A 16bit D/A interface with Sinc approximated semidigital reconstruction filter
7.6. Noise properties of the D/A interface
The semidigital FIR filter and the low pass analog filter will increase the amount of noise at the output. Another source of noise is the clock jitter. We are considering here only the noise of the current sources and the clock jitter. The opamp noise can be easily quantified. As long as the effect of the noise generated in the analog part of the D/A converter is larger than the quantization noise, the resolution of the converter will be impaired. One should be able to quantify those effects and to design a low noise analog interface.
Fig.7.16: Noise generated at the output 
In fig.7.16 the realization of the coefficient a _{i} has been considered. Due to the differential approach, the current sources I_{i} are both present when the coefficient a _{i} is active and gives a contribution to the output. The white and 1/f noise from the cascode transistors can be neglected. The noise sources which give a significant contribution at the output are shown explicitly.
7.6.1 White noise considerations
The effect of this noise at the output can be considered when the input PDM signal represents a pure sinusoidal waveform represented with ones and zeros with a density p and 1p respectively. During the period Ts, the coefficient a _{i} is active and the current Ii+noise flows in the parallel connection R and C. Because the signal is periodic, the switch will be on and off periodically. The noise transfer to the output, differs for pMOS and nMOS current sources as:
(7.22)
(7.23)
In the noise transfer appears the transconductance of the transistor configured as a current source and the pole determined by the RC combination in the low pass filter. The power of the output white noise due to the current Ii during the interval Ts can be determined from the noise bandwidth of the circuit and the power spectral density of the voltage noise V^{2}n_{ip} and V^{2}n_{in} respectively:
(7.24)
Denote f_{sig}=1/T_{sig} the audioband from 0 to 20KHz. From fig.7.16 we can find the relationship between the oversampling ratio OR=fs/2f_{sig} in the converter and the interval t:
(7.25)
During this interval, the current Ii can be dumped to the ground or can be used to make larger swing in the differential approach. The noise and the current Ii are available only a fraction (1t) from the total Tsig. The noise has to be considered in the audioband. Therefore, the power of the noise available at the output in the interval Tsig will be:
(7.26)
The pulse shown in fig.7.16 is not a singular pulse and will appear in the interval Tsig according to the density of ones in the representation of the sinusoidal signal in this interval. Given the fact that the density of ones and zeros is p and 1p respectively and the number of total samples is Q, the total noise generated by the pMOST and nMOST current sources due to the coefficient a_{i} in the Tsig becomes:
(7.27)
The total noise can be found by adding all the individual noise contributions of the coefficients a_{i }in the interval Tsig:
(7.28)
The current Ii is a fraction of the total current needed to bias the FIR filter Ii=a_{i}I_{TOT}. Now, the transconductances g_{mi,p} and g_{mi,n} can be replaced by:
(7.29)
where V_{GTin,p} are the effective gatesource voltages for the pMOS and nMOS branches. The voltages V_{GTi} are all equal in the case of pMOS and nMOS current sources respectively. In the FIR semidigital filter, the sum of the coefficients Sa_{i} is one. Hence, we can find a relationship between the total current I_{TOT} ,the density of ones and zeros in the representation of the signal, the oversampling ratio, the effective gatesource voltages and the total power spectral density of the noise voltage at the output.
(7.30)
As a conclusion, in order to minimize the white noise present at the output of the converter one has to increase the oversampling ratio and to decrease the total current I_{TOT} in the semidigital filter. Decreasing the value of the resistor R will give a negligible influence of the resistor noise at the output. On the other hand, the output stage of the opamp is rail to rail to ensure large swings of the signal. If the common mode voltage at the input of the opamp is V_{DD}/2, then V_{DD}=2I_{TOT}R in order to use the full swing at the output. In the particular case p=1/2 the density of ones and zeros in the signal representation are equal. Under this assumption and by denoting:
(7.31)
then the power spectral density at the output becomes:
(7.32)
7.6.2 1/f noise considerations
In the following paragraph we are considering the effects of the 1/f noise on the power spectral density of the output noise voltage. The noise transfer to the output has the same behaviour as the white noise transfer from (7.31) and (7.32). Given the power spectral density of the 1/f noise present at the gates of the pMOS and nMOS current sources of the coefficient a_{i}:
(7.33)
one can find the output noise power up to the corner frequency of the 1/f noise f_{c}. It can be approximated as follows:
(7.34)
This noise is present at the output only during the period Ts and should be weighted with the density of ones and/or zeros. By following the same pattern as in the case of the white noise, we can find the total noise power at the output as a function of oversampling ratio, signal statistics and the FIR filter structure:
(7.35)
The transconductances of the PMOS and NMOS transistors depend on the coefficient a_{i} and the total current needed to bias the semidigital filter as:
(7.36)
All the transistors of the current sources have the same length L_{p} and L_{n} respectively and different widths according to the filter coefficients. By using the property of the filter coefficients Sa_{i}=1 it is possible to rewrite (7.35) in the following form:
(7.37)
The total noise coming only from the semidigital FIR filter can be found by adding the contributions of the white noise and the 1/f noise. Besides, we have to take into account the noise of the opamp and the noise generated by the feedback resistor R.
