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Second-Order Delta–Sigma Modulation with Interfered Reference
Yu-Chung Huang and Wei-Shinn Wey
Abstract—A delta–sigma(16) modulator has been traditionally
an-alyzed by assuming its reference to be constant, but practically the refer-ence may be interfered and thus vary with time. For an interfered referrefer-ence modulator, the performance of quantization noise is degraded by quanti-zation noise leakage due to interfered feedback. In this paper, a systematic study for observing the behavior of a second-order modulator with an inter-fered references is presented, based on a linear modeling, spectral analysis, and behavioral simulations. An analytical form of the output of a16 mod-ulator with an interfered feedback is obtained and compared with behav-ioral simulation. Due to the agreement between the theoretical calculation and the behavioral simulation results, it is concluded that the quantization noise leakage should be considered for describing the behavior of the16 modulators more precisely.
Index Terms—Analog-to-digital converter (ADC), delta–sigma
modula-tion, interfered reference, quantization noise, quantization noise leakage, sigma–delta modulation, varying reference.
I. INTRODUCTION
Delta–sigma(16) modulators have become increasingly important in mixed-mode signal processing ICs. Conventionally,16 modula-tors are analyzed by assuming their reference inputs to be constant. In practice, the reference input may be interfered by deterministic signals (e.g., pickup of radio frequency, power lines, etc.) and/or by random noise (e.g., flicker, thermal noise, etc.), thus varying with time. Taking
Manuscript received February 2000; revised January 2001. This paper was recommended by Associate Editor R. Geiger.
The authors are with the Institute of Electronics, National Chiao-Tung Uni-versity, Hsin Chu 300 Taiwan, R.O.C. (e-mail: ychuang@cc.nctu.edu.tw).
Publisher Item Identifier S 1057-7130(01)03052-X.
account of the reference input, the signal-transfer characteristic of a 16 modulator becomes a ratiometric function. Hence, interference occurring on the reference will modulate with the signal on the input due to the ratiometric operation. Based on this ratiometric concept, the transfer characteristic of the modulator has been analyzed while the ref-erence interfered by deterministic signals [1] orkT=C noise [2]. How-ever, because16 modulators use feedback to lock onto a band-lim-ited input, interfered feedback incurs quantization leakage to the band of interest. Hence, not only the signal-transfer characteristics but also the quantization noise spectrum will be affected by the interfered ref-erence. Therefore, exploring how the quantization noise spectrum is influenced is relevant to the study of16 modulators.
In this paper, some aspects of the behavior of a second-order16 modulator with an interfered reference are studied. The main reason why16 modulators are difficult to analyze rigorously is the existence of a 1-bit analog-to-digital converter (ADC) in a feedback loop intro-ducing strong nonlinearity. The most popular approach to analyze16 modulators is to assume the quantization noise to be a signal-indepen-dent white random signal [3]. This model replaces an inherent non-linear modulator by a stochastic non-linear system, thereby permitting the use of linear systems methods to analyze16 modulators. Although this model cannot perfectly describe loop stability [4] and pattern noise [5], it predicts the in-band noise surprisingly well [6]–[8]. However, this model really cannot describe the interfered reference case since it assumes that the reference is constant. To capture the behavior of an interfered reference modulator, a modified model based on the same uncorrelated white noise assumption is proposed, which is presented in Section II. Based on this model, the transfer function of the inter-fered reference modulator can be derived. In Section III, assuming a stochastic noise interference, the quantization noise spectral density and cumulative power density are obtained by taking Fourier transfor-mation for the autocorrelation function of the modulator’s output. In Section IV, theoretical spectral densities are compared with simulated ones. These results provide a good match between the calculated and simulated power spectral densities. In Section V, some comments are made that relate a deterministic signal interfering to the modulator per-formance. Finally, conclusions are presented in Section VI.
II. ANALYTICALMODEL
A conventional linear model of a 1-bit ADC in a16 modulator is shown in Fig. 1(a), which considers the quantization process as an additive white noise sourcee and the output step size as 1. An analog inputx is assumed to be in the no-overload range of 61. Note here that the reference signal is normalized to unity for convenience since it is constant.
