When the sampling rate of a sigma-delta modulator far exceeds the frequencies of the input signal, its modulation noise is highly correlated with the amplitude of the input. We derive simple algebraic expressions for this noise and its spectrum in terms of the input amplitude. The results agree with measurements taken on a breadboard circuit. This work can be useful for designing oversampled analog to digital converters that use sigma-delta modulation for the primary conversion.

Oversampling and digital filtering have been used to design a per-channel voiceband codec with resolution that exceeds the typical transmission system requirement by more than 15 dB. This extended dynamic range will allow for the use of digital processing in the management of signal levels and system characteristics in many telecommunication applications. Digital filtering contained in the codec provides rejection of out-of-band inputs and smoothing of the analog output that is sufficient to eliminate the need for analog filtering in most telephone applications. Some analog filtering may be required only to maintain the expanded dynamic range in cases where there is a danger of large amounts of out-of-band energy on the analog input impairing the dynamic range of the modulator. The encoder portion of the oversampled codec comprises an interpolating modulator that samples at 256 kHz followed by digital filtering that produces a 16-bit PCM code at a sample rate of 8 kHz. In the decoder, digital processing is used to raise the sampling rate to 1 MHz prior to demodulation in a 17-level interpolating demodulator. The circuits in the codec are designed to be suitable for large-scale integration. Component matching tolerances required in the analog circuits are of the order of only ± 1 percent, While the digital circuits can be implemented with fewer than 5000 gates with delays on the order of 0.1 μs. In this paper the response of the codec is described mathematically and the results are confirmed by measurements of experimental breadboard models.