sparse frequency waveform design based on psd fitting - IEEE Xplore

SPARSE FREQUENCY WAVEFORM DESIGN BASED ON PSD FITTING Guohua Wang, Yilong Lu School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 ABSTRACT New method for designing sparse frequency waveform with sidelobe constraint is proposed in this paper. First, optimal power spectrum density (PSD) corresponding to minimum integrated sidelobe energy (ISE) is derived for sparse frequency waveform. Then, based on this optimal condition, the original problem dealing with both PSD requirement and sidelobe requirement is reformulated into a problem focusing only on PSD. In this way, the proposed method has a simpler optimization objective function and enjoys more efficiency than current methods focusing on both two requirements. Several optimization techniques are applied to this simpler design concept and numerical studies are provided to illustrate the effectiveness of it. Index Terms— Waveform design, sidelobe reduction, sparse frequency 1. INTRODUCTION Sparse frequency waveform has stopbands randomly distributed over a large spectrum. It is a desirable option for radar and communication systems operating in such highly congested bands as HF, VHF, and UHF bands to avoid spectrum interference from and/or to other customers while gaining performance improvement [1-7]. However, due to the sparsity in spectrum, the performance of sparse frequency waveform could be severely subject to the problem of high range sidelobes in autocorrelation function (ACF). Therefore, sparse frequency waveform design with sidelobe suppression has been a topic of interest in the past several years. Many works has been dedicated to the design of sparse frequency waveforms. Most of them were reviewed in [1], where sparse frequency transmit-receive waveform design was introduced. The method proposed in [1] is efficient in computation and flexible for stopband frequency suppression. However, the sidelobe property of the transmitting waveform is not considered. Paper [2-4] then proposed improved sparse frequency waveform design method with sidelobe reduction using stochastic evolutionary algorithms. More recently, papers [5-7] exploit different optimization approaches to design sparse frequency waveforms with sidelobe reduction. All these methods bring the sidelobe constraint into the transmit waveform design by optimizing a total objective function con-

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sisting of one power spectrum density (PSD) sub-objective function and one sidelobe sub-objective function. In this paper, we introduce a simpler new approach of designing sparse frequency waveform with sidelobe reduction by focusing on the PSD only. Firstly, we investigate the relationship between the sidelobe level and the power distribution over PSD. Optimal PSD corresponding minimum integrated sidelobe energy (ISE) is derived. Then, based on this optimal PSD, the original waveform design problem dealing with both the PSD constraint and the sidelobe constraint is casted as a problem focusing on only the PSD requirement. Mathematically, the equivalent problem is a PSD fitting problem, or more generally, a spectra factorization problem. With that, a new penalty function is constructed to find an waveform that has the closest PSD to the optimal one, automatically leading to the minimum ISE achievable in theory. To minimize the penalty function, we employ several different optimization techniques. Their performance are tested and compared with current methods. The designed waveform has constant modulus and is preferable for transmitter to consistently deliver maximum power. 2. MINIMUM ISE AND OPTIMAL PSD We focus on constant modulus digital waveforms in this paper. For such a waveform {x(n)}, n = 1, 2 · · · , N , the M point PSD of waveform (M ≥ N ) is defined by p(ωk ) = |

N 

xn e−jωk n |2 , ωk =

n=1

2π , k = 1, · · · , M M

(1)

And the aperiodic ACF is defined by rk =

N 

∗ xn x∗n−k = r−k , k = 0, · · · , N − 1

(2)

n=k+1

The ISE is defined as N −1 

ISE =

| rk |2

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rk ejkωm

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Based on the fact that p(ωm ) =

N −1  k=1−N

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We obtain the relationship between the PSD and the total energy in the ACF for M ≥ 2N like M 

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p(ωm ) = M

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N −1 

| rk |2

(5)

k=1−N

From this relationship, we can see that the total energy in the ACF is determined by the squared sum of the PSD. Also, from this relationship we can also easily derive the generalized ISE expression in [8]. In the case of sparse frequency waveform, the stopband power is larger than zero in practice although we may wish it to be zero. Thus, we assume that the average pass-to-stop band power ratio (APSPR) is r, that the total frequency samples in passbands is Np with total power ppass , and that the total frequency samples in stopbands is Ns with total power pstop , we then have the following relationship γ=

1−e ppass /Np ppass = pstop /Ns e M N − ppass

(6)

where e is the spectrum efficiency defined as Np /M . Now, let us investigate the optimal PSD corresponding to minimum ISE. From (5), we can further observe that M 

p(ωm )2 = M (ISE + N 2 )

(7)

m=1

M N  1  p(ωm ) = | xn |2 = E M m=1 n=1

(8)

To this point, the optimal PSD corresponding to the minimum ISE can be obtained through solving the following constrained problem min

