Sparse Channel EstimationYijun ShanDepartment of Electrical and Electronic Engineering, Imperial College London

AbstractCompressed sensing is a topic that has recently gained much attention in the applied mathematics and signal processing. In this project, various channel estimators are investigated to exploit channel sparsity in the time domain. Compressed sensing approaches are adopted in the form Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) algorithms. Numerical simulation of an OFDM system is used to evaluate the proposed algorithms in comparison to the conventional least-squares (LS) channel estimator. The observation is made that compressed sensing algorithms uniformly outperform the LS methods.

Least-Square (LS) EstimationLeast-Square (LS) estimator: . Where and . Specifically, the channel estimation is done based on the pilot location.

Where pilot signals inserted in X(k) and is the th pilot carrier.

Fig.4: (LEFT) Constellation diagram of 16QAM at transmitter and (RIGHT) Constellation at receiver.

Orthogonal Matching Pursuit: This greedy algorithm iteratively identifies one delay at a time and solves a constrained LS at each iteration to measure the fitting error and update the residual.

Fig. 5: Performance comparison between BP and LSE

No. subcarrier N 256Cyclic Prefix D 32

Sampling period 0.01msSymbol duration T 2.56msGuard Interval 0.32msTotal duration 2.88ms

64 subcarrier pilot symbol, distributed on every fourth

subcarrier.

References[1] C. R. Berger, S, Zhou, J. C. Preisig, P. Willett. ”Sparse Channel Estimation for Multicarrier Underwater Acoustic Communication: From Subspace Methods to Compressed Sensing”. IEEE Trans Signal Processing. 2010;58(3)

[2] C. R. Berger, Z. Wang, J. Huang, S. Zhou, “Application of compressive sensing to Sparse Channel Estimation”, IEEE Communication Magazine, 2010.

Multipath Model

Fig.1: Block diagram for OFDM system

Fig.3: OFDM signal spectrum simulation

The channel impulse response is considered as Dirac function with L taps, with , where . Mathematic model is described by:

A finite sum of Dirac impulses with complex magnitudes . The delay values are chosen uniformly at random from a continuous interval.

Fig.2:(LEFT) Multipath Model and (RIGHT) Channel impulse response with 5 taps.

Compressed SensingBasis Pursuit: As BP, the solution is denoted to the following convex optimization problem:

Where A is measurement matrix with discretization of estimated delay distribution at baseband sampling rate.

0 1 2 3 4 5 6 70.8

1

1.2

1.4

1.6

1.8

2

Frequency index

Mag

nitu

de

OFDM spectrum

5 10 15 20 25 3010

-3

10-2

10-1

100

SNR

BE

R

BPLSE

5 10 15 2010

-3

10-2

10-1

SNR

BE

R

OMP M=20OMP M=30OMP M=35OMP M=40

5 10 15 2010

-3

10-2

10-1

SNR

BE

R

OMPBPLSE

Fig.6: Block Diagram of Greedy algorithm

Fig.7: (LEFT) Performance of OMP with number of iterations to determine the best stopping time and (Right) Performance comparison between three channel estimators.

ConclusionClearly, all compressed sensing schemes outperform the simple least-square (LS) channel estimator, gaining about 1.5 dB, with BP having a slight edge over OMP. Intuitively, the advantage of sparse channel estimation relative to its LS counterpart comes from the fact that by exploiting sparsity in the estimate, sparse channel estimation can effectively reduce the number of unknowns.

A OFDM system with N subcarriers and cyclic prefix (CP) of D samples. By taking IFFT and adding CP, the modulated N-point data signal is transformed to OFDM symbol where is the FFT matrix.

The received signal is then written as:

Where n denotes noise samples, and H is the channel mixing-matrix. The estimated frequency responses can be collected into a vector h with diagonal entries of H.

Table.1: Numerical Simulation setup parameters.

OFDM Simulation