Question

A band limited message signal which is uniformly distributed in the range [- 5 V, + 5 V] is sampled at nyquist rate and quantized using constant step size. The quantized samples are encoded using an n-bit encoder. If the minimum signal to quantization noise ratio required is 31.8 dB. then the minimum value of 'n' required will be______.

• A. 6

Answer 1

The correct option is A 6
Message signal is uniformly distributed,

So, Signal power, $S={\int }_{-5}^5{x}^2{f}_X\left(x\right)dx$

$={\int }_{-5}^5\frac{1}{10}{x}^{2}dx=\frac{25}{3}W$


Quantization noise power, $N_Q=\frac{{\Delta }^2}{12}=\frac{(10/2^n{)}^2}{12}=\frac{100}{\left(12\right)\left(2^{2n}\right)}W=\frac{\left(\frac{25}{3}\right)}{2^{2n}}W$

So, $SQNR=\frac{S}{N_Q}=2^{2n}=6.02ndB$

$\left(6.02\right)n\ge 31.8$

$n\ge 5.282$

$n_{min}=6$