However, for an interfered reference modulator, this conventional model fails to capture the behavior related to the reference. In order to catch this behavior, the reference signal is represented explicitly by a variablew instead of unity, and the two possible output states of y are defined as61 instead of 61=2, as shown in Fig. 1(b). Without nor-malization (to the reference), analog feedback signalv has two pos-sible levels of6w, and the 1-bit ADC input x is in the range between 62w. Consequently, it can be found that the additive white noise model should be led by a factor of 1=w. Thus, the digital output y can be given by
y = xw+ " (1)
featuring a ratiometric function plus a white noise. If the white quan-tization error has equal probability of lying anywhere in the range61 1057–7130/01$10.00 © 2001 IEEE
Fig. 1. (a) Traditional linear model and (b) modified model.
and is sampled at frequencyfS = 1= , then the autocorrelation
func-tion can be given by [9]
Ef"n"kg = 23 n0k (2) wheren0kis the Kronecker delta
n0k 1; for n = k0; for n 6= k: (3)
Based on this model, a sampled-data equivalent circuit of a second-order16 modulator with an interfered reference can be illustrated in Fig. 2. The outputykcan be given by
yk= rk+ nk= xwk01 k + ("k0 2b1; k"k01+ b2; k"k02) (4) where b1; k= wwk01 k and b2; k= w k02 wk (5)
which shows a ratiometric functionrkand a noise equationnk. Thus, we have
nk= "k0 2b1; k"k01+ b2; k"k02: (6)
As viewed from quantization noise"k, the output noisenkdepending onwkis time variant. In a conventional case with a constant reference, b1; kandb2; kboth are unity, and thus the output noisenkreduces to a well-known second difference equation as follows [3]:
nk= "k0 2"k01+ "k02: (7) Comparing (6) with (7), it can be found that the varying reference makes the zeros of the noise transfer function be no longer at dc, and thus the signal-to-noise ratio (SNR) of the modulator is degraded by the quantization noise leakage in the band of interest. The more rapidly the reference varies, the more crucial the SNR degradation becomes. More-over, an analytical solution to describe the quantization noise power spectral density can be found by its autocorrelation function, which will be presented in next section.
where the constant voltage is normalized to unity for simplicity. Note here thatkis an interfering signal and has to be restricted to ensure wk’s satisfying (8). Thus, we also have
jkj < 1: (10)
III. SPECTRALANALYSIS
Spectral analysis of16 modulators can be formulated in the frame-work of quasi-stationary process as considered as in [11] and [12]. The classification of quasi-stationary processes forms a general class of de-terministic and random processes for which the first- and second-order moments are well defined and to which traditional system autocorre-lation and spectral analysis can be applied [13]. This class includes stationary as well as asymptotically mean stationary random processes [14].
A discrete time process"kis said to be quasi-stationary if there is a constantC such that
Ef"kg C; for allk Ef"kg = lim
N!1
1
N Ef"kg exists
jR"(n; k)j C; for alln; k where R"(n; k) Ef"n"kg (11)
and if for eachk the limit Ef"n"kg = lim N!1 1 N N n=1 R"(n; k) R"(k) (12) exists. The power spectrum of" is defined as the discrete Fourier trans-form of an autocorrelation function
S"(n; ej!) = 1 k=01
R"(n; k)e0j!k (13) which may depend on timen. If " is a quasi-stationary process, then for eachk the limit
S"(ej!)S"(n; ej!) = lim N!1 1 N 1 n=1 1 k=01 R"(n; k)e0j!k (14) exists. We can now compute the noise spectrum of the second-order modulator with its reference interfered by a signal.
From (4) and assuming that"n,xn, andnare uncorrelated with each other, we get
Efynyn+kg = Efrnrn+kg + Efnnnn+kg: (15) Considering the noise transfer characteristic, we have
Efnnnn+kg =2 3 (1 + 4Efb1; n+kb1; ng + Efb2; n+kb2; ng) k 02 (Efb1; ng + Efb1; n+kb2; ng) k+1 02 (Efb1; ng + Efb1; nb2; n+kg) k01 +Efb2; ng(k+2+ k02) = Rn(n; n + k): (16)
Plugging (16) into (13), we obtain Sn(n; ej!)
=2 3
1 + 4Efb2
1; ng + Efb22; ng
04 (Efb1; ng + Efb1; nb2; n+1g) cos(!)
+2Efb2; ng cos(2!)
Fig. 2. Discrete-time equivalent circuit of a second-order16 modulator.
Fig. 3. Simulation diagrams for observing (a) quantization noise leakage due to interfered feedback as well as interfered signal transfer characteristics, (b) interfered signal transfer characteristics regardless of interfered feedback induced quantization noise leakage, and (c) interfered noise transfer characteristics with compensated signal input.
