TESTING OF ADCS BY FREQUENCY-DOMAIN ANALYSIS IN MULTI-TONE MODE

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volue 5, Nuber /004, pp.000-000 TESTING OF ADCS BY FREQUENCY-DOMAIN ANALYSIS IN MULTI-TONE MODE Daniel BELEGA Politehnica University of Tiişoara Bv. V. Pârvan, Nr., 1900, Tiişoara, Roânia E-ail: belega@etc.utt.ro In this paper the testing of analog-to-digital converters (ADCs by frequency-doain analysis in ulti-tone ode is investigated. The theoretical expressions of soe of the ost iportant ADC dynaic paraeters are derived. Also, the theoretical expression of the difference between the ADC effective nuber of bits (ENOB obtained by testing in ulti-tone ode and the ENOB obtained by testing in the single-tone ode is derived. Fro the theoretical expression of this difference soe iportant conclusions concerning the accuracy in the evaluation of ENOB by testing in ulti-tone ode are drawn. Soe experiental results are presented in order to validate these conclusions. The experients were carried out using the test syste MULTI-TONE ADC TEST, specially developed for this purpose. Key words: Multi-tone ode, testing of ADCs by frequency-doain analysis, estiation of the effective nuber of bits of an ADC. 1. INTRODUCTION Due to its advantages the frequency-doain analysis ethods are ostly eployed to deterine the dynaic paraeters of an analog-to-digital converter (ADC. These ethods frequently use single-tone ode [1]-[]. In this ode the signal used to test the ADC is a high purity sinewave (i.e. the single-tone signal. Then, a certain algorith, based on the discrete Fourier transfor (DFT of the corresponding output codes, is used to estiate the dynaic paraeters of the ADC. In this paper the ADC characterization by frequency-doain analysis in ulti-tone ode is described. In ulti-tone ode the test signal is coposed of the su of ( high purity sinewaves with the sae aplitude (i.e. the -tone signal. First, the theoretical expressions of soe of the ost iportant dynaic paraeters of an ADC are derived. These paraeters are: signal-to-noise and distortion ration (SINAD, effective nuber of bits (ENOB, total haronic distortion (THD and interodulation distortion (IMD. The global dynaic perforances of an ADC are evaluated by eans of ENOB. Thus, it is very iportant to copare the ENOB obtained by testing in ulti-tone ode with the one obtained by testing in single-tone ode. Fro this reason, then, the theoretical expression of the difference between the values of ENOB, ENOB, obtained by testing in these odes is derived. The derived expressions represent a generalization of the expressions presented in [4] for the dual-tone ode (. Based upon the ENOB expression soe iportant conclusions concerning the accuracy in the evaluation of ENOB by testing in ulti-tone ode are drawn. In order to validate these conclusions soe experients were carried out. The experiental results were obtained by eans of the test syste naed MULTI-TONE ADC TEST, specially ipleented for this purpose.. ADC DYNAMIC PARAMETERS OBTAINED IN MULTI-TONE MODE A real ADC can be odeled by an ideal ADC with the sae resolution, followed by a nonlinear syste [4], [5]. The nonlinear syste is characterized by a polynoial transfer function given by Recoended by Ioan DUMITRACHE, eber of the Roanian Acadey

