127:. If the PAC capacity is 2, the PA output word must be truncated to M bits as shown in Figure 1. However, the truncated bits can be used for interpolation. The truncation of the phase output word does not affect the frequency accuracy but produces a time-varying periodic phase error which is a primary source of spurious products. Another spurious product generation mechanism is finite word length effects of the PAC output (amplitude) word.
1041:
noise is summed into the lower W+1 bits of the PA output word to linearize the truncation operation. Often the improvement can be achieved without penalty because the DAC noise floor tends to dominate system performance. Amplitude truncation spurs can not be mitigated in this fashion. Introduction of
667:
The number of phase accumulator bits of an NCO (N) is usually between 16 and 64. If the PA output word were used directly to index the PAC look-up table an untenably high storage capacity in the ROM would be required. As such, the PA output word must be truncated to span a reasonable memory space.
1008:
One way to reduce the truncation in the address lookup is to have several smaller lookup tables in parallel and use the upper bits to index into the tables and the lower bits to weigh them for linear or quadratic interpolation. Ie use a 24-bit phase accumulator to look up into two 16-bit LUTS.
623:
containing 2 contiguous samples of the desired output waveform which typically is a sinusoid. Often though, various tricks are employed to reduce the amount of memory required. This include various trigonometric expansions, trigonometric approximations and methods which take advantage of the
1261:
159:
configured as shown in Figure 1. Each clock cycle produces a new N-bit output consisting of the previous output obtained from the register summed with the frequency control word (FCW) which is constant for a given output frequency. The resulting output waveform is a staircase with step size
54:
Numerically controlled oscillators offer several advantages over other types of oscillators in terms of agility, accuracy, stability and reliability. NCOs are used in many communications systems including digital up/down converters used in 3G wireless and software radio systems, digital
934:
122:
which is then converted by the phase-to-amplitude converter (PAC) to a sampled sinusoid, where N is the number of bits carried in the phase accumulator. N sets the NCO frequency resolution and is normally much larger than the number of bits defining the memory space of the PAC
1009:
Address into the truncated 16 MSBs, and that plus 1. Linearly interpolate using the 8 LSBs as weights. (One could instead use 3 LUTs instead and quadratically interpolate). This can result in decreased distortion for the same amount of memory at the cost of some multipliers.
1242:
1004:
Another related spurious generation method is the slight modulation due to the GRR outlined above. The amplitude of these spurs is low for large N and their frequency is generally too low to be detectable but they may cause issues for some applications.
363:
130:
The frequency accuracy relative to the clock frequency is limited only by the precision of the arithmetic used to compute the phase. NCOs are phase- and frequency-agile, and can be trivially modified to produce a
761:
780:
999:
480:
651:
in the creation of the output waveform due to non-linear numerical effects in the signal processing chain. Only numerical errors are covered here. For other distortion mechanisms created in the
563:
98:
is used with the look-up table to provide better accuracy and reduce phase error noise. Other methods of converting phase to amplitude, including mathematical algorithms such as
609:
1259:, McCallister, R. D. & Shearer, D., "Numerically controlled oscillator using quadrant replication and function decomposition", published 12/04/1984
285:
254:
227:
393:
181:
1122:
While some authors use the terms DDS and NCO interchangeably, by convention an NCO refers to the digital (i.e. the discrete-time, discrete amplitude) portion of a DDS
1109:
1021:
of the sampled waveform contained in the PAC look up table(s). If the number of DAC bits is P, the AM spur level is approximately equal to −6.02 P − 1.76
1042:
noise into the static values held in the PAC ROMs would not eliminate the cyclicality of the truncation error terms and thus would not achieve the desired effect.
202:
of its operands exceeds its capacity (2−1). The overflow bit is discarded so the output word width is always equal to its input word width. The remainder
183:, the integer value of the FCW. In some configurations, the phase output is taken from the output of the register which introduces a one clock cycle
619:
The phase-amplitude converter creates the sample-domain waveform from the truncated phase output word received from the PA. The PAC can be a simple
293:
929:{\displaystyle \zeta _{max}=2^{-M}{\frac {\pi {\mbox{GCD}}(\Delta F,2^{W})}{\sin \left(\pi \cdot 2^{-P}{\mbox{GCD}}(\Delta F,2^{W})\right)}}}
682:
191:
1300:
945:
1148:
107:
1018:
405:
1310:
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1174:
1315:
629:
496:
676:
in proportion to the number of bits truncated. The number of spurious products created by this distortion is given by:
1057:
774:, we are interested in the spurious product with the largest amplitude relative to the carrier output level given by:
771:
652:
44:
656:
48:
939:
where P is the size of the phase-to-amplitude converter's lookup table in bits, i.e., M in Figure 1. For W >4,
1240:, Miller, B. M., "Numerically controlled oscillator and method of operation", issued October 14, 2008
1051:
369:
60:
1305:
1275:
184:
1215:
1256:
1237:
625:
257:
136:
95:
90:(PAC), which uses the phase accumulator output word (phase word) usually as an index into a waveform
576:
644:
152:
32:
64:
229:, called the residual, is stored in the register and the cycle repeats, starting this time from
83:(PA), which adds to the value held at its output a frequency control value at each clock sample.
