Question 4 (11 marks] The quantization signal to noise ratio is defined as Signal Power SNR = 10 log10 Noise Power where Vrms is the root mean square voltage of the signal and Vrms, noise is the root mean square voltage of the noise. Vrms 20 log10 Vrms, noise The expectation of the squared quantization noise, e is given by 4/2 E{e?} €²de -A/2 where A is the quantization step size equal to the LSB. (a) What is the RMS value of the noise of your A/D? (Give your answer to 3 signifi- cant figures) [6 marks] (b) Given the peak-to-peak voltage of the A/D is equivalent to 2v2Vrms. What is the RMS of the input voltage? [2 marks] (c) What is the SNR of your A/D? [2 marks] (d) The SNR of any A/D is also given by 6.02N + 1.76 dB, where N is the number of bits, calculate the SNR of your A/D using this formula [1 mark]

This problem deals with the quantization signal-to-noise ratio (SNR) for an Analog-to-Digital Converter (ADC). Let's break it down step by step.

### (a) RMS value of the noise of the A/D [6 marks]
The expectation of the squared quantization noise, \(E\{e^2\}\), is given by:

\[
E\{e^2\} = \frac{1}{A} \int_{-A/2}^{A/2} e^2 \, de
\]

where \( A \) is the quantization step size, which is equal to the Least Significant Bit (LSB). 

The RMS value of the quantization noise, \( V_{\text{rms, noise}} \), is the square root of this expectation:

\[
V_{\text{rms, noise}} = \sqrt{E\{e^2\}}
\]

Given that for a uniform distribution of the quantization error:

\[
V_{\text{rms, noise}} = \frac{A}{\sqrt{12}}
\]

So the RMS value of the noise is \( \frac{A}{\sqrt{12}} \). Since \( A \) is the quantization step size (LSB), you would need the specific value of \( A \) to calculate this.

### (b) RMS of the input voltage [2 marks]
Given the peak-to-peak voltage \( V_{\text{pp}} \) is equivalent to \( 2\sqrt{2} V_{\text{rms}} \):

\[
V_{\text{rms}} = \frac{V_{\text{pp}}}{2\sqrt{2}}
\]

So the RMS value of the input voltage \( V_{\text{rms}} \) is given by:

\[
V_{\text{rms}} = \frac{V_{\text{pp}}}{2\sqrt{2}}
\]

### (c) SNR of your A/D [2 marks]
The SNR can be calculated using the formula:

\[
\text{SNR} = 20 \log_{10} \left( \frac{V_{\text{rms}}}{V_{\text{rms, noise}}} \right)
\]

Substituting the values:

\[
\text{SNR} = 20 \log_{10} \left( \frac{V_{\text{rms}}}{\frac{A}{\sqrt{12}}} \right)
\]

### (d) SNR using the formula \(6.02N + 1.76 \text{ dB}\) [1 mark]
Given that the SNR is also given by:

\[
\text{SNR} = 6.02N + 1.76 \text{ dB}
\]

Where \( N \) is the number of bits of the ADC. Plug in the value of \( N \) to calculate the SNR.

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### Next Steps:
1. Clarify or find the value of the quantization step size \( A \) (LSB) to calculate the exact RMS noise value.
2. Identify the number of bits \( N \) in the ADC to compute the SNR using the formula in part (d).
3. Verify the peak-to-peak voltage \( V_{\text{pp}} \) to determine the RMS value of the input voltage.
4. Recalculate the SNR in part (c) with the correct \( V_{\text{rms}} \) and \( V_{\text{rms, noise}} \) values.
5. Consider how the uniform distribution of quantization noise affects the SNR.

### Tip:
For accurate calculations, always ensure the units and conversions are correct, especially when dealing with logarithmic scales like decibels (dB).

Learn how to calculate the RMS value of noise, RMS of input voltage, and SNR for an Analog-to-Digital Converter (ADC) using mathematical formulas. Understand the principles of quantization noise, SNR calculations, and logarithmic scales involved. Suitable for advanced undergraduate students studying electrical engineering or signal processing.

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