Daniel BELEGA ( t ax( t + bx ( t + cx ( t +! y (1 in which x is the input ter; y is the output ter; a, b, c, are the linear, squared, third, gain ters. The order of the polynoial transfer function is or 5 [4], [6]. In order to siplify the calculation a threeorder transfer function was considered. The -tone signal is coposed by a su of ( high purity sinewaves with the aplitude A and frequency f i (i 1,,, For testing of all ADC codes A is equal with x ( t A ( πf i t i 1 sin ( FSR A ( where FSR is the ADC full-scale range. By substituting ( in (1 the following haronic and interodulation coponents are obtained haronic coponents H H ( f i i 1,,! ( f, i 1,,! i ba ca 4 where H(f i, i 1,,,,, is the aplitude of the haronic coponent at frequency f i ; interodulation coponents IM IM IM ( f i ± f j ( f ± f i ba, j 1,,!, i < j 1,,!, i ca ( f ± f ± f, 1,,!, i < j < i j j ca, 4, where: IM(f i ± f j, j 1,,,, i < j is the aplitude of the interodulation coponent at frequency f i ± f j ; IM(f i ± f j, j 1,,,, i j is the aplitude of the interodulation coponent at frequency f i ± f j ; IM(f i ± f j ± f, 1,,,, i < j < is the aplitude of the interodulation coponent at frequency f i ± f j ± f ; The nuber of the coponents IM(f i ± f j is (-1, of the coponents IM(f i ± f j is (-1 and of the coponents IM(f i ± f j ± f is (-1(-/. Fro (4 THD in ulti-tone ode becoes j j (4 (5 THD ( H H ( fi H ( fi T + 4 b A c A + 4 16 A 4 i 1 i 1 b A c A T + ST ST 8 6 (6 in which S T is the effective value of the -tone signal, S T A /.

Testing of ADCs by frequency-doain analysis in ulti-tone ode Based on (5 IMD in ulti-tone ode is given by IMD T IM ( ( f i ± f j + IM ( f i ± f j + IM ( f i ± f j ± f IM ij T j 1 j 1 1 S T i < j i j 9 6 ( 1 ba + ( 1 c A + ( 1( c A 8 A 4 ( 1 b A + ( 1(4 5 c A. 4 Fro (6 and (7 we obtain ( 1(1 17 IMDT ( 1 THDT + c 4 Because the second ter of the expression (8 is positive it follows that S 6 T i < j < 4 4 A IMD THD, (9 T 4( 1 T which shows that IMD T is uch greater than THD T. The signal-to-noise ratio (SNR of an ideal n-bit ADC tested by frequency-doain in ulti-tone ode is SNR S 10 log σ 6.0n + 1.76 10 log( T T q where σ q is the variance of the quantization noise; σ q is given by q /1 (where q is the quantization step, q FSR/ n. In the case of a real ADC besides the quantization noise other noise sources can be appear such as haronic, interodulation and spurious coponents, jitter of sapling cloc, additive noise. Therefore, the ADC is characterized by another dynaic paraeter naed SINAD. In ulti-tone ode SINAD is given by SINAD T 10 log ( IM ij + ( H T S T T + σ q + σ ex. ( db ( db in which σ ex is the variance of the excess noise (jitter, additive noise,. ENOB is calculated fro (10 by replacing SNR T by SINAD T. Thus, it is obtained ENOB T ( db (7 (8 (10 (11 SINADT 1.76 + 10 log(. (1 6.0 ENOB is the ost iportant dynaic paraeter of an ADC because it evaluates the global dynaic perforances of the ADC. Fro this reason it is very iportant to now how the testing in ulti-tone affects the evaluation of the ENOB by coparison with the one obtained by testing in single-tone ode. The difference ENOB between the ENOB T and the ENOB obtained by testing in single-tone ode ENOB 1T is given by ENOB ENOB T ENOB 1T SINAD SINAD 1T is given by the following expression T ( db SINAD ( db 6.0 1T + 10 log(. (1

