What is Delta-Sigma modulation?
The Delta-Sigma (ΔΣ) modulation is a closed-loop oversampling technique involving a 1-bit ADC, an error integrator, and a 1-bit DAC. It is designed to sample a slowly varying input signal multiple times, a process known as oversampling. This setup continuously tracks the input signal to ensure that the time-averaged output of the DAC accurately reproduces the input signal. Due to closed-loop control, it suppresses the quantization noise within the signal band (low frequency) and increases the high-frequency quantization noise. The high-frequency increase in noise can be filtered using digital filters.
In this context, “Δ” represents the difference between the incoming analog signal and the DAC output, often referred to as the error (e[n]). Meanwhile, “Σ” signifies the accumulation of this error over time. These are popular for high-resolution and medium-to-low-speed applications such as high-quality digital audio and baseband processing in wireless systems. Delta-sigma modulation is also sometimes referred to as sigma-delta modulation.
Oversampling ADC without noise shaping (open-loop control)
If a signal is sampled at frequency significantly higher than Nyquist rate (2fm), the quantisation noise floor can be brought down, and the dynamic range is improved. This improvement in dynamic range allows us to detect a weaker signal in the spectrum. Let’s define oversampling ratio as the ratio between the sampling rate (fs) and Nyquist rate (2fm):
$$\text{OSR}=\cfrac{f_s}{2f_m}$$
The signal to quantisation noise ratio for an oversampling ADC can be rewritten as:
$$\text{SQNR}=6.02\text{N}+1.76+10\log{}_{10}(\text{OSR})$$
An 8-bit quantiser with OSR = 1 has a SQNR = 49.92dB. With OSR = 10, the SQNR is 59.92dB which is 10dB better than quantiser with OSR = 1. This improves the effective number of bits (ENOB).
$$\text{ENOB}=\cfrac{59.92-1.76}{6.02}=9.66\,\text{bits}$$
The ENOB is 9.66 bits which is 1.66 bits better than 8-bit ADC with OSR=1. We can increase the OSR further to get more ENOB however it will put strain on the 8-bit ADC because it has to sample faster.
Usually a 1-bit ADC is used in delta-sigma modulation as it is very simple to construct, linear and can achieve high sampling rate (much faster than 8-bit ADC). A 1-bit ADC has a SQNR of 6 dB. To achieve effective SQNR of 100 dB, the OSR should be:
$$\text{OSR}=10^{\left(\cfrac{100-6.02}{10}\right)}=2.5\times{}10^9$$
The OSR is coming very high. It means that to sample a signal at 1kHz, the sampling frequency should be 5000 GHz ! This clearly means that to obtain high dynamic range, we cannot use oversampling alone.
Oversampling with Noise shaping (closed-loop control)
With closed loop control, the quantisation noise can be suppressed significantly in the frequency-band of interest. With close loop control, the signal transfer function (y[n]/x[n]) can be written as:
$$S_{TF}(z)=\cfrac{H(z)}{1+H(z)}$$
and the noise transfer function (y[n]/q[n]) can be written as:
$$N_{TF}(z)=\cfrac{1}{1+H(z)}$$
Just to get a feel, it is reasonable to assume that H(z) has very high gain at low frequencies and zero gain at high frequencies. Therefore, at low frequency, signal transfer function is unity; which means entire signal is flowing to output without any attenuation. At high frequency, no signal is coming at the output.
In contrast, the value of noise transfer function is near zero (very small) at low frequency. This means the noise is suppressed significantly at low frequency. At high frequency value of H(z) drops to zero, which allows the NTF to grow to 1.
SQNR of first order delta-sigma modulator
The number of integrators in the delta-sigma modulator defines the order of delta-sigma modulator. For example, a first order modulator has only one integrator. The number of integrators signifies how steep the suppression of noise at low frequency.
SQNR of First-order Delta-Sigma modulator is:
$$\text{SQNR}=6.02\text{N}-3.41+30\log{}_{10}(\text{OSR})$$
We see that the SQNR is now a stronger function of oversampling (OSR).
SQNR of second order delta-sigma modulator
SQNR of Second-order Delta-Sigma modulator is:
$$\text{SQNR}=6.02\text{N}-11.14+50\log{}_{10}(\text{OSR})$$
For comparison with oversampling without noise shaping, the sampling frequency for 1-bit ADC to sample signal at 1kHz to achieve 100dB SQNR was coming out to be 5000 GHz. With first order noise shaping, the required sampling rate to achieve 100dB SQNR is 3.5 MHz which is much lower than 5000 GHz ! With second order noise shaping, the required sampling rate to achieve 100dB SQNR is 253 kHz which is even lower than 3.5 MHz. Higher order modulators are harder to stabilize so they are not very popular.
Decimation filter
A decimation filter has two purposes:
- It filters out-of-band noise. The noise-shaped quantisation noise after the Nyquist frequency is removed by the decimation filter.
- It reduces the data rate of the DSM output from Fs to the Nyquist rate, which is 2 times the filter bandwidth (2 x Fb = (Fs / OSR)).
Advantages of Delta-sigma modulation
- A lot of analog-dependent tasks are now done using digital circuits like low pass filtering. This makes it suitable for scaling with process technology.
- Higher linearity is attainable by employing low-bit Analog-to-Digital Converters (ADCs) and Digital-to-Analog Converters (DACs). For instance, a 1-bit DAC that only produces two values, a precise high voltage and a precise low voltage, is perfectly linear in principle. This approach ensures a faithful representation of the signal without distortion.
- Noise shaping is a technique used to relocate noise to higher frequencies, beyond the range of the signal of interest. This makes it easier to eliminate unwanted noise through low-pass filtering, resulting in cleaner and more accurate signal representation.
- Additionally, delta-sigma modulation reduces the steepness requirement for analog low-pass anti-aliasing filters. High-order filters with flat passbands and steep-rolloffs are costlier to implement.