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Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
2: Signals & Systems - I
Y. Yoganandam, Runa Kumari, and S. R. Zinka
Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus
August 5 & 7, 2015
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
First of all ... What is a Signal?
A signal is a set of information or data.
In most of the cases that we deal in this course, signals are functions of theindependent variable time. This is not always the case, however. Give me a
few examples which are not functions of time ...
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
First of all ... What is a Signal?
A signal is a set of information or data.
In most of the cases that we deal in this course, signals are functions of theindependent variable time. This is not always the case, however. Give me a
few examples which are not functions of time ...
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
First of all ... What is a Signal?
A signal is a set of information or data.
In most of the cases that we deal in this course, signals are functions of theindependent variable time. This is not always the case, however. Give me a
few examples which are not functions of time ...
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
What is a System?
A system is an entity that processes a set of signals (inputs) to yield anotherset of signals (outputs).
A system may be made up of physical components or it may be an algorithmthat computes an output from input signal.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
What is a System?
A system is an entity that processes a set of signals (inputs) to yield anotherset of signals (outputs).
A system may be made up of physical components or it may be an algorithmthat computes an output from input signal.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
What is a System?
A system is an entity that processes a set of signals (inputs) to yield anotherset of signals (outputs).
A system may be made up of physical components or it may be an algorithmthat computes an output from input signal.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy
Signal energy for a real signal is defined as
Eg =
ˆ +∞
−∞g2 (t) dt. (1)
The above definition can be generalized to a complex valued signal g (t) as
Eg =
ˆ +∞
−∞|g (t)|2 dt. (2)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy
Signal energy for a real signal is defined as
Eg =
ˆ +∞
−∞g2 (t) dt. (1)
The above definition can be generalized to a complex valued signal g (t) as
Eg =
ˆ +∞
−∞|g (t)|2 dt. (2)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy
Signal energy for a real signal is defined as
Eg =
ˆ +∞
−∞g2 (t) dt. (1)
The above definition can be generalized to a complex valued signal g (t) as
Eg =
ˆ +∞
−∞|g (t)|2 dt. (2)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Power
If the amplitude of g (t) does not→ 0 as t→ ∞, the signal energy is infinite.
So, a more meaningful measure would be the time average of the energy (if itexists), which is the average power Pg defined by
Pg = limT→∞
1T
ˆ T/2
−T/2g2 (t) dt. (3)
The above definition can be generalized to a complex valued signal g (t) as
Pg = limT→∞
1T
ˆ T/2
−T/2|g (t)|2 dt. (4)
Observe that the square root of Pg is closely related to root mean square (rms)value of g (t).
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Power
If the amplitude of g (t) does not→ 0 as t→ ∞, the signal energy is infinite.
So, a more meaningful measure would be the time average of the energy (if itexists), which is the average power Pg defined by
Pg = limT→∞
1T
ˆ T/2
−T/2g2 (t) dt. (3)
The above definition can be generalized to a complex valued signal g (t) as
Pg = limT→∞
1T
ˆ T/2
−T/2|g (t)|2 dt. (4)
Observe that the square root of Pg is closely related to root mean square (rms)value of g (t).
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Power
If the amplitude of g (t) does not→ 0 as t→ ∞, the signal energy is infinite.
So, a more meaningful measure would be the time average of the energy (if itexists), which is the average power Pg defined by
Pg = limT→∞
1T
ˆ T/2
−T/2g2 (t) dt. (3)
The above definition can be generalized to a complex valued signal g (t) as
Pg = limT→∞
1T
ˆ T/2
−T/2|g (t)|2 dt. (4)
Observe that the square root of Pg is closely related to root mean square (rms)value of g (t).
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Power
If the amplitude of g (t) does not→ 0 as t→ ∞, the signal energy is infinite.
So, a more meaningful measure would be the time average of the energy (if itexists), which is the average power Pg defined by
Pg = limT→∞
1T
ˆ T/2
−T/2g2 (t) dt. (3)
The above definition can be generalized to a complex valued signal g (t) as
Pg = limT→∞
1T
ˆ T/2
−T/2|g (t)|2 dt. (4)
Observe that the square root of Pg is closely related to root mean square (rms)value of g (t).
