To understand the algorithm first we need to specify what we have at the output of Q1 and Q2 represented by Y1 and Y2 respectively, the digital outputs can be represented as: Y_1= XSTF+E_1NTF ( 5 )

Y_2=XSTF-E_1 STF+E_2 ( 6 )
Fig. 7 is a representation of the relationship between Y1 and Y2. Assuming that only quantization is present in the system and that the gain of DAC1 is equal to 1, it can be seem that the system transfer function can be obtained by deconvolution in the time domain or division in the frequency domain as follow: ├ Y_2/Y_1 ┤|_(X→0)=〖-LF〗_1+E_2/(E_1 NFT)=(-STF+E_2/E_1 )/NTF ( 7 )

Fig. 7. Quantizer output relationship.
As shown is equation ( 7 ) the loop gain can be measured, with some small error introduced by the quantization …show more content…
8, in which all the data needed is obtained from the digital outputs, no approximations or analog to digital transfer functions matching is required, is clear that the algorithm is very similar to the implementation shown in Fig. 3 in which two digital filter are require, where H1 has gain of 1 or is simply a delay, the subtraction of the digital output will give us the quantization error from Q1 as shown in ( 8 ) which is exactly the same information that is going to be provided by Y2 in Fig. 3. Finally, H2 is represented by NTFAD, which is equal to the inverse of ( 7 ) and it represents the adaptive cancellation transfer function obtain from the digital output data. The output of the algorithm will yield to, ├ Y_ADMASH ┤|_(X→0)= E_1 NTF-(E_2-E_1 )[NTF/(-STF+E_2/E_1 )] = (-E_1 STFNTF+E_2 NTF-E_2 NTF+E_1 NTF)/(-STF+E_2/E_1 ) = (E_1 NTF(1-STF))/(-STF+E_2/E_1 ) ∴├ Y_ADMASH ┤|_(X→0)=(E_1 〖NFT〗^2)/(-STF+E_2/E_1 )
( 9 ) Fig. 8. Block diagram algorithm.

Since NTFAD is the inverse of ( 7 ) the operation that needs to be implemented in the frequency domain is the division of Q1 digital output over Q2 digital output as …show more content…
The reason of this relays on the information provided by NTFAD, since this equation is obtained by deconvolution or division in the frequency domain the error included by E2 will be measured and will be cancelled (similar effect to Wiener filter in Image processing). Consequently, if the final SNR is not limited by the resolution of Q2, the number of bits can be as low as the number of bits in Q1 plus 1. However, the quantization noise E2 will appear in the denominator of the algorithm output as a constant (again, similar to a Wiener filter), but it will not affect the final