SIGNALS BASICS (PART 2)

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UNIVERSIDAD YACHAY TECH
SCHOOL OF BIOLOGICAL SCIENCES AND ENGINEERING
BIOMEDICAL ENGINEERING
SIGNAL THEORY AND MEDICAL BIOCOMPUTATION
HIGHER MATHEMATICS FOR ENGINEERING
Ing. Diego Almeida Ph.D
June
2022
01
CLASS 08: SIGNALS BASICS (PART 2)
03
Basic signals in discrete 
time
Energy signals and power 
signals
Solving Task II
03
BASIC SIGNALS IN DISCRETE TIME
The unit impulse function, δ[n], also known as the Kronecker delta function, is
defined as:
01 02 03 04 05
Unit impulse function in discrete time :
𝜹 𝒏 = ቊ
1 𝑛 = 0
0 𝑛 ≠ 0
04
01
Remember: Delta de Dirac in continuous time:
Other cases
Area
• Parameterized function:
න
−∞
∞
𝜹(𝒕) = ቊ
1 𝑡 = 0
0 𝑡 ≠ 0
05
01
Unit impulse function in discrete time:
Meets the following properties:
06
01
Unit step function in discrete time:
The unit step function, u[n], is defined as:
𝓾 𝒏 = ቊ
1 𝑛 ≥ 0
0 𝑛 < 0
07
01
Decomposition of a discrete signal:
Suppose the signal x [n] given in the figure. We can decompose it as:
x 𝒏 = σ𝒌=−∞
∞ 𝒙 𝒌 𝜹 𝒏 − 𝒌
08
ENERGY SIGNALS AND POWER SIGNALS:
01 02 03 04 05
Energy Signal:
The energy (E) of a signal x (t) is defined as:
The signal x(t) is a finite energy signal (or simply energy signal) if and only if: 
0 < 𝐸𝑋(𝑡) < ∞ → 𝑃 = 0
09
Power signal:
02
The power (P) of a signal x (t) is defined as:
The signal x(t) is a finite energy signal (or simply energy signal) if and only if: 
0 < 𝑃𝑋(𝑡) < ∞ → 𝐸 = ∞
For periodic signals the power can be calculated by integrating the signal into a period:
010
Important notes:
• The signal x(t) that does not satisfy either of the two relationships is neither of finite
energy nor of finite power. Therefore, it is not a power or energy signal.
0 < 𝐸𝑋(𝑡) < ∞
0 < 𝑃𝑋(𝑡) < ∞
• Periodic signals are power signals.
• The unit step is a power signal (its energy is infinite).
• The unit impulse is an energy signal (its average power is 0).
• The increasing exponential signal is a signal with infinite power and energy.
• Finite signals are energy signals
02
011
Summary:
02
Continuous time Discrete time
Energy in a time interval: 
Energy in a time interval: 
Power in a time interval: 
Total energy of a signal: 
Total power of a signal: 
Power in a time interval: 
Total energy of a signal: 
Total power of a signal: 𝑁
012
Example 1:
Determine whether or not each the following signals is power or energy signals:
02
T
• Periodic signal. Period T=4
𝟎 < 𝑷𝑿(𝒕) < ∞ → Power signal
𝑷 =
𝟏
𝑻
׬
𝟎
𝑻
𝒙(𝒕) 𝟐𝒅𝒕
𝑃 =
1
𝑇
න
0
1
(−2) 2𝑑𝑡 +
1
𝑇
න
2
3
(2) 2𝑑𝑡
𝑃 =
1
4
0׬
1
4𝑑𝑡 +
1
4
2׬
3
4𝑑𝑡
𝑃 = 1 + 1 = 2
1)
013
Example 2:
02
2)
• Total Energy:
𝟎 < 𝑬𝑿(𝒕) < ∞ → Energy signal
𝑬 = ∞−׬
∞
𝒙(𝒕) 𝟐𝒅𝒕
𝐸 = ׬
0
∞
(2𝑒−𝑡) 2𝑑𝑡
𝐸 = ׬
0
∞
4𝑒−2𝑡𝑑𝑡
𝐸 = 4
4𝑒−2𝑡
−2
∞
0
𝐸 = 4 0 − −
1
2
= 2
𝑃 = lim
𝑇→∞
1
𝑇
𝐸 = 0
014
TASK II – SIGNALS AND NOISE:
01 02 03 04 05
Determine is the signal are analog, digital, continuous and discrete 
in time? :
015
Solving task II:
03
Determine is the signal are analog, digital, continuous and discrete 
in time? :
Continuous signal in time - Analog signal. Discrete signal in time - Digital signal.
