On the SFDR Upperbound in DDFS Utilizing Polynomial ...

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 5, MAY 2012 307On the SFDR Upperbound in DDFS UtilizingPolynomial Interpolation MethodsAshkan Ashrafi, Senior Member, IEEEAbstract—In this brief, the theoretical analysis that obtainsthe upperbound of the spurious free dynamic range (SFDR) ofthe direct digital frequency synthesizers (DDFSs) with phaseto-sinemapper (PSM) structured by polynomial interpolation isrevisited. A complete analysis of the SFDR upperbound of theseDDFS architectures was provided by Ashrafi and Adhami in2007, where the authors formulated the SFDR upperbound asthe Chebyshev minimax solution to an overdetermined system oflinear equations. However, some issues in finding the Chebyshevminimax solution were not made clear in that paper regarding theminimax error at the fundamental harmonic of the output sinusoid.These issues, along with the erroneous behavior of numericalcomputation packages used for solving the Chebyshev minimaxproblem, misled Chau and Chen to claim that the results of theaforementioned paper were wrong. This brief clarifies the issueswith the original paper, exhibits the problems with some of theavailable numerical computation packages, and refutes the claimsthat Chau and Chen made.Index Terms—Chebyshev minimax approximation, direct digitalfrequency synthesizers (DDFSs), spurious free dynamic range(SFDR).I. INTRODUCTIONSPURIOUS free dynamic range (SFDR) in a direct digitalfrequency synthesizer (DDFS) is the parameter that describesthe spectral purity of the output sinusoidal signal. TheSFDR is defined as the ratio of the magnitude of the fundamentalharmonic and the magnitude of the maximum spur ofthe output signal. The DDFS systems comprise an accumulator,phase-to-sine mapper (PSM), and digital-to-analog converter.The frequency of the sine output is controlled by a digital inputnumber F rf out = F r2 L f clk (1)where F r , L, and f clk are the input of the accumulator, theaccumulator wordlength, and the clock frequency, respectively[1]. The PSM could be as simple as a lookup table or ascomplex as a Coordinate Rotation Digital Computer (CORDIC)system [2]. Piecewise polynomial interpolation methods arealso used to approximate the sine signal by polynomials [3]–[6]. In this method, the first quadrant of the sinusoid is dividedinto s =2 u segments, and each segment is approximated byManuscript received November 10, 2011; revised January 14, 2012; acceptedFebruary 25, 2012. Date of publication April 13, 2012; date of current versionMay 16, 2012. This paper was recommended by Associate Editor A. Brambilla.The author is with the Department of Electrical and Computer Engineering,San Diego State University, San Diego, CA 92182 USA (e-mail: ashrafi@mail.sdsu.edu).Digital Object Identifier 10.1109/TCSII.2012.2190861a polynomial. Therefore, the output can be reconstructed byadditions and multiplications of the phase signal (the accumulatoroutput) along with the format conversion that generates theother quadrants of the sinusoid. The sequence that is producedby the PSM is called the signature sequence [7].Considering infinite output wordlength, there are two sourcesfor the output sinusoid spur harmonics. The first source is thephase truncation, which is necessary to reduce the hardwarecomplexity of the PSM. The effect of the phase truncation onthe SFDR is thoroughly studied in [7] and [8]. The secondsource of the output sinusoid spur harmonics is the nonidealitiesin the PSM that create an error in the signature sequence.It has been shown that the spur magnitudes and locationscaused by the phase truncation and nonidealities of the PSMare disjoint [7]. Therefore, by knowing the spectrum of thesignature sequence, one can exactly determine the magnitudesand locations of the spur harmonics at the sine output of theDDFS.The main goal in designing DDFS systems is to reduce thesize and power of the final chip. This goal can be reached bymodifying the architecture of the PSM that results in higherspur harmonics and lower SFDR. Therefore, we should strike abalance in the hardware complexity and the SFDR of the DDFS.To be able to do so, the theoretical upperbound of the SFDRshould be known for the desired sine approximation method.The SFDR upperbound can provide a meaningful criterionto stop the optimization algorithm targeted at achieving thehighest SFDR for the desired hardware complexity. A comprehensivestudy of the SFDR upperbound for the polynomialinterpolation methods is given in [9] where the SFDR upperboundwas found as the Chebyshev minimax solution to anoverdetermined system of equations.Although the method given in [9] is a universal method offinding the SFDR upperbound for any polynomial interpolationbased DDFS, it highly depends on the optimization algorithmemployed to find the Chebyshev minimax solution of theoverdetermined system of equations. Unfortunately, some ofthe available optimization packages do not provide a correctanswer and may lead to an incorrect SFDR upperbound. Thiserroneous behavior usually happens at the minimax error valuesof the solution, particularly at the minimax error value of thefirst harmonic. Unfortunately, Chau and Chen [10] used theincorrect results of their numerical calculations and challengedthe results of [9]. In this brief, a few issues that remainedunclear in [9] (including the fact that one of the minimax errorvalues always occurs at the first harmonic of the spectrum of theDDFS output and it is always a positive number) are addressed,and based on them, the results of [10] are refuted.1549-7747/$31.00 © 2012 IEEE

