# The basics of sigma delta analog-to-digital converters

For the benefit of software and hardware developers whose experience has been mainly in the digital domain, we provide a review of the basics of sigma delta (SD) analog to digital converters (ADCs). It will be useful to a designer as a review, whether implementing an ADC on a board to work with associated digital components, or in a more complex SoC environment. We explain the functioning of all components by using an analog input example. Many different parameters used with respect to sigma delta ADCs are also explained. (This article is meant to be read as a companion to **Mixed Signal Verification of Sigma Delta ADCs in an SoC environment** .

An SD-ADC has a modulator and a digital filter (also known as decimation filter) as shown in **Figure 1** . A modulator converts the input analog signal into digital bit streams (1s and 0s). One can observe a bit, either 1â€™b1 or 1â€™b0 coming at every clock edge of the modulator.

The decimation filter receives the input bit streams and, depending on the over sampling ratio (OSR) value, it gives one N-bit digital output per OSR clock edge. For example, if we consider OSR to be 64, then the Filter gives one N-bit output for every 64 clock edges (64 data outputs of the modulator). Here N is the resolution of the SD ADC.

How a modulator works

The working of a modulator can be explained using a conversion example. In Table 1 the headings X, B, C, D, and W correspond to points in the signal path of the block diagram in **Figure 2** . For this example, the input X is a DC input of 3/8. The resultant signal at each point in the signal path for each signal sample is shown in **Table 1** .

Note that a repetitive pattern develops every sixteen samples, and that the average of the signal W over samples 1 to 16 is 3/8, thus showing that the feedback loop forces the average of the feedback signal W to be equals to the input X.

**Figure 2: First order sigma delta block diagram** **Table 1 **

The data D is received by the decimation filter, which generates an N Bit output. In the above example, if the averaging (or OSR) is less than 16, then there will be a quantization error. This is because the feedback loop has not been giving sufficient time for the output to reach the value of the input. Hence, a sigma delta ADCâ€™s accuracy/SNR improves with increasing OSR value, provided the input frequency is very slow. Also, even if the OSR is greater than 16, if resolution of the SDADC (the value N) is less than 8, then there will be a finite quantization noise, the reason being the resolution of the ADC is less than the granularity of the Signal.

From a frequency domain perspective, when the input signal passes through the modulator, the white noise shifts to a high frequency noise in the frequency domain, as shown in **Figure 3** , but the signal frequency does not shift. The decimation filter, being a low pass filter, then cuts off the high frequency components.

*Figure 3: Showing how the quantization noise transforms into a high pass shape* **How a decimation filter works** There are two kinds of digitalfilters â€“ FIR (finite impulse response) and IIR (infinite impulseresponse) The filter most commonly used for the back end of a sigmadelta converter is the FIR because of its stability, ease ofimplementation, linear phase response, and the fact that decimation canbe incorporated into the filter itself.Â A signal flow diagram of theDecimation filter is shown in

**Figure 4.**

Figure 4: Signal flow diagram of the decimation filterFigure 4: Signal flow diagram of the decimation filter

An example of transfer function of an IIR Decimation Filter is:

**Formula 1**An example of transfer function of an FIR Decimation Filter with its block diagram is shown in **Figure 5** .

**Figure 5: FIR filter block diagram**Theprocess of decimation is used in a sigma delta converter to eliminateredundant data at the output. The sampling theorem tells us that thesample rate only needs to be two times the input signal bandwidth inorder to reliably reconstruct the input signal without distortion.

**Click on image to enlarge.**

**Figure 6: Decimation filter cutting off the high frequency components**

However,the input signal was grossly oversampled by the sigma delta modulatorin order to reduce the quantization noise. Therefore, there is redundantdata that can be eliminated without introducing distortion to theconversion result. The decimation process is shown in frequency domainin **Figure 6** . It shows that the decimation process simply reduces the output sample rate while retaining the necessary information.

**References1. **

*A Brief Introduction to Sigma Delta Conversion*by*David Jarman*

**Siddi Jai Prakash**is a mixed signal architecture engineer at Freescale Semiconductorinvolved in architecture and verification of a metering SoC and anautomotive SoC for radar applications.

**Kushal Kamal** has been involved in mixed signal SoC verification for the past threeyears at Freescale Semiconductor.Â He specializes in HDL modelling,mixed signal simulation at SoC/IP levels and full chip SPICE simulation.

**Kumar Abhishek** is Mixed Signal Architecture and Verification Lead at Freescale. He has been working in the area of mixed signal architecture and verification for the last 8 years