016
Solving task II: 
03
Determine whether or not each the following signals is periodic 
through using fundamental period:
a) 𝑥(𝑡) = cos(𝑤𝑜𝑡)
017
Solving task II: 
03
Determine whether or not each the following signals is periodic 
through using fundamental period:
a) 𝑥(𝑡) = cos(𝑤𝑜𝑡)
−
018
Solving task II: 
03
Determine whether or not each the following signals is periodic 
through using fundamental period:
b) 𝑥(𝑡) = 𝒆𝑗𝒘𝒐𝑡, use hint: 𝒆𝑗𝒘𝒐𝑡 = cos(𝑤𝑜𝑡) + 𝑗 sin(𝑤𝑜𝑡))
019
Solving task II: 
03
Determine whether or not each the following signals is periodic 
through using fundamental period:
b) 𝑥(𝑡) = 𝒆𝑗𝒘𝒐𝑡, use hint: 𝒆𝑗𝒘𝒐𝑡 = cos(𝑤𝑜𝑡) + 𝑗 sin(𝑤𝑜𝑡))
020
03
Solving task II: 
Determine whether or not each the following signals is periodic 
through using fundamental period:
c) (𝑡) = cos (𝑡 + π/4)
021
03
Solving task II: 
Determine whether or not each the following signals is periodic 
through using fundamental period:
c) (𝑡) = cos (𝑡 + π/4)
−
022
03
Solving task II: 
Determine whether the signal is periodic or aperiodic signal?
a. 𝑥(𝑡) = 𝑠𝑖𝑛 ( 
𝟐𝝅
𝟑
𝑡) 
023
03
𝑻𝟏
𝑻𝟐
=
𝒎
𝒏
→ 𝑻 = 𝑻𝟏 ∗ 𝒏 = 𝑻𝟐 ∗ 𝒎 𝒎,𝒏 ∈ 𝒁
Solving task II: 
Determine whether the signal is periodic or aperiodic signal?
b. 𝑥(𝑡) = cos ( 
𝝅
𝟑
𝑡) + sin ( 
𝝅
𝟒
𝑡) 
024
03
𝑻𝟏
𝑻𝟐
=
𝒎
𝒏
→ 𝑻 = 𝑻𝟏 ∗ 𝒏 = 𝑻𝟐 ∗ 𝒎 𝒎,𝒏 ∈ 𝒁
𝑿𝟏 𝒕 = sin
𝝅
𝟑
𝑡 → 𝑻𝟏 =
𝟐𝝅
𝝎𝒐
= 𝟔
𝑿𝟐 𝒕 = sin
𝝅
𝟒
𝑡 → 𝑻𝟐 =
𝟐𝝅
𝝎𝒐
= 𝟖
𝑿 𝒕 = 𝑿𝟏 𝒕 + 𝑿𝟐 𝒕 →
𝑻𝟏
𝑻𝟐
=
𝟔
𝟖
=
𝟑
𝟒
𝑋 𝑡 is periodic → 𝑇 = 𝑇1 ∗ 4 = 𝑇2 ∗ 3 = 24
Solving task II: 
Determine whether the signal is periodic or aperiodic signal?
b. 𝑥(𝑡) = cos ( 
𝝅
𝟑
𝑡) + sin ( 
𝝅
𝟒
𝑡) 
025
03
Solving task II: 
Given the signal x(t) shown in Figure, obtain: a) x(t + 1); 
b) x(-t + 1); and c) x(3/2 t).
026
03
Solving task II: 
Given the signal x(t) shown in Figure, obtain: a) x(t + 1); 
b) x(-t + 1); and c) x(3/2 t).
a. x(t + 1): 
b. x(-t + 1):
c. x(3/2 t):
027
Solving task II (power and energy signals):
01 02 03 04 05
Determine whether or not each the following signals is power, 
energy or neither:
a) 𝑥(𝑡) = 𝑒−𝑎𝑡 𝑢(𝑡) 
028
04
Solving task II (power and energy signals):
Determine whether or not each the following signals is power, 
energy or neither:
a) 𝑥(𝑡) = 𝑒−𝑎𝑡 𝑢(𝑡) • Total Energy:
𝟎 < 𝑬𝑿(𝒕) < ∞ → Energy signal
𝑬 = ∞−׬
∞
𝒙(𝒕) 𝟐𝒅𝒕
𝐸 = 0׬
∞
(𝑒−𝑎𝑡) 2𝑑𝑡
𝐸 = 0׬
∞
𝑒−2𝑎𝑡𝑑𝑡
𝐸 =
𝑒−2𝑎𝑡
−2𝑎
∞
0
𝐸 = 0 − −
1
2𝑎
=
1
2𝑎
𝑃 = lim
𝑇→∞
1
𝑇
𝐸 = 0
Muchas gracias
Diego Almeida