7.6.3 Noise generated by the clock jitter
Another contribution upon the total noise generated by the D/A interface would be the noise generated by the clock jitter. As long as the FIR semidigital filter is being implemented with a lot of coefficients, the area would be large and the clock distribution would be a problem. Going to high oversampling ratios OR, to be able to increase the accuracy of the converter we encounter the clock jitter and its consequences. The clock jitter can be reduced by using a crystal referenced clock. In order to consider those effects we have to take into account two possible realizations of the PDM codes, non return to zero (NRZ) and return to zero (RTZ) coding.
a. NRZ coding
Let’s assume an uncertainty Dt in the sampling moments of the clock. In the case of NRZ coding, the actual values of the output depend on the previous values generating intersymbol interference. In fig. 7.17.a, the currents flowing in the resistor R are shown. The uncertainty Dt in the sampling moments kT will generate an uncertainty DV in the lowpass filtered voltage at the output of the D/A. The output voltage V_{O} in the period [kT,(k+1)T] according to fig.7.17. b) and the time constant of the lowpass filter is:
(7.38)
From (7.38) it is obvious that NRZ coding generates intersymbol interference because of the correlation between samples. If the period of the signal PDM represented is T_{sig }, the period of the top samples T, and the number of samples Q, then the average of the output voltage V_{O}(t) can be approximated as:
(7.39)
Fig.7.17: NRZ coding 
The assumption would be that the lowpass filter has a larger time constant RC compared to the period of the top samples T. In order to find the effect of jitter on the output noise, consider (7.38) for t=(k+1)T under the assumption RC>>T:
(7.40)
The consequence of jitter will be the spread of the mean value of the voltage variation considered in (7.40). The mean value of this variation in the interval T_{sig} is:
(7.41)
and its spread as a function of the clock jitter :
(7.42)
This value represents actually the rms noise in a bandwidth 1/T_{sig}.
b. RTZ coding
Fig.7.18 illustrates the return to zero current top pulses and the output voltage V_{O}(t) exaggeratedly small for understanding purposes. In the case of return to zero, the top current pulses will go periodically to zero and the intersymbol interference will be reduced. This has consequences on signal dependent distortion too. However, we have to face another problem inherent to return to zero coding. Because current steps are larger now in comparison to NRZ coding, the errors made in the area of the top pulses by clock jitter are also larger, generating larger noise at the output.
Figure 7.18: RTZ coding and jitter 
The period of the output signal V_{O}(t) is denoted T_{sig}. The effect of the jitter on the output voltage has to be considered in a bandwidth given by 1/T_{sig} following the same pattern as in the case of noise calculation at the output. The average of the current top pulse I_{R}[kT] in the period T is:
(7.43)
The mean of the output voltage V_{O} in the interval [kT,(k+1)T] depends on the current I_{R}[kT]:
(7.44)
Clock jitter generates the spread of the T_{on} denoted sT_{on} which gives the spread of the output voltage V_{O} denoted sV_{O}. The relationship between s^{2}T_{on} and s^{2}V_{O} is given by:
(7.45)
This represents the power of the noise related to the bandwidth 1/T which has to be related now to the bandwidth 1/T_{sig}. The power of the noise in this bandwidth will be:
(7.46)
The interval in which the pulse is ON denoted T_{on} is related to T as T_{on}=l T. By using the oversampling ratio OR=T_{sig}/2T, the duty cycle l and (7.46), the power of the noise in bandwidth 1/T_{sig} is found to be:
(7.47)
The signal to noise ratio in the bandwidth 1/T_{sig} when only the clock jitter is taken in consideration can be determined from:
(7.48)
The area of the top pulse will be larger when T_{on} increases and the noise generated by jitter will have less influence on the total S/N. As a conclusion, the larger the ON time of the pulse, the better the S/N and larger clock jitter can be tolerated. When return to zero coding is being used the intersymbol interference will become less dominant as in the case of non return to zero coding. Breaking the correlation between the current top pulses, the signal dependent distortion will be improved.
As a final example, consider 1/T=2.8MHz, OR=64 and l =3/4. A clock jitter of sT_{ON}=100ps gives a S/N ratio of about 91dB corresponding to 15 bits resolution. In order to have a higher resolution than 15 bits a clock jitter of about 50ps or less will be required. This gives a design condition for the clock generation part of the interface.
Fig.7.19: Circuit diagram 