308 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 5, MAY 2012The rest of the brief is organized as follows. In Section II,the theoretical upperbound of the SFDR is revisited, and itis proved that the first minimax error value always occurs atthe first harmonic. In Section III, different computer packagesare used to find the SFDR upperbound. It is shown that mostof the computer packages have difficulties finding the correctminimax solution and the SFDR upperbound. In Section IV,the SFDR upperbound of the DDFS based on the distressedeven polynomials (DEPs) is calculated for different polynomialorders. In Section V, we discuss the results and show that theycontradict the results obtained in [10], which concludes that theresults in [10] are incorrect. In Section VI, the conclusions aredrawn.II. REVISITING THE THEORETICAL UPPERBOUNDOF THE SFDRConsider that the first quadrant of the sinusoidal signal( π)f(x) =sin2 x , 0 ≤ x ≤ 1 (2)is divided into s =2 u segments and each segment is approximatedby a polynomial P k (x) of order d, such thatP k (x) =d∑i=0c (k)i (x − x k ) i ,x k ≤ x ≤ x k+1 ; k =1,...,s (3)where x represents the phase of the DDFS (truncated output ofthe accumulator); c (k)i is the ith coefficient of the polynomial,which interpolates the kth segment x k ≤ x ≤ x k+1 ; and x k =(k − 1)/s is the starting point of the kth segment. To findthe spectrum of the entire signal, the Fourier series can beemployed. Because of the quadrature wave symmetry, the evenharmonics are zero, and the odd coefficients are obtained byb n =2∫ 10( nπ)P (x)sin2 x dx. (4)By substituting (3) into (4), one can obtains∑ d∑b n = a (k)ni c(k) i (5)whereBy defining(c k =a (k)ni =2c (k)i)c k ∈ R (d+1)×1∫x k+1x k(A k =k=1 i=0(x − x k ) i sina (k)ni)A k ∈ R N×(d+1)( nπ2 x )dx. (6)1 ≤ k ≤ s 0 ≤ i ≤ d n =1, 3,...,2N − 1A =[A 1 |A 2 |A 3 |···|A s−1 |A s ] A ∈ R N×(d+1)sz = [ c T 1 |c T 2 |c T 3 |···|c T s−1|c T ] Ts z ∈ R (d+1)s×1 (7)one can represent (5) in the matrix form ofb = Az (8)where b ∈ R N×1 is a vector containing the harmonic values ofthe output signal. For more calculation details, please see [9].One can use (8) as an equation in which the entries of the vectorb are predefined. The desired values of the entries of the vectorb areb ′ =[1 0 ··· 0] T , b ′ ∈ R N×1 (9)because we would like to have an ideal sinusoid (only thefundamental harmonic is nonzero). Therefore, the problem canbe simplified as the solution to the following matrix equation:Az = b ′ . (10)It should be mentioned that N>(d +1)s,asitisshownin[9];thus, (10) is an overdetermined matrix equation.The matrix equation (10) does not have any solution. Anapproximate solution that minimizes the infinity norm of theerror (‖η(z)‖ ∞ = ‖b ′ − Az‖ ∞ ) can be found, where η(z) =b ′ − Az is called the error vector. This solution is called theChebyshev minimax solution [11]. By defining the positive realparameter ρ as the upperbound of the error vector |η(ẑ)| =|b ′ − Az|, one can express the problem as minimizing the valueof ρ such that|b ′ − Az| ρ. (11)This inequality can be divided into two inequalitiesρ + Az b ′ρ − Az − b ′ .(12a)(12b)The only parameter in (12) that needs to be minimized is ρ;thus, (12) can be posed as a linear programming (LP) problemdefined as follows:minimizeρsubject to[ ]zvρ[+A 1N×1−A][ ] [ ]z b′1 N×1 ρ −b(13)where v =[0 (d+1) s 1].If the solution to (13) is ˜z (that minimizes the norm‖η(z)‖ ∞ = ‖b ′ − Az‖ ∞ ), then according to [9, Th. 1], theassociated magnitude error vector |η(˜z)| = |b − A˜z| has, atmost, N +1 components (minimax) equal to the minimizednorm ˜ρ = ‖η(˜z)‖ ∞ . It is also shown in [9] that if the matrixA is a Haar matrix, the solution ˜z is unique, and there areexactly N +1minimax values in the error vector. Otherwise,the problem has infinite solutions, and one of them is associatedto the error vector with N +1 minimax error values (see [9,Fig. 3]). We call this solution the reference solution ˜z r .Itisworth noting that an n × m matrix (n >m) is a Haar matrixwhen all of its m × m submatrices are full rank. The referencesolution can be obtained using the simplex algorithm, andit is employed to define the reference matrix, which can beused to find the solution associated to the sparsest error vector(see [9, Fig. 4]). We call this solution the strict solution ˜z s .

