Question

Document a complete and thorough answer to Class Discussion Item 8.2 Class Discussion Item 8.2 Sampling a Beat Signal What is the minimum sampling rate required to adequately represent the signal in Example &1? Example 8.1 Sampling Theorem and Aliasing Consider the function Fn- sin( ar) + sin( bry Using a trigonometric identity for the sum of two sinusoidal functions, we can rewrite FO as the following product: If frequencies a and b are close in value, the bracketed term has a very low frequency in comparisoo to the sinusoidaal term on the right. Therefore, the bracketed term modulates the amplitude of the higher frequency sinusoidal term. The resulting waveform exhibits what is called a beat trequency that is common in optics, mechanics, and acoustics when twoInternet Link waves clos in frequency add For more information and an audio example of beat frequency see Intemet Link 8.1 and Video Demo 8.3 8.1 Beat To illustrate aliasing associated with improper sampling, the wavefom is plotted in the requency following figures using two different sampling frequencies If a and b are chosen as rad sec then to sample the signal Fu) properly, the sampling rate must be more than twice the high- est frequency in the signal: Video Demo 8.3 Beat frequency from 左> 2a . 2Hz Therefore, the dme interval between samples (plotted points) must be mibding signals of simlar frequency A<0.5 sec The first data set is plotted with a time interval of 0.01 sec (100 Hz sampling rate), providing an adequate representation of the wavefomm. The second data set is plotted with a time interval of 0.75 sec (1.33 Hz sampling rate), which is less than twice the maximum fre- queacy of the waveforim (2 Hz). Therefore, the signal is undersampled, and aliasing results The sampled waveform is an incorrect representation, and its observed maximum frequency appears to be approximately 0.4 Hz because there are approximately four cycles over 10 sec.

Answer 1

Solution:

The given signal is F(t) = sin(at)+ sin(bt)

The frequency of the signal sin(at) is f1 = $\frac{a}{2\pi}$ and T1 = $\frac{2\pi}{a}$

The frequency of the signal sin(bt) is f2 =
b-2
and T2 = $\frac{2\pi}{b}$

The period of the sum signal is give by,

T1b T2

period is aT1 or bT2.

frequency is 1/aT1 or 1/bT2 and a and b are integers and not fractional numbers.

To avoid aliasing, sampling period must be less than aT1 or bT2.