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Power
If the amplitude of g (t) does not→ 0 as t→ ∞, the signal energy is infinite.
So, a more meaningful measure would be the time average of the energy (if itexists), which is the average power Pg defined by
Pg = limT→∞
1T
ˆ T/2
−T/2g2 (t) dt. (3)
The above definition can be generalized to a complex valued signal g (t) as
Pg = limT→∞
1T
ˆ T/2
−T/2|g (t)|2 dt. (4)
Observe that the square root of Pg is closely related to root mean square (rms)value of g (t).
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Let’s See a Few Signals ...
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Let’s See a Few Signals ...
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Let’s See a Few Signals ...
g(t)
t
g(t)
t
Energy is finite
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Let’s See a Few Signals ...
g(t)
t
g(t)
t
Energy is finite
Energy is infinite and power is finite
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy and Power – Physical Interpretation
+−g(t) R
g(t)R
g(t) R+
−Rg(t)
Energy dissipated =
ˆ +∞
−∞
g2 (t)R
dt =Eg
R(5)
If R = 1, the energy dissipated in the resistor is Eg. Similar observation appliesto signal power Pg.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy and Power – Physical Interpretation
+−g(t) R
g(t)R
g(t) R+
−Rg(t)
Energy dissipated =
ˆ +∞
−∞
g2 (t)R
dt =Eg
R(5)
If R = 1, the energy dissipated in the resistor is Eg. Similar observation appliesto signal power Pg.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy and Power – Physical Interpretation
+−g(t) R
g(t)R
g(t) R+
−Rg(t)
Energy dissipated =
ˆ +∞
−∞
g2 (t)R
dt =Eg
R(5)
If R = 1, the energy dissipated in the resistor is Eg. Similar observation appliesto signal power Pg.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Energy and Power – Physical Interpretation
+−g(t) R
g(t)R
g(t) R+
−Rg(t)
Energy dissipated =
ˆ +∞
−∞
g2 (t)R
dt =Eg
R(5)
If R = 1, the energy dissipated in the resistor is Eg. Similar observation appliesto signal power Pg.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Few Notes
• The measure of energy is indicative of the energy capability of thesignal, not the actual energy.
• So, the concept of conservation of energy should not be applied to themeasure of signal energy.
• Units of energy and power are not correct dimensionally and theydepend upon the type of the signal too.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Few Notes
• The measure of energy is indicative of the energy capability of thesignal, not the actual energy.
• So, the concept of conservation of energy should not be applied to themeasure of signal energy.
• Units of energy and power are not correct dimensionally and theydepend upon the type of the signal too.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Few Notes
• The measure of energy is indicative of the energy capability of thesignal, not the actual energy.
• So, the concept of conservation of energy should not be applied to themeasure of signal energy.
• Units of energy and power are not correct dimensionally and theydepend upon the type of the signal too.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Few Notes
• The measure of energy is indicative of the energy capability of thesignal, not the actual energy.
• So, the concept of conservation of energy should not be applied to themeasure of signal energy.