Daniel BELEGA 4 S 1T SINAD1T 10 log ( db (14 ( H 1T + σ q + σ ex where: S 1T is the effective value of the single-tone signal; (H 1T is the haronic in single-tone ode. To test all the ADC output codes the aplitude of the single-tone signal is equal with FSR/. So, S 1T is equal with A S 1T S T. (15 Fro (11, (14 and (15 ENOB becoes ENOB 10 6.0 log ( H 1T + σ q ( IM ij + ( H T For an ideal ADC the haronic and interodulation coponents and, also, the noise in excess do not exist. Thus, we obtain + σ T ENOB ENOB T ENOB 0 1T. For a real ADC the haronic and interodulation coponents are uch greater than the rest of the noise. In this case ENOB can be approxiated by the expression ENOB 10 6.0 log ex + σ ( H 1T ( IM ij + ( H T Moreover, fro (9 it follows that in ulti-tone ode the interodulation distortion is uch greater than the haronic distortion. Thus, (18 becoes Also, (19 can be written as ( H 1 ( IM ij q T + σ. ex. (16 (17 (18 10 T ENOB log. (19 6.0 T 10 THD1T ENOB log (0 6.0 IMDT in which THD 1T is the total haronic distortion in single-tone ode. The single-tone signal is x(t Asin(πft, where f is the frequency of the single-tone signal. By substituting this signal in (1 the following THD 1T is obtained 4 4 THD1 T b A + c A. (1 8 By replacing (8 and (1 in (0 and after soe algebra we have ENOB 10 6.0 log 4 + 10 6.0 log 8b ( 1 8b + ( 4 5 + c A c A. (

5 Testing of ADCs by frequency-doain analysis in ulti-tone ode If the second haronic distortion is the ost iportant, situation frequently encountered in practice, ( becoes 10 log ENOB 6.0 4( 1. ( In dual-tone ode ( fro ( it follows that the ENOB 0.5 bits, result which was obtained also in [4]. Fro ( it follows that the accuracy in the evaluation of ENOB increases as the nuber of tones increases. Another iportant conclusion deduced fro ( is that it is possible to use sinewave generators with distortion perforances inferior to those of the converter to estiate with high precision the ENOB. In this situation, nown the distortion perforances of the generators, based on (, we can establish for a given ADC resolution the nuber of the generators to be used. For exaple, according to (, by eans of four sinewave generators with distortion perforances that perit to test ADCs with resolution up to 9 bits it is possible to estiate with high accuracy the ENOB of the ADCs with resolution up to 10 bits. So, there is a 1-bit gain in resolution. In next section soe experiental results were carried out in order to validate the conclusions that were drawn above.. EXPERIMENTAL RESULTS The experiental results were obtained by eans of the test syste naed MULTI-TONE ADC TEST. This test syste has the following ey features: The acquisition syste is based on TMS0C5x DSK board [7]. Maxiu record length of 4096 saples. Maxiu sapling frequency of 0 MHz. Control of the signal generators via the IEEE-488 bus. The software is easy to use; it interacts with the user through ouse driven graphic interfaces. Designed for testing the ADCs when the sapling frequency is noncoherent with the sinewave input frequency [1] because this is the ost encountered situation in practical applications of ADCs. However, MULTI-TONE ADC TEST perits, also, the testing of ADCs when these frequencies are coherent. Four odes were eployed for testing: - single-tone ode; - dual-tone ode; - three-tone ode; - four-tone ode. For each ode two graphical pages are available, which provide a large aount of inforation concerning the sinewaves paraeters and the ADC dynaic perforances. Saving in ASCII forat data files of the paraeters that characterize the sinewaves and of the ADC dynaic paraeters. Possibility to process also data files obtained by siulation or by eans of other acquisition systes. The sinewave paraeters (aplitude, frequency and phase were deterined by eans of the interpolated fast Fourier transfor (IFFT [8]. The δ values corresponding to the sinewave frequencies were, also, estiated by IFFT [8]. The ADC dynaic paraeters estiated by MULTI-TONE ADC TEST were SINAD and ENOB. These paraeters were estiated by the algorith proposed in []. The bloc diagra of the MULTI -TONE ADC TEST is presented in Figure 1.

