of a delta modulator/demodulator. The linearity property of integration, , makes it possible to move the integrator, which reconstructs the analog signal in the demodulator section, in front of the delta modulator. Again, the linearity property of the integration allows the two integrators to be combined and a delta-sigma modulator/demodulator block diagram is obtained. Because the quantizer is not homogeneous, delta-sigma is inspired by delta modulation, but the two are distinct in operation. From the first block diagram in Figure 3, the integrator in the feedback path can be removed if the feedback is taken directly from the input of the low-pass filter. Hence, for delta modulation of input signal , the low-pass filter sees the signal However, sigma-delta modulation of the same input signal places at the low-pass filter In other words, delta-sigma and delta modulation swap the position of the integrator and quantizer. The net effect is a simpler implementation that has the added benefit of shaping the quantization noise to be mostly in frequencies above the signals of interest. This effect becomes more dramatic with increased oversampling, which allows for quantization noise to be somewhat programmable. On the other hand, delta modulation shapes both noise and signal equally. Additionally, the quantizer (e.g., comparator) used in delta modulation has a small output representing a small step up and down the quantized approximation of the input while the quantizer used in delta-sigma must take values outside of the range of the input signal, as shown in Figure 4. In general, delta-sigma has some advantages versus delta modulation: The structure is simplified as only one integrator is needed, the demodulator can be a simple linear filter (e.g., RC or LC filter) to reconstruct the signal and the quantizer (e.g., comparator) can have full-scale outputs The quantized value is the integral of the difference signal, which makes it less sensitive to the rate of change of the signal. Quantization theory formulas When a signal is quantized, the resulting signal has approximately the second-order statistics of a signal with independent white noise added. Assuming that the signal value is in the range of one step of the quantized value with an equal distribution, the root mean square value of this quantization noise is In reality, the quantization noise is, of course, not independent of the signal and this dependence results in limit cycles and is the source of idle tones and pattern