IEEE SIGNAL PROCESSING MAGAZINE110 SEPTEMBER 2004

n Part 1 of this article (inthe July 2004 issue of IEEESignal Processing Maga-zine), we presented an over-view of the basics of direct

digital frequency synthesis (DDS),simple formulas to compute boundsof the signal characteristics, and ascheme to improve the DDS spuri-ous free dynamic range (SFDR). Inthis Part 2, we discuss additionaltricks used to optimize DDS perfor-mance by maximizing SFDR.

Improving SFDR ThroughSpur-Reduction TechniquesThe easiest method to reduce thelevel of DDS spurs, discussed inPart 1, is to increase the accuracy ofthe phase to waveform converter.The limit of this approach has beenmentioned; it is mainly technologi-cal [lookup table (LUT) size].

We now review three simple andeffective methods to reduce the spurlevel of the sinewave DDS, alongwith the corresponding spectra com-puted from simulated DDS outputs.

The Odd-Number ApproachThe worst-case spur level is given by(9) in Part 1. Making �ACC an odd

number improves the SFDR by 3.9dB [14] in Part 1. The repetitionperiod of the accumulator TACC ,often referred to as the grand repeti-tion period, is given by

TACC = 2N

GCD(2N ,�ACC)(1)

where GCD(x , y ) stands for thegreatest common divisor of x and y[1]. When GCD(2N ,�ACC) = 1, asit will be whenever �ACC is an oddnumber, then TACC = 2N , whichspreads the spurs over the entirespectrum (otherwise they arealiased to a frequency within thespectrum, as described in [14] and[12] in Part 1). As an example, wecomputed the output spectra with afast Fourier transform, the lengthof which is equal to TACC, for thetwo cases of �ACC = 13, 248 and�ACC =13,249, with N = 16,

P = 9, and M = 8. The odd num-ber for �ACC leads to an increase of54.2 − 50.3 = 3.9 dB in SFDR.

The Phase-DitheringApproachIn another method to spread thespurs throughout the available

bandwidth, one can add a dither sig-nal [2] to the ACC phase values asshown in Figure 1.

The dither signal can be a pseu-do-random noise sequence (generat-ed, for example, with binary shiftregisters and exclusive-or gates, andhaving a repetition period muchgreater than the output signal peri-od) whose word width is B bits pro-viding noise values in the range of 0and 2B . Choosing B = N − P , thespurs do follow a 12-dB per-phasebit law [3] instead of the 6-dB per-phase bit of (9), thus allowing asmaller LUT for the same SFDR.

An output spectrum is givenwithout any dither signal in Figure2(a) with Fs = 44, 100 Hz,�ACC =1,657, N = 16, P = 5, M = 16,and the dither s ignal(B = N − P = 16 − 5 = 11 b)applied in Figure 2(b). The spectrahave been computed for ten outputsignals that have been averaged fol-lowing the method described in [3].The high-resolution, 16-b, LUT hasbeen chosen so as to focus only onthe 5 b of phase quantization, as in[3]. The drawback of this dithering

Lionel Cordesses

Direct Digital Synthesis: A Tool For Periodic Wave Generation (Part 2)

“DSP Tips and Tricks” introduces prac-tical tips and tricks of design andimplementation of signal processingalgorithms so that you may be able toincorporate them into your designs.We welcome readers who enjoy read-ing this column to submit their contri-butions. Contact Associate Editors RickLyons (r.lyons@ieee.org) or Amy Bell(abell@vt.edu).

▲ 1. DDS, including phase dither.

Dither Generator

PhaseGenerator

PhaseQuantization

∆ACCFo N – 1 N N

B

ACC

P

Φ

I

1053-5888/04/$20.00©2004IEEE

dsp tips & tricks

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IEEE SIGNAL PROCESSING MAGAZINESEPTEMBER 2004 111

method is the increase of the noisefloor, but that’s a small price to payfor such a large increase in SFDR.

Other dithering methods areavailable, such as amplitude dither-ing and both phase and amplitudedithering (see [3] and [4]). Thephase-dithering approach has alsobeen applied to squarewave signalDDS [5].

