RESOLVING AND PARAMETER ESTIMATION OF NON-SYNCHRONOUS SAMPLING PULSE SIGNALS BASED ON TAYLOR SERIES EXPANSIONBo Wu

National Mobile Communications Research Laboratory Southeast University, Nanjing 210096, P.R. China

Email: Wuboyx@gmail.com

INTRODUCTION

Overlapped pulse signals detection in white noise has been widely discussed, the well-known matched filter maximizes signal-to-noise ratio in the presence of Gaussian wideband noise.

To reduce the effects of masking nearby targets, Many efforts have been made. In general, there are two kinds of approaches, one widely accepted approach is to adopt different kinds of waveforms, Another type of approach is different estimation strategies.

INTRODUCTION

Non-synchronous sampling is shown in this figure.

ALGORITHM DESCRIPTION: PE

Considering the non-synchronous sampling, the receive signal can be represented as

By executing Taylor series expansion, we have

𝑌 (𝑡 )=∑𝑖=1

𝐼

𝛼𝑖∗h (𝑡+𝑡 𝑖 )+𝑣 (𝑡 ) , 𝑡∈ [𝑇1 𝑇 2 ]

h (𝑡+𝛥𝑡 )=h (𝑡 )+∑𝑛=1

∞ h(𝑛 ) (𝑡 )𝑛!

𝛥𝑡𝑛

ALGORITHM DESCRIPTION

The performance of parameter estimation with derivative is acceptable, Better performance is achieved by using higher derivatives but more consumption and calculation is needed. Simplified, we have

Expressing the above equation in vector form, we get

h (𝑛+𝛥𝑡 )=h (𝑛)+h′ (𝑛 ) 𝛥𝑡

𝑌=𝛼𝐻+𝛼𝐻 ′ 𝛥𝑡+𝑉

ALGORITHM DESCRIPTION

Our parameter estimation procedure is listed as follows ignoring the noise , we get the below tentative

estimates

Without considering the noise and using the tentative, we get

From the above equation, we obtain

𝛼= 𝐻𝐻𝑌𝐻𝐻𝐻

𝑌=𝛼𝐻+𝛼𝐻 ′ 𝛥𝑡

𝛥𝑡=  𝐻 ′ 𝐻(𝑌 −𝛼𝐻𝛼 )

𝐻 ′ 𝐻𝐻 ′

ALGORITHM DESCRIPTION

Our parameter estimation procedure is listed as follows

Refining by using the estimated

The sampling phase offset the amplitude can be achieved thought Grid searching instead of our tentative setting and calculating, but the dramatic increment in the computational complexity as the number of targets increasing makes it hard to be implemented.

𝛼= 𝑌𝐻𝑌𝑌𝐻 (𝐻+𝐻 ′𝛥𝑡 )

MODIFIED CLEAN ALGORITHM FOR DISCRETE POINT TARGETS

The proposed algorithm tentatively chooses the strongest target signal and estimates the parameter of the chosen target by the presented parameter estimation method based on Taylor series, then recovers the echo using the tentative estimator and subtracts it from original received signal. Without disturbing from the strongest echo, the output from estimator about rest targets' parameter is more reliable and accurate. Canceling the weaker echoes recovered using the above estimator, the strongest echo's parameter can be accurate.

MODIFIED CLEAN ALGORITHM FOR DISCRETE POINT TARGETS The novel method is detailed as follows:

Save the originally received signal in the proceeding window as OS and the initial signal is SC <= OS,

Get output from the conventional matched filter, find the maximum peak and its integer time delay index ,

Estimate timing phase and amplitudes of the strongest signal

Recover the strongest signal , cancel it and update Using matched filter processing , repeat steps (3) - (4),

until all observed target echoes are detected and canceled from and form the table of the targets' parameters,

Cancel all targets' echo from using the parameters' table except the strongest signal , repeat steps (3) - (5) until the estimation of parameters can't be improved obviously,

Output

COMPUTATIONAL COMPLEXITY ANALYSIS

the total consumption can be expressed as follows: Multiplication：

Addition：

𝑁 ·𝑅 · (𝐿 ·𝑆+ (𝐿+𝑆 ) ·3+(3 ·𝑆+5 ·𝑆+4 ·𝑆+3 ·𝑆 ) )

𝑁 ·𝑅 · (𝐿 ·𝑆+ (𝐿+𝑆 )+𝑆+𝑆+𝑆 )

SIMULATION RESULTS

This figure shows how our algorithm distinguishes the two adjacent targets. The signal amplitude from the stronger target is a little greater than another target.

SIMULATION RESULTS

The effect of different iterations on performance is given in this figure.

SIMULATION RESULTS The performance

of the algorithms is been raised as the increasing of time interval. Beyond 1.1 time of symbol length, the simulation's statistics are primarily affected by SNR instead of the targets’ distance.

CONCLUSION

Non-synchronous sampling inevitably exists because of the low SNR and persistently sampling time. Most of the exiting algorithms fail to separate the overlapped pulse signals in non-synchronous sampling. We propose a Taylor series based parameter estimation method which combines with the CLEAN. Simulation results demonstrate that our proposed algorithm can succeed in distinguishing two adjacent similar-power targets. Simultaneously, the proposed algorithm is effective in eliminating the sidelobe effect of nearby targets.