263:
232:
205:
1144:
640:
570:
569:
Equation (1) shows that the phase accumulator can be thought of as a programmable non-integer
375:
163:
156:
56:
669:
620:
140:
132:
115:
28:
395:
which for a high resolution NCO can be very long. Usually we are more interested in the
199:
1294:
124:
119:
91:
1185:
99:
624:
quadrature symmetry exhibited by sinusoids. Alternatively, the PAC may consist of
1034:
490:, defined as the smallest possible incremental change in frequency, is given by
190:
59:, radar systems, drivers for optical or acoustic transmissions, and multilevel
1175:"Low-Spur Numerically Controlled Oscillator Using Taylor Series Approximation"
673:
648:
40:
287:. The interval K is referred to as the grand repetition rate (GRR) given by
260:, eventually the residual at some sample K must return to the initial value
1033:
Phase truncation spurs can be reduced substantially by the introduction of
110:
Figure 1: Numerically controlled oscillator with optional quadrature output
358:{\displaystyle {\mbox{GRR}}={\frac {2^{N}}{{\mbox{GCD}}(\Delta F,2^{N})}}}
36:
106:
1038:
756:{\displaystyle n_{W}={\frac {2^{W}}{{\mbox{GCD}}(\Delta F,2^{W})}}-1}
1184:. Gliwice, Poland: Silesian University of Technology. Archived from
35:(i.e., clocked), discrete-time, discrete-valued representation of a
1089:
The IEEE Standard
Dictionary of Electrical and Electronics Terms
994:{\displaystyle \zeta _{max}\approx -6.02\cdot P\;{\mbox{dBc}}.}
1022:
372:
function. The GRR represents the true periodicity for a given
94:(LUT) to provide a corresponding amplitude sample. Sometimes
475:{\displaystyle F_{out}={\frac {\Delta F}{2^{N}}}F_{clock}}
187:
but allows the adder to operate at a higher clock rate.
139:
output by summation at the appropriate node, or provide
151:
A binary phase accumulator consists of an N-bit binary
982:
887:
823:
713:
321:
298:
198:
The adder is designed to overflow when the sum of the
1017:
Another source of spurious products is the amplitude
948:
783:
685:
672:
of the output sinusoid which introduces non-harmonic
579:
499:
408:
378:
296:
266:
235:
208:
166:
114:When clocked, the phase accumulator (PA) creates a
993:
928:
755:
603:
558:{\displaystyle F_{res}={\frac {F_{clock}}{2^{N}}}}
557:
474:
399:determined by the average overflow rate, given by
387:
357:
279:
248:
221:
175:
256:(see figure 2). Since a phase accumulator is a
194:Figure 2: Normalized phase accumulator output
8:
763: (3)
628:which can be filled as desired to create an
102:can be used, particularly in a software NCO.
43:. NCOs are often used in conjunction with a
1276:"The NCO as a Stable, Accurate Synthesizer"
1214:Murphy, Eva; Slattery, Colm (August 2004),
1104:
1102:
1100:
1098:
1112:. Lattice Semiconductor Corporation. 2009.
980:
981:
953:
947:
909:
886:
877:
845:
822:
816:
807:
788:
782:
767:where W is the number of bits truncated.
735:
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165:
75:An NCO generally consists of two parts:
1134:
1132:
1130:
1128:
189:
105:
1168:
1166:
1164:
1162:
1160:
1076:
1281:. Intersil Corporation. 1998. TB318.1.
1182:XI International PhD Workshop OWD 2009
1173:Popek, G.; Kampik, M. (October 2009).
1091:. New York, NY: IEEE Standards Office.
1082:
1080:
1209:
1207:
1205:
1141:Direct Digital Frequency Synthesizers
655:see the corresponding section in the
7:
1216:"All About Direct Digital Synthesis"
668:Truncation of the phase word causes
1110:"Numerically Controlled Oscillator"
1037:prior to truncation. The so-called
896:
832:
722:
580:
431:
379:
330:
167:
14:
21:numerically controlled oscillator
143:outputs as shown in the figure.
