1006 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 11, NOVEMBER 2013

Optimal Multiband Pulse Shaping andImproved Power Spectrum Estimation

Using Stopbands InformationAmir R. Forouzan, Senior Member, IEEE, and Mohammad Farzan Sabahi, Member, IEEE

Abstract—In this letter, we derive formulas for optimal dis-crete-time pulse shaping for communications in multibandchannels. Furthermore, we propose a technique for extending ashort signal segment for the purpose of power spectral estimationusing the available information about the signal stopbands. Thelater technique can be also used for minimizing the energy leakageinto stopbands of a signal, e.g., as in windowing for orthogonal fre-quency division modulation (OFDM). Simulation results show thatwhen the pulse duration is long enough, the stopband energy canbe practically reduced to zero using the proposed technique. Theproposed power spectral estimation technique can improve theresolution and reduce the energy leakage from signal passbandsinto adjacent stopbands. Moreover, it can reduce the out-of-bandpower leakage significantly when used for windowing in OFDM.

Index Terms—Bandstop filter design, DMT, OFDM, power spec-tral estimation, pulse shaping, VDSL2, windowing, Yule-WalkerAR method.

I. INTRODUCTION

P OWER spectral density (PSD) estimation has many ap-plications in various areas of signal processing, science,

and engineering. Many PSD estimation techniques have beenproposed since the last century and they continue to evolve withintroduction of new applications and more powerful digitalsignal processors. Some of the most popular PSD estimationtechniques include periodogram, Welch’s method, Bartless’smethod, Yule-Walker autoregresive (AR) technique, multi-taper, maximum entropy spectral estimation, and subspacemethods (Pisarenko harmonic decomposition, MUSIC, andESPRIT) [1]–[4]. Still considerable research is carried out onPSD estimation techniques for emerging applications such asspectrum sensing for cognitive radios [5], compressive sensing[6], speech source separation [7], direction of arrival estimation[8], phase correction filtering [9], and applications in genetics[10].Power spectral density (PSD) estimation techniques calculate

the spectrum of a sequence by using a finite-duration segmentof the signal. This involves assigning values to samples outsidethe available interval which is carried out by periodic exten-sion, windowing, or extrapolation [2]. Unfortunately, periodic

Manuscript received June 18, 2013; revised August 07, 2013; accepted Au-gust 11, 2013. Date of publication August 15, 2013; date of current version Au-gust 22, 2013. The associate editor coordinating the review of this manuscriptand approving it for publication was Prof. Arrate Munoz-Barrutia.The authors are with the Department of Electrical Engineering, Faculty

of Engineering, University of Isfahan, Isfahan, 81746-73441, Iran (e-mail:a.forouzan@eng.ui.ac.ir; sabahi@eng.ui.ac.ir).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2013.2278622

extension is not valid in the general case, windowing results inenergy leakage and/or loss of resolution, and available extrapo-lation techniques (also known as model based PSD estimation)require some form of sophisticated signal model [2]. The errorof all these techniques can be large when the available signalsegment length is too short1.In this letter, we propose an optimal technique for the

problem of extrapolating the samples outside a short signalsegment when there are known stopbands in the PSD. The PSDcan then be estimated using a model based or non-model basedPSD estimation technique [12]. Since, usually some informa-tion regarding the bandwidth of a signal or a random processis available, this technique can be applied in most applications.For example, discrete-time signals are usually obtained byoversampling the original continuous-time signal. This resultsin a low-pass signal. Moreover, in many applications, the signalis bandpass or has known stopbands. For example, VDSL2 sys-tems use discrete multi-tone (DMT) modulation2 and separatedupstream (US) and downstream (DS) bands. Since the DMTsignal has a finite duration, there is always some energy leakagefrom DS and US signals into the other direction’s band. Thisenergy leakage is reduced by adding a few windowed samplesat the two ends of each DMT symbol.In this letter, we formulate the problem and based on our for-

mulation we first solve the discrete pulse shaping problem inmultiband channels. The pulse shaping problem gives us moreinsight into the problem and helps in finding different solutionsfor the original problem by introducing an energy constraint. Byusing this technique, we can design pulses with zero energy instopband provided that the pulse duration is long enough. Whenapplied to the original PSD estimation problem, simulation re-sults show improvement in the resolution and reduction in theenergy leakage for the resulting PSD. Finally, our simulation re-sults show that the proposed technique can significantly reducethe out-of-band energy in OFDM.

