CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
1 SIGNALS AND SYSTEMS
1.0.
Introduction
1.1.
Continuous-Time and Discrete-Time Signals
1.1.1. Examples and Mathematical Representations
1.1.2. Signal Power and Energy
1.2.
Transformations of the Independent Variable
1.2.1. Examples of the Transformations of the Independent Variable
1.2.2. Periodic Signals
1.2.3. Even and Odd Signals
1.3.
Exponential and Sinusoidal Signals
1.3.1. Continuous-Time Complex Exponential and Sinusoidal Signals
1.3.2. Discrete-Tine Complex Exponential and Sinusoidal Signals
1.3.3. Periodicity Properties of Discrete-Time Complex Exponentials
1.4.
The Unit Impulse and Unit Step Functions
1.4.1. The Discrete-Time Unit Impulse and Unit Step Sequences
1.4.2. The Continuous-Time Unit Step and Unit Impulse Functions
1.5.
Continuous-Time and Discrete-Time Systems
1.5.1. Simple Examples of Systems
1.5.2. Interconnections of Systems
1.6.
Basic System Properties
1.6.1. Systems with and without Memory
1.6.2. Invertibility and Inverse Systems
1.6.3. Causality
1.6.4. Stability
1.6.5. Time Invariance
1.6.6. Linearity
1.7.
Summary
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
1.1.2
Signal Power and Energy
The total energy over the time internal 𝑡1 ≤ 𝑡 ≤ 𝑡2 in a continuous-time signal 𝑥(𝑡) is defined as
𝑡2
∫ |𝑥(𝑡)|2 𝑑𝑡
(1.4)
𝑡1
where |𝑥| denotes the magnitude of the (possibly complex) number 𝑥.
The time averaged power is given by
𝑡2
1
∫ |𝑥(𝑡)|2 𝑑𝑡
(𝑡2 − 𝑡1 )
(1.4-1)
𝑡1
Similarly, the total energy in a discrete-time signal 𝑥[𝑛] over the time interval 𝑛1 ≤ 𝑛 ≤ 𝑛2 is
defined as
𝑛2
∑ |𝑥[𝑡]|2
(1.5)
𝑛=𝑛1
The average power over the interval in this case is given by
𝑛2
1
∑ |𝑥[𝑡]|2
𝑛2 − 𝑛1 + 1
(1.5-1)
𝑛=𝑛1
It is interesting to find out the power and energy in a signal over an infinite time interval, i.e.,
for −∞ < 𝑡 < +∞ or for −∞ < 𝑛 < +∞. In these cases the total energy and power carried out
by the signal are given by:
For continuous-time (CT) case:
𝑇
Total Energy
∞
𝐸∞ ≜ lim ∫|𝑥(𝑡)|2 𝑑𝑡 = ∫ |𝑥(𝑡)|2 𝑑𝑡
𝑇→∞
−𝑇
Total Average Power
𝑇
(1.6)
−∞
1
∫|𝑥(𝑡)|2 𝑑𝑡
𝑇→∞ 2𝑇
𝑃∞ ≜ lim
(1.8)
−𝑇
For discrete-time (DT) case:
+𝑁
Total Energy
𝐸∞ ≜ lim ∑
𝑁→∞
𝑛=−𝑁
+∞
|𝑥[𝑛]|2
= ∑ |𝑥[𝑛]|2
(1.6)
𝑛=−∞
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
+𝑁
1
𝑃∞ ≜ lim
∑ |𝑥[𝑛]|2
𝑁→∞ 2𝑁 + 1
Total Average Power
(1.8)
𝑛=−𝑁
Three important cases can be identified:
Case 1: Signals with finite total energy, i.e., 𝑬∞ < ∞:
Such a signal must have zero average power. For example, in continuous case, if 𝐸∞ <
∞, then
𝐸∞
=0
𝑇→∞ 2𝑇
𝑃∞ = lim
An example of a finite-energy signal is a signal that takes on the value of 1 for 0 ≤ 𝑡 ≤ 1
and 0 otherwise. In this case, 𝐸∞ = 1 and 𝑃∞ = 0.
Case 2: Signals with finite average power, i.e., 𝑷∞ < ∞:
For example, consider the constant signal where 𝑥[𝑛] = 4. This signal has infinite
energy, as
+𝑁
𝐸∞ = lim ∑
𝑁→∞
𝑛=−𝑁
+𝑁
|𝑥[𝑛]|2
= lim ∑ 42 = ⋯ + 16 + 16 + 16 …
𝑁→∞
𝑛=−𝑁
However, the total average power is finite,
+𝑁
+𝑁
𝑛=−𝑁
𝑛=−𝑁
1
1
𝑃∞ ≜ lim
∑ |𝑥[𝑛]|2 = lim
∑ 42
𝑁→∞ 2𝑁 + 1
𝑁→∞ 2𝑁 + 1
+𝑁
16
= lim
∑ 1
𝑁→∞ 2𝑁 + 1
𝑛=−𝑁
16(2𝑁 + 1)
𝑁→∞
2𝑁 + 1
= lim
= lim 16 = 16
𝑁→∞
Case 3: Signals with neither 𝑬∞ nor 𝑷∞ finite:
A simple example of such a case could be 𝑥(𝑡) = 𝑡. In this case both 𝐸∞ and 𝑃∞ are infinite.
