1. Aliasing
Dr. Ali Hussein Muqaibel
ver. 4.4
G()
+2B

2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
G()
+2B
2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
Not only the above range is affected (damaged) by
aliasing but all of this frequency range
...
s–2B
...

2. Introduction
• To reconstruct the signal, 𝑓𝑠 > 2𝐵.
Dr. Ali Hussein Muqaibel 2
G()
+2B

2B s s
s
s s
s
A/Ts
...
...
s+2B
s–2B
s+2B
s–2B
LPF for reconstructing the origianl
signal from the sampled signal
Reconstructed Signal
+2B
2B s s
s
s s
s
A/Ts
Ts
Magnitude of LPF should be Ts to cancel
the scaling factor caused by sampling
s > 2(2B)  No interference between Images


3. Aliasing
G()
+2B

2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
G()
+2B
2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
Not only the above range is affected (damaged) by
aliasing but all of this frequency range
...
s–2B
... Under
sampled
spectrum
Dr. Ali Hussein Muqaibel 3

4. Aliasing and Nyquist Sampling Rate
• All practical signals are time-limited ⇒ Bandwidth unlimited.
• Two Problems:
1. Loss of the tail 𝐺(𝑓) beyond 𝑓 >
𝑓𝑠
2
𝐻𝑧.
2. The reappearance of this tail inverted or folded onto the spectrum
𝑓𝑠
2
=
1
2𝑇𝑠
is the folding frequency.
• 𝒇𝒔/𝟐 + 𝒇𝒙 shows up or "impersonate" a component at 𝒇𝒔/𝟐 − 𝒇𝒙
This tail inversion is known as spectral folding or aliasing
Dr. Ali Hussein Muqaibel 4
𝒇𝒔
𝟐
+ 𝒇𝒙
𝒇𝒔
𝟐
− 𝒇𝒙

5. Anti-Alias Filter
Anti-Aliasing
LPF
BW = s/2 rad/s
= fs/2 Hz
Sampler
Sampling Freq.
= fs samples/s
Reconstruction
LPF
BW = s/2 rad/s
= fs/2 Hz
g(t)
g*(t) g*(t)
gRecons(t)
[= g*(t)]
G()
+2B

2B s
s
/2
s
A
s
/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
G*()
+2B
2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
Frequency range of input signal
above half the sampling frequency
Because the original input signal was filtered, non of
images will intefer with adjacent images
...
s–2B
...
G*()
+2B

2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
This range of the input signal above
half the sampling frequency was
filtered out
Output of Anti-Aliasing Filter
bandlimited to s/2
(This is the signal input to the sampler)
GRecons()
+2B

2B s
s/2
s
A
s/2
Sampling
Frequency
Half the Sampling
Frequency
This part of the original
input signal is lost
To SAVE HALF of the signal in the frequency range
[𝒇𝒔 − 𝑩 , 𝑩]
solution (Anti-aliasing filter) use an ideal low pass
filter of bandwidth 𝑓𝑠/2 before sampling.
Practically, use a steep cutoff filter which leaves a
sharply attenuated residual after 𝑓𝑠/2.
Dr. Ali Hussein Muqaibel 5

6. Aliasing Examples in Real Life
• When video taping a TV
or a PC monitor.
• Rotating wheel or fan.
• 2D example
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Fun time!
Amazing Water Experiment https://www.youtube.com/watch?v=uENITui5_jU
Airplane fan https://www.youtube.com/watch?v=vIsS4TP73AU

7. Examples: Aliasing of Sinusoidal Signals
Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz
Dr. Ali Hussein Muqaibel 7
What is the reconstructed frequency?
900 Hz
What is the reconstructed frequency?

8. Example
• The following figure shows two sinusoidal
signals. One of them is the original signal and
the other one is the recovered signal after
sampling.
• Write a time domain expression to represent
the two signals
• The original signal =
• Signal recovered after sampling =
• Why the recovered signal does not equal to the
original signal?
• What do we call this
phenomena?.....................................
• Sketch the spectrum of the SAMPLED signal
Dr. Ali Hussein Muqaibel 8