Daniel BELEGA 6 BUS IEEE - 488 Input Sinewave #1 Input Sinewave # Input Sinewave # Input Sinewave #4 ADC Evaluation Board Sapling frequency Acquisition Syste (TMS0C5X DSK RSC GPIB Interface PC (Signal Processing Fig. 1 - Bloc diagra of the MULTI TONE ADC TEST. The TMS0C5x DSK is a low-cost, siple, stand-alone application board equipped with a 16-bit fixed-point digital signal processor (DSP TMS0C50 DSP. DSK contains an analog interface circuit (AIC - TLC040, which provides the necessary conversion between the analogue and digital doain. For this purpose TLC040 incorporates a band-pass antialiasing input filter, a 14-bit ADC, a serial port by which the AIC counicates with the TMS0C50 DSP, a 14-bit digital-to-analog converter (DAC and a lowpass output reconstruction filter. The DSK is connected to a PC via a RS interface. The data acquisition progras were written in C and in assebly language of the TMS0C5x DSK [9]. The data processing and the interactive graphical pages were realized by eans of the MATLAB 4.. Two ADCs were tested: TLC080 [10] and the ADC of the TLC040. TLC080 is a high-speed 8-bit unipolar half-flash converter, realized in LinCMOS technology, with a iniu access and conversion tie of 1.18 µs in the ost rapid write-read ode. The sinewave generators eployed for testing the ADCs were the prograable function generator Haeg - HM810 [11]. The total haronic distortion of the HM810 in the frequency doain used is saller than 0.5% (i.e. saller than 46 dbc. A sinewave with this total haronic distortion perits to test ADCs with a axiu resolution of 7 bits. TLC080 was tested at three frequency sets: a f 1 5.1 Hz, f 7. Hz, f 9.9 Hz, f 4 1.7 Hz; b f 1 16.1 Hz, f 18.8 Hz, f 1.9 Hz, f 4 5. Hz; a f 1 5.1 Hz, f 8.6 Hz, f 4. Hz, f 4 45.7 Hz. Because the coherent sapling relationships are not et [1] to eliinate the spectral leaage errors the -ter iniu energy error was used [1]. Figures -5 show the results obtained after the testing with these frequencies sets. (a (b

7 Testing of ADCs by frequency-doain analysis in ulti-tone ode (c (d (e (f (g Fig. - The results obtained after the testing of TLC080 with the first frequency set in: (a-(b single-tone ode; (c-(d dual-tone ode; (e-(f three-tone ode; (g-(h four-tone ode. (h

Daniel BELEGA 8 (a (b (c (d (e (f (g Fig. - The results obtained after the testing of TLC080 with the second frequency set in: (a-(b single-tone ode; (c-(d dual-tone ode; (e-(f three-tone ode; (g-(h four-tone ode. (h

9 Testing of ADCs by frequency-doain analysis in ulti-tone ode (a (b (c (d (e (f (g Fig. 4 - The results obtained after the testing of TLC080 with the third frequency set in: (a-(b single-tone ode; (c-(d dual-tone ode; (e-(f three-tone ode; (g-(h four-tone ode. (h

Daniel BELEGA 10 If the first frequency set is eployed, the ENOB estiates obtained in the tested odes were practically the sae. This behavior is achieved because the distortion perforances of the sinewaves at the considered frequencies were uch higher than those of the converter. In this situation it is sufficient to use a single-tone signal for the estiation with high precision of the ENOB. Fro the results obtained after the testing with the second and the third frequency sets it is obvious that the accuracy in the estiation of ENOB increases when the nuber of tones increases. When the second frequency set is used the ENOB estiates obtained in the three and four tone odes are practically the sae. Moreover, these ENOB estiates are practically equal with those obtained after testing with the first frequency set. Fro these results it follows that it is sufficient to test the converter with three-tone signal to obtain high accuracy ENOB estiates. In the analyzed situation the distortion perforances of the sinewaves are poorer, but very close to those of the converter. In the case of testing with the third frequency set the ENOB estiates in the three and four tone odes are not close. Fro this reason we conclude that is necessary to use a test signal with ore than four tones to estiate with high accuracy the ENOB. If the accuracy of the ENOB estiates is the sae with that obtained after testing with the first frequency set it is sufficient to use a five-tone signal for estiation with high accuracy the ENOB. In Figure 5 the results obtained after testing of the TLC040 s ADC are presented. Because the coherent sapling relationships are not et, to eliinate the leaage errors the 4-ter iniu error energy window was eployed [1]. (a (b (c (d

