noise in sigma-delta converters. Quantization noise may be reduced by increasing the oversampling ratio (OSR) defined by where is the sampling frequency and is Nyquist rate. The RMS noise voltage within the band of interest () can be expressed in terms of OSR Oversampling ΔΣ modulation is a technique of oversampling to reduce the noise in the band of interest (green in Figure 5), which avoids the use of high-precision analog circuits for the anti-aliasing filter. The total quantization noise is the same both in a Nyquist converter (in yellow) and in an oversampling converter (in blue), but it is distributed over a different spectrum. In ΔΣ converters, noise is further reduced at low frequencies, which is the band where the signal of interest is, and it is increased at the higher frequencies, where it can be filtered out. This technique is known as noise shaping. For a first-order delta-sigma modulator, the noise is shaped by a filter with transfer function . Assuming that the sampling frequency is large compared to a signal frequency of interest, , the quantization noise in the desired signal bandwidth can be approximated as: . Similarly for a second-order delta-sigma modulator, the noise is shaped by a filter with transfer function . The in-band quantization noise can be approximated as: . In general, for a -order ΔΣ modulator, the variance of the in-band quantization noise is: . When the sampling frequency is doubled, the signal-to-quantization-noise ratio is improved by dB for a -order ΔΣ modulator. The higher the oversampling ratio, the higher the signal-to-noise ratio and the higher the resolution in bits. Another key aspect given by oversampling is the speed/resolution tradeoff. The decimation filter put after the modulator not only filters the whole sampled signal in the band of interest (cutting the noise at higher frequencies), but also reduces the frequency of the signal, increasing its resolution. This is obtained by a sort of averaging of the higher-data-rate bitstream. Naming The technique was first presented in the early 1960s by professor Yasuhiko Yasuda while he was a student at the University of Tokyo. The name delta-sigma comes directly from the presence of a delta modulator and an integrator, as firstly introduced by Inose et al. in their patent application. That is, the name comes from integrating or summing differences, which, in mathematics, are operations usually associated with Greek letters sigma and delta respectively. Both