• Units of energy and power are not correct dimensionally and theydepend upon the type of the signal too.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Classes of Signals
• Continuous-time and discrete-time signals
• Analog and digital signals
• Periodic and aperiodic signals
• Energy and power signals
• Deterministic and probabilistic signals
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Classes of Signals
• Continuous-time and discrete-time signals
• Analog and digital signals
• Periodic and aperiodic signals
• Energy and power signals
• Deterministic and probabilistic signals
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Continuous-Time and Discrete-Time Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Continuous-Time and Discrete-Time Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Continuous-Time and Discrete-Time Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Analog and Digital Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Analog and Digital Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Analog and Digital Signals
x(t)
1234567
t
x(t)
1234567
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Periodic and Aperiodic Signals
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Periodic and Aperiodic Signals
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Periodic and Aperiodic Signals
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Periodic and Aperiodic Signals
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy and Power Signals
g(t)
t
g(t)
t
Energy is finite
Energy is infinite and power is finite
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy and Power Signals
g(t)
t
g(t)
t
Energy is finite
Energy is infinite and power is finite
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Deterministic and Probabilistic Signals
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Deterministic and Probabilistic Signals
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Deterministic and Probabilistic Signals
g(t)
t
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Shifting
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Shifting
g(t)
t
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Shifting
g(t)
t
g(t-T)
tT
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Shifting
g(t)
t
g(t-T)
t
g(t+T)
t
T
T
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Scaling
g(t)
tT1 T2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Scaling
g(t)
tT1 T2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Scaling
g(t)
t
g(2t)
t
T1 T2
T1/2 T2/2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Scaling
g(t)
t
g(2t)
t
g(t/2)
t
T1 T2
T1/2 T2/2
2T1 2T2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Inversion
g(t)
t0-2 5
-1
2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Inversion
g(t)
t0-2 5
-1
2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Time Inversion
g(t)
t0-2 5
-1
2
g(−t)
t02-5
-1
2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse or Dirac Delta Function
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2 -1 0 1 2
The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,
δ (x) =
{+∞, x = 00, x 6= 0
(6)
and which is also constrained to satisfy the identityˆ +∞
−∞δ (x) dx = 1. (7)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse or Dirac Delta Function
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2 -1 0 1 2
The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,
δ (x) =
{+∞, x = 00, x 6= 0
(6)
and which is also constrained to satisfy the identityˆ +∞
−∞δ (x) dx = 1. (7)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse or Dirac Delta Function
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2 -1 0 1 2
The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,
δ (x) =
{+∞, x = 00, x 6= 0
(6)
and which is also constrained to satisfy the identityˆ +∞
−∞δ (x) dx = 1. (7)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse or Dirac Delta Function
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2 -1 0 1 2
The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,
δ (x) =
{+∞, x = 00, x 6= 0
(6)
and which is also constrained to satisfy the identityˆ +∞
−∞δ (x) dx = 1. (7)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse Function – A Few Properties
• δ (−x) = δ (x) (Symmetry Property)
• ´ +∞−∞ δ (αx) dx =
´ +∞−∞ δ (u) du
|α| =1|α| (Scaling Property)
• f (x) δ (x− x0) = f (x0) δ (x− x0)
• ´ +∞−∞ f (x) δ (x− x0) dx = f (x0) (Sampling or Sifting Property)
• δ (x)⇔ 1
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Impulse Function – A Few Properties
• δ (−x) = δ (x) (Symmetry Property)
• ´ +∞−∞ δ (αx) dx =
´ +∞−∞ δ (u) du
|α| =1|α| (Scaling Property)
• f (x) δ (x− x0) = f (x0) δ (x− x0)
• ´ +∞−∞ f (x) δ (x− x0) dx = f (x0) (Sampling or Sifting Property)
• δ (x)⇔ 1
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signals and Vectors
Signals are not just like vectors. Signals are vectors.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signals and Vectors
Signals are not just like vectors. Signals are vectors.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
g
e1 = g− c1x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
g
x
e1 = g− c1x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
ge1
c1x x
e1 = g− c1x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
ge2
c2x x
e2 = g− c2x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
ge
cx x
e = g− cx
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
ge
cx x
e = g− cx
⇒ e · x = g · x− cx · x
⇒ 0 = g · x− c |x|2
⇒ c =g · x|x|2
(8)
If the vectors g and x are orthogonal to each other, then g · x = 0 and c = 0.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Vector
ge
cx x
e = g− cx
⇒ e · x = g · x− cx · x
⇒ 0 = g · x− c |x|2
⇒ c =g · x|x|2
(8)
If the vectors g and x are orthogonal to each other, then g · x = 0 and c = 0.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x c =
(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x c =
(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x c =
(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x
c =(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x c =
(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal
Consider the problem of approximating a real signal g (t) in terms of anotherreal signal x (t) over an interval [t1, t2]:
g (t) ' cx (t) t1 ≤ t ≤ t2 (9)
By analogy, the value of c which minimizes the error signal e (t) = g (t)− cx (t)is:
c =g · xx · x c =
(g, x)(x, x)
where the inner product (g, x) is defined as (g, x) =´ t2
t1g (t) x (t) dt.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Signal ... Proof
Now, let’s see the proof. We need to minimize the energy of the error signal
Ee =
ˆ t2
t1
[g (t)− cx (t)]2 dt. (10)
So, the necessary condition is
⇒ ddc
ˆ t2
t1
[g (t)− cx (t)]2 dt = 0
⇒ˆ t2
t1
ddc
[g2 (t) + c2x2 (t)− 2cg (t) x (t)
]dt = 0
⇒ cˆ t2
t1
x2 (t) dt−ˆ t2
t1
g (t) x (t) dt = 0
⇒ c =
´ t2t1
g (t) x (t) dt´ t2t1
x2 (t) dt. (11)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Complex Signal
For complex signals, we can generalize the results as shown below:
⇒ c =
´ t2t1
g (t) x∗ (t) dt´ t2t1|x (t)|2 dt
(12)
For a detailed derivation of the above expression, please see Ch. 2 [T1].
