Oversampled Analog-to-Digital and Digital-to-Analog Converters:Alternative Sigma-Delta Architectures

Alternative Sigma-Delta Architectures

Eq. (59.20) appears to indicate that the order of the modulator, L, can be any value, and that increasing L would be beneficial. However, one further problem with two-level sigma-delta modulators is that two- level noise-differencing modulators of order greater than 2 can exhibit unstable behavior [10]. For this reason, only first- and second-order modulators were discussed in Section 59.2. Nonetheless, there have been acceptably stable practical alternative architectures that achieve quantization noise shaping that is superior to a second-order modulator. Two such architectures, high-order and cascaded modulators, are discussed in this section.

Another assumption in the previous section was that the noise-shaped region in a sigma-delta modulator is centered around DC. This is not necessarily the case; sigma-delta modulators with noise-shaped regions at frequencies other than near-DC are called bandpass modulators and are discussed at the end of this section.

High-Order Modulators

A high-order modulator is a modulator such as that depicted in Figure 59.13 in which there are more than two zeros in the noise transfer function. As stated earlier, if two-level quantization is employed, a simple noise-differencing series of integrators cannot be used, as such architectures produce unstable oscillations with large inputs that do not recover when the input is removed. To overcome this problem, high-order modulators use forward and feedback transfer functions that are more complex than the noise-differencing functions in Eq. (59.14) and Eq. (59.15) [24–26].

The general rule of thumb in the design of high-order modulators is that the modulator can be made stable if

and the integrator outputs are clipped and scaled to prevent self-sustaining instability [26,27]. The maximum error gain A is about 1.5, but the value used represents a trade-off between noise attenuation and modulator stability. These rules cover a broad class of filter types and modulator architectures, and the type of filter used generally follows the traditions of the previous designers in an organization. As an example, consider a fourth-order modulator with a highpass Butterworth error transfer function having a maximum gain, A, of 1.5, and a cutoff frequency set such that Eq. (59.24) is satisfied. The error spectrum of the Butterworth filter is shown in Figure 59.22, along with the error transfer function of an ideal fourth-order difference. While the Butterworth filter holds the maximum gain to 1.5 (3.5 dB), and while both filters have a fourth-order noise-shaping slope in the baseband (27 dB/octave), the error power

in the baseband is 44 dB higher with the Butterworth filter than with the ideal noise-differencing filter. This error penalty is typical of high-order designs; there is usually a direct trade-off between stability and noise reduction.

Consider the more general case of an Lth-order highpass Butterworth error transfer function. The error transfer function of such a filter around the unit circle is

where the values for bN are tabulated in Table 59.1. The loss in dynamic range relative to an ideal noise- differencing modulator, given by A2/(2b)2L, is also tabulated. In spite of this loss, high-order modulators can still achieve better noise performance than second-order modulators. However, because of the compromise in dynamic range required to stabilize high-order modulators, third-order modulators are generally not worth the effort. More common are fourth- and fifth-order modulators.

The noise penalty required to stabilize high-order modulators can be mitigated to some extent by alternate zero placement [25]. Classic noise-differencing modulators place all of the zeros of the error transfer function at DC (z = 1). This causes most of the noise power to be concentrated at the highest baseband frequencies. If, instead, the zeros are distributed throughout the baseband, the total noise in the baseband can be reduced, as illustrated in Figure 59.23. The amount by which zero placement can improve the noise transfer function is summarized in Table 59.1. Also tabulated is the net loss in dynamic range of a high-order Butterworth modulator that uses zero placement relative to an ideal noise-differencing modulator that has zeros at DC.

Cascaded Modulators

Cascaded, or multistage, architectures are an alternative means of achieving higher-order noise shaping without the stability problems of the high-order modulators described in the previous section [29,30]. In a cascaded modulator, two or more stable first- or second-order modulators are connected in series, with the input of each stage being the error from the previous stage, as illustrated in Figure 59.24.

Referring to this illustration, the first stage of the cascade has two outputs, y1 and e1. The output y1 is an estimate of the input x. The error in this estimate is e1. The second stage has as its input the error from the first stage, e1, and its outputs are y2 and e2. The second-stage output y2 is an estimate of the first-stage error e1. By subtracting this estimate of the first-stage error from the output of the first stage, y1, only the second-stage error remains. Thus, the error cancellation network uses the output of one stage to cancel the error in the previous stage.

