Find out number of quantization steps in conversion process

Question

An analogue audio signal is changed to digital form for transmission over a wireless headphone system. There receiver within the headphones changes the signal back to analogue form giving 1.5V peak amplitude to drive the headphones. There audio signal reproduced at the headphones should have an accuracy of ±0.01% while compared along with the original input analogue signal. Find out the number of quantization steps within the conversion procedure, the bit resolution required in the converters and the value of the quantum voltage included into the conversion.

Answer 1

Within this system the analogue signal is initially converted from analogue to digital form for transmission and after that reconverted back again from digital to analogue for reproduction within the headphones. 

The conversion will contain an error of ± ½ levels within both the AD Convertor and the DA Convertor.

This should be taken in account as finding the overall accuracy of ±0.01%.

 Hence in each converter the accuracy is as given below:

Δ = 0.005% FS or 0.00005

After that for every converter 1 level = 2Δ FS = 0.0001 FS

The number of levels required in every converter is then:

 L = 1/2Δ = 1/0.0001= 10,000

The number of bits is specified by the power of the radix 2 that offers a decimal value of greater than or identical to 10,000. 

 It can be seen that 2¹⁴ = 16,384. Thus, the number of bits needed in both the DAC and ADC is as:

 N = 14 bits.

During this case the actual number of levels will be 16,384 therefore the quantum level voltage can be determined as:

V_Q=V_FS/(L - 1)

= 1.5V/16383

=91.68mV