Which one of the following represents the output signal-to-noise ratio of a uniform quantizer? (where P denotes average power of the message signal m(t) and R denotes number of bits per sample)

Concept:

Signal-to-quantization-noise ratio (SQNR) is a widely used quality measure in analyzing digitizing schemes such as pulse-code modulation (PCM). The SQNR reflects the relationship between the maximum nominal signal strength and the quantization error (also known as quantization noise) introduced in the analog-to-digital conversion.

The SQNR formula is derived from the general signal-to-noise ratio (SNR) formula:

\(SQNR=\frac{P_{signal}}{P_{noise}}\)

Calculation:

Given the average power of the message signal

P = m_max²(t)

Noise power:

P_noise = Δ²/12 = m_max2(t)/3L²

Signal to noise ratio:

\(\mathrm{SQNR}=\frac{P}{m_{max}^{2}(t)/3L^{2}}=\frac{3P}{m_{max}^{2}(t)}L^{2}=\frac{3P}{m_{max}^{2}(t)}L^{2}=\left ( \frac{3P}{m_{max}^{2}(t)} \right )2^{2R}[\because \,L=2^{R}]\)