Find out the sampling frequency on disk

Question

The output signal of an audio amplifier consisting of a bandwidth of 20Hz to 20 kHz is to be digitized for storage upon a digital audio CD. There signal is sampled along with an accuracy of ±0.05% as well as a margin of 25% of the baseband spectrum is permitted for anti-aliasing filtering. Find out the sampling frequency needed and the estimated file size upon disk of a song lasting 3 min.

Answer 1

The highest baseband frequency is   f_M = 20 kHz

The lowest image frequency is f_S - f_M = f_M + 0.25f_M 

While permitting for the anti-aliasing filtering

In that case the sampling rate needed is as follows: 

 f_S = 2f_M + 0.25f_M = 2.25f_M = 2.25 x 20kHz = 45kHz

In that case the Data Conversion Rate is as 45x10³ samples/ second certainly.

An accuracy of as ±0.05% within sampling means a fractional accuracy of 5 parts within 10⁴.

In that case Δ = 5/10⁴

The number of quantization levels needed is the here L = 1/2Δ = 10⁴/(2x5) = 1000

The near binary integer power to that is 1024 levels that corresponds 2¹⁰ and hence:   

Number of Bits needed = 10 bits.

It provides a data transfer rate to disk of:

Data Transmission Rate is as= f_S x N = 45 x 10³ x 10 = 45 x 10⁴ bits/s

For a song ending 3 minutes the whole amount of data transferred provides the estimated file size as: 

File Size = Data Transmission Rate x Duration of Song

= 45 x 10⁴ x 3 x 60 bits

= 81 x 10⁶ bits

= (81x 10⁶)/8 = 10.125 x 10⁶ Bytes

= (10.125 x 10⁶)/( 1.048576 x 10⁶) = 9.66MBytes