EstimationMulti Frequency Hopping Signals Based on Compressive Spatial Time-frequency Joint Analysis

Parameter Estimation of Multi Frequency Hopping

Signals Based on Compressive Spatial

Time-frequency Joint Analysis

Chunlei Zhang

I.T. Institute of Zhengzhou Zhengzhou, Henan Province, China

chunlei927@http://wendang.chazidian.com

Abstract—In order to estimate the parameters of multi Frequency-Hopping(FH) signals in the condition of non-cooperation and overcome the bottleneck of huge data processing, a parameter estimation method based on compressive saptial time-frequency joint analysis is proposed. First the arbitrary compressive array structure is analyzed, then based on this structure we propose a method to estimate the direction of arrivals (DOAs) with only a small number compressive samplings by exploiting the spatial sparsity of multi FH signlas. Then by exploiting the sparsity in frequency domain, a spectrogram estimation algorithm is proposed by using the same compressive sampling. Simulation results show that this algorithm can effectively estimate multi Frequency-Hopping signals’ DOAs and specrtograms with a tiny samplings. This algorithm is lower in computation complexity, and can be very practical in real-time Frequency-Hopping signal processing.

Keywords-Compressive sensing; Parameter estimation; Multi Frequency-Hopping signal; Spatial time-frequency joint analysis

Lichun Li

I.T. Institute of Zhengzhou Zhengzhou, Henan Province, China

leetracy@http://wendang.chazidian.com

original FH signal, which can greatly reduce the computation complex and computation time. However, some prior parameters, such as hop period have to be known, which can be a limitation in practical using. References [4]-[6]have studied the compressive beaming method for far field acoustic signal , but only spatial sparsity is taken into consideration.

Inspired by all these work listed above, this paper investigates a new joint parameter estimation method for FH signals based on compressive spatial time-frequency(CS-STF) analysis, i.e. DOAs of multi FH source signal and carrier frequencies without any prior knowledge. This method has several advantages over traditional approaches, since these traditional methods require Nyquist sampling at the array sensors while CS-STF just needs a few compressive samplings. Simulations show that this proposed algorithm can greatly reduce the amount of data, and solve the computation time.

II.

BACKGROUND OF COMPRESSIVE SENSING

The theory of CS[7]-[9] was first introduced by Candès, Tao and Donoho in 2004, published in 2006. According to this theory, any sparse signal or compressive signal can be sampled by a much lower rate than Nyquist sampling theorem and be reconstructed by nonlinear optimal algorithms. So CS consists of three aspects: the sparse representation of signal, the linear projection of the signal and the reconstruction of the origin signal.

A. Sparse representation

Consider a finite-length, discrete-time signalx, which can be seen as a N?1vector with elementsx?n?,

I. INTRODUCTION

With a lot of advantages such as high security, good anti-jam performance and low probability of interception, Frequency Hopping (FH) signal has been widely used in many key areas, especially in military field. Therefore, parameter estimation of FH signal becomes more and more important such as the direction of arrivals (DOAs) and spectrogram. The most common used methods to estimate the DOAs include multiple signal classification[1] (MUSIC), ESPRIT[2] and other method like Root-MUSIC. And the most common methods to estimate the spectrogram is to use short time Fourier transform (STFT),Wigner Ville distribution (WVD), wavelet transform and so on. But all this methods listed above have the same disadvantage that the source signal should be acquired at the Nyquist rate at each array, which can be great pressure on analog-to-digital convertor(ADC) since the frequency range of FH signal has a trend of becoming large and large.

To overcome this disadvantage, a new theory called compressive sensing (CS) has been proposed to reduce the sampling cost, since CS can sample the sparse or compressive signals at a sampling rate far below than the Nyquist sampling rate .So we can use CS technology to sample the FH signal since FH is a sparse signal in a certain hopping period. Reference [3] have studied the estimation of FH signal just with the compressive measurement, without reconstructing the

n?1,2,,N.Using a N?N basis matrix Ψ???1?2to represent the signalx, so it can be expressed as ?

x??siψi

i?1N

?N?

orx?Ψs?????

