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The effect of quantization noise and distortion in a first-order converter circuit
is modeled as shown in Fig. B.3. The frequency response of
the integrator is
.
The integrator distortion and the quantization error are
modeled as a signal e(t), which is added to the output.
Figure B.3:
First-order sigma-delta converter (noise model)
|
The unfiltered output Y(s) is then
![\begin{displaymath}
Y(s) = E(s)\frac{1}{1+H(s)} + X(s)\frac{H(s)}{1+H(s)} =
E(s)\frac{s\tau_0}{1+s\tau_0} + X(s)\frac{1}{1+s\tau_0}
.
\end{displaymath}](img814.gif) |
(B.2) |
Thus, the converter forms a high-pass filter for the quantization
error and distortions E(s) and transmits the input signal X(s)
up to a frequency
.
The power spectral density of the error
![\begin{displaymath}
E(s) = \frac{\left<\left\vert e(t)\right\vert^2\right>}{\ensuremath{f_{\mathit{os}}}\xspace }
\end{displaymath}](img816.gif) |
(B.3) |
is constant because of the decorrelation through the randomness of the
digital output.
Especially, all distortions from the non-linearities of the integrator
and the quantizer are spread into noise, which is shifted towards
higher frequencies.
The in-band noise can be calculated as
![\begin{displaymath}
N = \frac{\left<\left\vert e(t)\right\vert^2\right>}{\ensur...
...suremath{f_{\mathit{s}}}\xspace /2}{\frac{1}{H(2\pi j f)}df}
.
\end{displaymath}](img817.gif) |
(B.4) |
The resulting SNR is then
![\begin{displaymath}
S/N = \frac{\left<\left\vert x(t)\right\vert^2\right>}{\lef...
...ce } \frac{8}{2\pi\tau_0\ensuremath{f_{\mathit{s}}}\xspace }
.
\end{displaymath}](img818.gif) |
(B.5) |
The lower limit to
is set by the oversampling frequency
.
Next: B.2.1.1 Finite Integrator Gain
Up: B.2 Noise and Distortion
Previous: B.2 Noise and Distortion
G. Schrom