# Oversampling and Noise Shaping in Delta-Sigma Modulation

### Key Takeaways

• Delta-sigma modulation is used in analog-to-digital converters and digital-to-analog converters.

• The advantage of oversampling in delta-sigma modulation is that the quantization noises are spread over a larger frequency range, reducing the quantization noise spectral density.

• In delta-sigma analog-to-digital converters, the digital decimation filter increases data resolution and removes the quantization noise that is outside the frequency band of interest. It also determines the signal bandwidth, settling time, and stopband rejection. Oversampling and noise shaping results in the efficient digital transmission of signals

Oversampling and noise shaping focuses on achieving high signal-to-noise-ratio (SNR) in a limited frequency band without reducing the data rate of the signal. This technique results in the efficient digital transmission of signals. The combination of oversampling and noise shaping is the basic underlying principle of delta-sigma modulation.

Delta-sigma modulation is used in analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). In  ADCs, the delta-sigma modulator features all the merits of oversampling and noise shaping along with a digital decimation filter.

Let’s take a closer look at the concept of delta-sigma modulation.

## The Basic Principles of Delta-Sigma Modulation

Delta-sigma modulation, otherwise called sigma-delta modulation, is used in digital signal processing and signal conversion to improve the resolution and SNR of signals. These techniques are extensively used in portable audio playback devices, mobile phones, analog-to-digital converters, and digital-to-digital converters. The basic principles of delta-sigma modulation are oversampling and noise shaping.

### Oversampling

The sampling process follows the Nyquist-Shannon criteria, where the sampling frequency should be at least twice the maximum analog signal frequency. In the oversampling process, samples per second are more than what is required, according to Nyquist-Shannon criteria. Allowing more sample rates does not influence the signal power and total quantization noise power. Therefore, the SNR remains unchanged.

The advantage of oversampling is that the quantization noises are spread over a larger frequency range. This reduces the quantization noise spectral density (the SNR over the frequency of interest is increased). Comparing normal sampling and oversampling, the quantization noise power is reduced by 3 dB for every doubling of oversampling ratio (OSR), where OSR is the ratio of sampling frequency to twice the frequency of the signal.

### Noise Shaping

Noise shaping is the second step of delta-sigma modulation. This is where the signal to quantization noise ratio is increased by altering the frequency distribution. The frequency band is reduced to the signal band so that the quantization noise density is reduced and the SNR is increased in the low-frequency area of the spectrum. Noise shaping increases the noise density at frequencies outside the signal band.

In delta-sigma modulators, noise shaping is realized with an error minimizing the feedback loop. It minimizes the error between the input signal (x) and the quantized output signal (y). The feedback loop compares the input signal and the quantized output signal, and the difference between them that lies within the signal band is passed to the output side without attenuation. The out-of-the-band differences are suppressed using a filter. The result of the weighing is passed to the quantizer, which generates the next output value. The output signal generated is again fed back to the loop for the next comparison, and this continues until a close match between the input signal and the output signal is obtained in the signal passband.

## Digital and Decimation Filters in Delta-Sigma ADCs

In delta-sigma modulators used for analog-to-digital conversion, the digital and decimation filter extracts information from the sampled data and reduces the data rate to a more useful range. It increases the data resolution and removes the quantization noise that is outside the frequency band of interest. It is the decimation filter block that determines the signal bandwidth, settling time, and stopband rejection in delta-sigma ADCs.

Oversampling and noise shaping in any delta-sigma modulation increases its efficiency, as the quantization noise is outside the scope of the signal band of interest. Delta-sigma modulation offers high resolution, high SNR, low power consumption, and low-cost ADCs and DACs for applications like process control and physical quantity measurements such as temperature and pressure.
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