Wheresis the coefficient vector consist weightsi?x,ψi, and the ?1?2?Ncan be some basis vectors or some frames. The key point of spare representation is that if onlyKof the si coefficients are nonzero and the other ?N?K? are zero, then we can say that signal x is compressible or K-sparse.

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B. Linear Projection Consider a M?N

matrix whose rows are

???

j

Mj?1

?

(M?N), process some linear projection

?cos ssin?s?1T?

?i??s???i?sin ssin?s???c

??cos?s??

????

asyj?x,?j. Arrange the measurement vectorsyj in a

M?1 vectory. Then y can be written as

?

y?Φs??????

Here the coherence between matrix Φand Ψmust be

small, in other words, the matrix Φ must obey the restricted isometry property (RIP). In fact, form Φ by I.I.D. entries from the normal distribution or symmetric Bernoulli distribution obeys the RIP with overwhelming probability. C. Signal Recovery

If the RIP holds, then the recovery of signal can be seen as a problem like this: ?

Since different DOAs of the source can lead to different time delays compared with the signal received at the origin sensor, so we discretize to produce a bearing-angle dictionary and can get a sparse representation in this dictionary at each sensor. Generate a set of Nangles pairs????1,?2,,?N?, and Ndetermines the angle domain resolution. Let bdetermines the bearing pattern,so the signal received at sensor i?i can be expressed as: ?

?i?Ψib?

??1??i???i?t0?,?i?t0??,

Fs????

?N?1??

,?i?t0?t???

Fs????

T

????

?????

minxN

x?R

l0

subjecttoΦx?y??????

Where Fsrepresents the Nyquist sampling rate, each column

of Ψi represents the different time delay signal of souce

There are two main recovery methods, one is the Orthogonal Matching Pursuit (OMP)[10] and other methods based on OMP, the other is to convert the problem tol1norm minimization[11] and use convex optimization to solve it. Both of the methods need large number of computations, so if we can find some method which can directly deal with compressive samplingsyto detect weather x exists or not without reconstructing original signalx, a lot of computation burden can be saved.

III. PARAMETER ESTIMATION OF FH SIGNAL BASED ON

CS-STF A. Single FH signal estimation model

Suppose there are Ldifferent sensors and the sensors’ positions are assumed known and given by

T

?i??xi,yi,zi?，i?1,2,,L. If there is a known signal s?t? far from this array, then the ith sensor receives a time-delay and attenuated version of s?t?: ?Where

s?t?corresponding to the angle-pair ?j:

?

?Ψi?j

'

??st0??i??j?,?

??

,st

?

'K?1

??i??j??

???

T

????

Where t'?t?

R

.The sampling rate Fs is typically high c

since FH signal can be seen as a wide-band signal, and there is huge date to be processed. So, we proposed a new data sampling model based on CS which can lower the sampling rate effectively. Let the matrix Φibe the compressive sensing matrix at the ith sensor, then the compressive samplings received at the ith sensorβican be written as:

?

βi?Φi?i?ΦΨiib?

????

?i?t???s?t??i??s??

?

?R???c?

????

For all the Ldifferent sensors, the sparsity pattern of beaming estimation can be solved byl1 minization problem: ?Where

??argminbs.t.Ab?β?bTβ????1,T

T

?,?L?

T

?????and

? denotes the attenuation factor and kept

constant,?s?? s,?s?represents the DOA of s?t?（ srepresents the angle with Zaxis,?srepresents the angle

,,ΦL?.

