A 16-bit D/A interface with Sinc approximated semidigital reconstruction filter

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7.3. S-D modulators and noise shaping

The specifications of the reconstruction filter are related to the properties of the noise-shaper. In this section the performance of the noise shaper with respect to the in-band noise is discussed.

7.3.1. Noise model

The quantizing error which is introduced into the signal is modeled by the addition of white noise Eqn as illustrated in fig.7.2. The one bit quantizer maps any non-negative input value onto A and any negative input onto -A. So the amplitude of the output signal is fixed and not dependent on the input signal level. In the noise model for the one bit quantizer the signal dependent gain of the quantizer is

Fig.7.2: Noise model for the one bit quantizer

represented by the gain constant cg. As the quantization step q is equal to 2A the quantization noise power Pq of the one bit quantizer is given by:


Within the noise model the noise is not correlated with the signal and the noise PSD of the noise Ndq introduced by the quantizer is uniform distributed in the fundamental interval as shown in fig.7.3 and given by:


Fig.7.3: Noise density

7.3.2. Sigma-delta modulator

A one bit code can be generated by means of a sigma-delta modulator . In a sigma-delta modulator the loop filter G is placed in the path of the input signal (see fig.7.4). If G(z) is the transfer function of the loop filter G we have:


Fig.7.4: S-D modulator


Eq. (7.3) shows that the signal transfer of the sigma-delta modulator is:


which approximates 1 in the signal band where |cgG(z)| >> 1. The noise PSD at the output is inversely proportional to |1 + cgG(z)|2 and the noise contribution of the modulator vanishes at those frequencies for which |G(z)|à ¥ . In the implementation of the modulator, an integrating loop filter is applied which results in minimal noise density at DC. The output noise density is shaped by means of feedback and the noise transfer is:


The overall gain of the signal that results from quantization is equal to one. The value of the gain constant cg can be obtained from the calculation of the power at the output of the quantizer, which is based on the integration of the power spectrum of the output noise. The total noise power PNt at the output of the noise shaper is obtained from the integration of the power spectral density:


7.3.3. Noise transfer

There is no difference between a noise shaper and a sigma-delta modulator. However, in the realization, the place of the loop filter is the only distinction. For this D/A converter a sigma-delta modulator has been used since the loop filter function G(q ) of a properly working device was available. The design of the modulator will not be discussed here since it is a separate topic. For the following sections it is important to know only the transfer function of the loop filter G(q ). The order of the modulator is a trade-off between accuracy and stability. Large order modulators give more attenuation for the noise in the baseband but stability becomes worse. A third order modulator will be the choice for this design. The loop filter has a transfer (see reference [12]) given by:





Fig.7.5: Noise transfer of the S-D modulator

For the constants of eq. (7.7), the following values have been used: k = 1.5, r = 0.763 and t = 0.0303. The gain constant cg has been numerically computed and its value is 0.95. Now, we have the transfer function of the noise as:


The noise transfer plotted in the fundamental interval is illustrated in fig.7.5 and it will be used in sizing the coefficients of the FIR filter. More about noise shapers can be found in reference [13].

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