A-level Physics (Advancing Physics)/Digitisation/Worked Solutions

1. Take samples for the signal below every 0.1ms, and then produce a reconstructed signal. How does it differ from the original?

The high frequency elements of the signal have been lost.

2. A signal is sampled for 5 seconds at a sampling rate of 20 kHz. How many samples were taken?

$20\times 10^{3}={\frac {\mbox{No. of samples}}{5}}$

${\mbox{No. of samples}}=20\times 10^{3}\times 5=100000$

3. Most sounds created by human speech except for 'ss' and 'ff' have a maximum frequency of 4 kHz. What is a suitable sampling rate for a low-quality telephone?

$4\times 2=8{\mbox{ kHz}}$

4. Using a sampling rate of 20 kHz and 3 bits, sample the following signal, and then produce a reconstructed signal. What is the maximum frequency that can be perfectly reconstructed using this sampling rate?

First, calculate the length of each sample, by letting the number of samples equal 1:

$20\times 10^{3}={\frac {1}{\mbox{Length of one sample}}}$

${\mbox{Length of one sample}}={\frac {1}{20\times 10^{3}}}=0.00005{\mbox{ s}}=0.05{\mbox{ ms}}$

Then, we can sample the waveform and create a reconstructed signal:

To calculate the maximum frequency that can be perfectly reconstructed using this sampling rate (20 kHz):

${\frac {20}{2}}=10{\mbox{ kHz}}$