2. Even/Odd Signals
Even
Odd
Any signal can be discomposed into a sum of an
even and an odd
]
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[
2
1
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,
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[
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x
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x
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x
]
[
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,
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( n
x
n
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x
]
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x
n
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4. 30 March 2023
4
Even and Odd Signals
Even
Functions xe(t)=xe(-t)
Odd
Functions xo(t)=-xo(-t)
2A
0
-1
-2
-A
1 2 t
xe(t)
A
2A
0
-1
-2
-A
1 2 t
xo(t)
A
5. Even and Odd Signals
Odd Signal
Even Signal
Signals & Systems Lecture 3
5
6. 30 March 2023
Veton Këpuska
6
Even and Odd Signals
Any signal can be expressed as the sum of even part
and on odd part:
)
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2
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2
1
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(
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
o
e
o
e
o
e
o
e
7. Find Even and Odd Components
Expression?
Veton Këpuska 30 March
2023 7
10. • Are there sets of “basic” signals, xk[n], such that:
We can represent any signal as a linear combination (e.g, weighted sum) of these
building blocks? (Hint: Recall Fourier Series.)
The response of an LTI system to these basic signals is easy to compute and provides
significant insight.
• For LTI Systems (CT or DT) there are two natural choices for these building blocks:
Later we will learn that there are many families of such functions: sinusoids,
exponentials, and even data-dependent functions. The latter are extremely useful in
compression and pattern recognition applications.
Exploiting Superposition and Time-Invariance
DT LTI
System
[ ] [ ]
k k
k
x n a x n
k
k
k n
y
b
n
y ]
[
]
[
DT Systems:
(unit pulse)
CT Systems:
(impulse)
0
t
t
0
n
n
12. Convolution
• Mixing of Two Signals
• Convolution is a mathematical operation on two functions
(f and g) that produces a third function
Applications:
i. Edge Detection
ii. Region Detection
iii. Radar Technology
Signals & Systems Lecture 3
12
13. Response of a DT LTI Systems – Convolution
• Define the unit pulse response, h[n], as the response of a DT LTI system to a unit
pulse function, [n].
• Using the principle of time-invariance:
• Using the principle of linearity:
• Comments:
Recall that linearity implies the weighted sum of input signals will produce a
similar weighted sum of output signals.
Each unit pulse function, [n-k], produces a corresponding time-delayed version
of the system impulse response function (h[n-k]).
The summation is referred to as the convolution sum.
The symbol “*” is used to denote the convolution operation.
DT LTI
k
k
k n
x
a
n
x ]
[
]
[
k
k
k n
y
b
n
y ]
[
]
[
n
h
]
[
]
[
]
[
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[ k
n
h
k
n
n
h
n
]
[
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[
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[
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[
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[
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[
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[
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[ n
h
n
x
k
n
h
k
x
n
y
k
n
k
x
n
x
k
k
convolution sum
convolution operator
14. LTI Systems and Impulse Response
• The output of any DT LTI is a convolution of the input signal with the unit pulse
response:
• Any DT LTI system is completely characterized by its unit pulse response.
• Convolution has a simple graphical interpretation:
DT LTI
]
[n
x ]
[
*
]
[
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[ n
h
n
x
n
y
n
h
]
[
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[
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[ n
h
n
x
k
n
h
k
x
n
y
k
n
k
x
n
x
k
k
15. Discrete Time Convolution
Two Main Methods
1. Graphical Method
2. Summation Method
3. Checking Method
Signals & Systems Lecture 3
15
20. Representation of output signal y[n]
1. Draw the output signal
2. Represent the signal Mathematically
y[n]= [1, 2, 3,0,0,0,0,0,0,1]
Signals & Systems Lecture 3
20
30. • Convolution method used for both the discrete
and continuous time signals
• The methods (graphical and summation) cannot
be employed when one or both of signals
involving in convolution sum approaches infinity
• Analytical convolution specializes in convolving
the two signals in which one or both the signals
approaches infinity
Signals & Systems Lecture 3
30
Analytical Evaluation of the Convolution
31. 31
Analytical Evaluation of the Convolution
otherwise
N
n
N
n
u
n
u
n
h
0
1
0
1
Convolve the following two signals.
k
y n x k h n k
Find the output at index n
n
u
a
n
x n
input
1
a
32. 3/30/2023
32
Analytical Evaluation of the Convolution
otherwise
N
n
N
n
u
n
u
n
h
0
1
0
1
h(k)
0
n
u
a
n
x n
input
1
a
33. Steps:
1. Draw both the signals carefully
2. Understand the formula
3. Change the domain of signals from “n” to k (replace any
“n” in amplitude of the signal with “k”)
4. Flip signal of your choice? Why [Commutative]
5. Shift the flipped signal “n” locations to make it, say x[n-k]
6. Start shifting the flipped signal
7. Whenever the two signals overlap, calculate convolution
sum
Signals & Systems Lecture 3 33
Analytical Evaluation of the Convolution
k
y n x k h n k
34. 34
0
，
otherwise
N
n
n
h
0
1
0
1
n
u
a
n
x n
h(k)
0
0
h(n-k)
x(k)
h(-k)
0
0
n ，
k
y n x k h n k
35. 3/30/2023
35 Zhongguo Liu_Biomedical Engineering_Shandong Univ.
1 0
0, 0 1
n n N n N
a
a
a
n
k
h
k
x
n
y
n
n
k
k
n
k
1
1 1
0
0
h(-k)
0
h(k)
0
h(n-k)
x(k)
36. 36
1 1
n n
k
k n N k n N
y n x k h k n a
h(-k)
0
h(k)
0
1 0 1
n N n N
1 1
1 1
1 1
n N n N
n N
a a a
a
a a
37. Analytical Evaluation of Convolution Sum
Determine the output of Linear Time Invariant System if the
input x[n] and h[n] are shown below:
Signals & Systems Lecture 3
37
𝑥[𝑛 = 𝜇[𝑛 ℎ[𝑛 = 𝑎𝑛𝜇[−𝑛 − 1
𝑥[𝑛 = 𝜇[𝑛 − 4 ℎ[𝑛 = 2𝑛
𝜇[−𝑛 − 1
![Even/Odd Signals
Even
Odd
Any signal can be discomposed into a sum of an
even and an odd
]
)
(
)
(
[
2
1
)
(
,
)]
(
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(
[
2
1
)
( 2
1 t
x
t
x
t
x
t
x
t
x
t
x
]
[
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[
,
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(
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( n
x
n
x
t
x
t
x
]
[
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[
,
)
(
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( n
x
n
x
t
x
t
x
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