Digital Signal Processing Reference
In-Depth Information
Folding occurs when the sampled signal has a sinusoidal component that is
greater than f
s
=2, but less than f
s
.
In general, we can replace some frequency
f with (f
s
f
0
) to give:
cos(2fnT
s
+ )
=
cos(2(f
s
f
0
)nT
s
+ )
=
cos(2f
s
nT
s
2f
0
nT
s
+ )
=
cos(2n2f
0
nT
s
+ )
=
cos(2f
0
nT
s
+ )
=
cos(2f
0
nT
s
):
The value of f
0
must be less than the original f, otherwise the folded frequency
will be outside the spectrum. For example, if we start with cos(2400nT
s
+), with
T
s
= 1000Hz, and replace 400 with 1000600, we would end up with cos(2600nT
s
). This would not show up in the spectrum (fromf
s
=2 to f
s
=2), except as
cos(2400nT
s
+ ), which is what we started with.
5.4.3
Locations of Replications After Sampling
Once a signal is sampled, an innite number of replications of frequency components
will appear at regular intervals. This is based on the periodicity of the sinusoids,
that is, cos(2ft) = cos(2ft + 2n), where n is an integer, just as cos(=6) =
cos(=6 + 2) = cos(=6 + 22). This means that sampling a signal will put all
frequency content in the range 0 .. f
s
.
Since n and k are both integers by denition, their product nk is also an integer.
cos(2f
1
nT
s
) = cos(2f
1
nT
s
+ 2nk)
f
1
f
s
+
f
s
= cos
2n
f
s
k
Thus, cos(2f
1
nT
s
) is indistinguishable from cos(2(f
1
+ kf
s
)nT
s
), for all integers
k. So when sampling at f
s
samples per second, cos(2f
1
t) is indistinguishable from
cos(2(f
1
+ kf
s
)t), meaning that sampling creates a replica of a frequency at every
integer multiple of the sampling frequency plus that frequency.
We will look at
negative values of k next.
The cosine function is an even one, meaning that cos() = cos(), or that
cos(2f
1
t) = cos(2f
1
t):
