{"id":442,"date":"2008-11-23T21:57:52","date_gmt":"2008-11-24T02:57:52","guid":{"rendered":"https:\/\/www.circuitdesign.info\/blog\/2008\/11\/unity-stf-a-sigma-delta-linearization-method\/"},"modified":"2017-05-23T00:42:09","modified_gmt":"2017-05-23T05:42:09","slug":"unity-stf-a-sigma-delta-linearization-method","status":"publish","type":"post","link":"https:\/\/www.circuitdesign.info\/blog\/2008\/11\/unity-stf-a-sigma-delta-linearization-method\/","title":{"rendered":"Unity STF | A sigma-delta linearization method"},"content":{"rendered":"

In a previous post<\/a>, 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\u2019s. However, it is more powerful with continuous-time sigma-delta because it enables the active-RC configuration<\/a>.<\/p>\n

<\/p>\n

Band-limited STF<\/h2>\n

Consider the conventional sigma-delta architecture shown below:<\/p>\n

\"scan0103a\"<\/a><\/p>\n

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.<\/p>\n

Typically, the quantizer is modeled as an additive noise term N:<\/p>\n

\"scan0103b\"<\/a><\/p>\n

\n

The response due to the input X and this additive quantization noise N is:<\/p>\n

Y = X \u00d7 H\/(1+H) + N \u00d7 1\/(1+H)<\/div>\n

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).<\/p>\n

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:<\/p>\n

\"scan0105\"<\/a><\/p>\n

The noise-shaping filter H operates on the term X \u2013 Y which equals:<\/p>\n

X \u2013 Y = \u2013X\/(1+H) \u2013 N\/(1+H)<\/div>\n

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

Unity STF<\/h2>\n

Consider instead what happens when we feed forward a signal term right before the quantizer:<\/p>\n

\"scan0103c\"<\/a><\/p>\n

Once again, modeling the quantizer as an additive noise N:<\/p>\n

\"scan0103d\"<\/a><\/p>\n

We find that the transfer terms are:<\/p>\n

Y = X + N\u00d71\/(1+H)<\/div>\n

That is the STF is one for all frequencies (unity)\u2014thus my nomenclature for this topology. The NTF remains unchanged. The STF to the point Z is also one for all frequencies.<\/p>\n

Let\u2019s now consider what the input to H looks like:<\/p>\n

X \u2013 Y = \u2013N\/(1+H)<\/div>\n

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).<\/p>\n

How does this improve the linearity requirements on the noise-shaping filter? Well, since the noise-shaping filter isn\u2019t processing the input signal, it cannot have terms related to the input signal (x2<\/sup>, x3<\/sup>, etc). Of course, this assumes N is not correlated with the input\u2013N is white.<\/p>\n

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).<\/p>\n

Implementation<\/h3>\n

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<\/a>:<\/p>\n

\"scan0101\"<\/a><\/p>\n

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.<\/p>\n

Trade-Off\u2019s<\/h2>\n

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.<\/p>\n

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\u2019ll explain in a future post the exact nature of this problem as it applies to radio receivers.<\/p>\n

Interested? Consider a free subscription by email<\/a> or RSS<\/a>.<\/p>\n

Reference<\/h2>\n

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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[48,104,26,44],"class_list":["post-442","post","type-post","status-publish","format-standard","hentry","category-analog-pro","tag-continuous-time","tag-continuous-time-sigma-delta","tag-linearity","tag-sigma-delta"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/poCEy-78","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/posts\/442","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/comments?post=442"}],"version-history":[{"count":12,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/posts\/442\/revisions"}],"predecessor-version":[{"id":1079,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/posts\/442\/revisions\/1079"}],"wp:attachment":[{"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/media?parent=442"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/categories?post=442"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.circuitdesign.info\/blog\/wp-json\/wp\/v2\/tags?post=442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}