Calculated Quantizing Noise of Single-Integration Delta-Modulation Coders

Abstract

We calculate the granular quantizing noise for a delta modulator that has unequal positive and negative step sizes. The asymmetry leads to a highly colored noise spectrum. We perform this calculation by adding a ramp function of time to the input of a symmetrical coder. The resulting formulas can also be used for uniform DPCM and PCM coders. The idle‐channel spectrum consists of discrete lines which scatter somewhat irregularly in amplitude and frequency, they can be regarded as the result of sampling (aliasing) a sawtooth wave. These lines are phase‐modulated by a coder input. For a sinusoidal input, discrete side frequencies are produced which again have an irregular progression of amplitudes. Gaussian inputs lead, to gaussian line shapes; the lines broaden as input power is increased. A totally white spectrum (as is often assumed in connection with delta‐modulation‐system considerations) cannot be attained, however, before the onset of slope overload. We give a numerical example that uses a coder suitable for telephone applications. One can see that step asymmetry can be very advantageous in attaining low noise.

Sigma-delta modulators operating at a limit cycle

“…For a second-order system, (4) is evaluated as (5) From (5), it follows that decreases with 15 dB for each doubling of . Secondly, the quantization noise spectrum that arises from the sampling of a square wave with frequency (the limit-cycle frequency) with clock frequency , is evaluated [12], [13] as (6) When we take into account that the loop operates at a fraction of the clock frequency, from (4), is…”

“…However, in [13], it is shown that for high clock speeds it is very difficult to achieve the theoretically expected performance with real circuit implementations. Even some recent designs [14], [15], display performance well below the theoretically achievable one for the implemented filter order and the applied clock speed.…”

Hexagonal<tex>$Sigma Delta$</tex>Modulators in Power Electronics

“…This work drew from the basic reference in communications for the behavior of quantization noise with dc input of Candy and Benjamin [3]. They extended methods of Iwersen [22] from modulation. Their approach was based on an approximate continuous time model for a modulator and their results well matched experimental results (for a continuous time system).…”

“…They applied their approximations to evaluate the mean squared error when an ideal low-pass filter is used as a decoder. Iwersen applied Fourier series expansion of the quantizer error function to a specific input to obtain a Fourier series for the error sequence from which the spectrum was deduced [22].…”