The Noise-ShapingApproachThe key idea of the noise-shapingapproach, to improve our DDSSFDR, is to filter out the quantiza-tion noise introduced by the phasequantization step in Figure 3. Thisquantization can be viewed as a spe-cial case of noise addition [6], asdepicted in Figure 3(a). The quanti-zation noise signal n can be recov-ered from the following equations:

� = n + ACCeQ = ACC − �. (2)

Thus, eQ = −n. In the noise-shap-ing approach, this quantizationnoise signal −n is fed back, througha filter G, to the ACC signal asshown in Figure 3(b). The transferfunction of interest is the one fromn to �, as the noise added to thephase signal � will eventually lead tophase noise. The phase signal is then

� = n(1 − G ) + ACC. (3)

From (3), one can infer that thephase signal � is affected by the fil-tered noise signal n, (1−G) beingthe transfer function of the filter.The choice of G will lead to differ-ent results, as we shall see.

A first-order noise-shapingapproach is to use the simple trans-fer function G proposed in [10] inPart 1 as the finite impulse response(FIR) filter G = z −1. Here z is thesymbol of the z-transform used fordiscrete-time systems. The function

G can be implemented as a singledelay register, and 1 − G has a 0 atz = 0 (0 Hz); it acts as a discretetime differentiator. The system fil-ters out the low-frequency compo-nents of the noise signal n buthigh-frequency signals, greater than8 kHz, are amplified. This simpleapproach prevents the filter from

rejecting high-frequency compo-nents of the noise signal n, thus jus-tifying the statement that oneshould use this filter for low-fre-quency Fo signals [7].

An example of the output spec-trum is given in Figure 4(a). [Theparameters are the same as inFigure 2(a).] The SFDR is greater

▲ 2. DDS output spectra: (a) without dither and (b) with dithering.

0

–10

–20

–30

–40

–50

–60

0

–10

–20

–30

–40

–50

–60

Amplitude (dBc) Amplitude (dBc)

0 Frequency(a)

Fs/2 0 Frequency(b)

Fs/2

▲ 3. Quantization and noise shaping: (a) quantization model and (b) noise-shapingimplementation.

Phase Quantization

PhaseQuantizationACC

N

n

P

Φ

+ –

eQ = –n

ACC

N P

Φ

+ –

–nG

(a) (b)

▲ 4. DDS spectra with first-order noise shaping and G = z−1 : (a) �ACC = 1, 657 and (b)�ACC = 13, 249.

0

–10

–20

–30

–40

–50

–60

0

–10

–20

–30

–40

–50

–60

Amplitude (dBc) Amplitude (dBc)

0 Frequency(a)

Fs/2 0 Frequency(b)

Fs/2

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dsp tips & tricks continued

IEEE SIGNAL PROCESSING MAGAZINE112 SEPTEMBER 2004

than 60 dB near the carrier, but itdecreases with the frequency andeventually reaches 46.8 dBc. Theoverall behavior is compliant withthe above G = z −1 analysis. At ahigher frequency (�ACC = 13249),the noise close to the carrier is less fil-tered, as one can see in Figure 4(b).

To implement higher-order noiseshaping, a more complex filter canbe used instead of G = z −1. A sec-ond-order FIR filter has been pro-posed in [8]:

1 − G = 1 + b1z −1 + b2z −2. (4)

Careful choice of b1 and b2 can leadto a double zero at 0: 1 − G =1 − 2z −1 + z −2 = (1 − z −1)2, whichimproves the rejection at 0 Hz.When the noise shaping is applied tothe amplitude signal instead of thephase signal, other values (often inte-ger values, so as to ease implementa-tion, see [8]) are preferred, and thisfilter can even be tuned online.

Multiple-zeros filters are also of

interest, for example, when onewants to reject a known frequencysuch as 2 × Fo [7]. A tunable notchfilter is added at the expense of amore complex feedback structure.The transfer function becomes

1 − G = (1 − z −1)(1 + b z −1 + z −2)

= 1 − (1 − b )z −1(1 − b )z −2

− z −3 (5)

with b = −2cos [2π(2Fo/Fs)] . Atlow frequencies, as shown in Figure5(a), the spectrum looks like theone with the first-order noise shap-ing. But at a higher frequency [thesame as in Figure 4(b)], theimprovement close to the carrier isclear, as shown in Figure 4(b).

Note that the same filter (withzero at a specific frequency) can beimplemented by feeding the errorsignal back to the accumulator. Thisstructure, proposed and patented in[7], is presented in Figure 6. F isthe transfer function of the accumu-

lator [an integrator: ACC(k) =ACC(k − 1) + �ACC (k− 1)], given by

F = z −1

1 − z −1 (6)

with the output phase � given by� = (1 − FG )n + F�ACC. One canchose G so as to ensure (1 − FG ) =(1 − z −1)(1 + b z −1 + z −2) as in (5).