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
1.2 Transformations of the Independent Variable
The transformation of a signal is one of the central concepts in the field of signals and systems.
We will focus on a very limited but important class of signal transformations that involves the
modifications of the independent variable, i.e., the time axis.
1.2.1
Examples of Transformations of the Independent Variable
Time Shift
The original and the shifted signals are identical in shape, but are displaced or shifted along the
time-axis with respect to each other. Signals could be termed as delayed or advanced in this
case.
Signals
Discrete-Time (DT)
Continuous-Time (CT)
𝒙[𝒏]
𝒙(𝒕)
𝒙[𝒏 − 𝒏𝟎 ]
𝒙(𝒕 − 𝒕𝟎 )
Original
Delayed
𝟎
𝒏
𝒙[𝒏 + 𝒏𝟎 ]
𝟎
𝒕
𝒙(𝒕 + 𝒕𝟎 )
Advanced
𝟎
𝒏
𝟎
𝒕
Such signals arise in applications such as radar, sonar and seismic signal processing. Several
receivers placed at different locations receive the time shifted signals due to the transmission
time they take while passing through a medium (air, water or rock etc.).
𝟎
𝒏
𝟎
𝒕
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
Time Reversal (Reflection)
In this case, the original signal is reflected about the time = 0. For example, if the original signal
is some audio recording, then the time reversed signal would be the audio recording played
backward.
Signals
Discrete-Time (DT)
Continuous-Time (CT)
𝒙[𝒏]
𝒙(𝒕)
𝒙[−𝒏]
𝒙(−𝒕)
Original
Time Reversed
𝟎
𝒏
𝟎
𝒕
Time Scaling
𝟎
𝒕
In this case, if the original signal
is 𝑥(𝑡), the time𝒏 variable is multiplied𝟎 with a constant to
get a
time-scaled signal, e.g., 𝑥(2𝑡), 𝑥(5𝑡), or 𝑥(𝑡⁄2). If we think of the signal 𝑥(𝑡) as audio
recording, then 𝑥(2𝑡) is the audio recording played at twice the speed and 𝑥(𝑡⁄2) is the
recording played at half of the speed.
Signals
Continuous-Time (CT)
𝒙(𝒕)
Original
𝒙(𝒕⁄𝟐)
Stretching
𝟎
𝒙(𝟐𝒕)
𝒕
𝟎
𝒕
Compressing
𝟎
Page 5
𝒕
Department of Computer Engineering
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
General Case of the Transformation of the Independent Variable
A general case for the transformation of independent variable is the one in which for the
original signal 𝑥(𝑡) is changed to the form 𝑥(𝛼𝑡 + 𝛽), where 𝛼 and 𝛽 are given numbers. It has
the following effects on the original signal:






The general shape of the signal is preserved.
The signal is linearly stretched if |𝛼| < 1.
The signal is linearly compressed if |𝛼| > 1.
The signal is delayed (shifted in time) if 𝛽 < 0.
The signal is advanced (shifted in time) if 𝛽 > 0.
The signal is reversed in time (reflected) if 𝛼 < 0.
Example 1.1
Given the signal 𝑥(𝑡) as shown in the figure below, the signal 𝑥(𝑡 + 1) can be obtained by
shifting the given signal to the left by one unit (as depicted in the following figure).
𝒙(𝒕)
Original Signal
−𝟏
𝟎
𝟏
𝟐
𝒕
𝒙(𝒕 + 𝟏)
Time Shifted Signal
−𝟏
𝟎
𝟏
𝟐
𝒕
The original signal can be written mathematically as:
0
1
𝑥(𝑡) = {
2−𝑡
0
𝑖𝑓
𝑡<0
𝑖𝑓 0 ≤ 𝑡 < 1
𝑖𝑓 1 ≤ 𝑡 < 2
𝑡≥2
𝑖𝑓
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
To determine 𝑥(𝑡 + 1), the above definition of the original signal can be used. For example, to
calculate the value of 𝑥(𝑡 + 1) at 𝑡 = 0, we determine 𝑥(1), which according to the above
definition is 𝑥(1) = 1, therefore, 𝑥(𝑡 + 1) at 𝑡 = 0 is equal to 1.