So, we define the signals g (t) and x (t) to be orthogonal over the interval [t1, t2]if ˆ t2
t1
g (t) x∗ (t) dt = 0. (13)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Complex Signal
For complex signals, we can generalize the results as shown below:
⇒ c =
´ t2t1
g (t) x∗ (t) dt´ t2t1|x (t)|2 dt
(12)
For a detailed derivation of the above expression, please see Ch. 2 [T1].
So, we define the signals g (t) and x (t) to be orthogonal over the interval [t1, t2]if ˆ t2
t1
g (t) x∗ (t) dt = 0. (13)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Component of a Complex Signal
For complex signals, we can generalize the results as shown below:
⇒ c =
´ t2t1
g (t) x∗ (t) dt´ t2t1|x (t)|2 dt
(12)
For a detailed derivation of the above expression, please see Ch. 2 [T1].
So, we define the signals g (t) and x (t) to be orthogonal over the interval [t1, t2]if ˆ t2
t1
g (t) x∗ (t) dt = 0. (13)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy of Sum of Orthogonal Signals
If vectors x and y are orthogonal, and if z = x + y, then
|z|2 = |x|2 + |y|2 . (14)
If signals x (t) and y (t) are orthogonal over an interval [t1, t2], and if z (t) =x (t) + y (t) , then
Ez = Ex + Ey. (15)
Try to derive the above expression by yourself using the definition of signalenergy.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy of Sum of Orthogonal Signals
If vectors x and y are orthogonal, and if z = x + y, then
|z|2 = |x|2 + |y|2 . (14)
If signals x (t) and y (t) are orthogonal over an interval [t1, t2], and if z (t) =x (t) + y (t) , then
Ez = Ex + Ey. (15)
Try to derive the above expression by yourself using the definition of signalenergy.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy of Sum of Orthogonal Signals
If vectors x and y are orthogonal, and if z = x + y, then
|z|2 = |x|2 + |y|2 . (14)
If signals x (t) and y (t) are orthogonal over an interval [t1, t2], and if z (t) =x (t) + y (t) , then
Ez = Ex + Ey. (15)
Try to derive the above expression by yourself using the definition of signalenergy.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Energy of Sum of Orthogonal Signals
If vectors x and y are orthogonal, and if z = x + y, then
|z|2 = |x|2 + |y|2 . (14)
If signals x (t) and y (t) are orthogonal over an interval [t1, t2], and if z (t) =x (t) + y (t) , then
Ez = Ex + Ey. (15)
Try to derive the above expression by yourself using the definition of signalenergy.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Best Friends, Worst Enemies, and Complete Strangers
g x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Best Friends, Worst Enemies, and Complete Strangers
g x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Best Friends, Worst Enemies, and Complete Strangers
g x
g x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Best Friends, Worst Enemies, and Complete Strangers
g x
g x
g
x
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Vector Correlation
g
x
C = cos θ =g · x|g| |x| (16)
−1 ≤ C ≤ 1 (17)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Vector Correlation
g
x
C = cos θ =g · x|g| |x| (16)
−1 ≤ C ≤ 1 (17)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Vector Correlation
g
x
C = cos θ =g · x|g| |x| (16)
−1 ≤ C ≤ 1 (17)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Vector Correlation
g
x
C = cos θ =g · x|g| |x| (16)
−1 ≤ C ≤ 1 (17)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Correlation
Using analogy, signal correlation is defined as
C =(g, x)√
(g, g) (x, x)=
´ ∞−∞ g∗ (t) x (t) dt√´ ∞
−∞ |g (t)|2 dt´ ∞−∞ |x (t)|
2 dt
=
´ ∞−∞ g∗ (t) x (t) dt√
EgEx. (18)
Again, it can be shown using Schwarz’s inequality principle that
−1 ≤ C ≤ 1. (19)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Correlation
Using analogy, signal correlation is defined as