II. PROBLEM FORMULATION AND ANALYSIS

We assume a short segment of a real random sequence oflength ( ) is available. We know that the PSDof is zero in stopbands. We denote the -th stopband by

where

. We aim to achieve a betterestimation of the signal spectrum by extending the sequenceat left by samples and at right by samples using the

1See [11] for an uncertainty assessment of PSD estimation based on finitesecond-order statistics.2DMT is a type of orthogonal frequency division modulation (OFDM) in

which a cyclic-prefix or a cyclic-suffix is added to each symbol for easier equal-ization

1070-9908 © 2013 IEEE

FOROUZAN AND SABAHI: OPTIMAL MULTIBAND PULSE SHAPING 1007

available knowledge about the signal’s stopbands (where usu-ally ). Let ( ) denotethe sequence obtained by extending where for

and ’s for andare the added samples.The PSD of is defined as where

(1)

The added samples are then calculated by minimizing the totalenergy in the stopbands, i.e.,

(2)

where is the union of the stopbands and we haveused the assumption that is real and therefore is aneven function of . The solution is found by differentiatingw.r.t. for and and settingthe result equal to zero. Equivalently, we set the gradient ofequal to zero. Note that

(3)

where

(4)

Then, (3) can be rearranged into the following matrix form

(5)

where and .

A. Discrete Pulse Shaping Problem

For now let us assume that all elements of are unknown.In this case, (5) is an standard quadratic problem and the so-lution for a given positive definite matrix is the eigenvectorcorresponding to the minimum eigenvalue [13]. In this section,we briefly study the solution as it gives us more insight to theproblem in the general case (i.e., when the central elements ofare given). First, without loss of generality, we impose the fol-lowing energy constraint on :

(6)

This defines a hypersphere in the dimensionalspace with radius . Assume is the

minimum of (5), then the gradient of , i.e., , isperpendicular to the tangent of hypersurface at . For a hyper-sphere the gradient is in the direction of the radius connectingthe center of the hypersphere to the solution point, i.e.,

(7)

where is a constant. Obviously, is an eigenvector of andis the corresponding eigenvalue.

B. Extrapolation of Short Signal Segments for MinimizingEnergy Leakage Into the Stopbands

In this section we use our technique to extend a short signalsegment using the available information about the signalstopbands. Assume elements to of the sequenceare given. We would like to extend the sequence at left by

symbols and at right bysymbols such that the resultingsequence has the minimum energy in the given stopbands. Inthis case, the gradient in (7) should be merely calculated w.r.t.the remaining elements of , i.e., the elements toand to . This is equal to eliminating the centralrows of and corresponding to to . If we splitaccordingly as follows:

(8)

we obtain

(9)

where , , and. Equation (9) can be rearranged as

(10)

where is the identity matrix of size .Theoretically, it is possible to calculate and by calcu-

lating the inverse of the matrix on the left hand side of (10) insymbol . Then by substituting them into the energy constraint(6), we obtain a polynomial equation of degreein . Since the degree of this equation is even, the equationmight not have any solution when is too small, i.e., smallerthan , the energy of . Otherwise, we generally have sev-eral solutions for , corresponding to the global minimum, theglobal maximum, and a few saddle points.Amore practical technique for finding the solution is by using

a numerical solution. For a given , and can be calculatedfrom (10). Parameter should be chosen to satisfy the energyconstraint . The constraint is generally met at severalvalues of and the one corresponding to the minimum sum en-ergy in the stopbands should be selected. Finding the solutionsfor can be done relatively easily. Plotting the sum energy vs.shows that the sum energy goes to infinity at a few points. These

points correspond to the eigenvalues of ,

which is because the matrix at the right-hand side of (10), i.e.,, is rank deficient at the eigenvalues. In fact, it is easy

to show that the energy is a rational function of with dominatorequal to the squared characteristic polynomial of . Searching

1008 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 11, NOVEMBER 2013

Fig. 1. Bandstop filters with a single stopband at designed using theproposed pulse shaping technique for and in (a) time and (b)frequency domains.

for is made easier by examining the intervals between succes-sive eigenvalues separately.

III. SIMULATIONS

Fig. 1 shows sample bandstop filters obtained by using thepulse shaping technique proposed in Section II-A forand 25, and a single stopband (i.e., highstop filters withcutoff frequency ) in time and frequency domains. For, the null space of is non-empty. As a result the resulting

filter gain in the stopband is zero to the accuracy of double pre-cision arithmetic. For , the null space of is empty.Therefore, the solution is the eigenvector corresponding to theminimum eigenvalue of . As it can be seen, the filter’s gainin the stopband is a few orders of magnitude greater than thatfor the longer filter. Obviously, the longer filter has more de-grees of freedom which can help in achieving a zero gain in thestopband. These filters do not show ideal characteristics in thepassband (e.g., flat gain and linear phase). However, they pro-vide the maximum attenuation in the stopband. Therefore, theyare ideal in pulse shaping and filtering in communication sys-tems where the inter-channel interference should be kept to aminimum.Simulation results for the proposed PSD estimation tech-

nique are presented in Fig. 2. The considered signal ispassband and consists of two equi-power sinusoids plusAWGN background noise. More specifically the signal isgiven by ,where is white and Gaussian with zero mean and variance

