A study of excess loop delay in tunable continuous-time bandpass delta–sigma modulators using RC-resonators

Abstract

Excess loop delay is one of the most critical non-idealities of continuous-time delta–sigma modulators as it leads to degradation of the signal-to-noise-ratio or even instability. A comprehensive study of the impact of excess loop delay on tunable continuous-time bandpass delta–sigma modulators using RC-resonators is performed in this paper, both analytically and by simulations. Moreover, a detailed analysis of the conventional compensation techniques for single-band continuous-time bandpass modulators as well as their adaptability to tunable bandpass modulators is performed. The results indicate that only tuning of the scaling coefficients is suitable to compensate for excess loop delay in high-speed tunable bandpass modulators. Based on this result, an approach to the compensation of excess loop delay is proposed which maps the poles of the noise transfer function (NFT) to almost ideal and thus stable positions. Excess loop delay equal to one clock cycle may thus be compensated while the available tuning range of the center frequency depends on the order and the out-of-band-gain of the NFT. A prototype implemented on a printed circuit board proves the feasibility of the proposed approach.

Appendix

1.1 Equations for the $4{th}$-order CT BP ${\Delta }{\Sigma }$ modulator

The DT equivalent of the ELD afflicted CT $4{th}$-order loop filter including the additional compensation path scaled by $b_5$ yields

$$\begin{aligned} H_D[z]=\frac{h_{4d}z^{4}+h_{3d}z^{3}+h_{2d}z^{2}+h_{1d}z+h_{0d}}{z\left( z^{2}-2z\cos \omega _c+1\right) ^2} \end{aligned}$$

(33)

where

$$\begin{aligned} h_{0d}&= k_0-k_1\cos x_1-k_2\sin x_1 \end{aligned}$$

(34)

$$\begin{aligned} h_{1d}&= -4k_0\cos x_0 \nonumber \\ & \quad +k_1\left[ \cos x_1+2\cos \left( x_0-x_1\right) \right] \nonumber \\ & \quad +k_2\left[ \sin x_1-2\sin \left( x_0-x_1\right) \right] \nonumber \\ & \quad+k_3\cos \left( x_0+x_1\right) \nonumber \\ & \quad - k_4\sin \left( x_0+x_1\right) \end{aligned}$$

(35)

$$\begin{aligned} h_{2d}&= 2k_0\left[ 2+\cos \left( 2x_0\right) \right] \nonumber \\ & \quad-k_1\left[ \cos \left( 2x_0-x_1\right) +2\cos \left( x_0-x_1\right) \right] \nonumber \\ & \quad+k_2\left[ \sin \left( 2x_0-x_1\right) +2\sin \left( x_0-x_1\right) \right] \nonumber \\& \quad-k_3\left[ 2\cos x_1+\cos \left( x_0+x_1\right) \right] \nonumber \\ & \quad+k_4\left[ 2\sin x_1+\sin \left( x_0+x_1\right) \right] \end{aligned}$$

(36)

$$\begin{aligned} h_{3d}&= -4k_0\cos x_0 \nonumber \\& \quad +k_1\cos \left( 2x_0-x_1\right) \nonumber \\ & \quad -k_2\sin \left( 2x_0-x_1\right) \nonumber \\& \quad +k_3\left[ 2\cos x_1+\cos \left( x_0-x_1\right) \right] -k_4\left[ 2\sin x_1-\sin \left( x_0-x_1\right) \right] \end{aligned}$$

(37)

$$\begin{aligned} h_{4d}&= k_0-k_3\cos \left( x_0-x_1\right) -k_4\sin \left( x_0-x_1\right) \end{aligned}$$

(38)

with

$$\begin{aligned} k_0&= \frac{b_1+\omega _c^2\left( b_3+b_5\omega _c^2\right) }{\omega _c^4} \end{aligned}$$

(39)

$$\begin{aligned} k_1&= \frac{2b_1-\omega _c^2\left( b_2\tau _d-2b_3\right) }{2\omega _c^4} \end{aligned}$$

(40)

$$\begin{aligned} k_2&= \frac{b_1\tau _d+b_2+2b_4\omega _c^2}{2\omega _c^3} \end{aligned}$$

(41)

$$\begin{aligned} k_3&= \frac{2b_1+\omega _c^2\left[ \left( 1-\tau _d\right) b_2+2b_3\right] }{2\omega _c^4} \end{aligned}$$

(42)

$$\begin{aligned} k_4&= \frac{b_1\left( 1-\tau _d\right) -b_2-2b_4\omega _c^2}{2\omega _c^3} \end{aligned}$$

(43)

and

$$\begin{aligned} x_0&= \omega _c \end{aligned}$$

(44)

$$\begin{aligned} x_1&= \tau _d\omega _c. \end{aligned}$$

(45)

For $b_5$ and $\tau _d$ equal to zero, (34)–(45) simplify to the equations of the ideal fourth-order modulator.

Equivalent to (21), the coefficient $b_5$ results in

$$\begin{aligned} b_5&= \frac{2\left( b_1+b_3\omega _c^2\right) +\left[ 2b_1+\left( 2b_3+b_2\tau _d\right) \omega _c^2\right] \cos \left( \tau _d\omega _c\right) }{2\omega _c^4} \nonumber \\& \quad +\frac{\omega _c\left( b_2-b_1\tau _d+2b_4\omega _c^2\right) \sin \left( \tau _d\omega _c\right) }{2\omega _c^4} \end{aligned}$$

(46)

Replacing the CT scaling coefficients $b_i$ by their corresponding equations based on the original DT coefficients $a_i$ and setting $\tau _d$ equal to 1, (46) simplifies to $a_3+a_4$. As in case of the second-order modulator, $a_3+a_4$ and thus $b_5$ yields zero only for a center frequency equal to $f_s/4$.

Scaling coefficients for a fourth-order NTF with an OOBG equal to 1.6 are summarized in Table 3.

Table 2 CT $2{nd}$-order scaling coefficients

Table 3 CT $4{th}$-order scaling coefficients