Aliasing example
As an example, the certain base band signal be band limited to 8 Hz and let the sampling frequency be Fs = 16 Hz. We don't expect frequencies higher than 8 Hz. Then the base band frequency of, say, F1 = 4 Hz, the aliases, F2, can be given by
F2 = kFs - F1, k = 1, 2, ...
Setting k = 1 gives the lowest alias of F1, that is, F2 = 1 x 16 - 4 = 12 Hz. The next alias is 2 x 16 - 4 = 28 Hz. Consider two waveforms with frequencies of 4 Hz and 12 Hz given by
x1(t) = 1 cos 2π4t and x2(t) = 1 cos 2π12t
The sampled versions are
x1(n) = cos 2π4nT = cos 2π4n(1/16) = cos (nπ/2) and
x2(n) = cos 2π12nT = cos 2π12n(1/16) = cos (n3π/2)
whose 1st few samples are tabulated as follows:
|
n
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
|
x1(n)
|
1
|
0
|
-1
|
0
|
1
|
0
|
-1
|
0
|
|
x2(n)
|
1
|
0
|
-1
|
0
|
1
|
0
|
-1
|
0
|
These 2 sequences are seen to have the identical digital frequency and cannot be distinguished from each other as the frequency is concerned. When passed through the filter, they will both seem as 4 Hz signals. This is true if the amplitudes of the underlying analog waveforms are different.
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