{p(ωm )}

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3. SPARSE FREQUENCY WAVEFORM DESIGN VIA PSD FITTING 3.1. New design objective function From the analysis in Section 2, we know that designing sparse frequency waveforms with minimum energy in sidelobe is equivalent to design sparse frequency waveforms with a PSD function that has uniform power distribution over both stopbands and passbands. With that, one can expect to design optimal sparse frequency waveforms by spectral factorization technique. However, practical waveform design problem always has on the waveform further constraints such as constant modulus, and finite quantization on phase values, etc, which are not compatible with conventional spectral factorization. Thus, alternative methods of PSD fitting should be applied to design the waveform with required properties. With above mentioned optimal PSD, we propose to design sparse frequency waveform with sidelobe constraint via PSD fitting through a compact objective function like:  min ||F ({x(n)})|2 − popt (ω)|2 (12) {x(n)}

(7) implies that the smaller the squared sum of PSD, the smaller the ISE. Thus, if we want to suppress the sidelobe level in the ACF, it is equivalent for us to minimize the squared sum of the PSD. Meanwhile, based on the Parseval’s theorem, the total energy in PSD is

M 

As indicated by (11), when the power distribution is uniform in both passbands and stopbands, the ISE is optimally suppressed, a same conclusion as that in [11] but with different derivation.

2

p(ωm )

  p(ωh ) = ppass ωh ⊂Ωp s.t. ωl ⊂Ωs p(ωl ) = pstop

{x(n)}

(9)

ω

where F (·) stands for the discrete Fourier transform. To solve the minimization problem in (12), we can alternatively solve a closet point problem in the waveform space approaching to a waveform with the optimal PSD [9]. That is, with the optimal PSD, we can obtain a Fourier transform xopt (jω) corresponding to a waveform that has the minimum sidelobe energy. After that, by seeking for a waveform closest to this given waveform determined by xopt (jω), we can design a sparse frequency waveform with sidelobe level ’optimally’ suppressed. Thus, the equivalent problem to (12) is  min |F ({x(n)}) − xopt (jω)|2 (13) ω

Or equivalently, (13) can be written as min Ax − v2

where Ωp and Ωs stand for the frequency sets of passband and stopband, respectively. To solve the constrained optimization problem in (9), we may apply the Lagrangian multiplier technique. The resulting Lagrangian function is M 2 L({p(ω m) −  m )}, λp , λs ) = m=1 p(ω λp ( ωh ⊂Ωp p(ωh ) − ppass ) − λs ( ωl ⊂Ωs p(ωl ) − pstop ) (10) From (10), we can easily obtain the optimal PSD as  popt (ωh ) = ppass /Np , ωh ⊂ Ωp popt (ω) = (11) popt (ωl ) = pstop /Ns , ωl ⊂ Ωs

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x

(14)

where A is the M -points FFT matrix, and v is the optimal Fourier transform of the desired sparse frequency waveform x. (14) can be a generalized form for the waveform design problem discussed in [8], and can be taken as one variety of the phase retrieval problem solved in [12]. In this way, the original problem is transformed into a simpler problem described in (14), a ’least square problem’ subject to some design constraints such as the constant modulus. To (14), several optimization techniques can be applied to get solutions. We in the following apply Alternating projection method (AP), gradient descent method (GD)to solve the problem.

3.3. Gradient Descent Algorithm

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where  denotes the Hadmard Product. To (15), we can apply the GD method like in [1] to solve for the optimal solution. For brevity, we will not discuss this point. It is worthy of noting that although the two GD based methods from this paper and paper [6] share a similar computation procedure, the proposed new method directly come out with a solution that suppresses the stopbands while keeping the sidelobe reduced as heavy as possible rather than need a further parameter to balance the performance of the PSD and that of the ACF like that in [6]. This argument is important in the sense that in realtime design, one may desire a solution that inherently meets one requirement while keeping the other satisfied as much as possible. 4. SIMULATION STUDY We start by evaluating the effectiveness of the concept of proposed method by comparing it with the performance of those method considering both ACF and PSD besides constant modulus. In this case study, the waveform has N = 200 samples with constant modulus. The interested normalized frequency region of stopband is [0.25, 0.375] over [0, 1]. The

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Autocorrelation (dB)

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Alternating projection is a method capable of finding a point that is nearest to two or more different sets with each corresponding to one distinct property. With the proposed concept, two waveform properties should be satisfied. The first one is that the spectrum should have uniform distribution over passbands and stopbands; the second one is that the modulus of waveform should be constant (or within certain dynamic range). We suppose two sets, S and Y correspond to the two requirements, respectively. AP method can iteratively find the optimal waveform that is closest to these two sets. The detailed design procedure for AP method can be referred to [5]. Notice that, for conventional design method, at least three properties should be considered, incurring much heavier computation load. Meanwhile, for practical waveform design, as the PSD distribution can not be exactly uniform and stopbands are always required as low as possible, we may set an upper and a lower bound for passbands while setting a low level for stopbands to replace the expected property. Detailed discussion on the performance and implementation of AP method can be found in [5, 9]. Also, the Cyclic Algorithm proposed in [8] is in fact a variety of the AP method.