Similarly, the signal 𝑥(−𝑡 + 1) can be obtained using the mathematical definition or figure of
the original signal 𝑥(𝑡). If we use the mathematical definition, then making the following table
could be useful.
𝒕
−𝟐
−𝟏. 𝟓
−𝟏
−𝟎. 𝟓
𝟎
𝟎. 𝟓
𝟏
𝟏. 𝟓
𝟐
𝟐. 𝟓
𝟑
−𝒕 + 𝟏
3.0
2.5
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
−1.5
−2.0
𝒙(−𝒕 + 𝟏)
0
0
0
0.5
1
1
1
0
0
0
0
Plotting of this signal would give us the following figure:
𝒙(𝒕)
Original Signal
−𝟏
𝟎
𝟏
𝟐
𝒕
𝒙(−𝒕 + 𝟏)
Time Reversed Signal
−𝟏
𝟎
𝟏
𝟐
𝒕
In MATLAB®, the original signal can be written as an inline function. This function can then be
used to plot the original signal, the shifted signal and the time-reversed signal using the
following MATLAB® code.
>> g = inline(' ((t>=0)&(t<1)) + (2-t).*((t>=1) & (t<2))','t');
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
>> t = -3:0.001:3;
>> subplot(3,1,1), plot(t,g(t)), axis([-3 3 -0.1 1.1]), title('Original Signal')
>> subplot(3,1,2), plot(t,g(t+1)), axis([-3 3 -0.1 1.1]), title('Time-Shifted Signal')
>> subplot(3,1,3),plot(t,g(-t+1)),axis([-3 3 -0.1 1.1]), title('Time-Reversed Signal')
Original Signal
1
0.5
0
-3
-2
-1
0
1
2
3
1
2
3
1
2
3
Time-Shifted Signal
1
0.5
0
-3
-2
-1
0
Time-Reversed Signal
1
0.5
0
-3
-2
-1
0
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
1.2.2
Periodic Signals
A periodic continuous-time signal 𝑥(𝑡) is defined as
𝑥(𝑡) = 𝑥(𝑡 + 𝑇)
where 𝑇 is a positive number called the period.
A typical example is that of a sinusoidal signal 𝑥(𝑡) = sin(𝑡) for −∞ < 𝑡 < +∞. This is shown
in the figure below:
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
For the above signal, the period is 𝑇 = 2𝝅. It can be noticed that for any time 𝑡:
𝒔𝒊𝒏(𝒕 + 𝟐𝝅) = 𝒔𝒊𝒏(𝒕)
Also
𝒔𝒊𝒏(𝒕 + 𝒎𝟐𝝅) = 𝒔𝒊𝒏(𝒕)
where 𝒎 is an integer.
The fundamental period 𝑇0 of 𝑥(𝑡) is the smallest positive value of 𝑇 for
which the equation 𝑥(𝑡) = 𝑥(𝑡 + 𝑇) holds.
Periodic signals are defined analogously in discrete-time.
A discrete-time signal 𝑥[𝑛] is periodic with period 𝑁, where 𝑁is a positive
integer, if it is unchanged by a time-shift of 𝑁, i.e., if
𝑥[𝑛] = 𝑥[𝑛 + 𝑁]
for all values of 𝑛.
The fundamental period 𝑁0 of 𝑥[𝑛] is the smallest positive value of 𝑁 for
which the equation 𝑥[𝑛] = 𝑥[𝑛 + 𝑁0 ] holds.
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
An example of a discrete-time periodic signal is given in the following diagram.
𝒙[𝒏]
𝒏
1.2.3
Even and Odd Signals
A signal 𝑥(𝑡) or 𝑥[𝑛] is defined as an even signal if it is identical to its time-reversed
counterpart, i.e., with its reflection about the origin.
𝒙(−𝒕) = 𝒙(𝒕)
𝒙[−𝒏] = 𝒙[𝒏]
Even continuous-time Signal
Even Discrete-time Signal
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
A signal 𝑥(𝑡) or 𝑥[𝑛] is defined as an odd signal if,
𝒙(−𝒕) = −𝒙(𝒕)
Odd continuous-time Signal
𝒙[−𝒏] = −𝒙[𝒏]
Odd Discrete-time Signal
As a special case, the odd signal must be zero at 𝑡 = 0 or 𝑛 = 0. Examples of odd signals are
given in the following figure.
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
An important fact is that any signal (continuous-time or discrete-time) can be broken into a sum
of two signals, one of which is even and one of which is odd.