ASHRAFI: ON THE SFDR UPPERBOUND IN DDFS UTILIZING POLYNOMIAL INTERPOLATION METHODS 309It is also shown that the first minimax error value occursat the first harmonic; thus, the SFDR upperbound can becalculated bySFDR upperbound = 1 − ˜ρ˜ρ . (14)The aforementioned analysis was supported by empirical data,and it was never proved in [9]. To prove that the first minimaxerror value occurs at the first harmonic, we can use the dual LPproblem associated to (13)maximizewsubject to[+b T − b T ]w[ ] [ ]11×N 1 1×N1+A T −A T w = ,0 (d+1)sw ≥ 0 (15)where w ∈ R 2N×1 . It has been proved that the solution to(15) ˜w has at most N +1nonzero entries [12]. If the matrix Ais a Haar matrix, then there are exactly N +1nonzero entriesin ˜w. The nonzero entries of ˜w can be used to find the strictsolution [12]. Moreover, it is also shown that the entries of˜w corresponding to strict inequalities in the LP (13) are zero[12]. Let ˜w =[p T |q T ] T , where p, q ∈ R N×1 .Iftheith entryof q (q i ) is nonzero, then the ith entry of p (p i ) is definitelyzero ([13, Lemma 4.3]). Since p corresponds to A in (13) andq corresponds to −A in (13), thus, the vector u =(p − q)provides the exact locations of the minimax values, and thesigns of the entries of u are the signs of their correspondingminimax components in the error vector associated to the strictsolution ˜z s .It is well known that when both the linear program (13) andits dual (15) have optimized solutions, their objective functionsare equal [14], i.e.,[˜z˜ρ]v =[+b T − b T ] ˜w. (16)According to the definitions of v, b, w, p, and q, we cansimplify (16) to˜ρ = p 1 − q 1 . (17)We know that ˜ρ >0 and ˜w ≥ 0. Thus, p 1 ≥ 0, and q 1 ≥ 0.On the other hand, either p 1 =0or q 1 =0. Therefore, it canbe easily seen that q 1 =0because both sides of (17) shouldbe positive. This means that the first minimax value occursat the first harmonic, and its value is positive. Therefore,η 1 (˜z) =b 1 − a 1˜z =˜ρ, where a 1 is the first row of the matrixA. Knowing the fact that b 1 =1, we can conclude that themagnitude of the first harmonic of the output signal is 1 − ˜ρ,and thus, the SFDR upperbound can be obtained by (14).If the algorithm utilized to solve the LP problem (13) doesnot converge to the correct solution, the SFDR upperboundobtained by (14) will be wrong. The behaviors of several optimizationpackages are investigated in the next section, and itis shown that some of them cannot be trusted when a Chebyshevminimax solution of an overdetermined system of equations issought.Fig. 1. Inconsistencies between the error vectors of the output of a DDFS withs =8and d =3obtained with different packages.TABLE ICOMPARISON BETWEEN THE FIRST MINIMAX ERROR(×10 −6 ) OBTAINED BY DIFFERENT PACKAGESIII. NUMERICAL COMPUTATIONSFinding the SFDR upperbound of the output of a polynomialbasedDDFS architecture is not straightforward. The SFDRdepends on the minimax errors, and they are usually smallnumbers. Thus, high-accuracy algorithms are required. Mistakescould be easily made in finding the SFDR upperbound(for example, see [15] and [16]). These mistakes can be misleadingand, if not verified with mathematical facts, will lead toincorrect conclusions. We have used several numerical analysispackages to solve the LP problem (13). These packages areMATLAB LP algorithm, IBM CPLEX Studio, MOSEK, andCVX that uses SeDuMi and SDPT3 engines [17].Fig. 1 illustrates the inconsistencies between different methodsof solving the LP problem. The goal was to find the strictsolution to the problem. Although the inconsistencies exist forall minimax error values, it is more significant for the firstminimax error that occurs at the first harmonic. Table I showsthe magnitudes of the first minimax error values obtained by theaforementioned packages for d =2and different values of s.It can be seen from the table that the LP methods provide thebest results. However, the IBM CPLEX package showed to bemore accurate than MATLAB and MOSEK. It should be notedthat the LP problem (13) is degenerate because of the fact thatthe matrix A is not a Haar matrix [13]. This degeneracy may bethe cause of the inconsistencies between different packages.

