MT-022 Datasheet, PDF (8/12 Pages) Analog Devices – ADC Architectures III: Sigma-Delta ADC Basics

MT-022
X –Y
X
∑
+
_Y
1 _(X–Y)
f
ANALOG FILTER
H(f) = ¹
f
Q=
QUANTIZATION
NOISE
Y
∑
Y=
1 _(X–Y)
f
+Q
REARRANGING, SOLVING FOR Y:
Y=
X+
f+1
Qf
f+1
SIGNAL TERM
NOISE TERM
Figure 6: Simplified Frequency Domain Linearized Model
of a Sigma-Delta Modulator
The integrator in the modulator is represented as an analog lowpass filter with a transfer function
equal to H(f) = 1/f. This transfer function has an amplitude response which is inversely
proportional to the input frequency. The 1-bit quantizer generates quantization noise, Q, which is
injected into the output summing block. If we let the input signal be X, and the output Y, the
signal coming out of the input summer must be X – Y. This is multiplied by the filter transfer
function, 1/f, and the result goes to one input of the output summer. By inspection, we can then
write the expression for the output voltage Y as:
Y = ¹ (X − Y) + Q .
f
Eq. 2
This expression can easily be rearranged and solved for Y in terms of X, f, and Q:
Y = ^X + ^Q⋅f .
f +1 f +1
Eq. 3
Note that as the frequency f approaches zero, the output voltage Y approaches X with no noise
component. At higher frequencies, the amplitude of the signal component approaches zero, and
the noise component approaches Q. At high frequency, the output consists primarily of
quantization noise. In essence, the analog filter has a lowpass effect on the signal, and a highpass
effect on the quantization noise. Thus the analog filter performs the noise shaping function in the
Σ-Δ modulator model. For a given input frequency, higher order analog filters offer more
attenuation. The same is true of Σ-Δ modulators, provided certain precautions are taken.
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