Signal
Component
Mathematical Form
Even Part
1
ℰ𝑣 {𝑥(𝑡)} = [𝑥(𝑡) + 𝑥(−𝑡)]
2
1
𝒪𝑑 {𝑥(𝑡)} = [𝑥(𝑡) − 𝑥(−𝑡)]
2
1
ℰ𝑣 {𝑥[𝑛]} = [𝑥[𝑛] + 𝑥[−𝑛]]
2
1
𝒪𝑑 {𝑥[𝑛]} = [𝑥[𝑛] − 𝑥[−𝑛]]
2
Continuous-time Signal 𝒙(𝒕)
Odd Part
Even Part
Discrete-time Signal 𝒙[𝒏]
Odd Part
A Discrete-Time Signal
𝟏, 𝒏 ≥ 𝟎
𝒙[𝒏] = {
𝟎, 𝒏 < 0
𝟏
−𝟑 – 𝟐 − 𝟏 𝟎 𝟏
𝟐 𝟑
𝒏
Odd Part
Even Part
𝟏⁄𝟐 , 𝒏 < 0
ℰ𝑣 [𝒏] = { 𝟏, 𝒏 = 𝟎
𝟏⁄𝟐 , 𝒏 > 0
𝟏
−𝟏⁄𝟐 , 𝒏 < 0
𝒪𝑑 [𝒏] = { 𝟎, 𝒏 = 𝟎
𝟏⁄𝟐 , 𝒏 > 0
𝟏
𝟏
𝟐
−𝟑 – 𝟐 − 𝟏 𝟎 𝟏
−𝟑 − 𝟐 – 𝟏
𝟐 𝟑
𝟏
𝟐
𝟎 𝟏
𝒏
𝟐 𝟑
𝒏
𝟏
−
𝟐
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
1.3 Exponential and Sinusoidal Signals
1.3.1 Continuous-Time Complex Exponential and Sinusoidal
Signals
A continuous-time complex signal 𝑥(𝑡) can be written as
𝑥(𝑡) = 𝐶𝑒 𝑎𝑡
where 𝐶 and 𝑎 are, in general, complex numbers.
(1.20)
Real Exponential Signals
In this case both 𝐶 and 𝑎 are real numbers, and 𝑥(𝑡) is called a real exponential.
Continuous-time Real Exponential with a>0
Continuous-time Real Exponential with a<0
x(t)=Ceat, C>1, a>1
x(t)=Ceat, C>0, a<0
C
C
t
t
Periodic Complex Exponential and Sinusoidal Signals
Now we consider the case of complex exponentials where 𝑎 is purely imaginary. More,
specifically, we consider:
𝑥(𝑡) = 𝑒 𝑗𝜔0 𝑡
(1.21)
An important property of this signal is that it is periodic. Let’s explore this.
By definition 𝑥(𝑡) is periodic, if
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
𝑥(𝑡) = 𝑥(𝑡 + 𝑇)
for some real value of 𝑇. Therefore,
𝑒 𝑗𝜔0 𝑡 = 𝑒 𝑗𝜔0 (𝑡+𝑇)
Or,
𝑒 𝑗𝜔0 𝑡 = 𝑒 𝑗𝜔0 𝑡 𝑒 𝑗𝜔0 𝑇
⇒ 𝑒 𝑗𝜔0 𝑇 = 1
(1.23)
This equation can be true,
1
2
If, 𝜔0 = 0, then 𝑥(𝑡) = 1, which is periodic for any value of 𝑇.
If, 𝜔0 ≠ 0, then the fundamental period 𝑇0 of 𝑥(𝑡), i.e. the smallest value of 𝑇 for
which equation (1.23) holds, is
𝑇0 =
2𝜋
|𝜔0 |
(1.24)
Replacing the value of 𝑇with this 𝑇0 in equation (1.23), and using Euler’s formula,
that is,
𝑒 𝑗𝜔0 𝑇 = cos(𝜔0 𝑇) + 𝑗sin(𝜔0 𝑇)
We get
𝑒 𝑗𝜔0 𝑇 = cos(2𝜋) + 𝑗 sin(2𝜋) = 1 + 𝑗0 = 1
Therefore, the signal 𝑥(𝑡) as given by equation (1.21) is a periodic signal. Similarly, it can be
shown that the signal 𝑥(𝑡) = 𝑒 −𝑗𝜔0 𝑡 has the same fundamental period.
