Table 2 ADC angle quantization \(\theta _{\text {ADC},n,m}\) and ADC signal quantization \(s_{\text {ADC},n,m}\) (first quadrant)
From: Quantization error for weak RF simultaneous signal estimation
| Â | \(\left( {\varvec{x}_{\varvec{n}} = \varvec{n} + \frac{{\mathbf{1}}}{{\mathbf{2}}},\varvec{y}_{\varvec{m}} = \varvec{m} + \frac{{\mathbf{1}}}{{\mathbf{2}}}} \right)\) | \(\varvec{s}_{{{\mathbf{ADC}},\varvec{n},\varvec{m}}} = \sqrt {(\varvec{x}_{\varvec{n}} )^{{\mathbf{2}}} + (\varvec{y}_{\varvec{m}} )^{{\mathbf{2}}} }\) | \(\varvec{\theta }_{{{\mathbf{ADC}},\varvec{n},\varvec{m}}} = {\mathbf{arctan}}\left( {\frac{{\varvec{y}_{\varvec{m}} }}{{\varvec{x}_{\varvec{n}} }}} \right)\) | |||
|---|---|---|---|---|---|---|
\(N=1\) | 1-bit ADC | n | m | \((x_n,y_m)\) | \(s_{\text {ADC},n,m}\) | \(\theta _{\text {ADC},n,m}\) |
(0,0) | 0 | 0 | \(\{0.5, 0.5 \}\) | \(s_{\text {ADC},1,1} = \sqrt{(0.5)^2 + (0.5)^2}\) | \(\theta _{\text {ADC},1,1} \simeq 0.2500 \pi\) | |
\(N=2\) | 2-bit ADC | n | m | \((x_n,y_m)\) | \(s_{\text {ADC},n,m}\) | \(\theta _{\text {ADC},n,m}\) |
(1,0) | 1 | 0 | \(\{1.5, 0.5 \}\) | \(s_{\text {ADC},1,0} = \sqrt{(1.5)^2 + (0.5)^2}\) | \(\theta _{\text {ADC},1,0} \simeq 0.1024 \pi\) | |
(1,1) | 1 | 1 | \(\{1.5, 1.5 \}\) | \(s_{\text {ADC},1,1} = \sqrt{(1.5)^2 + (1.5)^2}\) | \(\theta _{\text {ADC},1,1} \simeq 0.2500 \pi\) | |
fv | (0,1) | 0 | 1 | \(\{0.5, 1.5 \}\) | \(s_{\text {ADC},0,1} = \sqrt{(0.5)^2 + (1.5)^2}\) | \(\theta _{\text {ADC},0,1} \simeq 0.3976 \pi\) |
\(N=3\) | 3-bit ADC | n | m | \((x_n,y_m)\) | \(s_{\text {ADC},n,m}\) | \(\theta _{\text {ADC},n,m}\) |
(3,0) | 3 | 0 | \(\{3.5, 0.5 \}\) | \(s_{\text {ADC},3,0} = \sqrt{(3.5)^2 + (0.5)^2}\) | \(\theta _{\text {ADC},3,0} \simeq 0.0455 \pi\) | |
(3,1) | 3 | 1 | \(\{3.5, 1.5 \}\) | \(s_{\text {ADC},3,1} = \sqrt{(3.5)^2 + (1.5)^2}\) | \(\theta _{\text {ADC},3,1} \simeq 0.1289 \pi\) | |
(3,2) | 3 | 2 | \(\{3.5, 2.5 \}\) | \(s_{\text {ADC},3,2} = \sqrt{(3.5)^2 + (2.5)^2}\) | \(\theta _{\text {ADC},3,2} \simeq 0.1974 \pi\) | |
(2,2) | 2 | 2 | \(\{2.5, 2.5 \}\) | \(s_{\text {ADC},2,2} = \sqrt{(2.5)^2 + (2.5)^2}\) | \(\theta _{\text {ADC},2,2} \simeq \frac{\pi }{4}\) | |
(2,3) | 2 | 3 | \(\{2.5, 3.5 \}\) | \(s_{\text {ADC},2,3} = \sqrt{(2.5)^2 + (3.5)^2}\) | \(\theta _{\text {ADC},2,3} \simeq 0.3026 \pi\) | |
(1,3) | 1 | 3 | \(\{1.5, 3.5 \}\) | \(s_{\text {ADC},1,3} = \sqrt{(1.5)^2 + (3.5)^2}\) | \(\theta _{\text {ADC},1,3} \simeq 0.3711 \pi\) | |
(0,3) | 0 | 3 | \(\{0.5, 3.5 \}\) | \(s_{\text {ADC},0,3} = \sqrt{(0.5)^2 + (3.5)^2}\) | \(\theta _{\text {ADC},0,3} \simeq 0.4545 \pi\) | |