Multi-Antenna Synthetic Aperture Radar

8.4.3.3  Analog-Digital-Converter (ADC)

ADC quantization errors may result in what appears to be a white noise floor, but is actually a “sea” of very finely spaced discrete spurs. The amplitude quantization errors can be assumed to be totally uncorrelated and uniformly distributed within each quantization step

$-\frac{{\Delta }_A}{2}\le e_A\le \frac{{\Delta }_A}{2}$

(8.85)

For a D-bit ADC, the quantization step size is

${\Delta }_A=\frac{1}{2^{D-1}}$

(8.86)

Then the amplitude error power is [25]

$E\left(e_A^2\right)=\frac{1}{{\Delta }_A}{\displaystyle {\int }_{-\frac{{\Delta }_A}{2}}^{\frac{{\Delta }_A}{2}}{e}_A^{2}d}{e}_A=\frac{{\Delta }_A^2}{12}$

(8.87)

Correspondingly, the noise variance is

$\sigma =\sqrt{\frac{{\Delta }_A^2}{12}}=\frac{{\Delta }_A}{2\sqrt{3}}$

(8.88)

Since the signal power is

$S={\left(2^D{\Delta }_A\right)}^2$

(8.89)

The noise-signal-ratio can then be expressed as

$\frac{N}{S}=10\log \frac{{\sigma }^2}{S}=20\log \frac{{\text{Δ}}_A/2\sqrt{3}}{2^D{\text{Δ}}_A}$

(8.90)

Suppose the sampling noise bandwidth is B_n, we then have

$L_{{\phi }_{AD}}\left(f\right)=\frac{N}{S}.$