310 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 5, MAY 2012Fig. 2. Error vector of the DDFS output based on the DEP interpolation for different polynomial degrees m.TABLE IISFDR UPPERBOUNDS AND THEIR CORRESPONDING COEFFICIENTS FOR THE DDFS BASED ON THE DISTRESSED EVEN POLYNOMIALSIn [10], the first harmonic is fixed to the unity by includinga linear equality constraint to the LP problem (13). This willtake care of the inconsistency regarding the first harmonic.However, it cannot solve the problem of other inconsistencieswhich arise with the other minimax error values because theseinconsistencies are caused by the fact that the matrix A is nota Haar matrix. The added linear equality constraint just reducesthe order of the reference matrix by one. On the other hand,this constraint pushes up the other spurs in such a way thatthe SFDR upperbound will remain the same. Therefore, nomeaningful benefit can be gained by adding this constraint.IV. SFDR OF THE DDFS ARCHITECTURESBASED ON THE DEPSOne way to avoid segmentation of the first quadrant of thesinusoidal signal is using the DEPs of order mD(x) =c 0 + c 2 x 2 + c 4 x 4 + c 6 x 6 + ...+ c 2m x 2m . (18)Utilizing the DEP of order 2 is thoroughly studied in [9] and[18]. In [10], the SFDR upperbounds and their correspondingoptimum coefficients are found, but the results are incorrect.To have the correct SFDR upperbounds and their correspondingcoefficients, we have solved the LP problem (13) for thepolynomial configuration of (18). The matrix A in this problemis a Haar matrix; thus, the Chebyshev minimax solution isunique and can be easily calculated by the simplex algorithm.Fig. 2 shows the error vectors and the minimax values of theDDFS designed based on the DEPs of orders 1, 2, 3, and 4.Table II shows the SFDRs, the corresponding coefficients, andthe values of ˜ρ for the DEPs of orders 1, 2, 3, and 4. It shouldbe noted that the values of ˜ρ are all positive, which affirms themathematical argument given in Section III.V. D ISCUSSIONIn the previous sections, it is mathematically proved thatthe first minimax error value corresponding to the coefficientsthat give the SFDR upperbound always occurs at the firstharmonic. The conclusions drawn in [10] are entirely based onthe incorrect assumption that the error of the first harmonic isdifferent than the minimax error. To show the discrepancies in[10], one can compare the SFDR upperbound obtained in [10]for the fourth-order DEP (155.38 dBc) and the one obtainedhere and shown in Table I for the same polynomial interpolation(158.14 dBc), which is obviously higher. In addition, one canclearly see that the coefficients of the DEPs are incorrectly

ASHRAFI: ON THE SFDR UPPERBOUND IN DDFS UTILIZING POLYNOMIAL INTERPOLATION METHODS 311obtained in [10] (see [10, Table II]) by comparing them to thecoefficients given in Table II of this brief. Moreover, [10, Fig. 3]clearly illustrates that the SFDR upperbounds obtained in [9]are less than 0.1% smaller that the ones obtained in [10]for polynomial interpolations of different orders. These smalldifferences fall in the numerical computation error range, yetthe authors in [10] argued that their method provides a betterSFDR upperbound due to these small differences.VI. CONCLUSIONFinding the SFDR upperbound of the DDFSs with architecturesbased on the polynomial interpolation method hasbeen revisited in this brief. The Chebyshev minimax solutionof the related overdetermined linear equations determines theSFDR upperbound. The Chebyshev minimax problem can besolved by an LP problem. It has been shown that most of theavailable computer packages fail to solve this problem becausethe matrix of the equation is not a Haar matrix. This has ledother authors to incorrectly deduce that the first minimax errorvalue does not occur at the first harmonic. In this brief, it hasbeen mathematically proved that the first minimax error valuealways occurs at the first harmonic by using the dual of the LPproblem. This has been used to show that the results obtained byChau and Chen, where they claimed that the first minimax errorvalue does not occur at the first harmonic, are in fact incorrect.This method has been also used to find the SFDR upperbound ofthe DDFS based on DEP interpolation. A Comparison betweenthe obtained SFDR upperbounds with those reported by Chauand Chen can clearly show the incorrectness of the latter.REFERENCES[1] J. Vankka and K. Halonen, Direct Digital Synthesizers, Theory, Designand Applications. Norwell, MA: Kluwer, 2001.[2] J. 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