A closely associated signal is the sinusoidal signal
𝑥(𝑡) = 𝐴 cos(𝜔0 𝑡 + 𝜙)
(1.25)
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 02
Date: June 11, 2012
Continuous-Time Sinusoidal Signal
x(t)=A cos( 0 t+  )
A
A cos(  )
t
Using Euler’s formula, the complex exponential of equation (1.21) can be written as
𝑥(𝑡) = 𝑒 𝑗𝜔0 𝑡 = cos(𝑗𝜔0 𝑡) + 𝑗 sin(𝑗𝜔0 𝑡)
(1.26)
Similarly, the sinusoidal signal of equation (1.25) can be written as
𝐴 cos(𝜔0 𝑡 + 𝜙) = 𝐴 (
𝑒 𝑗(𝜔0 𝑡+𝜙) + 𝑒 −𝑗(𝜔0 𝑡+𝜙)
𝐴
𝐴
) = 𝑒 𝑗𝜙 𝑒 𝑗𝜔0 𝑡 + 𝑒 −𝑗𝜙 𝑒 −𝑗𝜔0 𝑡
2
2
2
(1.27)
It can noted that the two exponentials in equation 1.27 have complex amplitudes. If we define
ℜℯ{𝑐} as the real component of a complex number 𝑐, and ℑ𝓂{𝑐} as the imaginary part of 𝑐,
then
𝐴 cos(𝜔0 𝑡 + 𝜙) = 𝐴 ℜℯ{𝑒 𝑗(𝜔0 𝑡+𝜙) }
𝐴 sin(𝜔0 𝑡 + 𝜙) = 𝐴 ℑ𝓂{𝑒 𝑗(𝜔0 𝑡+𝜙) }
(1.28)
(1.29)
Frequency and Period
From equation 1.24, it is clear that the fundamental period 𝑇0 of a continuous-time sinusoidal
or a periodic complex exponential signal, is inversely proportional to the |𝜔0 |, which is called
the fundamental frequency. If we decrease the value of the magnitude of 𝜔0 , we slow down the
rate of oscillations and hence increase the period 𝑇0 . Alternatively, if we increase the value of
the magnitude of 𝜔0 , we increase the rate of oscillations and hence decrease the period 𝑇0 .
Energy and Power
Over the one fundamental period 𝑇0 of a continuous-time sinusoidal or a periodic complex
exponential signal, the signal energy and power can be determined as:
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Date: June 11, 2012
𝑇0
𝑇0
𝐸𝑝𝑒𝑟𝑖𝑜𝑑 = ∫ |𝑒
𝑇0
𝑃𝑝𝑒𝑟𝑖𝑜𝑑 =
𝑗𝜔0 𝑡 2
(1.30)
| 𝑑𝑡 = ∫ 1𝑑𝑡 = 𝑇0
0
0
𝑇0
1
1
𝑇0
2
∫ |𝑒 𝑗𝜔0 𝑡 | 𝑑𝑡 = ∫ 1𝑑𝑡 = = 1
𝑇0
𝑇0
𝑇0
0
(1.31)
0
As there are an infinite number of periods as 𝑡 ranges from −∞ to +∞, the total energy
integrated over all time is infinite. The total average power is however remains 1, as by
definition,
𝑇
1
1
2
𝑃∞ = lim
∫|𝑒 𝑗𝜔0 𝑡 | 𝑑𝑡 = lim
2𝑇 = 1
𝑇→∞ 2𝑇
𝑇→∞ 2𝑇
(1.32)
−𝑇
Harmonics of a Periodic Complex Exponential
We have noted that,
𝑒 𝑗𝜔𝑇0 = 1
(1.33)
which implies that 𝜔𝑇0 is a multiple of 2𝜋, i.e.,
𝜔𝑇0 = 2𝜋𝑘,
where 𝑘 = 0, ±1, ±2, ⋯
(1.34)
This shows that 𝜔 must be an integer multiple of 𝜔0 , i.e., the fundamental frequency. We can
therefore, write
𝜙𝑘 (𝑡) = 𝑒 𝑗𝑘𝜔0 𝑡 ,
where 𝑘 = 0, ±1, ±2, ⋯
(1.36)
This is called the k-harmonic of the complex exponential signal.
General Complex Exponential Signals
The general complex exponential signals are of the form
𝑥(𝑡) = 𝐶𝑒 𝑎𝑡
Where both 𝐶 and 𝑎 are complex numbers. Let us represent them as
𝐶 = |𝐶|𝑒 𝑗𝜃
𝑎 = 𝑟 + 𝑗𝜔0
Thus 𝐶 is written in polar form and 𝑎 is written in Cartesian form, then
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𝑥(𝑡) = 𝐶𝑒 𝑎𝑡 = |𝐶|𝑒 𝑗𝜃 𝑒 (𝑟+𝑗𝜔0 )𝑡 = |𝐶|𝑒 𝑟𝑡 𝑒 𝑗(𝜔0 𝑡+𝜃)
(1.42)
Using Euler’s formula, it can be written as
𝑥(𝑡) = 𝐶𝑒 𝑎𝑡 = |𝐶|𝑒 𝑟𝑡 cos(𝜔0 𝑡 + 𝜃) + 𝑗|𝐶|𝑒 𝑟𝑡 sin(𝜔0 𝑡 + 𝜃)
1
2
3
(1.43)
For 𝑟 = 0, the real and imaginary parts of a complex exponential are sinusoidal.
For 𝑟 > 0, they correspond to sinusoidal signals multiplied with growing exponential.