C =(g, x)√
(g, g) (x, x)=
´ ∞−∞ g∗ (t) x (t) dt√´ ∞
−∞ |g (t)|2 dt´ ∞−∞ |x (t)|
2 dt
=
´ ∞−∞ g∗ (t) x (t) dt√
EgEx. (18)
Again, it can be shown using Schwarz’s inequality principle that
−1 ≤ C ≤ 1. (19)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Signal Correlation
Using analogy, signal correlation is defined as
C =(g, x)√
(g, g) (x, x)=
´ ∞−∞ g∗ (t) x (t) dt√´ ∞
−∞ |g (t)|2 dt´ ∞−∞ |x (t)|
2 dt
=
´ ∞−∞ g∗ (t) x (t) dt√
EgEx. (18)
Again, it can be shown using Schwarz’s inequality principle that
−1 ≤ C ≤ 1. (19)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Application to Signal Detection
g(t)
t0 1
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Application to Signal Detection
g(t)
t0 1
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Application to Signal Detection
g(t)
t0 1
z(t)
t0 T T+1
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Cross-Correlation
So, cross-correlation between g (t) and z (t) is defined as
ψgz (τ) =
ˆ ∞
−∞g∗ (t) z (t + τ) dt. (20)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Cross-Correlation
So, cross-correlation between g (t) and z (t) is defined as
ψgz (τ) =
ˆ ∞
−∞g∗ (t) z (t + τ) dt. (20)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Auto-Correlation
The correlation of a signal with itself is called auto-correlation and defined as
ψg (τ) =
ˆ ∞
−∞g∗ (t) g (t + τ) dt. (21)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Auto-Correlation
The correlation of a signal with itself is called auto-correlation and defined as
ψg (τ) =
ˆ ∞
−∞g∗ (t) g (t + τ) dt. (21)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Correlation Functions of Power Signals
Cross-correlation between two power signals g (t) and z (t) is defined as
Rgz (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) z (t + τ) dt. (22)
The auto-correlation of a power signal is defined as
Rg (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) g (t + τ) dt. (23)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Correlation Functions of Power Signals
Cross-correlation between two power signals g (t) and z (t) is defined as
Rgz (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) z (t + τ) dt. (22)
The auto-correlation of a power signal is defined as
Rg (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) g (t + τ) dt. (23)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Correlation Functions of Power Signals
Cross-correlation between two power signals g (t) and z (t) is defined as
Rgz (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) z (t + τ) dt. (22)
The auto-correlation of a power signal is defined as
Rg (τ) = limT→∞
1T
ˆ T/2
−T/2g∗ (t) g (t + τ) dt. (23)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Outline
1 Signal Energy & Power
2 Classification of Signals
3 Signal Operations
4 Signals & Vectors
5 Signal Correlation
6 Orthogonal Signal Spaces
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
g
c1x1
c2x2e
g ' c1x1 + c2x2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
g
c1x1
c2x2e
g ' c1x1 + c2x2
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
g
c1x1
c2x2
c3x3
g = c1x1 + c2x2 + c3x3
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
Thus, the vectors x1, x2, and x3 form a complete orthogonal set in three dimen-sional space.
To summarize, if a set of vectors {xi} is mutually orthogonal, that is, if
xm · xn =
{0 m 6= n|xm|2 m = n
,
and if this basis set is complete, a vector g in this space can be expressed as
g = c1x1 + c2x2 + c3x3 + · · · ,
where the constants ci are given by
ci =g · xixi · xi
, i = 1, 2, 3, · · · . (24)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
Thus, the vectors x1, x2, and x3 form a complete orthogonal set in three dimen-sional space.