It can be assumed thatnis a zero-mean stochastic signal with root-mean-square (rms) power of2and an instantaneous value far less than the reference voltage such thatjnj 1. Thus, we have
Efng = Efn0ig = 0;
Efn2g = 2 forn = 1; 2; 3; . . . ; and i < n: (18)
By using this property, it can be further obtained that Ef2l01
n g = 0 forl = 1; 2; 3; . . . : (19)
Then we can take
Efb1; ng = Efb2; ng = 1 + 2
Efb2
1; ng = Efb22; ng = 1 + 42
Efb1; nb2; n+1g = 1 + 32 (20)
(in the Appendix). All of them depend only on the rms power, regard-less of which kind of stochastic processn. Hence, substituting (20) into (17), the noise spectrum of the second-order modulator with ref-erence interfered by a random noise is given by
Sn(ej!) =23 6 + 2020 8(1 + 22) cos(!)
+ 2(1 + 2) cos(2!) (21)
which depends on the rms power of the random processn. Assuming a signal in frequency band0 f < f0, the oversampling ratio is defined as
OSR = f2fS
0 = 12f0 = !0: (22)
For a sufficiently high oversampling ratio, we have Sn(ej!) 23 !4+ 2 6 + 4!2+23!4 ;
where!
1: (23)
The noise power in the signal band can be approximately given by n20= f 0 Sn(f)df 154 OSR1 5 + 2 OSR2 + 49 2 OSR3 + 245 4 OSR5 : (24) It appears to be the power of the second-order shaped noise plus quan-tization noise leakage. The leakage power is proportional to the power of the interfering noise and depends on the oversampling ratio.
IV. SIMULATIONRESULTS
In this section, simulation results of the second-order 16 modu-lator with the reference interfered by a zero-mean stochastic process are presented. The simulation results obtained are compared with the analytical results of the previous section.
The output of16 modulators can be represented by a combination of a signal function and a noise function as (4). It is useful to inves-tigate the influences of the interfered reference on signal function, on noise function, and on both of them. The simulation diagrams for in-vestigating these influences are illustrated by Fig. 3. Based on the di-agrams, the cumulative power density as well as the power spectral density of the three different cases can be obtained by simulation. Note that the cumulative power density, the integral of the power spectral, of the modulator’s output represents the in-band noise power.
The simulation diagram of a16 modulator with reference interfered by a signal is shown as Fig. 3(a). The signal and reference inputs are, respectively, assumed by
xk= sin(!0k)
Fig. 4. Output spectrums of the simulation results of Fig. 3(a)–(c) while the interfering process is uniformly distributed and has an rms power of 1/1000.
Fig. 5. Simulated and theoretical calculated power spectral density while the interfering process is uniformly distributed with rms power of 1/1000.
wherekis a zero-mean stochastic process and should be restricted to satisfy (8). In this case, the interfered reference has influence on both signal function and noise function.
In traditional analysis based on the ratiometric concept, the effect of interfered reference on a signal function has been considered [1], [2], but the quantization noise leakage incurred by the interfered feedback was ignored. This case can be modeled by assuming
xk= sin(!1 + 0k) k
wk= 1 (26)
as shown in Fig. 3(b).
As deduced above, not only the signal function but also the quantiza-tion noise equaquantiza-tion is affected by the interfered reference, resulting in quantization noise leakage to the band of interest, as expressed by (21).
In order to compare this theoretical prediction directly with a simula-tion result, the signal input should be multiplied by the reference input, as shown in Fig. 3(c). Then, from (4), the output functionyk can be given by
yk= xwk01
k + ("k0 2b1; k"k01+ b2; k"k02)
= sin (!0(k 0 1)) + ("k0 2b1; k"k01+ b2; k"k02): (27) The signal function is reduced to a sinusoidal signal whose Fourier transform will appears to be a single spectral line. Thus, how the quan-tization noise spectrum is affected by the interfered reference can be observed directly and clearly from the output spectrum.
Output spectra of the simulation results are shown in Fig. 4, where the zero-mean interfering process k has uniform distribution with power of 0.001 and the signal frequency is fs=64. It can be found
Fig. 6. Simulated and theoretical calculated power spectral density while the interfering signal is a sinusoidal wave.
that the cumulative power spectrum of ignoring the quantization noise leakage, case (b), is approximately 15 dB lower than that of the practical interfered case (a). Besides, the output spectrum of only considering noise leakage is plotted by curve (c), which is much closer to curve (a) than curve (b). Therefore, we can conclude that to describe the behavior of the modulator more precisely, the quantization noise leakage should be taken into account for a modulator with an interfered reference.
