S/N and power in fixed point digital filters
The power of the signal at the output P_{y} for every possible architecture can be computed by knowing the transfer function H(e^{jq}) and the input signal power P_{x}:
(A1.1)
S/N and power for FIR DSP
Given the need for rounding after multiplication, the equivalent structure with quantization noise sources is presented in fig.A1.1. Assume that the noise sources e[n] are mutually uncorelated and uncorelated with the signal. Another assumption would be that the input signal is random. The noise power at the output, for the m coefficient structure from fig.A1 is P_{noise}=mq^{2}/12. Now the signal to noise ratio S/N can be found:
Fig.A1.1: FIR DSP with noise sources 
(A1.2)
A FIR filter has mfold pole in origin. Denote the integral term from (A1.2) as W_{FIR} . The integral term can be computed from the theorem of Cauchy by taking into account the residues in z=0:
(A1.3)
A typical value for the overflow factor k is 0.25. From (A1.2) and (A1.3) one can be able to find a relation between the number of bits B and the S/N.
(A1.4)
Considering only the computation power P_{COMP }and neglecting the overhead from memory and I/O, the power needed for FIR DSP unit can be found by replacing the number of bits B from (A1.4) in the computational power:
(A1.5)
S/N and power for IIR DSP
For an IIR DSP unit, the noise power at the output is found by adding the noise sources after every multiplier. The IIR2 is more efficient from power point of view. That is why consider only the case of fig.2.7. If D(e^{jq }) represents the denominator in the filter transfer function, the noise power at the output will be:
(A1.6)
In the case of IIR filters the difficulty comes from the integral term which cannot be evaluated easily without knowing the structure of the filter. From (A1.2) and (A1.6) we get:
(A1.7)
By following the same pattern as in the case of the FIR filters we are able to calculate the computational power as a function of S/N:
(A1.8)
Given the structure of the multiplier (k_{mult}), the structure of the filter (W_{IIR}) and the desired S/N one can be able to find the computational power. Sometimes it is more important to make relative comparisons between FIR and IIR DSP instead of computing absolute values. The comparison between the IIR and FIR structures can be done under equal S/N condition. Though, the structure of the filters is different, they have the same transfer function. From (A1.2) and (A1.7) we find:
(A1.9)
In (A1.9) é xù rounds the result to the closest largest integer. For accurate comparisons, a correction factor should be added in order to take into account the difference between the approximation of the transfer ç H(e^{jq })ç in FIR and IIR situations.