A?ΦΨ

with Xaxis）,Ris the distance from the signal source to the array,cis the signal transmitting speed ,?i??s? is the relative time delay at the ith sensor with bearing ?s compared with the origin of the array. From the geometry aspect we can conclude that the relationship between time delay ?i??s? with ?s:

TT

?Ψ=?Ψ，，Ψ1L??, Φ=diag?Φ1,

When s?t?is unknown to us, sensor should sample the signal

at the Nyquist rate or even higher, this high-rate sensor’s location should be seen as the origin of this array, while the other sensors can sample the signal at a lower sampling rate required by CS theory.

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When we take noise into consideration , (10) can be written as: ?

??argminbs.t.AT?β?Ab?b?

??1?

?????

(11),only the large terms inAT?andATAbbe cancelled can

statisty this condition. With these assumptions, the two largest elements inAT?occur when ?n??1and?n??2. So inATAbwhen?n??1,that is ?r??1in

Where ?1 represents the noise level. The optimization problem (11) can be solved by Dantzig selector[12]。 B. Multi FH signals estimation model

Suppose there is another far-field FH signals2?t?,and its’ bearing is ?2,these two signals are not relative. So the signal received at the reference sensor is: ?

b

,the term

R11??i??n?,???1??in AT? can be canceled；Likewise

when ?n=?r??2, the term R22??i??n?,???1?? in

AT? can also be canceled[6].

All in all, by optimization method Dantzig selector we can solve the DOAs estimation problem of multi FH signals with a tiny compressive samplings.

Since FH signals is sparse in frequency domain in specific time which is far short than one hopping period, so we can get: ?

?0?t??s1?t??s2?t??

A possible scenario is shown in Fig.1:

?????

β=ΦΨfx?

?????

Where Ψf is a discrete Fourie basis,xdenotes the sparse cofficients of FH signal in frequency domain which can be estimate the carrier frequency of multi FH signals,we can use OMP to estimate its spectrogram.

For multi frequency hopping signal based on compressive signals, we can use that the the sparsity of array compressive signals in the spatial and frequency domain is the same, so combine (11) and (16) we can get:

?

Figure 1. A model for compressive beamforming

Suppose the two signals have the same amplitude, then the signal received at ith sensor is: ?

?b??argminbs.t.AT?β?Ab?

?

?

??argminx?0s.t.β=ΦΨfx??x?

?=x?0b?0?

?

??1

?

?????

?i?t??s1?t??i??1???s2?t??i??2???

?????

Where ?i??1?and?i??2?represents the time delay for signal bearing ?1and?2at ith sensor。AT?andATA are usually

auto- and cross-correlations, For AT?, the nth element is:

Convex optimization and OMP algorithm are used to

?andx?,to get the DOAs and estimate the parameter b

spectrogram of multi FH signals.

IV. SIMULATION RESULTS AND ANALYSIS

A. DOAs estimation for multi FH signals

In this section, we compare our method with traditional DOA estimation method, MUSIC. We adopt a total of L?8 uniform linear array with an RS sensor added at the origin with a Nyquist sampling rate Fs=100MHz,while the other sensor working at the compressive sampling rate Fc?Fs/8?12.5MHz, and the sensor space d?60m.The source FH signals are placed at 31o and 142o,the SNR at each source is 10dBand their hoping rang is between 12.5MHzand 37.5MHz. Compared the proposed CS-STF algorithm with MUSIC, and the angle domain resolutionN?180, and the simulation results are as follows:

?

R11??i??n?,???1???R12??i??n?,???2???R12??i??n?,???1???R22??i??n?,???2??

??????

For ATA,the element in nth row and rth column is:

?

R11??i??n?,???r???R12??i??n?,???r???R12??i??n?,???r???R22??i??n?,???r??

??????

Since s1?t?ands2?t?are not relative,the corss correlation R12?,

? is very small. Then we examine the constraint in

551

DOA（°）

that is to say the compressive sampling contains more

information about original signal, so we can get a more accurate results by using CS-STF; When the SNR increases, the OMP has a higher probability to choose the exact coefficient in frequency domain, so CS-STF can also get a more accurate spectrogram estimation.

