Digital Sampling Errors
• To avoid amplitude ambiguity, set the sample period equal to the least
common multiple of all of the signal’s contributory periods.
The least common multiple or lowest common multiple or
smallest common multiple of two integers a and b is the
smallest positive integer that is a multiple of both a and b. Since
it is a multiple, a and b divide it without remainder. For
example, the least common multiple of the numbers 4 and 6 is
12. (Ref: Wikipedia)
• For a signal with only one period, T1, set T = mT1 = m/f1.
Recall T = Nδt = N/fs. → fs = (N/m) f1.= N f1/m = N/T
Illustration of Correct Sampling
y(t) = 5sin(2πt)
→ f = 1 Hz
with fs = 8 Hz
• correct frequency because
fs (=8) > 2f = 2
• correct amplitude because
fs /f = 8/1 = integer
Figure 10.10
Aliasing of sin(20πt)
y(t) = sin(20πt)
→ f = 10 Hz
with
fs = 12 Hz
digital signal
appears to
have frequency
of 2 Hz
Figure 10.5 modified
The Folding Diagram
To determine the
aliased frequency, fa:
1. Determine k, where
k = f/fN = 2f/fs
2. Using the folding
diagram, find the ka,
into which k folds.
3. Calculate the aliased
frequency,,where
fa = kafN =kafs/2
Example: f = 10 Hz; fs = 12 Hz
Figures 10.3 and 10.4
Aliasing of sin(20πt)
y(t) = 5sin(2πt)
→ f = 1 Hz
fs = 1.33 Hz
Aliasing occurs
because of
low sample rate
Figure 10.12
In-Class Example
• At what cyclic frequency will y(t) = 3sin(4πt) appear if
fs = 6 Hz?
fs = 4 Hz ?
fs = 2 Hz ?
fs = 1.5 Hz ?
Correct Sample Time Period
y(t) =
3.61sin(4πt+0.59)
+ 5sin(8πt)
f’s of 2 Hz and 4 Hz
→
T’s = 0.5 s and 0.25 s
→
use T = m(0.5) s
m = 1 or 2 or 3 …
Figure 10.15
Sampling with Aliasing
y(t) = 5sin(2πt)
→ f = 1 Hz
fs = 1.33 Hz
Sample ended
at cycle’s end
→ correct
amplitude
BUT aliasing
occurs because
of low sample rate
Figure 10.12
Sampling with Amplitude Ambiguity
y(t) = 5sin(2πt)
→ f = 1 Hz
fs = 3.33 Hz
Sample NOT
ended at
cycle’s end
→ false
amplitudes
Figure 10.11
In-Class Example
y(t) = 6 + 2sin(πt/2) + 3cos(πt/5) +4sin(πt/5 + π) – 7sin(πt/12)
fi (Hz):
Ti (s):
Smallest sample period that contains all integer multiples of the Ti’s:
→ T can be set at 120, 240, 360, … to avoid amplitude ambiguity
Smallest sampling to avoid aliasing (Hz):