To summarize, if a set of vectors {xi} is mutually orthogonal, that is, if
xm · xn =
{0 m 6= n|xm|2 m = n
,
and if this basis set is complete, a vector g in this space can be expressed as
g = c1x1 + c2x2 + c3x3 + · · · ,
where the constants ci are given by
ci =g · xixi · xi
, i = 1, 2, 3, · · · . (24)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
Thus, the vectors x1, x2, and x3 form a complete orthogonal set in three dimen-sional space.
To summarize, if a set of vectors {xi} is mutually orthogonal, that is, if
xm · xn =
{0 m 6= n|xm|2 m = n
,
and if this basis set is complete, a vector g in this space can be expressed as
g = c1x1 + c2x2 + c3x3 + · · · ,
where the constants ci are given by
ci =g · xixi · xi
, i = 1, 2, 3, · · · . (24)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Vector Space
Thus, the vectors x1, x2, and x3 form a complete orthogonal set in three dimen-sional space.
To summarize, if a set of vectors {xi} is mutually orthogonal, that is, if
xm · xn =
{0 m 6= n|xm|2 m = n
,
and if this basis set is complete, a vector g in this space can be expressed as
g = c1x1 + c2x2 + c3x3 + · · · ,
where the constants ci are given by
ci =g · xixi · xi
, i = 1, 2, 3, · · · . (24)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Signal Space
If a set of signals {xi (t)} is mutually orthogonal over the interval [t1, t2], thatis, if
(xm, xn) =
ˆ t2
t1
xm (t) x∗n (t) dt =
{0 m 6= nEm m = n
,
and if this basis set is complete, a signal g (t) in this space can be expressed as
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · · ,
where the constants cn are given by
cn =(g, xn)
(xn, xn)=
´ t2t1
g (t) x∗n (t) dt´ t2t1|xn (t)|2 dt
=
´ t2t1
g (t) x∗n (t) dtEn
, n = 1, 2, 3, · · · . (25)
If En = 1 for all n, then the set {xn (t)} is called an orthonormal set.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Signal Space
If a set of signals {xi (t)} is mutually orthogonal over the interval [t1, t2], thatis, if
(xm, xn) =
ˆ t2
t1
xm (t) x∗n (t) dt =
{0 m 6= nEm m = n
,
and if this basis set is complete, a signal g (t) in this space can be expressed as
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · · ,
where the constants cn are given by
cn =(g, xn)
(xn, xn)=
´ t2t1
g (t) x∗n (t) dt´ t2t1|xn (t)|2 dt
=
´ t2t1
g (t) x∗n (t) dtEn
, n = 1, 2, 3, · · · . (25)
If En = 1 for all n, then the set {xn (t)} is called an orthonormal set.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Signal Space
If a set of signals {xi (t)} is mutually orthogonal over the interval [t1, t2], thatis, if
(xm, xn) =
ˆ t2
t1
xm (t) x∗n (t) dt =
{0 m 6= nEm m = n
,
and if this basis set is complete, a signal g (t) in this space can be expressed as
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · · ,
where the constants cn are given by
cn =(g, xn)
(xn, xn)=
´ t2t1
g (t) x∗n (t) dt´ t2t1|xn (t)|2 dt
=
´ t2t1
g (t) x∗n (t) dtEn
, n = 1, 2, 3, · · · . (25)
If En = 1 for all n, then the set {xn (t)} is called an orthonormal set.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Signal Space
If a set of signals {xi (t)} is mutually orthogonal over the interval [t1, t2], thatis, if
(xm, xn) =
ˆ t2
t1
xm (t) x∗n (t) dt =
{0 m 6= nEm m = n
,
and if this basis set is complete, a signal g (t) in this space can be expressed as
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · · ,
where the constants cn are given by
cn =(g, xn)
(xn, xn)=
´ t2t1
g (t) x∗n (t) dt´ t2t1|xn (t)|2 dt
=
´ t2t1
g (t) x∗n (t) dtEn
, n = 1, 2, 3, · · · . (25)
If En = 1 for all n, then the set {xn (t)} is called an orthonormal set.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Orthogonal Signal Space
If a set of signals {xi (t)} is mutually orthogonal over the interval [t1, t2], thatis, if
(xm, xn) =
ˆ t2
t1
xm (t) x∗n (t) dt =
{0 m 6= nEm m = n
,
and if this basis set is complete, a signal g (t) in this space can be expressed as
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · · ,
where the constants cn are given by
cn =(g, xn)
(xn, xn)=
´ t2t1
g (t) x∗n (t) dt´ t2t1|xn (t)|2 dt
=
´ t2t1
g (t) x∗n (t) dtEn
, n = 1, 2, 3, · · · . (25)
If En = 1 for all n, then the set {xn (t)} is called an orthonormal set.