11 Testing of ADCs by frequency-doain analysis in ulti-tone ode (e (f (g (h Fig. 5 - The results obtained after the testing of the TLC040 s ADC in: (a-(b single-tone ode; (c-(d dual-tone ode; (e-(f three-tone ode; (g-(h four-tone ode. Figure 5 shown that the precision in the estiation of the ENOB increases as the nuber of tones increases. However, the ENOB estiates are not accurate because the distortion perforances of the generators are uch lower than those of the converter. For estiating ENOB with high accuracy a test signal with a high nuber of tones is necessary. This leads to a prohibitive nuber of sinewave generators. This drawbac can be overcoe by using sinewave generators with better distortion perforances. 4. CONCLUSION Testing of ADCs by frequency-doain analysis in ulti-tone ode has been investigated. The theoretical expressions of the THD, IMD, SINAD and ENOB dynaic paraeters were derived. Also, the perforance concerning the evaluation of ENOB obtained by testing in ulti-tone ode was copared this the one obtained by testing in single-tone ode. For this purpose, the theoretical expression of the difference between the values of ENOB, ENOB, obtained by testing in these odes was derived. Fro the ENOB expression it follows that the accuracy in the evaluation of ENOB increases as the nuber of odes used in the test signal increases. Another iportant conclusion, based upon the ENOB expression, is that the ENOB can be estiated with high accuracy even when using sinewave generators with distortion perforances poorer than the ones of the converter under test. The experiental results confir these conclusions. Future wor will focus on the ipleentation of accurate -tone signal by nuerical ethods. This overcoes the necessity to use sinewave generators, and so, lowers significantly the cost of the test syste.

Daniel BELEGA 1 REFERENCES 1. IEEE Std. 141, Standard for Terinology and Test Methods for Analog-to-Digital Converters, 000.. Linnenbrin, T.,E., Tilden, S., J., Miller, M., T., ADC Testing with IEEE Std. 141, Instru. and Meas. Technology Conference, Budapest, pp. 1986-1991, May 1-, 001.. Benetazzo, L., Narduzz C, Offell C., and Petr D., A/D Converter Perforance Analysis by a Frequency-Doain Approach, IEEE Trans. Instru. Meas., vol. 41, pp. 84-89, Feb. 199. 4. Benaïs, M., Le Masson, S., Marchegay, P., A/D Converter Characterization by Spectral Analysis in Dual-Tone Mode, IEEE Trans. Instru. Meas., vol. 44, pp. 940-944, Oct. 1995. 5. Jenq, Y-C., Measuring Haronic Distortion and Noise Floor of an A/D Converter Using Spectral Averaging, IEEE Trans. Instru. Meas., vol. 7, pp. 55-58, Dec. 1988. 6. Giacoin J., D., Most ADC Syste Require Interodulation Testing, Electronic Design, pp. 57-65, Aug. 199. 7. Texas Instruents, TMS0C5X DSP Starter Kit, User s Guide, 1994. 8. Offell C, Petr D., Interpolation Techniques for Real-Tie Multifrequency Wavefors Analysis, IEEE Trans. Instru. Meas., vol. 9, pp. 106-111, Feb. 1990. 9. Belega, D., Applications with thetms0c5x Board (in Roanian, pp. 70-95, Ed. Politehnica of Tiişoara, 00. 10. Texas Instruents, Data Acquisition Circuits. Data Conversion and DSP Analog Interface, Data Boo, ch., pp..61-.70, 1998. 11. Haeg Instruents, Manual for Function Generator HM810, 1996. 1. Belega, D., Optial Choice of the Windows for ADC Characterization by Spectral Analysis in Single-Tone and Dual-Tone Modes, subitted for publication in Revue Rouaine des Sciences Techniques. Serie Electrotehnique et Energetique, 00. Received Septeber, 00