Optimal Envelope Approximation in Fourier Basis with Applications in TV White Space

06/03/2017 ∙ by Animesh Kumar, et al. ∙ IIT Bombay 0

Lowpass envelope approximation of smooth continuous-variable signals are introduced in this work. Envelope approximations are necessary when a given signal has to be approximated always to a larger value (such as in TV white space protection regions). In this work, a near-optimal approximate algorithm for finding a signal's envelope, while minimizing a mean-squared cost function, is detailed. The sparse (lowpass) signal approximation is obtained in the linear Fourier series basis. This approximate algorithm works by discretizing the envelope property from an infinite number of points to a large (but finite) number of points. It is shown that this approximate algorithm is near-optimal and can be solved by using efficient convex optimization programs available in the literature. Simulation results are provided towards the end to gain more insights into the analytical results presented.

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1 Introduction

This work introduces a fundamental topic in some Electrical Engineering applications—envelope approximations. First, this problem is motivated. TV white space devices are required to consult a TV white space database [1], which in turn computes the protection region of the TV transmitters. The TV white space database service providers are licensed by a regulatory body such as the FCC in United States. The protection regions for the TV transmitters is smooth and can be non-circular in shape; for example, see Fig. 1, which illustrates protection regions obtained from the iconectiv website [2] for Channel 22 in the New York region. iconectiv is one of the database service providers licensed by the FCC. Observe that protection regions such as 2 and 3 are non-circular.

Figure 1: The TV protection regions for Channel 22 near New York, from the website of a TV white space service provider iconectiv in the United States [2], are shown. Protection regions labeled 2 and 3 are non-circular in shape.

The protection region signifies a closed region where only the licensed TV transmitter can use the TV channel frequencies For example, in Fig. 1, in the region labeled 2 only licensed user can operate in Channel 22 of the TV band. If the TV white space database wishes to communicate the protection region by using a lowpass (sparse or rate-efficient) approximation, it needs to calculate a lowpass representation of shapes such as 2 and 3

. While performing the approximation, there are two possible errors: (i) a point in TV protection region is declared as unprotected; and (ii) a point in unprotected region is declared as TV protection region. To protect the licensed operation of TV transmitters type (i) errors are

not allowed . So, any lowpass representation of TV protection region must have an “enveloping” structure. To address such problems, envelope approximations are studied in this work.

Figure 2: A protection region can be viewed as a one-dimensional signal with respect to the angle as shown.

Consider a smooth TV protection region as depicted in Fig. 2. Let its centroid be the origin. Then, the protection region can be parametrized by a periodic signal as shown in Fig. 2. With the knowledge of origin (the center), the signal is equivalent to the protection region. This periodic and smooth signal can be approximated by orthogonal basis in a linear space; for example, Fourier series can be used [3]. In this work, will be approximated to a bandlimited Fourier series , where has only harmonics in its Fourier series. The envelope constraint requires for all , while minimizing a desirable cost function. For TV protection region approximation, the area enclosed by should be minimized, which means

minimize
subject to (1)

This is the core problem addressed in this work.

Main result: An approximate algorithm, which banks upon convex optimization program, is developed to address the optimization problem in (1). The approximate algorithm has two features: (i) it is provably near-optimal to the best solution of optimization in (1); and (ii) the nearness to optimality can be controlled by choosing the complexity of solving the approximate algorithm.

Related work: As far as we know, the topic of envelope approximation, subject to a cost function, has not been addressed in the literature. This is a fundamentally new topic. The topic of approximation or greedy approximation in linear basis, on the other hand, is classically well known [4, 5].

Organization: Section 2 discusses the signal and approximation model, and introduces the cost function. Section 3 presents the approximate algorithm for finding a near-optimal envelope of a signal. Section 4 presents simulation results while conclusions are in Section 5.

2 Modeling assumptions

A finite support real-valued field will be considered, where , without loss of generality. It will be assumed that a periodic repetition of , that is , is differentiable in so that Fourier basis is sparse for the signal [3, 4]. It will be assumed that

(2)

for some constant . With Fourier basis, the pointwise representation for differentiable signals is given by [3]

(3)

where the Fourier series coefficients are given by Since is real-valued, conjugate symmetry implies . In general, to specify completely, infinite number of coefficients have to be specified.

In this work, an optimal envelope approximation of will be designed. Let be any -complex coefficient based envelope approximation. Any -coefficient envelope approximation will have the following form:

(4)

where the envelope approximation will satisfy:

(5)

Since is real-valued, the coefficients and are related by conjugate symmetry, that is . The approximation is specified by coefficients . For compact notation let

(6)

where

is a column vector.

For the protection region approximation, a mean-squared cost will be minimized. The cost is defined by

(7)

where is the envelope approximation of . This cost represents the white space area lost as protection region.

