A 12-bit ADC with V_REF=3.300 V and V_LSB=V_REF/4096 is read sixteen times in…

A 12-bit ADC with \(V_{\mathrm{REF}}=3.300\,\mathrm{V}\) and \(V_{\mathrm{LSB}}=V_{\mathrm{REF}}/4096\) is read sixteen times in succession on a DC input. The sixteen right-aligned codes, in the order taken, are \[ \begin{array}{cccccccc} 1550 & 1553 & 1549 & 1552 & 1551 & 1554 & 1550 & 1552\\ 1551 & 1553 & 1548 & 1552 & 1551 & 1550 & 1553 & 1551. \end{array} \] Software oversampling forms the integer sum \(S\) of the sixteen codes and the 14-bit result \[ N_{14}=\bigl\lfloor S/4\bigr\rfloor. \] The reconstructed oversampled voltage is \(\hat{v}_{14}=N_{14}\cdot V_{\mathrm{LSB}}/4\). The arithmetic mean of the sixteen reconstructed 12-bit voltages is \(\bar{v}=(S/16)V_{\mathrm{LSB}}\). Ignore missing codes. (a) Compute \(S\), \(N_{14}\), \(\hat{v}_{14}\), and \(\bar{v}\). Report voltages in millivolts. (b) Compute the unique 12-bit code \(N_{\mathrm{med}}\) obtained by sorting the sixteen samples and averaging the eighth and ninth in the ordered list (use the arithmetic mean, which may be a half-integer), and the corresponding voltage \(N_{\mathrm{med}}V_{\mathrm{LSB}}\). (c) Determine the sample standard deviation of the sixteen codes using the divisor \(15\), and convert that standard deviation to millivolts. Then compute the standard error of the mean \(s/\sqrt{16}\) in millivolts. (d) A second routine reports \(\hat{v}'=N_{14}\cdot V_{\mathrm{REF}}/16384\). Decide whether \(\hat{v}'=\hat{v}_{14}\) for every integer \(N_{14}\), and evaluate both reconstructions on the \(N_{14}\) of part (a).

Oversampling here is an integer sum followed by a power-of-two shift, not a floating-point average. The median of sixteen sorted samples sits between positions eight and nine.