MT-022

∑

ANALOG FILTER

H(f) = 1

f

∑

X

Y

+

_

X – Y

1

f

( X – Y )

Q =

QUANTIZATION

NOISE

Y =

1

f

( X – Y ) + Q

REARRANGING, SOLVING FOR Y:

Y =

X

f + 1

+

Q f

f + 1

SIGNAL TERM

NOISE TERM

Y

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:

Q

)

Y

X

(

f

1

Y

+

−

=

.

Eq. 2

This expression can easily be rearranged and solved for Y in terms of X, f, and Q:

1

f

f

Q

1

f

X

Y

+

⋅

+

+

=

.

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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