Question

2. A digital signal processing system with analog audio signal input in the range of 0-20 Khz uses oversampling techniques and a second order sigma delta modulator to convert the analog signal into a digital bit stream at a rate of 3.072 Mhz.

i) Explain how the digital bit stream may be converted in to a digital multibit stream for x(n).

ii) Determine the overall improvement in signal to quantization noise ratio made possible by oversampling and noise shaping and hence estimate the effective resolution in bits of the digitizer                                                                                                  

Answer 1

The single-bit stream is converted to multibit words by decimation (down sampling process). The output of the SDM contains very small in-band quantization noise, but very large out-of-band noise. The out-of-band noise is removed by lowpass digital filtering. Because of the high sampling rate, direct use of a digital filter is impractical. Instead. filtering is achieved by decimation which also serves to reduce the rate to the desired value. Typically, a two-stage decimator would be used (factors of 16 and 4). After filtering, the resulting signal is B-bit quantized data. The filtering serves to average out the high quantization noise. Typically, the FIR coefficients of the decimating filter are represented by 16—24 bits.

Fig: z plane model of second order SDM

Fig: Effect of noise shaping on quantization noise

(2) An estimate of the effective resolution can be obtained by a simplified analysis as follows.
The noise power is reduced by oversampling and noise shaping. The reduction in noise power, due to oversampling. is given by the oversampling ratio.
The oversampling ratio is

That is a reduction of 18 dB in the quantization noise power.
From the z-plane model of the sigma delta modulator, the transfer function seen by the quantization noise is given by
N(z)=(1-z^-1)²
This is essentially a highpass filter, with a double zero at d.c. It attenuates the noise component at the low frequency end as depicted in figure. The magnitude response is given by

This offers a reduction in SQNR of 52.35 dB. The effective wordlength of the ADC is determined mainly by the signal-to-noise ratio achieved through oversampling and noise shaping.

The overall reduction in SQNR is 70.41 dB.

This corresponds to an effective ADC resolution of 11.4 bits ( from SQNR 6.02 B + 1.77 dB).