In a previous post, I discussed the trade-offs in linearity of several continuous-time sigma-delta schemes. In this post, I will describe a method that linearizes the sigma-delta noise-shaping filter (NSF). That is, the scheme presented in this article greatly suppresses the linearity requirements on the noise-shaping filter. This method applies to both discrete-time and continuous-time sigma-delta ADC’s. However, it is more powerful with continuous-time sigma-delta because it enables the active-RC configuration.

## Band-limited STF

Consider the conventional sigma-delta architecture shown below:

I have shown it with one feedback path from the quantized output back to the input. However, the same result holds true if multiple feedback paths are provided.

Typically, the quantizer is modeled as an additive noise term N:

The response due to the input X and this additive quantization noise N is:

Thus, the signal-transfer function (STF) is H/(1+H) and the noise-transfer function (NTF) is 1/(1+H). The signal transfer function to the point Z is also H/(1+H).

I informally call this a band-limited STF since the signal-transfer function is band-limited by H: when H is low (there is no gain in the noise-shaping filter), the term H/(1+H) is also low. As a result, H/(1+H) follows approximately the same roll-off characteristics as H:

The noise-shaping filter H operates on the term X – Y which equals:

As one can see, this term includes both a signal component [X/(1+H)] and a noise component [N/(1+H)].

## Unity STF

Consider instead what happens when we feed forward a signal term right before the quantizer:

Once again, modeling the quantizer as an additive noise N:

We find that the transfer terms are:

That is the STF is one for all frequencies (unity)—thus my nomenclature for this topology. The NTF remains unchanged. The STF to the point Z is also one for all frequencies.

Let’s now consider what the input to H looks like:

That is, the noise-shaping filter operates on a term that only depends on the additive noise. It is no longer signal dependent (at least not directly).

How does this improve the linearity requirements on the noise-shaping filter? Well, since the noise-shaping filter isn’t processing the input signal, it cannot have terms related to the input signal (x^{2}, x^{3}, etc). Of course, this assumes N is not correlated with the input–N is white.

In reality, N will be dependent on the the input X, since N originates from quantization and a strong component of X appears at the quantizer. However, whatever N’s dependence on the input, it is less than X itself. As a result, the linearity requirements on the noise-shaping filter have been reduced (although not completely eradicated).

### Implementation

One easy way to implement the unity STF structure is with an active-RC. We basically add an extra feedback resistor R2 (mistakenly labeled R1 in parallel with C2 in the picture) to the first stage of the active-RC structure that I discussed previously:

This causes not just an integral of the input, but a proportion of the input itself to pass through to the output of the filter.

## Trade-Off’s

The main trade-off of this Unity STF method is that it relieves the requirements on the noise-shaping filter. However, for the quantizer, the requirements are worse.

With the band-limited STF, the quantizer saw a signal term with a gain of H/(1+H). With the unity STF method, the signal term at the input of the quantizer is unity. As a result, there is no roll-off and the entire signal (including all out-of-band interference terms) hit the quantizer right at its input. I’ll explain in a future post the exact nature of this problem as it applies to radio receivers.

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## Reference

This idea (or at least my first encounter with it) appeared in “Wideband low-distortion delta-sigma ADC topology” by J. Silva, U. Moon, J. Steensgaard and G.C. Temes.