SummaryDDS is a useful tool for generatingperiodic waveforms. In this two-partarticle, we presented the basic idea ofthis synthesis technique and thenfocused on the quality of the sinewavea DDS can create, introducing theSFDR quality parameter. Next wepresented effective methods toincrease the SFDR through sinewaveapproximations, hardware schemessuch as dithering and noise shaping,and an extensive list of references.When the desired output is a digitalsignal, the signal’s characteristics canbe accurately predicted using the for-mulas given in this article. When thedesired output is an analog signal, thereader should keep in mind that theperformance of the DDS is eventuallylimited by the performance of thedigital-to-analog converter and thefollow-on analog filter [9].

We hope that this article willincite engineers to use DDS; eitherintegrated-circuits DDS or soft-ware-implemented DDS. From theauthor’s experience, this techniquehas proved valuable when frequen-cy resolution is the challenge, par-ticularly when using low-costmicrocontrollers.

AcknowledgmentsI would like to thank Rick Lyons, whohelped improve this article throughmany comments and a thorough read-ing of the original manuscript.

▲ 5. DDS spectra with higher-order noise shaping: (a) same parameters as Figure 2(a)and (b) same parameters as Figure 4(b).

▲ 6. Noise shaping: Analog Devices’ approach.

PhaseQuantization

∆ACC

N – 1

–n

N

ACC

P

ΦF

G

+ –

0

–10

–20

–30

–40

–50

–60

0

–10

–20

–30

–40

–50

–60

Amplitude (dBc) Amplitude (dBc)

0 Frequency(a)

Fs/2 0 Frequency(b)

Fs/2

(continued on page 117)

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IEEE SIGNAL PROCESSING MAGAZINESEPTEMBER 2004 117

Single Instrument Solutionfor WLAN DevicesThe MT8860A is Anritsu Compa-ny’s first fully integrated single-instrument test solution for WLANdevices that can conduct bothtransmitter and receiver measure-ments based on the IEEE 802.11standard. With the ability to per-form these measurements ten timesfaster than existing test alternatives,the MT8860A is a markedimprovement over existing rack-and-stack test solutions that arebased on multiple instruments plus

“golden” radios from variousWLAN chip manufacturers.

The MT8860A covers both the2.4 GHz and 4–6 GHz WLAN bandsand supports all 802.11 WLAN stan-

dards with options. Accurate high-speed transmitter power, frequency,carrier suppression, and harmonicmeasurements can be made with thetest set. Each of the measurementscan be conducted on all frequencychannels and all specified power lev-els. A PCI bus design makes it simpleto add measurement capability.

Anritsu Company, CorporateOffices, 490 Jarvis Dr., Morgan Hill,CA 95037-2809 USA.Tel: +1 800 267 4878Fax: +1 408 776 1744URL: http://www.us.anritsu.com

Lionel Cordesses is a control engineer at the Techno-centre Renault (Guyancourt, France), the researchcenter of the Renault company. His technical interestsinclude analog electronics, digital signal processing,and embedded controllers. He is a member of theIEEE and the ACM.

References[1] J.F. Garvey and D. Babitch. “An exact spectral analysis of a number con-

trolled oscillator based synthesizer,” in Proc. 44th Annu. Symp. FrequencyControl, 1990, pp. 511–521.

[2] L. Schuchman, “Dither signals and their effect on quantization noise,”IEEE Trans. Commun., vol. 12, no. 4, pp. 162–165, 1964.

[3] M.J. Flanagan and G.A. Zimmerman, “Spur-reduced digital sinusoid synthe-sis,” IEEE Trans. Commun., vol. 43, no. 7, pp. 2254–2262, 1995.

[4] J. Vankka, “Spur reduction techniques in sine output direct digital synthe-sis,” in Proc. 1996 IEEE Frequency Control Symp., 1996, pp. 951–959.

[5] C.E. Wheatley, III and D.E. Phillips, “Spurious suppression in direct digi-tal synthesizers,” in Proc. 35th Annu. Frequency Control Symp., 1981, pp.428–435.

[6] H. Spang, III and P. Schultheiss, “Reduction of quantizing noise by use offeedback,” IEEE Trans. Commun., vol. 10, no. 4, pp. 373–380, 1962.

[7] D.B. Ribner and S. Kidambi, “Direct-digital synthesizers,” EuropeanPatent EP1037379, 2000.

[8] J. Vankka, “A direct digital synthesizer with a tunable error feedbackstructure,” IEEE Trans. Commun., vol. 45, no. 4, pp. 416–420, 1997.

[9] T.M. Higgins, “Analog output system design for a multifunction synthesiz-er,” Hewlett-Packard J., vol. 40, no. 1, pp. 66–69, 1989.

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