FFT point is M = 400. We also take the method considering only stopband suppression and constant modulus as a benchmark case for comparison. We take the best results (in terms of both the PSD and ACF performance) of each method from 10 trials. Each trial, the iteration number is 200. The PSD and ACF comparison of these three methods is illustrated in Fig.1. The APSPR is 20.3 dB, 39.5 dB, and 21.4 dB for these three methods, respectively. The ACF comparison is illustrated in Fig. 1(b). The Integrated Sidelobe Level (ISL) for these three method is -8.8 dB, 10.3 dB, and -8.6 dB, respectively. And the Peak-to-Sidelobe Level (PSL) is -28.5 dB, -7.1 dB, and -27.6 dB, respectively.

PSD (dB)

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(b) ACF Fig. 1. Performance comparision. Then, we deal with a practical problem using the abovementioned different approaches, i.e., the AP method, the GD method,compared with a former iterative method proposed in [6] considering both ACF and PSD in optimization. The sparse frequency stopbands distribution is from a Canadian Report [10]. The frequency region of interest is [0, 710] KHz. We choose five stopbands and six passbands to test the performance of these methods. The stopbands are [30, 150] KHz, [160, 180] KHz, [200, 230] KHz, [250, 300] KHz, [320, 350]

KHz. The PSD of different method are shown in Fig. 2(a). The APSPR is 28.5 dB for the GD method, 13.9 dB for the AP method, and 28.7 dB for the iterative method. Sidelobe performance is shown in Fig. 2(b). The PSL is 14.5 dB for the Gradient descent method, 15.2 dB for the AP method, and 15.7 dB for the iterative method. The ISL is -4.0 dB, -7.0 dB, and -4.8 dB. Based on this result, the proposed method by Gradient descent optimization can achieve comparable performance with those methods considering both ACF and PSD in optimization.

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5. CONCLUSIONS In this paper, a relationship between the sidelobe energy in ACF and the PSD in frequency domain was studied for sparse frequency waveform. It was found that the minimum sidelobe energy of a sparse frequency waveform corresponds to a uniform power distribution in both stopbands and passbands. With this property, sparse frequency waveform design with sidelobe suppression can be taken as a PSD fitting problem. Based on that, new method of sparse frequency waveform design was proposed. The proposed method in this paper is simpler in implementation compared to the current method and enjoys efficiency in computation, while keeping the resulting waveform with almost comparable performance. The simulation studies demonstrated the concept and the effectiveness of the proposed method.

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6. REFERENCES

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[1] M. J.Lindenfield, “Sparse frequency transmit and receive waveform design,” IEEE Trans. on AES, vol.40, Jul 2004, pp.851-861.

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[2] W. Liu, Y. Lu, M. Lesturgie, “Optimal sparse waveform design for HFSWR system,” Proc. WDD 2007, May 2007, pp.127130.

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[3] G. Wang, Y. Lu, “Sparse Frequency transmit waveform design with soft-power constraint by using PSO algorithm,” IEEE Radar 2008, May 2008, pp.127-130.

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[4] G. Wang, Y. Lu, W. Liu, “Sparse Frequency transmit waveform design using PSO algorithm,” Review in IEEE AES.

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[5] R. Kassab, M. Lesturgie, and J. Fiorina, “Alternating projections technique for radar waveform design,” Proce. Radar 2009, Oct 2009, pp. 1-5.

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[6] G. Wang, Y. Lu, “Designing single/multiple sparse frequency transmit waveform with sidelobe constraint,” To appear in IET RSN.

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[7] H. He, P. Stoica, “”waveform design with stopband and correlation constraints for cognitive radar,” Proce. 2nd Inter. Worksh. Cogni. Infor. Process., Jun 2010, pp.1-5.

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[8] P. Stoica, H. He, J. Li, “New algorithm for designing unimodular sequences with good correlation properties,” IEEE Trans. SP, vol. 57, no. 4, Apr 2009, pp.1415-1425.

Fig. 2. Performance comparision.

[9] J. Topp, I. Dhillon, R. Heath, and T. Strohmer, “Design structured tight frames via an alternating projection method,” IEEE Trans. IT, vol. 51, no. 4, Apr 2005, pp.188-209. [10] H. W. H Leong and B. Dawe, “Channel availability for east coast high frequency surface wave radar sys-tems,” Defence RD Canada, Technical report, DREO TR 2001-104, Nov 2001.

Finally, let us investigate the effect of different weighting. We strengthen the weighting on the 5-th stopband 10 times heavier than others. The resulting APSPR is 16.3 dB for all and over 40 dB for the 5-th stopbands. The sidelobe performance is 6.2 dB for ISL, and 17.1 dB for PSL. Based on our simulation study, we can even suppress the 5-th stopband much heavier, but the total suppression performance may then sacrifice, which is a conclusion similar to that in [1].

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[11] G. Wang, Y. Lu, “Bounds on generalized integrated sidelobe level in waveforms with stopbands,” To appear in Electronics Letters. [12] R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik vol 35, pp.237-246,1972.