feedback loops outside the accumulators) A very popular decimation structure is the sinc filter. For second-order modulators, the sinc3 filter is close to optimum. Example of decimation Given an 8:1 decimation filter and a 1-bit bitstream: the sample frequency is reduced by a factor of eight the serial (1-bit) input bus becomes a parallel (3-bits) output bus. For example, the input stream 10010110 contains 4 1s. The decimation result is 4/8 = 0.5. This result can be represented by the 3-bit binary number 100, which corresponds to half of the largest possible number. Once decimation is applied, if the n-bit codes are transmitted, the signal becomes PCM. Decimation is strongly associated with delta-sigma modulation but is distinct. Variations There are many kinds of ADC that use this delta-sigma structure. The above analysis focuses on the simplest 1st-order, 2-level, uniform-decimation sigma-delta ADC. Many ADCs use a second-order 5-level sinc3 sigma-delta structure. Much of what follows uses an arcane shorthand using symbols representing operational functions with analysis given in terms of Laplace transforms, etc. This is the lingua franca of the data transmission industry and it does not communicate to the general public. If a fuller documentation of a particular method is needed look no further than the patents. (Patent examiners generally require full disclosure.) An excellent history is "The Evolution of Oversampling Analog-to-Digital Converters" by Bruce A. Wooley which gives many references to the relevant patents. Second-order and higher-order modulator The number of integrators, and consequently, the numbers of feedback loops, indicates the order of a ΔΣ modulator; a second-order ΔΣ modulator is shown in Figure 2. First-order modulators are unconditionally stable, but stability analysis must be performed for higher-order modulators. 3-level and higher quantizer The modulator can also be classified by the number of bits it has in its output, which strictly depends on the output of the quantizer. The quantizer can be realized with a N-level comparator, thus the modulator has log2N-bit output. A simple comparator has 2 levels and so is 1 bit quantizer; a 3-level quantizer is called a "1.5" bit quantizer; a 4-level quantizer is a 2-bit quantizer; a 5-level quantizer is called a 2.5-bit quantizer. Relationship to delta modulation Delta-sigma modulation is inspired by delta modulation, as shown in Figure 3. If quantization were homogeneous (e.g., if it were linear), the following would be a sufficient derivation of the equivalence: Start with a block diagram