Figure 1 : Conversion flow
64x Upsampler converts 44.1kHz PCM to 2.8MHz PCM.
(Original sample values are represented as floating point value ranging from -1.0 to 1.0. Noise shaping filter becomes unstable if input signal is large magnitude, therefore PCM sample value range is reduced to value range from -0.5 to +0.5 somewhere before Noise shaping.)
Noise shaping filter converts 2.8MHz PCM to 2.8MHz 1bit SDM. Noise shaper output is a stream of two symbols: -1.0 and +1.0.
Obtain noise transfer function (NTF) of noise shaping filter using Matlab Delta Sigma Toolbox synthesizeNTF function.
Input parameter is order of NTF and over sample rate = 64.
Output is position of NTF zeros and poles. (Figure 10 and Figure 11)
Result: https://sourceforge.net/p/playpcmwin/code/HEAD/tree/PlayPcmWin/WWDigitalFilter/NTFHzcoeffs.cs#l122
■ Calculate g_k, a_k, b_k parameters of CRFB circuit ■
I chose CRFB structure for noise shaping filter.
First calculate g_k.
NTF zeros z_k is placed on the unit circle of gauss plane with ±ω_k angle from real axis. Parameter g_k is calcuated by
solving equation 1-g_k/2 = cosω_k
Then calculate a_k.
Multiply all NTF poles p_k with Σ(z - p_k) and expand it to get polynomial coefficients d_k.
Build equation of NTF polynomial and reshape it to get unknown a_m values fron known d_n, g_o (Figure. 2 to Figure 9)
Calculate a_k values using this equation and known d_k and g_k values.
Then b_k := a_k.
Calculated result g_k, a_k, b_k is shown on Figure 2, 4, 6 and 8.
Example of CRFB noise shaping filter implementation : https://sourceforge.net/p/playpcmwin/code/HEAD/tree/PlayPcmWin/WWDigitalFilter/LoopFilterCRFB.cs
[WWOfflineResampler] uses 5th order CRFB to create CD quality 2.8MHz 1bit SDM data.
Figure 2 : 2nd order CRFB Noise shaper schematics.
Figure 3 : 2nd order CRFB NTF equation solving to calculate parameter a0, a1
Figure 4 : 3rd order CRFB Noise shaper schematics.
Figure 5 : 3rd order CRFB NTF equation
Figure 6 : 4th order CRFB Noise shaper schematics.
Figure 7 : 4th order CRFB NTF equation
Figure 8 : 5th order CRFB Noise shaper schematics.
Figure 9 : 5th order CRFB NTF equation
Figure 10 : synthesizeNTF frequency response curve.
Figure 11 : 5th order noise shaping filter NTF pole zero plot.
Figure 12: 6th order CRFB NTF equation
Figure 13: 7th order CRFB NTF equation
Understanding Delta-Sigma Data Converter, Richard Schreier and Gabor C. Temes ,Wiley-IEEE Press, 2004
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