Decimation is an important component of oversampled analog-to-digital conversion. It transforms the digitally modulated signal from short words occurring at high sampling rate to longer words at the Nyquist rate. Here we are concerned with the initial stage of decimation, where the word rate decreases to about four times the Nyquist rate. We show that digital filters comprising cascades of integrate-and-dump functions can match the structure of the noise from sigma delta modulation to provide decimation with negligible loss of signal-to-noise ratio. Explicit formulas evaluate particular tradeoffs between modulation rate, signal-to-noise ratio, length of digital words, and complexity of the modulating and decimating functions.

Interpolative digital-to-analog converters generate an output that has only a few analog levels. They provide fine resolution by oscillating rapidly between these levels in such a manner that the average output represents the value of the applied code. Here we describe an improved method of interpolating that results in reduced noise in the signal band. A theory of the interpolation, confirmed by experiments, demonstrates that switching between only two levels at 1.3 mHz could provide 16 bit resolution for telephone signals.

Sigma delta modulation is viewed as a technique that employs integration and feedback to move quantization noise out of baseband. This technique may be iterated by placing feedback loop around feedback loop, but when three or more loops are used the circuit can latch into undesirable overloading modes. In the desired mode, a simple linear theory gives a good description of the modulation even when the quantization has only two levels. A modulator that employs double integration and two-level quantization is easy to implement and is tolerant of parameter variation. At sampling rates of 1 MHz it provides resolution equivalent to 16 bit PCM for voiceband signals. Digital filters that are suitable for converting the modulation to PCM are also described.

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.

We present and analyze a method of interpolation that improves the amplitude resolution of an analog-to-digital converter. The technique requires feedback around a quantizer that operates at high speed and digital accumulation of its quantized values to provide a PCM output. We show that use of appropriate weights in the accumulation has important advantages for providing finer resoution, less spectral distortion, and white quantization noise. The theoretical discussion is supplemented by the report of a practical converter designed especially to show up the strengths and weaknesses of the technique. This converter comprises a sigma-delta modulator operating at 8 MHz and an accumulation of the 1-bit code with triangularly distributed weights. 13-bit resolution at 8 kwords/s is realized by periodically dumping the accumulation to the output. We present a practical method for overcoming a thresholding action that distorts low-amplitude input signals.

This paper describes a companded analog-to-digital (A/D) converter for voiceband signals that is simple and potentially inexpensive. The converter uses only 18 coarsely spaced analog levels. Fine resolution is obtained by oscillating between these levels at an increased speed and averaging the result over a Nyquist interval. The companding used in the converter is effectively the same as that of μ-255 pulse-code modulation (PCM). In the encoding process a one-bit code is generated at 256 000 samples/s. This 1-bit per sample signal can be transmitted and decoded directly, or a simple digital circuit will produce a 13-bit, 8-kHz linear PCM signal that can be compressed to 8-bit companded PCM format. In this paper the basic operation of the 1-bit coder is described and its performance when connected to a 1-bit decoder is illustrated. Methods for obtaining both linear and compressed PCM are then presented, and the properties of these PCM signals with respect to noise, gain tracking, and harmonic content are described. Relative insensitivity to circuit component variations, absence of analog gates, along with the need to generate only a few analog levels, make the coder especially well suited to integrated circuit realization.

High quality analog-to-digital conversions are obtained using simple and inexpensive circuits that require no high-precision components. Samples of the analog signal are cycled rapidly through a coarse quantizer while the roundoff error is fed back and subtracted from the input. By means of this feedback, the coarse quantizations are caused to oscillate between levels, keeping their running average representative of the input. A binary coding of the quantized values, summed over Nyquist intervals, provides a high resolution PCM output. The precision is determined by a product of the cycle rate and the spacing of the coarse quantization levels. The system is surprisingly tolerant of inaccuracies in gains and threshold settings; indeed, it has many of the desirable properties of classical feedback servomechanisms. An 8-bit limit cycling converter intended for 1-MHz signal bandwidths has been fabricated of standard components that, in total, cost less than $150.

Interpolative digital-to-analog (D/A) converters produce a final output via a two-step process. First, each digital input word is used to control a circuit whose output oscillates rapidly (i.e., many times faster than new digital input values are provided) between coarsely spaced analog values (i.e., many times coarser than the resolution specified by the input word). Second, the oscillating analog signal is low-pass filtered to give the final output. The oscillation pattern is chosen to produce an average value that corresponds to the fine resolution specified by the input word and to ensure that the power of the error (the difference between the oscillating signal and the desired fine resolution output) occurs predominantly out of band. By this means, high-speed operation reduces the need for many finely spaced analog signal amplitudes, a tradeoff which is especially desirable for integrated circuit implementation. In this paper, the basic operation of interpolative D/A converters is described. Three alternative means of generating patterns are compared with respect to circuit complexity, and amount of baseband distortion introduced. The relative insensitivity of these converters to circuit value variations is emphasized. Applications of the interpolative technique to decoding digital words in both linear and piecewise linearly companded formats are given.