Page 68 - Mechatronics with Experiments

54 MECHATRONICS
where w ave = (w + w )∕2, w beat = (w − w )∕2. Let us show the above equality directly
2
1
2
1
using trigonometric relations. The following trigonometric relations are used in the
derivation,
cos( + ) = cos( ) cos( ) −sin( ) sin( ) (2.19)
2 2
sin + cos = 1 (2.20)
1 + cos 2
2
cos = (2.21)
2
Then,
( )
( w − w 1 ) w + w 2
1
2
u(t) = 2 A ⋅ cos t ⋅ cos t (2.22)
2 2
(
w t w t w t w t
1
2
2
1
= 2 A ⋅ cos cos +sin sin + (2.23)
2 2 2 2
( ))
w t w t w t w t
2
1
1
2
⋅ cos cos −sin sin (2.24)
2 2 2 2
( )
w t 2 w t 2 w t 2 w t
2
1
1
2
2
= 2 A ⋅ cos cos −sin sin (2.25)
2 2 2 2
( ( )( ))
w t w t w t w t
1
1
2
2
= 2 A ⋅ cos 2 cos 2 − 1 − cos 2 1 − cos 2 (2.26)
2 2 2 2
(
w t 2 w t
1
2
2
= 2 A ⋅ cos cos − (2.27)
2 2
( ))
w t w t w t w t
2
1
2
1
1 − cos 2 − cos 2 + cos 2 cos 2 (2.28)
2 2 2 2
( ( ))
w t 2 w t
2
1
2
= 2 A ⋅ − 1 − cos − cos (2.29)
2 2
( )
1 + cos w t 1 + cos w t
= 2 A ⋅ 2 + 1 − 1 (2.30)
2 2
( )
cos w t cos w t
2
1
= 2 A ⋅ + (2.31)
2 2
= A ⋅ (cos w t + cos w t) (2.32)
1
2
which shows that the addition of two cosine functions of two discrete frequencies can also
be expressed as multiplication of two cosine functions with two frequencies that are the
average and difference of the original two frequencies. When the two frequencies are close
to each other, then this results in the beat phenomenon.
If w is very close to w , that is w = 5, w = 5.2, then the so called beat phenomenon
2
1
2
1
occurs,
u(t) = A cos(w t) ⋅ cos(w t) (2.33)
beat ave
where w beat = (w − w )∕2, w ave = (w + w )∕2. The same effect occurs when we sample a
2
1
2
1
signal with a frequency which is very closed to the minimum sampling frequency required,
yet the sampling theorem is not violated.
Let us consider a sinusoidal signal,
y(t) =sin(2 0.9t) (2.34)
and sample it with a sampling frequency of w = 1.9 Hz, which satisfies the sampling
s
theorem requirements. However, since sampling a signal effectively shifts the original signal