For 𝑟 < 0, they correspond to sinusoidal signals multiplied with decreasing exponentials.
This is shown graphically on the next page.
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x(t)= |C| ertcos(0 t+  )
t
x(t)= |C| e-rtcos(0 t+  )
t
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1.2.4
Periodic Signals (Recap)
A periodic continuous-time signal 𝑥(𝑡) is defined as
𝑥(𝑡) = 𝑥(𝑡 + 𝑇)
where 𝑇 is a positive number called the period.
A typical example is that of a sinusoidal signal 𝑥(𝑡) = sin(𝑡) for −∞ < 𝑡 < +∞. This is shown
in the figure below:
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
For the above signal, the period is 𝑇 = 2𝝅. It can be noticed that for any time 𝑡:
𝒔𝒊𝒏(𝒕 + 𝟐𝝅) = 𝒔𝒊𝒏(𝒕)
Also
𝒔𝒊𝒏(𝒕 + 𝒎𝟐𝝅) = 𝒔𝒊𝒏(𝒕)
where 𝒎 is an integer.
The fundamental period 𝑇0 of 𝑥(𝑡) is the smallest positive value of 𝑇 for
which the equation 𝑥(𝑡) = 𝑥(𝑡 + 𝑇) holds.
Periodic signals are defined analogously in discrete-time.
A discrete-time signal 𝑥[𝑛] is periodic with period 𝑁, where 𝑁is a positive
integer, if it is unchanged by a time-shift of 𝑁, i.e., if
𝑥[𝑛] = 𝑥[𝑛 + 𝑁]
for all values of 𝑛.
The fundamental period 𝑁0 of 𝑥[𝑛] is the smallest positive value of 𝑁 for
which the equation 𝑥[𝑛] = 𝑥[𝑛 + 𝑁0 ] holds.
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Lecture No. 02
Date: June 11, 2012
An example of a discrete-time periodic signal is given in the following diagram.
𝒙[𝒏]
𝒏
1.2.5
Even and Odd Signals (Recap)
A signal 𝑥(𝑡) or 𝑥[𝑛] is defined as an even signal if it is identical to its time-reversed
counterpart, i.e., with its reflection about the origin.
𝒙(−𝒕) = 𝒙(𝒕)
𝑥[−𝑛] = 𝑥[𝑛]
Even continuous-time Signal
Even Discrete-time Signal
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
A signal 𝑥(𝑡) or 𝑥[𝑛] is defined as an odd signal if,
𝒙(−𝒕) = −𝒙(𝒕)
Odd continuous-time Signal
𝑥[−𝑛] = −𝑥[𝑛]
Odd Discrete-time Signal
As a special case, the odd signal must be zero at 𝑡 = 0 or 𝑛 = 0. Examples of odd signals are
given in the following figures.
𝒙(𝒕)
−𝟔𝝅
−𝟒𝝅
−𝟐𝝅
𝟎
𝟐𝝅
𝟒𝝅
𝟔𝝅
𝒕
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Lecture No. 02
Date: June 11, 2012
An important fact is that any signal (continuous-time or discrete-time) can be broken into a sum
of two signals, one of which is even and one of which is odd.
Signal
Component
Mathematical Form
Even Part
1
ℰ𝑣 {𝑥(𝑡)} = [𝑥(𝑡) + 𝑥(−𝑡)]
2
1
𝒪𝑑 {𝑥(𝑡)} = [𝑥(𝑡) − 𝑥(−𝑡)]
2
1
ℰ𝑣 {𝑥[𝑛]} = [𝑥[𝑛] + 𝑥[−𝑛]]
2
1
𝒪𝑑 {𝑥[𝑛]} = [𝑥[𝑛] − 𝑥[−𝑛]]
2
Continuous-time Signal 𝒙(𝒕)
Odd Part
Even Part
Discrete-time Signal 𝒙[𝒏]
Odd Part
A Discrete-Time Signal
𝟏, 𝒏 ≥ 𝟎
𝒙[𝒏] = {
𝟎, 𝒏 < 0
𝟏
−𝟑 – 𝟐 − 𝟏 𝟎 𝟏
𝟐 𝟑
𝒏
Odd Part
Even Part
𝟏⁄𝟐 , 𝒏 < 0
ℰ𝑣 [𝒏] = { 𝟏, 𝒏 = 𝟎
𝟏⁄𝟐 , 𝒏 > 0
𝟏
−𝟏⁄𝟐 , 𝒏 < 0
𝒪𝑑 [𝒏] = { 𝟎, 𝒏 = 𝟎
𝟏⁄𝟐 , 𝒏 > 0
𝟏
𝟏
𝟐
−𝟑 – 𝟐 − 𝟏 𝟎 𝟏
−𝟑 − 𝟐 – 𝟏
𝟐 𝟑
𝟏
𝟐
𝟎 𝟏
𝒏
𝟐 𝟑
𝒏
𝟏
−
𝟐
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1.3.2
Discrete-Time Complex Exponential and Sinusoidal
Signals
A discrete-time complex exponential signal or sequence 𝑥[𝑛] can be written as
𝑥[𝑛] = 𝐶𝛼 𝑛
where 𝐶 and 𝛼 are, in general, complex numbers. This could also be written as
(1.44)
𝑥[𝑛] = 𝐶𝑒 𝛽𝑛
(1.45)
where
𝛼 = 𝑒𝛽
The form given in equation (1.44) is more often used in discrete-time signal processing.