Furthermore, the output spectrum of the16 modulator with its ref-erence interfered by different stochastic processes having different rms power is obtained by behavioral simulation and compared with theo-retical results, as shown in Fig. 5. The interfering process is normal (Gaussian) distribution with rms power of 1/1000. This result provides a good match between the calculated and simulated power spectral den-sities. With this approach, the analytical description matches the simu-lation results with a difference of just a few decibels for all signal levels, oversampling ratios, and interfering stochastic processes.
V. DETERMINISTICSIGNALINTERFERING
In practice, we are also interested to know the performance impact of the modulator with reference interfered by a deterministic signal. A sinusoidal interfering signal is taken here as an example to observe this phenomenon. The signal and reference inputs are, respectively, as-sumed by
xk= A sin(!0k)
wk= 1 + sin(!dk) (28)
where sin(!dk) is the interfering signal. From (4), the impact on the
linearity of this modulator can be found by rk= xwk01
k = A sin(! 0k)
1 + sin(!dk): (29)
It can be extended that
rk= A sin(!0k) 0 A sin(!0k) sin(!dk)
+ A 2sin(!
0k) sin2(!dk) 0 1 1 1
= A 1 + 22 + 1 1 1 sin(!0k)
+ A 2 [cos ((!d+ !0)k) 0 cos ((!d0 !0)k)]
0 A 42[sin ((2!d+!0)k)0sin ((2!d0!0)k)] + 1 1 1 (30) where the first term is the signal tone including aliasing components and the second and third terms are the signal tones modulated by the interfering signal and by its harmonics. Assuming thatA = = 0:1 and!d= 4!0 = 2=64, the simulated results are shown in Fig. 6. It
can be found that the simulated amplitude of the signal and harmonics tones agrees with the calculated results as (30).
Furthermore, the noise spectrum of this interfering can be derived by plugging (28) into (17) and (14). As with the stochastic case, this deterministic interfering also incurs quantization leakage to the band of interest. However, in contrast to that case, the values ofEfbi; ng,
Efb2
i; ng, and Efb1; nb2; ng depend not only on signal power but also
on its waveform and are obtained by numerical computation here since they are difficult to obtain by hand calculation. Finally, ifEfbi; ng,
Efb2
i; ng, and Efb1; nb2; ng exist, there is a good match between the
analytical and simulated power spectral densities even if the interfering signal is deterministic, as shown in Fig. 6.
VI. CONCLUSION
A systematic study of the16 modulator with an interfered reference has been carried out. It reveals that the interfered feedback incurs quan-tization noise leakage to the band of interest and degrades the quanti-zation noise performance. An analytical model has been proposed to obtain the output function of the modulator with an interfered feed-back. The output function shows that the interfered reference makes the zeros of the noise equation be no longer at dc, and thus the SNR of the16 modulator is degraded. Based on the quasi-stationary ap-proximation, the quantization noise spectrum and the in-band quanti-zation noise power have been derived by taking the Fourier transform of the autocorrelation function of the interfered output function. For a stochastic interfering, the in-band power of quantization noise leakage is proportional to the power of the interfering noise and also depends
pact of the modulator with an interfered reference can be quickly eval-uated, and the simulation items about the reference interfering could be decreased or even omitted.
APPENDIX
From (5) and (8), the expected value ofb1; nandb2; ncan be written as
Efbi; ng = E 1 + 1 + n0i
n fori = 1; 2: (A.1)
Sincejnj < 1, the Taylor expansion of this equation exists; then we have = (1 + Efn0ig) 1 + 1 l=1 (01)lEfnlg = 1 + 1 l=1 (01)lEfl ng + Efn0ig 1 + 1 l=1 (01)lEfl ng :
Using (18) and (19), we can be take
= 1 0 1 l=1 Ef2l01 n g + 1 l=1 Ef2l ng = 1 + 1 l=1 Ef2l ng = 1 + 1 l=1 (2l 0 1)!!2l 1 + 2 (A.2)
Similarly, the expected value ofb2i; ncan be written by
Efb2i; ng = E 1 + 2n0i+ 2 n0i
1 + 2n+ n2 : (A.3)
To ensure the Taylor expansion of this equation existing, we assume that
jnj <p2 0 1; for alln: (A.4) Hence, we have
1 + Ef2n0i+ n0i2 g 1 + 1 l=1 (01)lE [2 n+ 2n]l = 1 + Ef2 ng + 1 l=0 (01)l l m=0 l m 2mEfn2l0mg 1 + 42 : (A.5)
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