A 16bit D/A interface with Sinc approximated semidigital reconstruction filter
7.3. SD modulators and noise shaping
The specifications of the reconstruction filter are related to the properties of the noiseshaper. In this section the performance of the noise shaper with respect to the inband noise is discussed.
7.3.1. Noise modelThe quantizing error which is introduced into the signal is modeled by the addition of white noise E_{qn} as illustrated in fig.7.2. The one bit quantizer maps any nonnegative 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 c_{g}. As the quantization step q is equal to 2A the quantization noise power P_{q} of the one bit quantizer is given by:
(7.1)
Within the noise model the noise is not correlated with the signal and the noise PSD of the noise N_{dq} introduced by the quantizer is uniform distributed in the fundamental interval as shown in fig.7.3 and given by:
(7.2)
Fig.7.3: Noise density 
A one bit code can be generated by means of a sigmadelta modulator . In a sigmadelta 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: SD modulator 
(7.3)
Eq. (7.3) shows that the signal transfer of the sigmadelta modulator is:
(7.4)
which approximates 1 in the signal band where c_{g}G(z) >> 1. The noise PSD at the output is inversely proportional to 1 + c_{g}G(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:
(7.5)
The overall gain of the signal that results from quantization is equal to one. The value of the gain constant c_{g} 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 P_{Nt} at the output of the noise shaper is obtained from the integration of the power spectral density:
(7.6)
7.3.3. Noise transfer
There is no difference between a noise shaper and a sigmadelta modulator. However, in the realization, the place of the loop filter is the only distinction. For this D/A converter a sigmadelta 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 tradeoff 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:
(7.7)

Fig.7.5: Noise transfer of the SD 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 c_{g} has been numerically computed and its value is 0.95. Now, we have the transfer function of the noise as:
(7.8)
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].