. Ignoring the noise, this signal is pass-

band with two stopbands and . Asegment length of , ,has been considered and the PSD of the signal is estimatedusing the Yule-Walker autoregressive (AR) method [3],[14]. Then the signal is extended by 10 symbols at each

Fig. 2. PSD estimation using the proposed scheme (a) Estimated PSD usingYule-Walker technique for an -point signal segment, an -pointsignal segment, and an -point signal segment extrapolated topoints using the proposed technique (b) Total and stopbands energies vs. .

end (i.e., ) using the proposed methodand the PSD is estimated again using the Yule-WalkerAR method assuming two stopbandsand . For comparison, the PSD is esti-mated using the -point signal segment

as well. The ARmodel order is set to be five in all cases. The estimated PSDsare labeled “ -point segment”, “Proposed Technique”, and“ -point segment” in Fig. 2(a), respectively. Theactual PSD of the signal is also plotted with black solid lineswithout labeling.As can be seen, the Yule-Walker AR method is incapable

of resolving the two tones in the PSD using the -point and-point segments of the signal. However, when the

-point segment is extended to points using theproposed technique, the estimated PSD shows two peaks at thetones. In the stopbands, the PSD is estimated more accuratelyin this case and is much closer to the background noise level.Obviously, the proposed technique has improved the resolutionand reduced the energy leakage into the stopbands considerably.The total energy and the stopbands’ energy

of the extended vector are plotted vs. in Fig. 2(b). Thehorizontal dotted line shows the points at which the energyis equal to the expected signal energy . In this simulationis estimated using the available signal segment energy as

. The line intersects theenergy curve at two points. The point at which the stopbandenergy is less gives the desired solution for (indicated by thevertical dotted line). As expected the energy goes to infinity at

FOROUZAN AND SABAHI: OPTIMAL MULTIBAND PULSE SHAPING 1009

eigenvalues of . The distinct eigenvalues of are 0.198,0.216, 0.795, 0.808, 0.989, 0.994, and 1.000, which match thepeaks of Fig. 2(b).Finally, we have used our technique for reducing

the energy leakage of an orthogonal frequency divi-sion multiplexing (OFDMA) signal. Consider

for to be asingle symbol of a baseband OFDM signal with ,where is the tone spacing, is the symbolduration, and and are the data symbols on sub-carrierwhich are chosen randomly with normal distribution here. TheFourier transform (FT) of the symbol is

where , denotes Dirac’s deltafunction, and is the convolution operator. The bandwidth of

is limited to , however, convolution withresults in slowly decaying side lobes out-

side the target band.The proposed technique can be used to obtain a

faster decaying spectrum by adding a window withminimum length. We assume that the signal is over-sampled by an oversampling ratio (OSR) of 8 to obtain

,where is the sampling period and

. We have chosen and. That is the added

samples to the sequence would have half the power of thesequence on average. Finally, as it is oversampled with anOSR of 8, the sequence is lowpass with a single stopband

.The proposed technique has been applied on to obtain the

sequence with duration 1152. The resulting signal is illustratedin Fig. 3(a) and the upper envelope of the resulting signal’s FTis shown in Fig. 3(b). As it can be seen the FT of the extendedsignal is attenuated by 20 to 35 dBs in the stopband relativeto the original signal. By relaxing the power constraint we canreach higher power attenuations in the stopband. However, thisincreases the power and potentially the peak-to-average powerratio of the signal.

IV. CONCLUSION

In this letter, we proposed a new technique for extending shortsignal segments based on the signal’s stopbands information forthe purpose of PSD estimation. When the signal is extendedusing our method, a general PSD estimation scheme such as theYule-Walker AR method can be used to estimate the PSD of theextended signal. Our simulation results show that the proposedtechnique can lead to a higher resolution and a smaller amountof energy leakage to stopbands. The proposed technique can beused for optimal windowing of OFDM signals and our simula-tion results show significant reduction in the out of band energyleakage. As a byproduct of our analysis, we derived formulasfor optimal discrete pulse shaping in multiband channels. Thepulses obtained by our technique have the minimum sum out ofband energy, which can reach exactly zero if the pulse durationis long enough.

Fig. 3. Optimal windowing for an OFDM signal with 64 tones and an OSRof 8. The original and the extended signals (a) in the time and (b) frequencydomains.

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