# 2次デルタシグマ変調のまとめ

D級アンプ設計のリファレンスとして、

IRAUDAMP7Dを利用していますが、

この設計のPWM変調は自励発振によるアナログ入力の2次シグマデルタ変調になっています。

1次のフィルタとどう違ってくるのか調べてみました。

Wikipediaのエントリを見ていくと、

# Delta-sigma modulation

## Oversamplingのところにグラフと数式でわかりやすい説明があるので Let’s consider a signal at frequency f 0 and a sampling frequency of f s {\displaystyle \scriptstyle f_{\mathrm {s} }} much higher than Nyquist rate (see fig. 5). ΔΣ modulation is based on the technique of oversampling to reduce the noise in the band of interest (green), which also avoids the use of high-precision analog circuits for the anti-aliasing filter. The quantization noise is the same both in a Nyquist converter (in yellow) and in an oversampling converter (in blue), but it is distributed over a larger spectrum. In ΔΣ-converters, noise is further reduced at low frequencies, which is the band where the signal of interest is, and it is increased at the higher frequencies, where it can be filtered. This technique is known as noise shaping.

For a first-order delta-sigma modulator, the noise is shaped by a filter with transfer function H n ( z ) = [ 1 − z − 1 ] {\displaystyle \scriptstyle H_{n}(z)\,=\,\left[1-z^{-1}\right]} . Assuming that the sampling frequency f s ≫ f 0 {\displaystyle \scriptstyle f_{s}\,\gg \,f_{0}} , the quantization noise in the desired signal bandwidth can be approximated as:

n 0 = e rms π 3 ( 2 f 0 τ ) 3 2 {\displaystyle \mathrm {n_{0}} =e_{\text{rms}}{\frac {\pi }{\sqrt {3}}}\,(2f_{0}\tau )^{\frac {3}{2}}} .

Similarly for a second-order delta-sigma modulator, the noise is shaped by a filter with transfer function H n ( z ) = [ 1 − z − 1 ] 2 {\displaystyle \scriptstyle H_{n}(z)\,=\,\left[1-z^{-1}\right]^{2}} . The in-band quantization noise can be approximated as:

n 0 = e rms π 2 5 ( 2 f 0 τ ) 5 2 {\displaystyle \mathrm {n_{0}} =e_{\text{rms}}{\frac {\pi ^{2}}{\sqrt {5}}}\,\left(2f_{0}\tau \right)^{\frac {5}{2}}} .

In general, for a N {\displaystyle \scriptstyle \mathrm {N} } -order ΔΣ-modulator, the variance of the in-band quantization noise:

n 0 = e rms π N 2 N + 1 ( 2 f 0 τ ) 2 N + 1 2 {\displaystyle \mathrm {n_{0}} =e_{\text{rms}}{\frac {\pi ^{N}}{\sqrt {2N+1}}}\,\left(2f_{0}\tau \right)^{\frac {2N+1}{2}}} .

When the sampling frequency is doubled, the signal to quantization noise is improved by 10 log 10 ⁡ ( 2 ) ( 2 N + 1 ) d B {\displaystyle \scriptstyle 10\log _{10}(2)(2N+1)\,\mathrm {dB} } for a N {\displaystyle \scriptstyle \mathrm {N} } -order ΔΣ-modulator. The higher the oversampling ratio, the higher the signal-to-noise ratio and the higher the resolution in bits.

Another key aspect given by oversampling is the speed/resolution tradeoff. In fact, the decimation filter put after the modulator not only filters the whole sampled signal in the band of interest (cutting the noise at higher frequencies), but also reduces the frequency of the signal increasing its resolution. This is obtained by a sort of averaging of the higher data rate bitstream.

サンプリング周波数帯域内での量子化ノイズの分布が低周波数側が低ノイズ、高周波数側が高ノイズになるので、

D級GaN MOSFETアンプは1Mhzの自励発振周波数による2次デルタシグマ変調になっていますが、

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