3 Optimal Envelope Approximation

This section presents a framework to obtain a near-optimal envelope approximation for a smooth (differentiable) signal . The cost function is assumed to be

(8)
(9)

where . Given a signal , its energy is fixed. The envelope approximation problem is equivalent to finding

subject to (10)

The signal is fixed in the above optimization. For any fixed the constraint is linear since

(11)

where is a vector of phasors. If the constraint in (10) was restricted to a finite number of points in , then the optimization in (10) can be solved as a quadratic program with linear constraints.111The reader would notice that is complex-valued, while quadratic program works with real valued linear constraints. If , then conjugate symmetry implies that . These complex valued linear constraints can be re-cast into real valued linear constraints in terms of . The details are omitted for simplicity of the exposition and due to space constraints. The quadratic program is solvable using classical methods [6]. So, the difficulty in solving the optimization in (10) is an infinite number of constraints.

The signal has been assumed to be differentiable. The approximation will be infinitely differentiable due to bandlimitedness. This smoothness of suggests that if , it will be positive or near-zero in a small interval around . This intuition motivates an -point approximation to the constraint of optimization problem in (10). Consider the following optimization, which is an -point approximation to the optimization in (10)

(12)

The above optimization program has a quadratic cost with linear constraints and it is solvable by a convex program solver [6]. Let and be the unique arguments for which (10) and (12) are minimized. Then can be solved with a convex program. It is expected, though unproved so far, that and will be “close” as becomes large. Their closeness is established next.

First note that there are more constraints in (10) than in (12). Therefore,

(13)

A sub-optimal approximation with Fourier series will be constructed using such that for all . Assume that , where is a finite constant, which is proved later in this section. Since is obtained by solving optimization in (12), therefore

(14)

because of constraint equation. For any point

(15)
(16)

where follows by and for , follows by for , and follows by (14). The above inequality holds for every (uniformly), so

(17)

Define

(18)

which means

(19)

From (17) and (18), it follows that

(20)

or satisfies the constraint in (10), which means

(21)

From (13) and (21), we get the following key inequality

By this inequality,

(22)
(23)
(24)
(25)
(26)

Similarly,

(27)
(28)
(29)

The above results guarantee that the cost obtained by , a suboptimal solution to the envelope approximation problem obtained through , is at-most away from the true optimum. If is large-enough, the above discussion guarantees that a near-optimal solution to the envelope approximation problem can be obtained in an efficient way.

It remains to show that . First note that

(30)
(31)
(32)
(33)

Next, note that is always a part of the constraint set in (10) and (12). Therefore,

(34)

Substitution of (34) in (33) results in

(35)

This concludes the proof of . Simulations are presented next.

4 Simulations on TV protection region

To test the optimal envelope approximation method of the previous section, TV protection regions in Channel  of United States were examined using the TV white spaces US Interactive Map of Spectrum Bridge. Across United States, there are protected service contours. One of these contours was hand-picked and its protection region was segmented (using image processing techniques) to obtain . This protection region was picked since it has points with sudden change in derivative, and would be difficult to approximate. Then, the Fourier basis based envelope approximation technique was applied (see (19)). The results are shown in Fig. 3 and Fig. 4. In Fig. 3, and is increased from onwards to obtain . It is observed that nearly converges for . In Fig. 4, is increased to . It is observed that with Fourier series coefficients, the envelope is proximal to the original signal , except near derivative discontinuity.

Figure 3: The sub-optimal envelope approximation is obtained as a function of . Here . It is observed that is proximal to optimal for . By design, the approximations are larger than for each value of .
Figure 4: The optimal envelope approximation, a solution to (10), is illustrated for and . By design, the approximation is larger than . This property also ensures that the approximate protection region is a superset of the actual protection region.

5 Conclusions

An approximate algorithm for finding a signal’s envelope, while minimizing a mean-squared cost function, was detailed. A near-optimal envelope approximation was found in Fourier basis using linear space properties and efficient solvability of quadratic optimization subject to linear constraints. The approximate algorithm when subjected to -constraints resulted in a near-optimal envelope signal with a gap of in the cost function from the optimum. The results were verified with simulations on TV white space protection region.

References

  • [1] D. Gurney, G. Buchwald, L. Ecklund, S.L. Kuffner, and J. Grosspietsch, “Geo-location database techniques for incumbent protection in the TV white space,” in Proc. of IEEE Symposium on Dynamic Spectrum Access Networks. Oct. 2008, pp. 1–9, IEEE, New York.
  • [2] ,” https://spectrum.iconectiv.com/main/home/contour_vis.shtml.
  • [3] Stéphane Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press, Burlington, MA, USA, 2009.
  • [4] Ronald A. DeVore and George G. Lorentz, Constructive Approximation, Springer-Verlag, 1993.
  • [5] Vladimir Temlyakov, Greedy Approximation, Cambridge, New York, USA, 2011.
  • [6] Stephen Boyd and Lieven Vandenberghe, Convex optimization, Cambridge University Press, 2004.