For example, in a cascaded architecture comprising a second-order noise-differencing modulator followed by a first-order noise-differencing modulator, the transforms of the output of the two stages, as given by Eq. (59.16), are

The final output of this cascaded modulator is third-order noise shaped. As a general rule, the noise shaping of a cascaded architecture is comparable to a single-stage modulator whose order is the sum of all the orders in the cascade.

The extent to which the errors in a cascaded modulator can be cancelled depends on the matching between the stages. The earliest multistage modulators were cascades of three first-order stages, often called the MASH architecture [29]. The disadvantage of this structure is that to achieve third-order per- formance, the error in the first stage, which is only first-order-shaped, must be cancelled. Cancelling this relatively large error places a stringent requirement on inter-stage matching. An alternative architecture that has much more relaxed matching requirements is the cascade of a second-order modulator followed by a first-order modulator. This architecture, like the MASH, ideally achieves third-order noise shaping. Its advantage is that the matching can be 100 times worse than a MASH and still achieve better noise- shaping performance [31].

An additional benefit of cascaded modulators is improved tone performance. It has been shown both analytically and experimentally that the error spectra of the second and subsequent stages in a cascade are not plagued by the spectral tones that can exist in single-stage modulators [19,32]. To the extent that the first-stage error is cancelled, any tone in the first-stage error spectrum is attenuated, and the final output of the cascaded modulator is nearly tone-free.

Bandpass Modulators

The aforementioned sigma-delta architectures, called herein baseband modulators, all have zeros at or near DC, that is, at frequencies much less than the modulator sampling rate. It is also possible to group these zeros at some other point in the sampling spectrum; such architectures are called bandpass mod- ulators. Bandpass architectures are useful in systems that need to quantize a narrow band signal that is centered at some frequency other than DC. A common example of such a signal is the intermediate frequency (IF) signal in a communications receiver.

The simplest method for designing a bandpass modulator is by applying a transformation to an existing baseband modulator architecture. The most common transformation is to replace occurrences of z with -z2 [2]. Such an architecture has zeros at fS /4 and is stable if the baseband modulator is stable [33]. A comparison of the error transfer function of a baseband and bandpass modulator is shown in Figure 59.25. Note that a bandpass modulator generated through this transformation has twice the order of its equivalent baseband counterpart. For example, a fourth-order bandpass modulator is comparable to a second-order baseband modulator.

The noise-shaping properties of a bandpass modulator generated through the –z2 transformation are equivalent to the baseband modulator that was transformed. Thus, the approximation in Eq. (59.20) can be used where L is the order of the baseband modulator that was transformed and M is the effective oversampling ratio, which in a bandpass modulator is the sampling rate divided by the signal bandwidth. There are advantages and disadvantages to bandpass modulators when compared with traditional down-conversion and baseband modulation. One advantage of the bandpass modulator is its insensitivity to 1/f noise. Since the signal of interest is far from DC, at high frequencies where the factor 1/f is small, 1/f noise if often insignificant. Another advantage of bandpass modulation applies specifically to band- pass modulators having zeros at fS /4 that are used in quadrature I and Q demodulation systems. If the narrowband IF signal is to be demodulated by a cosine and sine waveform, as shown in Figure 59.26, the demodulation operation becomes a simple multiplication by 1, -1, or 0 when the demodulation frequency is fS /4 [34]. Furthermore, because a single modulator is used, the bandpass modulator is free of the I/Q path mismatch problems that can exist in baseband demodulation approaches.

Two disadvantages of bandpass modulators involve the sampling operation. Sampling in a bandpass modulator has linearity requirements that are comparable to a Nyquist rate converter sampling at the same IF frequency; this is much more severe than the linearity requirements of the sampling operation in a baseband converter with the same signal bandwidth. Also, because of the higher signal frequencies, the sampling in bandpass modulators is much more sensitive to clock jitter. To date, the state of the art in bandpass modulators has about 20 dB less in dynamic range than comparable baseband modulators [2]. While the remainder of this chapter focuses once again on baseband modulators, many of the techniques are applicable to bandpass modulators as well.