2) Sampling of a Vibrating String (5 points) A plucked electric guitar string ex

Question

2) Sampling of a Vibrating String (5 points) A plucked electric guitar string experiences a vibration with a fundamental frequency of 50 Hz and with odd harmonic frequencies (e.g. 3rd at 150 Hz, 5th at 250 Hz, 7h at 350 Hz...). You decide a good sample of data is obtained by setting the A/D board upper and lower limits at ±0.25v and taking 1000 samples at 1000 samples/sec. a) What is the voltage resolution of the 12-bit A/D board given these voltage limits? b) What is the minimum frequency that could be resolved from this data? c) What is the Nyquist frequency? d) What is the lowest-order string harmonic that would be aliased?

Explanation / Answer

a) Resolution of A/D converter is given by

Resolution = Range / 2n              ( where n = number of bits of A/D converter )

Resolution = +/- 0.25 V / 212

                = +/- 0.25 V / 4096

Resolution = 0.0000061 V

Resolution = 61 micro volts               ..........1

This is the required resolution of A/D converter

b) Sampling frequency is given as 1000 samples/second

   i.e. Sampling frequency is 1000 Hz

According to Shannon's Sampling theorem sampling frequency has to be at least twice the input frequency.

So maximum frequency sampled is 1000 /2 = 500 Hz.

However only fundamental and odd harmonics are present.

minimum frequency that can be sampled = 1*50 = 50 HZ                      ..........2

This is the minimum frequency resolved fro this data

3) Nyquist frequency fn is defined as the half of the sampling rate.

fn = sampling frequency / 2

fn = 1000 / 2

fn = 500 Hz                     ................3

This is the required Nyquist frequency

4)

According to Shannon's Sampling theorem sampling frequency has to be at least twice the input frequency.

So maximum frequency sampled is 1000 /2 = 500 Hz.

However only fundamental and odd harmonics are present.

maximum frequency that can be sampled = 9*50 = 450 HZ   i.e 9th harmonic

Upto 450 Hz i.e. 9 th harmonic signal can be properly sampled

Next odd Harmonic is 11 and frequency is 11*50 = 550 Hz at which aliasing would be present as sampling frequency is not twice this frequency.

Hence 11th is the lowest string harmonic that would be aliased.

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