# Delta-sigma ADC basics: How the digital filter works

This month we have guest blogger Joseph Wu from TI

Designing with a delta-sigma analog-to-digital converter can help achieve the highest possible resolution in your system. But to maximize this architecture’s benefits, it helps to understand how the delta-sigma modulator and digital filter in the ADC combine to achieve greater measurement resolution. On the Precision Hub blog on the TI E2E Community, I explained that the analog-to-digital converter (ADC) is constructed from a ΔΣ modulator and digital filter.

Using the delta-sigma (ΔΣ) modulator, the ADC converts the input into a digital pulse stream with a 1’s density proportional to the input divided by the reference. Let’s take a practical example to show how the digital filter converts the pulse stream into a data word representing the conversion. (For more on how the delta-sigma ADC is constructed or how the modulator works, see my recent Precision Hub post on the TI E2E Community).

Figure 1 shows a modulator. Let’s set the input to 0.15V and the reference to 1V so that the input range is +/-1V. After running the modulator for seven cycles, the bitstream will look something like Figure 1:

Figure 1

A first-order modulator running for seven cycles

If you add the number of times the modulator output became a 1, you would get as few as zero and as many as seven, with eight possible outcomes. The digital filter accumulates the modulator outputs over a number of modulator cycles to produce a digital representation of the input. If you plot the transfer function of the modulator output, it would look like Figure 2:

Figure 2

Transfer function for the modulator in Figure 1

With a full-scale range from -1V to 1V and eight possible outputs, the least-significant-bit (LSB) size is 0.25V, as shown in the step size in Figure 1. Using the transfer function of the modulator, you can construct the output in Table 1:

Table 1

Modulator output and output code from the modulator in Figure 1

Since the modulator output contains four 1s, this would correspond to a quantization value of 0V and a three-bit output code of 100.

If you extend the modulator output run for the next eight outputs for a total of 15 cycles, you would have the output shown in Figure 3:

Figure 3

A first-order modulator running for 15 cycles

Using 15 modulator cycles, we divide the full-scale range from -1V to 1V into 16 equal parts, giving an LSB size of 0.125V per bit. This is shown in Figure 3 and Table 2:

Figure 4

Transfer function for modulator in Figure 3

Table 2

Modulator output and output code from the modulator in Figure 3

Accumulating the first 15 pulses, you get nine 1s and a quantization value of 0.125V, with a four-bit output code of 1001.

In both examples, we oversample the input at a higher frequency and put out a data word at a lower data rate. In the first example, we sample the input voltage seven times to get one data word of three bits (from 23 -1 pulses). In the second example, we sample the input voltage 15 times to get one data word of four bits (from 24 -1 pulses). With more modulator iterations, we will get a result with more resolution but at a slower data rate.

When we change the data output rate, we change the digital filter to collect a different amount of modulator outputs. Digital filters can have different implementations with weighted averaging and convolution over multiple samples. With higher-order modulators, we can get more resolution with fewer modulator samples. With more modulator samples, we lower the frequency of the filter and increase the output resolution.

That’s how a basic ΔΣ data converter works. The modulation will take the quantization noise and shape it to have higher-frequency content. The digital filter removes high frequency noise while retaining the original input signal. The modulator and digital filter combine to give high-resolution measurements by oversampling the input. With a small sacrifice of speed, we can achieve greater resolution in the measurement.

For more in-depth information, see the following resources:

TI’s broad range of delta-sigma ADCs with resolutions ranging from 12 to 24 bits and speeds from 5SPS to 10MSPS

## 4 comments on “Delta-sigma ADC basics: How the digital filter works”

1. garyZLTK
February 11, 2015

Thanks for the explanation of sigma-delta A/Ds.

Going down to the bit level helps understanding.

But there is one thing which I cannot quite get about S-D A/Ds. Why does the noise in a S-D A/D get moved to the high frequencies? Many explanatory articles (including this one), simply make the stetement that the noise is shaped to higher frequenices, but don't exactly say why. For example, this article has the statement:
“The modulation will take the quantization noise and shape it to have higher-frequency content.”

Could someone explain exactly why this is..?

2. NelliTokleh
February 13, 2015

3. harryhoidini
February 19, 2015

Thanks Nelli for the information about the ti blog article on Sigma Delta ADCs.

Unfortunately, that ti blog, like the original article, also simply says that the “the quantization error spectral density is shaped by pushing it to higher frequencies”. In other words, like many other descriptions about S-D ADCs, that ti blog also simply says that the noise is pushed to higher frequencies without any further explanation.

My original question was about trying to understand WHY the noise gets pushed to the higher frequencies, in other words, I would like to understand why the noise shaping  actually occurs.

Luckily, since I asked the original question, I have found a couple of very good articles (and thought a lot about it), and discovered the reason.

Without going into detail, because it's the difference which gets propogated around the loop, and the one time interval delay, the integrator and the S-D loop effectively acts as a low pass filter for the signal, but a high pass filter for quantization noise – actually explaining why high frequency componets get accentuated (ie pushed to higher frequencies).

4. MelBrandle
August 21, 2018

Users would most often than not overlook this important component of how our digital usage works behind the scenes. We would enjoy high quality graphics and texts without even thinking how data is converted into the screen that we are seeing. This is how technology has managed to support our daily lifestyles as much as we can afford. We just see how much we can really enjoy what is being presented in front of us, it is good to know what happens before that too for a change.

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