Signals and Systems
Lecture 19:
Chapter 4
Sampling & Aliasing
Today's lecture
 Concept of Aliasing
 Spectrum for Discrete Time Domain
 Spectrum for Discrete Time Domain
– Oversampling
– Under=sampling




Sampling Theorem
Aliasing
Reconstruction
Folding
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Sampling Sinusoidal Signals
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Figure 4-3
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The Concept of Aliasing
Two different cosine signals can be drawn through the
same samples
x1[n] = cos(0.4πn)
x2[n] = cos(2.4πn)
x2[n] = cos(2πn + 0.4πn)
x2[n] = cos(0.4πn)
x2[n] = x1[n]
The smallest of all aliases is called the principal alias
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Reconstruction? Which one?
Figure 4-4
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Exercise 4.2
 Show that 7cos (8.4πn - 0.2π) is an alias of
7cos
(0.4πn - 0.2π). Also find two more frequencies that are
aliases of 0.4π rad.
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General Formula for Frequency Aliases
 Adding any integer multiple of 2π gives an alias
= 0.4 π + 2 πl l = 0,1,2,3,…..
 Another alias
x3[n] = cos(1.6πn)
x3[n] = cos(2πn - 0.4πn)
x3[n] = cos(0.4πn)
Since cos (2πn - θ) = cos (θ )
 All aliases maybe obtained as
,
+ 2 πl , 2 πl l = integer
o
o
o
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Folded Aliases
 Aliases of a negative frequency are called folded aliases
Acos (2πn - o n - θ) = Acos ((2π - o )n- θ)
= Acos (- o n- θ)
= Acos ( o n + θ)
 The algebraic sign of the phase angles of the folded
aliases must be opposite to the sign of the phase angle of
the principal alias.
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Spectrum of a Discrete-Time Signal
y1[n] = 2cos(0.4πn)+ cos(0.6πn)
y2[n] = 2cos(0.4πn)+ cos(2.6πn)
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