Real Exponential Signals
In this case both 𝐶 and 𝛼 are real numbers, and 𝑥[𝑛] is called a real exponential. We see
different types of behaviors, as shown below in the figure.
Discrete-time Real Exponential with 0<  <1
x[n]=C  n, C=1,  =0.9
Discrete-time Real Exponential with  >1
x[n]=C  n, C=1,  =1.1
n
n
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Discrete-time Real Exponential with -1<  <0
x[n]=C  n, C=1,  =-0.9
Discrete-time Real Exponential with  <-1
x[n]=C  n, C=1,  =-1.1
n
n
Sinusoidal Signals
Another important complex exponential is obtained by using the form given in equation 1.45,
and by considering 𝛽 to be purely imaginary (so that |𝛼| = 1). More, specifically, we consider:
𝑥[𝑛] = 𝑒 𝑗𝜔0 𝑛
(1.46)
A closely associated signal is the sinusoidal signal
𝑥[𝑛] = 𝐴 cos(𝜔0 𝑛 + 𝜙)
(1.47)
Using Euler’s formula, the complex exponential of equation (1.47) can be written as
𝑥[𝑛] = 𝑒 𝑗𝜔0 𝑛 = cos(𝜔0 𝑛) + 𝑗 sin(𝜔0 𝑛)
(1.48)
Similarly, the sinusoidal signal of equation (1.25) can be written as
𝑒 𝑗(𝜔0 𝑛+𝜙) + 𝑒 −𝑗(𝜔0 𝑛+𝜙)
𝐴
𝐴
𝐴 cos(𝜔0 𝑛 + 𝜙) = 𝐴 (
) = 𝑒 𝑗𝜙 𝑒 𝑗𝜔0 𝑛 + 𝑒 −𝑗𝜙 𝑒 −𝑗𝜔0 𝑛
2
2
2
(1.49)
It can noted that the two exponentials in equation 1.27 have complex amplitudes. If we define
ℜℯ{𝑐} as the real component of a complex number 𝑐, and ℑ𝓂{𝑐} as the imaginary part of 𝑐,
then
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𝐴 cos(𝜔0 𝑛 + 𝜙) = 𝐴 ℜℯ{𝑒 𝑗(𝜔0 𝑛+𝜙) }
𝐴 sin(𝜔0 𝑛 + 𝜙) = 𝐴 ℑ𝓂{𝑒 𝑗(𝜔0 𝑛+𝜙) }
(a) x[n]=cos(2 n/12)
n
(b) x[n]=cos(8 n/31)
n
(c) x[n]=cos(n/6)
n
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General Complex Exponential Signals
The general discrete-time complex exponential signals are of the form
𝑥[𝑛] = 𝐶𝛼 𝑛
where both 𝐶 and 𝛼 are complex numbers. Let us represent them as
𝐶 = |𝐶|𝑒 𝑗𝜃
𝛼 = |𝛼|𝑒 𝑗𝜔0
Thus 𝐶 and 𝛼 are written in polar form, then
𝑥[𝑛] = 𝐶𝛼 𝑛 = |𝐶|𝑒 𝑗𝜃 |𝛼|𝑛 𝑒 𝑗𝜔0 𝑛 = |𝐶||𝛼|𝑛 𝑒 𝑗(𝜔0 𝑛+𝜃)
Using Euler’s formula, it can be written as
𝑥[𝑛] = 𝐶𝛼 𝑛 = |𝐶||𝛼|𝑛 cos(𝜔0 𝑛 + 𝜃) + 𝑗|𝐶||𝛼|𝑛 sin(𝜔0 𝑛 + 𝜃)
4
5
6
(1.50)
For |𝛼| = 1, the real and imaginary parts of a complex exponential are sinusoidal.
For |𝛼| > 1, they correspond to sinusoidal signals / sequences multiplied with growing
exponential.
For |𝛼| < 1, they correspond to sinusoidal signals / sequences multiplied with decreasing
exponentials.
This is shown graphically on the next page.