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Caution About the Completeness
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · ·
The equality in the above equation is not an equality in the ordinary sense,but in the sense that the error energy, that is, the energy of the difference
between the two sides of the above equation approaches zero.
So, what happens to the Fourier series when g (t) has a jump discontinuity?
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Caution About the Completeness
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · ·
The equality in the above equation is not an equality in the ordinary sense,but in the sense that the error energy, that is, the energy of the difference
between the two sides of the above equation approaches zero.
So, what happens to the Fourier series when g (t) has a jump discontinuity?
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Caution About the Completeness
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · ·
The equality in the above equation is not an equality in the ordinary sense,but in the sense that the error energy, that is, the energy of the difference
between the two sides of the above equation approaches zero.
So, what happens to the Fourier series when g (t) has a jump discontinuity?
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
A Caution About the Completeness
g (t) = c1x1 (t) + c2x2 (t) + c3x3 (t) + · · ·
The equality in the above equation is not an equality in the ordinary sense,but in the sense that the error energy, that is, the energy of the difference
between the two sides of the above equation approaches zero.
So, what happens to the Fourier series when g (t) has a jump discontinuity?
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
Recall that the energy of the sum of orthogonal signals is equal to sum of theirenergies. Therefore, energy of the signal g (t) is given as
Eg = |c1|2 E1 + |c2|2 E2 + |c3|2 E3 + · · · = ∑n|cn|2 En. (26)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
Recall that the energy of the sum of orthogonal signals is equal to sum of theirenergies. Therefore, energy of the signal g (t) is given as
Eg = |c1|2 E1 + |c2|2 E2 + |c3|2 E3 + · · · = ∑n|cn|2 En. (26)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Basis Signals for Generalized Fourier Series
0.2
0.4
0.6
1.0
0.8
0.6
0.4
0.2
0.0
−0.2
−0.4
0 5 10 15 20
J (x)0
J (x)1
J (x)2
x−0.5 0.0 0.5 1.0
T
0.0
−0.5
−1.0
−1.0
0.5
1.0
x
n=1
n=4
n=0
n=3
(x)
n
n=2
n=5
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
x
P (
x)n
P₀(x)P₁(x)P₂(x)P₃(x)P₄(x)P₅(x)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Basis Signals for Generalized Fourier Series
0.2
0.4
0.6
1.0
0.8
0.6
0.4
0.2
0.0
−0.2
−0.4
0 5 10 15 20
J (x)0
J (x)1
J (x)2
x−0.5 0.0 0.5 1.0
T
0.0
−0.5
−1.0
−1.0
0.5
1.0
x
n=1
n=4
n=0
n=3
(x)
n
n=2
n=5
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
x
P (
x)n
P₀(x)P₁(x)P₂(x)P₃(x)P₄(x)P₅(x)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Basis Signals for Generalized Fourier Series
• Trigonometric functions
• Exponential functions
• Walsh functions
• Bessel functions
• Legendre polynomials
• Jacobi polynomials
• Hermite polynomials
• Chebyshev polynomials
• Laguerre functions
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Basis Signals for Generalized Fourier Series
• Trigonometric functions
• Exponential functions
• Walsh functions
• Bessel functions
• Legendre polynomials
• Jacobi polynomials
• Hermite polynomials
• Chebyshev polynomials
• Laguerre functions
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
Since exponential Fourier series are the most commonly used Fourierseries, let’s study them in detail here ...
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
Since exponential Fourier series are the most commonly used Fourierseries, let’s study them in detail here ...
