The present disclosure describes an improved multi-stage noise shaping (MASH) analog-to-digital converter (ADC) for converting an analog input signal to a digital output signal. In particular, a full delta-sigma (ΔΣ) modulator is provided at the front-end of the MASH ADC, and another full ΔΣ modulator is provided at the back-end of the MASH ADC. The front-end ΔΣ modulator digitizes an analog input signal, and the back-end ΔΣ modulator digitizes an error between the output of the front-end ΔΣ modulator and the (original) analog input signal. In this configuration where the back-end modulator digitizes the error of the (full) front-end modulator, some design constraints of the front-end are relaxed. These design constraints include thermal noise, digital noise cancellation filter complexity (the quantization noise of the front-end is already shaped by the noise transfer function of the front-end), and/or non-linearity.