EE422G
Signals and Systems Laboratory
Sampling and Quantization
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Sampling Function
A link between the analog and discrete worlds can be analyzed
using a train of Dirac Delta functions:
∞
s t = ∑ t−nT 
n=−∞
where the impulses are separated by T seconds (sampling
interval), which corresponds to a sampling frequency of:
 s=2  F s=
1
2
T
s(t)
…
…
-3T
-2T
-T
0
T
2T
3T
t
FT of Sampling Function
Recall from the FS review, that the FS of this train of impulse
functions is:
∞
st =
∑
k =−∞
1
exp j 2 k F s t 
T
The Fourier Transform (FT) of s(t) can then be written as:
∞
S  f =
∑
k=−∞
1
T
…
-3Fs
-2Fs
-Fs
0
1
  f −k F s 
T
Sˆ ( f )
…
Fs
2Fs
3Fs
f
Bandlimited Signal
Consider a bandlimited signal xc(t) with spectrum shown below:
1
Xc(f)
f
-FN
FN
Note that all frequency content is less than FN.
If xc(t) is multiplied by the sampling function s(t) to isolate
discrete values, the multiplication corresponds to convolution in
the frequency domain:
st xc t ⇔ S  f ∗X c  f 
Aliasing
Aliasing is the result of convolving the signal spectrum by
the sampling function spectrum:
∞
∞
1
∑ T   f −−k F s  X c d 
−∞ k=−∞
X s  f = S  f ∗ X c  f = ∫
∞
X s  f =
∑
k=−∞
1 
X c  f −k F s 
T
Example: Aliased spectra when FN < Fs/2 (Fs/2 is referred to as
the Nyquist frequency)
Xs(f)
…
…
-2FS
- FS
-FN
FN
FS
2 FS
f
Reconstructing Analog Signal
The original analog signal in theory can be reconstructed
from its samples without error as long as Fs > 2FN using
an ideal low-pass filter to filter out the non-interfering
aliased spectra, the original continuous time signals can be
recovered without distortion
Ideal low-pass filter with
cut-off frequency greater
than FN and less than FS - FN
Xs(f)
…
…
-2FS
- FS
-FN
FN
FS
2 FS
f
Sampling Above and Below Nyquist Rate
Sampling with Nyquist Frequency greater than FN (Fs > 2FN)
Xs(f)
…
…
-2FS
-FN
- FS
FN
FS
f
2 FS
Sampling with Nyquist Frequency below FN (Fs < 2FN ), where
distortion arises due to overlapping spectra, the signal cannot
be recovered without error using low-pass filtering.
Xs(f)
…
…
-3FS
-2 FS
-FS
FS
FS-FN
2 FS
3 FS
f
Aliasing Example
Given x t=3 cos2 15k t sampled at Fs = 8kHz. Find the
frequency after sampling aliased to the range 0 to Fs/2.
 f − f 0  f  f 0

x
t
=
A
cos
2
f
t

Note FT of
is X  f = A
0
2
which actually consists of 2 frequencies ± f 0.
Now apply aliasing formula:
∞
1
X s  f = ∑ X  f −n F s 
n=−∞ T
Find n and f so that arguments of the Dirac deltas are 0 and f is between ±F s / 2
∞
3
X s  f =8000 ∑   f −n 8 k−15 k  f −n 8 k15 k
2 n=−∞
For n = 2 (using appropriate sign for each Dirac delta argument), f =
 1000 Hz, so aliased frequency is 1000 Hz
Aliasing The Movie I
The following movie superimposes an input sinusoid in red with its
output after sampling in blue. The sampling frequency is 200 Hz and
the input frequency ranges from 50 Hz to 150 Hz. If program will
not play movie in browser, you can download avi movie file from
http://www.engr.uky.edu/~donohue/ee422/alias1.avi
Aliasing The Movie II
The following movie superimposes an input sinusoid in red with its
output after sampling in blue. The sampling frequency is 200 Hz and
the input frequency ranges from 350 Hz to 450 Hz. If program will
not play movie in browser, you can download avi movie file from
http://www.engr.uky.edu/~donohue/ee422/alias2.avi