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aliasing frequency that shows up on the samples of the signal is
( ) ( )
| w s w s |
∗
w = | w + mod (w ) − | (2.14)
s
1
| 2 |
| 2 |
where w is the frequency content of the original signal. For simplicity, we consider a
1
specific frequency content for the original signal. If the sampling theorem is violated (the
sampling frequency is less than twice the highest frequency content of the original signal),
1. reconstruction of the original signal from its samples is impossible,
2. high frequency components look like low frequency components.
Let us consider two sinusoidal signals with frequencies 0.1 Hz and 0.9 Hz. If we sample
both of the signals at w = 1 Hz, the sampling theorem is not violated in sampling the first
s
signal, but it is violated in sampling the second signal. Due to the aliasing, the samples of
the 0.9 Hz sinusoidal signal will look like the samples of the 0.1 Hz signal (Figure 2.6).
Figure 2.12 shows the two cases,
1. continuous signal sin(2 (0.9)t) and sampling frequency w = 1.0Hz,
s
2. continuous signal sin(2 (0.1)t) and sampling frequency w = 1.0Hz.
s
The high frequency signal 0.9 Hz looks like a 0.1 Hz signal when sampled at the 1.0 Hz
rate as a result of the aliasing,
( w ) ( w )
| s s |
∗
w = | w + mod(w ) − |
s
1
| 2 |
| 2 |
∗
w = |(0.9 + 0.5)mod(1.) − 0.5| = |0.4 − 0.5|
∗
w = 0.1 Hz (2.15)
Another example is a sinusoidal signal with w = 3 Hz, and sampled values of it at w =
1
s
4 Hz. The sampling theorem is violated. The samples will show a 1 Hz oscillation which
does not exist in the original signal.
In summary, if the sampling theorem is violated, high frequency content of a signal
shows up as low frequency (aliasing frequency) content in the sampled signal as a result of
aliasing. The aliasing frequency is given by Eqn. 2.14.
(ii) Hidden oscillations: If the original signal has a frequency content which is an
exact integer multiple of the sampling frequency (sampling theorem is violated), then there
could be hidden oscillations. In other words, the original signal would have high frequency
oscillations, whereas the sampled signal would not show them at all (Figure 2.13). If
w = n ⋅ w ; n = 1, 2, ….
signal s
1
When n = ; with correct phase of the sampling time to the oscillation frequency,
2
hidden oscillations are also possible.
(iii) Beat Phenomenon: The beat phenomenon is observed when two signals with
very close frequency content with very close magnitudes are added. The result looks like
two signals (one slowly varying, the other fast varying) are multiplied.
This phenomenon occurs as a result of the sampling operation when the sampling
frequency is just a little larger than twice the highest frequency content of the signal
(Figure 2.14). Notice that the sampling theorem is not violated. Consider the following
signal,
u(t) = A ⋅ (cos(w t) + cos(w t)) (2.16)
1 2
( )
w − w 1 w + w 2
( )
1
2
= 2 A ⋅ cos t ⋅ cos t (2.17)
2 2
u(t) = 2 A ⋅ cos(w beat t) ⋅ cos(w ave t) (2.18)