y1,SD and y2,SD [7].

The delay-Doppler pair for an MPC due to an SD scatterer can therefore appear also outside the U-shaped

region in Fig. 4. However, since the SD scatterers are static, the Doppler parameter is in the interval

[−⌫S,⌫S], i.e., the same interval as for the diffuse scatterers.

Mobile Discrete Scatterers: We assume that no vehicle travels with an absolute speed exceeding

vmax. It then follows from Eq. (11) that the Doppler due to a mobile discrete (MD) scatterer is in the

interval [−⌫max,⌫max], where⌫max = 4vmax

λ⌫. For example, in Fig. 4, the Doppler shift⌫p is due to an MD

scatterer (vehicle) that travels in the oncoming lane (vP < 0).

Based on the analysis above, we can conclude that the delay-Doppler parameters for the multipath

October 10, 2014 DRAFT

• line-of-sight (LOS) • Discrete Components (a) Mobile Discrete (MD) components (e.g.: other vehicles, ….) (b) Static Discrete (SD) Components (e.g.: Large traffic signs, ….) • Diffuse Components (all other components)

Channel Ingredients :

Differences:

• Higher speeds • Geometry dependence

Motivation: V2V channels Estimation

Application:

• Traffic safety • Intelligent transportation

Prior Arts: • Least-Square (Unstructured) • Adaptive Low Rank Wiener Filtering (High Complexity) • Compressed Sensing using basis pursuit (Mismatching, …) • Hybrid Sparse diffuse model (Information complexity)

Nested Sparse Approximation: Structured Estimation of V2V Channels Using Geometry-Based Stochastic Channel Model

Sajjad Beygi, , Erik G. Ström*, Urbashi Mitra

Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA *Chalmers University of Technology, Dept. of Signals and Systems, Gothenburg, Sweden

{beygihar, ubli}@usc.edu and erik.strom@chalmers.se

USC UNIVERSITY

OF SOUTHERN

CALIFORNIA