V.

CONCLUSION

Based on CS, this paper presents a new FH signal estimation method only with a tiny number of compressive measurements. This algorithm (CS-STF) fully exploits the sparse property in spatial and frequency domain of the FH signal. What’s more, as we only use the time delay of reference sensor, the geometry of the array can be arbitrarily, which is more flexible in practical using. Meanwhile, as only using the compressive samplings, the proposed method can greatly reduce the estimation complexity and time.

REFERENCES

[1] Schmidt R O. Multiple emitter location and signal parameter estimation.

Antennas and Propagation, IEEE Transactions on, 1986, 34(3): 276-280. [2] Capon J. High-resolution frequency-wavenumber spectrum analysis.

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[3] Yuan J, Tian P and Yu H. “Subspace compressive frequency estimation

of frequency hopping signal.” Wireless Communications, Networking and Mobile Computing, 2009. WiCom'09. 5th International Conference on. IEEE, 2009: 1-4.

[4] Gurbuz A C, McClellan J H and Cevher V. “A compressive

beamforming method. Acoustics,” Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on. IEEE, 2008: 2617-2620.

[5] Cevher V, Gurbuz A C, McClellan J H and Chellappa R. “Compressive

wireless arrays for bearing estimation.” Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on. IEEE, 2008: 2497-2500.

[6] Gurbuz A C, Cevher V and McClellan J H. “Bearing estimation via

spatial sparsity using compressive sensing.” Aerospace and Electronic Systems, IEEE Transactions on, 2012, 48(2): 1358-1369. [7] Candès E and Tao T. “Near-optimal signal recovery from random

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signal reconstruction from highly incomplete frequency information.” IEEE Transactions on Information Theory, 2006, 52(2): 489-509.

[10] Tropp J A and Gilbert A C. “Signal recovery from random

measurements via orthogonal matching pursuit. Information Theory,” IEEE Transactions on, Vol.53(2007),No.12,p. 4655-4666.

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Theory, IEEE Transactions on, Vol.51(2005), No.12, p. 4203-4215.

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p is much larger than n.” The Annals of Statistics, 2007: 2313-2351.

Figure 2. The DOA estimation for MUSIC and CS-STF algorithm

In Fig.2,MUSIC take 256 measurements at Nyquist rate, while CS-ST only take 16 measurements. In CS-STF, each using random measurement matricesΦ?Φ?CM?N? for each sensor drawn independently from I.I.D N?0,1?. Note that the CS-STF algorithm’s results are sparse compared with MUSIC as expected since CS-STF use the characteristic that FH signals is sparse in spatial.

B. Spectrogram estimation for multi FH signals

Using the same array structure as listed above, we estimate the spectrogram of two frequency-hopping signals Fh1 and Fh2. Fh1 and Fh2 are both working at the frequency band 12.5MHzand 37.5MHzand modulated by FSK, and the minimum frequency interval is 2.5MHz. The hopping rate is 2000 hop/s and 4000 hop/s. As shown in Fig. 3,we can get the spectrogram estimation under different SNRs and compression rate. The hopping frequency collection of Fh1 is ?35MHz,30MHz?,and the hopping collection of FH2 is

37.5MHz,15MHz?, the observation time is ?37.5MHz,20MHz，

1ms.

(A)

(B)

frequency(MHz)

time(ms)

frequency(MHz)

time(ms)

frequency(MHz)

time(ms)

frequency(MHz)

time(ms)

Figure 3. different spectrogram estimation results under different SNRs and M/N.(A)SNR=-5dB,M/N=0.25;(B)SNR=0dB,M/N=0.25;(C)SNR=-5dB,M/N

=0.125;(D)SNR=0dB,M/N=0.125

From Figure 3. , as the M/N increase, the CS measurement matrix Φ has a higher probability to satisfy the RIP condition,

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