TCAS-I 2013 paper by Callegari and Bizzarri

This section illustrates how to replicate the results presented in the paper [Cal13a].

[Cal13a]

Sergio Callegari, Federico Bizzarri “Output Filter Aware Optimization of the Noise Shaping Properties of ΔΣ Modulators via Semi-Definite Programming”, IEEE Transactions on Circuits and systems - Part I: Regular Papers, Vol. 60, N. 9, pp. 2352-2365. Sept. 2013. DOI: 10.1109/TCSI.2013.2239091. Pre-print available on arXiv.

To this aim, some sample code is provided in the directory Examples/TCAS1-2013.

The research paper provides 3 examples, in Sections V-a, V-b and V-c.

Example in section V-a

This example refers to the design of a Digital ΔΣ modulator that is followed by a low pass filter in charge of removing the quantization noise.

The corresponding code is provided in the file demo-lp.py. Once PyDSM and all its pre-requisites are installed, this can be started directly by opening a shell (command prompt) and typing:

python demo-lp.py

Alternatively, the script can be opened in Spyder and launched from there.

The code runs, showing some intermediate output from the optimizer. Then it provides the graphical output that is reported in Figures 7 and 9 in the TCAS-I paper.

The code also provides a further graph, obtained by the time domain simulation of the modulator. This includes 3 curves:

1. The output of a ΔΣ modulator passed through the filter, where the modulator is designed by the technique in the TCAS-I paper.
2. The output of a ΔΣ modulator passed through the filter, where the modulator is designed by the synthesizeNTF function from the DELSIG toolbox.
3. The input of the modulator passed through the filter.

This latter plot is meant to provide a graphical illustration that the modulator obtained by the proposed design methodology is actually working correctly. All the three plots should overlap almost perfectly. However, zooming in, little differences should become apparent, with the curve from the optimized modulator following the curve given by the input signal slightly better.

The code also provides some numerical output, corresponding to the SNR values reported in the research paper, obtained both with the linearized model and with the actual time-domain simulation of the modulator.

Note that the code does not provide any equivalent of Figure 8. For this, it is necessary to introduce a loop so that the NTF is optimized for different modulator orders. This is easy to achieve.

Neither the current code provides an equivalent of Figure 10. This is anyway easy to achieve by using the plotPZ function in the pydsm.delsig module.

As a final remark, the proposed example code is not particularly elegant, but should be rather easy to read, also thanks to the many comments.

Example in section V-b

This example refers to the design of a Digital ΔΣ modulator that is followed by a bandpass filter in charge of removing the quantization noise.

The corresponding code is provided in the file demo-bp.py. Once PyDSM and all its pre-requisites are installed, this can be started directly by opening a shell (command prompt) and typing:

python demo-bp.py

Alternatively, the script can be opened in Spyder and launched from there.

The code runs, showing some intermediate output from the optimizer. Then it provides the graphical output that is reported in Figures 11 and 13 in the TCAS-I paper.

The code also provides a further graph, obtained by the time domain simulation of the modulator. This includes 3 curves:

1. The output of a ΔΣ modulator passed through the filter, where the modulator is designed by the technique in the TCAS-I paper.
2. The output of a ΔΣ modulator passed through the filter, where the modulator is designed by the synthesizeNTF function from the DELSIG toolbox.
3. The input of the modulator passed through the filter.

This latter plot is meant to provide a graphical illustration that the modulator obtained by the proposed design methodology is actually working correctly. All the three plots should overlap almost perfectly. However, zooming in, little differences should become apparent, with the curve from the optimized modulator following the curve given by the input signal slightly better.

The code also provides some numerical output, corresponding to the SNR values reported in the research paper, obtained both with the linearized model and with the actual time-domain simulation of the modulator.

Note that the code does not provide any equivalent of Figure 12. For this, it is necessary to introduce a loop so that the NTF is optimized for different modulator orders. This is easy to achieve.

Nor the current code provides an equivalent of Figure 10. This is anyway easy to achieve by using the plotPZ function in the pydsm.delsig module.

As a final remark, the proposed example code is not particularly elegant, but should be rather easy to read, also thanks to the many comments.

Warning The optimization of the modulator NTF can be particularly time consuming in Windows, where only a 32 bit computation environment is available and where some math libraries are not fully optimized to the specific hardware platform in CVXOPT. For maximum performance, use a modern 64 bit Linux distribution on a multi core CPU with fast floating point math and instructions set with SIMD (SSEx) extensions and possibly Advanced Vector Extensions (AVX).

Example in section V-c

This example refers to the design of a Digital ΔΣ modulator that is followed by a two band filter in charge of removing the quantization noise.

The corresponding code is provided in the file demo-multiband.py. Once PyDSM and all its pre-requisites are installed, this can be started directly by opening a shell (command prompt) and typing:

python demo-multiband.py

Alternatively, the script can be opened in Spyder and launched from there.

The code runs, showing some intermediate output from the optimizer. Then it provides the graphical output that is reported in Figures 15 and 17 in the TCAS-I paper.

Note that the sample code provides no comparison to Nagahara’s min-max design strategy.

The code also provides a further graph, obtained by the time domain simulation of the modulator. This includes 2 curves:

1. The output of a ΔΣ modulator passed through the filter, where the modulator is designed by the technique in the TCAS-I paper.
2. The input of the modulator passed through the filter.

This latter plot is meant to provide a graphical illustration that the modulator obtained by the proposed design methodology is actually working correctly. The two plots should overlap almost perfectly.

The code also provides some numerical output, corresponding to the SNR values reported in the research paper, obtained both with the linearized model and with the actual time-domain simulation of the modulator.

Note that the code does not provide any equivalent of Figure 16. For this, it is necessary to introduce a loop so that the NTF is optimized for different modulator orders. This is easy to achieve.

Nor the current code provides an equivalent of Figure 18. This is anyway easy to achieve by using the plotPZ function in the pydsm.delsig module.

As a final remark, the proposed example code is not particularly elegant, but should be rather easy to read, also thanks to the many comments.

Warning The optimization of the modulator NTF can be particularly time consuming in Windows, where only a 32 bit computation environment is available and where some math libraries are not fully optimized to the specific hardware platform in CVXOPT. For maximum performance, use a modern 64 bit Linux distribution on a CPU with fast floating point math and instructions set with SIMD (SSEx) extensions and possibly Advanced Vector Extensions (AVX).