47:(DAC) at the output to create a
1064:Digitally controlled oscillator
915:
893:
851:
829:
741:
719:
604:{\displaystyle \Delta F/2^{N}}
349:
327:
1:
630:arbitrary waveform generator
615:Phase-to-amplitude converter
88:phase-to-amplitude converter
1058:Digital-to-analog converter
772:spurious-free dynamic range
653:digital-to-analog converter
643:products are the result of
45:digital-to-analog converter
1332:
1013:Amplitude truncation spurs
657:direct-digital synthesizer
49:direct digital synthesizer
1301:Digital signal processing
280:{\displaystyle \phi _{0}}
249:{\displaystyle \phi _{n}}
222:{\displaystyle \phi _{n}}
67:modulators/demodulators.
1052:Direct digital synthesis
388:{\displaystyle \Delta F}
176:{\displaystyle \Delta F}
16:Digital signal generator
565: (2)
482: (1)
370:greatest common divisor
1311:Electronic oscillators
1139:Kroupa, V. F. (1999).
995:
930:
757:
663:Phase truncation spurs
605:
559:
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359:
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1029:Mitigation techniques
996:
931:
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477:
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109:
1035:white gaussian noise
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781:
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626:random access memory
577:
497:
488:frequency resolution
406:
376:
294:
264:
258:finite state machine
233:
206:
164:
1316:Digital electronics
1087:Radatz, J. (1997).
770:In calculating the
397:operating frequency
137:frequency-modulated
991:
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472:
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57:phase-locked loops
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636:Spurious products
571:frequency divider
553:
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368:where GCD is the
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147:Phase accumulator
120:sawtooth waveform
81:phase accumulator
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1226:, Analog Devices
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670:phase modulation
647:or non-harmonic
621:read only memory
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573:of divide ratio
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31:which creates a
29:signal generator
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1220:Analog Dialogue
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133:phase-modulated
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27:) is a digital
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1143:. IEEE Press.
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200:absolute value
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1191:on 2011-08-20
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1150:0-7803-3438-8
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125:look-up table
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96:interpolation
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92:look-up table
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1306:Synthesizers
1288:
1270:
1251:
1232:
1223:
1219:
1193:. Retrieved
1186:the original
1181:
1140:
1118:
1088:
1032:
1019:quantization
1016:
1007:
1003:
938:
769:
766:
666:
639:
618:
568:
487:
485:
396:
367:
197:
150:
129:
113:
100:power series
87:
80:
74:
53:
24:
20:
18:
33:synchronous
1295:Categories
1257:US 4486846
1238:US 7437391
1195:2010-02-12
1071:References
674:distortion
649:distortion
141:quadrature
41:sinusoidal
39:, usually
975:⋅
969:−
966:≈
951:ζ
897:Δ
879:−
871:⋅
868:π
860:
833:Δ
820:π
809:−
786:ζ
748:−
723:Δ
659:article.
581:Δ
432:Δ
380:Δ
331:Δ
269:ϕ
238:ϕ
211:ϕ
168:Δ
71:Operation
1046:See also
645:harmonic
641:Spurious
157:register
37:waveform
185:latency
51:(DDS).
1263:
1244:
1147:
1039:dither
155:and a
116:modulo
1279:(PDF)
1189:(PDF)
1178:(PDF)
1066:(DCO)
1060:(DAC)
1054:(DDS)
153:adder
1145:ISBN
972:6.02
486:The
1023:dBc
984:dBc
889:GCD
857:sin
825:GCD
715:GCD
323:GCD
300:GRR
135:or
118:-2
65:PSK
61:FSK
25:NCO
1297::
1224:38
1222:,
1218:,
1204:^
1180:.
1159:^
1127:^
1097:^
1079:^
1025:.
632:.
611:.
86:A
79:A
19:A
1198:.
1153:.
989:.
978:P
961:x
958:a
955:m
920:)
916:)
911:W
907:2
903:,
900:F
894:(
882:P
875:2
864:(
852:)
847:W
843:2
839:,
836:F
830:(
812:M
805:2
801:=
796:x
793:a
790:m
751:1
742:)
737:W
733:2
729:,
726:F
720:(
707:W
703:2
697:=
692:W
688:n
597:N
593:2
588:/
584:F
549:N
545:2
539:k
536:c
533:o
530:l
527:c
523:F
517:=
512:s
509:e
506:r
502:F
468:k
465:c
462:o
459:l
456:c
452:F
444:N
440:2
435:F
426:=
421:t
418:u
415:o
411:F
383:F
350:)
345:N
341:2
337:,
334:F
328:(
315:N
311:2
305:=
273:0
242:n
215:n
171:F
63:/
23:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