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
It can be shown that the set of signals{
ejnω0t}
is mutually orthogonal overthe interval [T, T + 2π/ω0], that is, if
(ejmω0t, ejnω0t
)=
ˆ T+ 2πω0
Tejmω0te−jnω0tdt =
{0 m 6= n2πω0
m = n. (27)
A signal g (t) over an interval of duration T0 = 2πω0
can be expressed as anexponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t, (28)
where the constants Dn are given by
Dn =1
T0
ˆ T+ 2πω0
Tg (t) e−jnω0tdt, n = 1, 2, 3, · · · . (29)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
It can be shown that the set of signals{
ejnω0t}
is mutually orthogonal overthe interval [T, T + 2π/ω0], that is, if
(ejmω0t, ejnω0t
)=
ˆ T+ 2πω0
Tejmω0te−jnω0tdt =
{0 m 6= n2πω0
m = n. (27)
A signal g (t) over an interval of duration T0 = 2πω0
can be expressed as anexponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t, (28)
where the constants Dn are given by
Dn =1
T0
ˆ T+ 2πω0
Tg (t) e−jnω0tdt, n = 1, 2, 3, · · · . (29)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
It can be shown that the set of signals{
ejnω0t}
is mutually orthogonal overthe interval [T, T + 2π/ω0], that is, if
(ejmω0t, ejnω0t
)=
ˆ T+ 2πω0
Tejmω0te−jnω0tdt =
{0 m 6= n2πω0
m = n. (27)
A signal g (t) over an interval of duration T0 = 2πω0
can be expressed as anexponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t, (28)
where the constants Dn are given by
Dn =1
T0
ˆ T+ 2πω0
Tg (t) e−jnω0tdt, n = 1, 2, 3, · · · . (29)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Exponential Fourier Series
It can be shown that the set of signals{
ejnω0t}
is mutually orthogonal overthe interval [T, T + 2π/ω0], that is, if
(ejmω0t, ejnω0t
)=
ˆ T+ 2πω0
Tejmω0te−jnω0tdt =
{0 m 6= n2πω0
m = n. (27)
A signal g (t) over an interval of duration T0 = 2πω0
can be expressed as anexponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t, (28)
where the constants Dn are given by
Dn =1
T0
ˆ T+ 2πω0
Tg (t) e−jnω0tdt, n = 1, 2, 3, · · · . (29)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
We have seen that a signal g (t) over an interval of duration T0 = 2πω0
can beexpressed as an exponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t. (30)
Power corresponding to the component Dnejnω0t is given as
Pn =1
T0
ˆ T+T0
TDnejnω0tD∗ne−jnω0tdt = |Dn|2 . (31)
Therefore, the power of g (t) is given by
Pg =∞
∑n=−∞
|Dn|2 . (32)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
We have seen that a signal g (t) over an interval of duration T0 = 2πω0
can beexpressed as an exponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t. (30)
Power corresponding to the component Dnejnω0t is given as
Pn =1
T0
ˆ T+T0
TDnejnω0tD∗ne−jnω0tdt = |Dn|2 . (31)
Therefore, the power of g (t) is given by
Pg =∞
∑n=−∞
|Dn|2 . (32)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
We have seen that a signal g (t) over an interval of duration T0 = 2πω0
can beexpressed as an exponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t. (30)
Power corresponding to the component Dnejnω0t is given as
Pn =1
T0
ˆ T+T0
TDnejnω0tD∗ne−jnω0tdt = |Dn|2 . (31)
Therefore, the power of g (t) is given by
Pg =∞
∑n=−∞
|Dn|2 . (32)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
Signal Energy & Power Classification of Signals Signal Operations Signals & Vectors Signal Correlation Orthogonal Signal Spaces
Parseval’s Theorem
We have seen that a signal g (t) over an interval of duration T0 = 2πω0
can beexpressed as an exponential Fourier series
g (t) =∞
∑n=−∞
Dnejnω0t. (30)
Power corresponding to the component Dnejnω0t is given as
Pn =1
T0
ˆ T+T0
TDnejnω0tD∗ne−jnω0tdt = |Dn|2 . (31)
Therefore, the power of g (t) is given by
Pg =∞
∑n=−∞
|Dn|2 . (32)
2: Signals & Systems - I Communication Systems, Dept. of EEE, BITS Hyderabad
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