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Growing Sinusoidal Signal
n
Decaying Sinusoidal Signal
n
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1.3.3
Periodicity Property of Discrete-Time Complex
Exponential
There are many similarities between continuous-time and discrete-time signals. But also there
are many important differences. One of them is related with the discrete-time exponential
signal 𝑒 𝑗𝜔0 𝑛 .
The following properties were found with regard to the continuous-time exponential signal
𝑒 𝑗𝜔0 𝑡 :
1. The larger the magnitude of 𝜔0 , the higher is the rate of oscillations in the signal;
2. 𝑒 𝑗𝜔0 𝑡 is periodic for any value of 𝜔0 .
Let us now see how these properties are different in the discrete-time case:
1. To see the difference for the first property, consider the discrete-time complex
exponential:
𝑒 𝑗(𝜔0 +2𝜋)𝑛 = 𝑒 𝑗2𝜋𝑛 𝑒 𝑗𝜔0 𝑛 = 𝑒 𝑗𝜔0 𝑛
(1.51)
This shows that the exponential at 𝜔0 + 2𝜋 is the same as that at frequency 𝜔0 .
This is very different when compared with the continuous-time exponential case, in
which the signals 𝑒 𝑗𝜔0 𝑡 are all distinct for distinct values of 𝜔0 .
 In discrete-time, these signals are not distinct. In fact, the signal with
frequency 𝜔0 is identical to signals with frequencies 𝜔0 ± 2𝜋, 𝜔0 ± 4𝜋 and
so on. Therefore, in considering discrete-time complex exponentials, we
need only consider a frequency interval of size 2𝜋. The most commonly used
2𝜋 intervals are 0 ≤ 𝜔0 ≤ 2𝜋 or the interval −𝜋 ≤ 𝜔0 ≤ 𝜋.
 Due to equation 1.51, as 𝜔0 is gradually increased, the rate of oscillations in
the discrete-time signal does not keep on increasing. If 𝜔0 is increased from
0 to 2𝜋, the rate of oscillations first increase and then decreases. This is
shown in the figure 1.27 below.
 Note in particular that for 𝜔0 = 𝜋 or for any odd multiple of 𝜋,
𝑛
(1.52)
𝑒 𝑗𝜋𝑛 = (𝑒 𝑗𝜋 ) = (−1)𝑛
so that the signal oscillates rapidly, changing sign at each point in time (see
figure 1.27).
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x[n] = cos( n/8)
x[n] = cos(0*n)
x[n] = cos( n/4)
1
1
1
0.5
0
0
0
0
10
20
30
-1
0
x[n] = cos( n/2)
10
20
30
-1
1
1
0
0
0
0
10
20
30
-1
0
x[n] = cos(7 n/4)
10
20
30
-1
1
0
0
0.5
10
20
30
-1
0
10
20
20
30
10
20
30
x[n] = cos(2 n)
1
0
0
x[n] = cos(15 n/8)
1
-1
10
x[n] = cos(3 n/2)
1
-1
0
x[n] = cos( n)
30
0
0
10
20
30
Figure 1.27: Discrete-time sinusoidal sequences for several different frequencies
2. The second property is the periodicity of the discrete-time complex exponential
signals.
𝑒 𝑗𝜔0 (𝑛+𝑁) = 𝑒 𝑗𝜔0 𝑛
(1.53)
Or equivalently,
𝑒 𝑗𝜔0 𝑁 = 1
Equation 1.54 could only be true if 𝜔0 𝑁 is a multiple of 2𝜋
𝜔0 𝑁 = 2𝜋𝑚
(1.54)
(1.55)
Or
𝜔0 𝑚
(1.56)
=
2𝜋 𝑁
This equation (1.56) means that the discrete-time signal 𝑒 𝑗𝜔0 𝑛 is periodic only when
𝜔0
is a rational number (i.e. it could written as a fraction).
2𝜋
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Table 1.1 Comparison of the signals 𝒆𝒋𝝎𝟎 𝒕 and 𝒆𝒋𝝎𝟎 𝒏 .
𝒆𝒋𝝎𝟎𝒕
𝒆𝒋𝝎𝟎𝒏
Distinct signals for distinct values of 𝝎𝟎 .
Periodic for any choice of 𝝎𝟎 .
Identical signals for values of 𝜔0 separated by
multiples of 2𝜋.
Periodic only if 𝜔0 = 2𝜋𝑚⁄𝑁 for some
integer 𝑁 > 0 and 𝑚.
Fundamental frequency 𝝎𝟎 .
Fundamental period
Fundamental frequency 𝜔0 /𝑚.
Fundamental period
𝝎𝟎 = 𝟎: undefined
𝟐𝝅
𝝎𝟎 ≠ 𝟎: 𝝎
𝜔0 = 0: undefined
2𝜋
𝜔0 ≠ 0: 𝑚 (𝜔 )
𝟎
0
Department of Computer Engineering
Page 28
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia