Shifting & Scaling Properties of Signals | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


 
 
  
 
 
What is a Signal  
Anything which contains some information is known as a signal. A signal may be a 
function of one or more independent variables like time, pressure, distance, position, etc. 
For electrical purpose, signal can be current or voltage which is function of time as the 
independent variable. 
Signals can be classified into two broad categories. These are  
1. Continuous Time Signals  
2. Discrete Time Signals  
Continuous Time Signals  
A continuous signal may be defined as a continuous function of independent variable. In 
case of continuous time signal, the independent variable is time. Signals are continuous 
function of time. They can also be termed as Analog Signals. 
Discrete Time Signals  
For discrete time signals, the independent variable is discrete. So, they are defined only 
at certain time instants. These signals have both discrete amplitude and discrete time.  
They are also known as Digital Signals.  
 
Signal Energy and Power  
Another important parameter for a signal is the signal energy and power.  
Energy of a signal ? ?
|x(t)|
2
dt
8
-8
                       for continuous time signals  
                                   ? ? |x[n]|
2 8
?? =-8
                     for discrete time signals  
Power of a signal ? lim
T?8
1
T
?
|x(t)|
2
dt
T
2
-
T
2
             for continuous time signals  
                                   ? lim
N?8
1
2N+1
? |x[n]|
2 N
?=-N
    for discrete time signals  
If we take an example of an electrical circuit given as follows 
Page 2


 
 
  
 
 
What is a Signal  
Anything which contains some information is known as a signal. A signal may be a 
function of one or more independent variables like time, pressure, distance, position, etc. 
For electrical purpose, signal can be current or voltage which is function of time as the 
independent variable. 
Signals can be classified into two broad categories. These are  
1. Continuous Time Signals  
2. Discrete Time Signals  
Continuous Time Signals  
A continuous signal may be defined as a continuous function of independent variable. In 
case of continuous time signal, the independent variable is time. Signals are continuous 
function of time. They can also be termed as Analog Signals. 
Discrete Time Signals  
For discrete time signals, the independent variable is discrete. So, they are defined only 
at certain time instants. These signals have both discrete amplitude and discrete time.  
They are also known as Digital Signals.  
 
Signal Energy and Power  
Another important parameter for a signal is the signal energy and power.  
Energy of a signal ? ?
|x(t)|
2
dt
8
-8
                       for continuous time signals  
                                   ? ? |x[n]|
2 8
?? =-8
                     for discrete time signals  
Power of a signal ? lim
T?8
1
T
?
|x(t)|
2
dt
T
2
-
T
2
             for continuous time signals  
                                   ? lim
N?8
1
2N+1
? |x[n]|
2 N
?=-N
    for discrete time signals  
If we take an example of an electrical circuit given as follows 
 
 
  
 
 
 
Instantaneous power is P(t) = V(t)I(t) =
1
R
V
2
(t) = I
2
(t). R  
Energy dissipated in this circuit, E = ? P(t)dt
8
-8
  
=
1
R
? V
2
(t)dt
8
-8
   
The total energy dissipated in the time interval t
1
= t = t
2
 is  
? P(t)dt =
1
R
? V
2
(t)dt
t
2
t
1
t
2
t
1
   
Average power other this time interval , P
av
=
1
t
2
-t
1
?
1
R
t
2
t
1
V
2
(t)dt  
 
Signal Transformations through Variations of the 
Independent Variable  
A signal can undergo several transformations some of which are:  
1. Time Shifting  
2. Time Scaling  
3. Time Inversion / Time Reversal  
 
Time Shifting  
Page 3


 
 
  
 
 
What is a Signal  
Anything which contains some information is known as a signal. A signal may be a 
function of one or more independent variables like time, pressure, distance, position, etc. 
For electrical purpose, signal can be current or voltage which is function of time as the 
independent variable. 
Signals can be classified into two broad categories. These are  
1. Continuous Time Signals  
2. Discrete Time Signals  
Continuous Time Signals  
A continuous signal may be defined as a continuous function of independent variable. In 
case of continuous time signal, the independent variable is time. Signals are continuous 
function of time. They can also be termed as Analog Signals. 
Discrete Time Signals  
For discrete time signals, the independent variable is discrete. So, they are defined only 
at certain time instants. These signals have both discrete amplitude and discrete time.  
They are also known as Digital Signals.  
 
Signal Energy and Power  
Another important parameter for a signal is the signal energy and power.  
Energy of a signal ? ?
|x(t)|
2
dt
8
-8
                       for continuous time signals  
                                   ? ? |x[n]|
2 8
?? =-8
                     for discrete time signals  
Power of a signal ? lim
T?8
1
T
?
|x(t)|
2
dt
T
2
-
T
2
             for continuous time signals  
                                   ? lim
N?8
1
2N+1
? |x[n]|
2 N
?=-N
    for discrete time signals  
If we take an example of an electrical circuit given as follows 
 
 
  
 
 
 
Instantaneous power is P(t) = V(t)I(t) =
1
R
V
2
(t) = I
2
(t). R  
Energy dissipated in this circuit, E = ? P(t)dt
8
-8
  
=
1
R
? V
2
(t)dt
8
-8
   
The total energy dissipated in the time interval t
1
= t = t
2
 is  
? P(t)dt =
1
R
? V
2
(t)dt
t
2
t
1
t
2
t
1
   
Average power other this time interval , P
av
=
1
t
2
-t
1
?
1
R
t
2
t
1
V
2
(t)dt  
 
Signal Transformations through Variations of the 
Independent Variable  
A signal can undergo several transformations some of which are:  
1. Time Shifting  
2. Time Scaling  
3. Time Inversion / Time Reversal  
 
Time Shifting  
 
 
  
 
Time shifting is a very basic operation that you never stop to come across if you are 
handling a signals and systems problem. We seek to settle all doubts regarding it for one 
last time.   
Consider that we are given a signal x(t) then how do you implement time shifting and 
scaling to obtain the signal x(–at – ß), x( –at + ß), x(at – ß) or x(at + ß), where a and ß 
are both positive quantities. The first thing we would want to clear forever is that a 
negative time shift implies a right shift and a positive time shift implies a left shift.  
Remember it by the thinking of creating an arrow out of negative sign, - to ? which 
implies right shift for negative time shift. The other (+ time shift) would obviously mean 
a left shift.  
Returning to our original agenda, the next thing to know is that time shifting and scaling 
can be implemented, starting from both the left and right side. Since, we are discussing 
time shifting, we set a = 1 which is responsible for scaling.   
Time Shift - Working from the Right 
This is general method which always works.  
 
Let, x(t) = u(t) – u(t – 1)  
 
Then, to implement x(–t –3), working from the right, we first implement right shift by 3 
(due to -3) and then do time reversal (due to -1 coefficient of t) .  
Page 4


 
 
  
 
 
What is a Signal  
Anything which contains some information is known as a signal. A signal may be a 
function of one or more independent variables like time, pressure, distance, position, etc. 
For electrical purpose, signal can be current or voltage which is function of time as the 
independent variable. 
Signals can be classified into two broad categories. These are  
1. Continuous Time Signals  
2. Discrete Time Signals  
Continuous Time Signals  
A continuous signal may be defined as a continuous function of independent variable. In 
case of continuous time signal, the independent variable is time. Signals are continuous 
function of time. They can also be termed as Analog Signals. 
Discrete Time Signals  
For discrete time signals, the independent variable is discrete. So, they are defined only 
at certain time instants. These signals have both discrete amplitude and discrete time.  
They are also known as Digital Signals.  
 
Signal Energy and Power  
Another important parameter for a signal is the signal energy and power.  
Energy of a signal ? ?
|x(t)|
2
dt
8
-8
                       for continuous time signals  
                                   ? ? |x[n]|
2 8
?? =-8
                     for discrete time signals  
Power of a signal ? lim
T?8
1
T
?
|x(t)|
2
dt
T
2
-
T
2
             for continuous time signals  
                                   ? lim
N?8
1
2N+1
? |x[n]|
2 N
?=-N
    for discrete time signals  
If we take an example of an electrical circuit given as follows 
 
 
  
 
 
 
Instantaneous power is P(t) = V(t)I(t) =
1
R
V
2
(t) = I
2
(t). R  
Energy dissipated in this circuit, E = ? P(t)dt
8
-8
  
=
1
R
? V
2
(t)dt
8
-8
   
The total energy dissipated in the time interval t
1
= t = t
2
 is  
? P(t)dt =
1
R
? V
2
(t)dt
t
2
t
1
t
2
t
1
   
Average power other this time interval , P
av
=
1
t
2
-t
1
?
1
R
t
2
t
1
V
2
(t)dt  
 
Signal Transformations through Variations of the 
Independent Variable  
A signal can undergo several transformations some of which are:  
1. Time Shifting  
2. Time Scaling  
3. Time Inversion / Time Reversal  
 
Time Shifting  
 
 
  
 
Time shifting is a very basic operation that you never stop to come across if you are 
handling a signals and systems problem. We seek to settle all doubts regarding it for one 
last time.   
Consider that we are given a signal x(t) then how do you implement time shifting and 
scaling to obtain the signal x(–at – ß), x( –at + ß), x(at – ß) or x(at + ß), where a and ß 
are both positive quantities. The first thing we would want to clear forever is that a 
negative time shift implies a right shift and a positive time shift implies a left shift.  
Remember it by the thinking of creating an arrow out of negative sign, - to ? which 
implies right shift for negative time shift. The other (+ time shift) would obviously mean 
a left shift.  
Returning to our original agenda, the next thing to know is that time shifting and scaling 
can be implemented, starting from both the left and right side. Since, we are discussing 
time shifting, we set a = 1 which is responsible for scaling.   
Time Shift - Working from the Right 
This is general method which always works.  
 
Let, x(t) = u(t) – u(t – 1)  
 
Then, to implement x(–t –3), working from the right, we first implement right shift by 3 
(due to -3) and then do time reversal (due to -1 coefficient of t) .  
 
 
  
 
 
 
 
Time Shift - Working from the Left  
When we work from left side, first we take common everything that is coefficient of t. So, 
x(± at ± ß) becomes x[± a(t ± ß/a) ]. For x(-t-3), we get x[– (t+3)]. So, we first implement 
time inversion [due to -1 getting multiplied with (t + 3)] and then do left shift by 3 (due 
to +3).  
 
 
This method works always for continuous time signals but not always for discrete time 
signals. For discrete time signals, when taking common creates a fraction inside the 
bracket, the method fails. The last thing to verify that the final results from both 
approaches is same.  
Origin Shifting  
Suppose we want to shift the origin from (0, 0) and (0, a) (let a be positive), then it is a 
right shift for axis, or left shift for the signal relative to the axis. And hence x(t) change 
to x(t + a) contrary to what we intuitively expect it to be x(t – a) due to right shift for  
axis. Similarly, if we want to shift origin to (0, –a) then this is left shift for axis but a right 
Page 5


 
 
  
 
 
What is a Signal  
Anything which contains some information is known as a signal. A signal may be a 
function of one or more independent variables like time, pressure, distance, position, etc. 
For electrical purpose, signal can be current or voltage which is function of time as the 
independent variable. 
Signals can be classified into two broad categories. These are  
1. Continuous Time Signals  
2. Discrete Time Signals  
Continuous Time Signals  
A continuous signal may be defined as a continuous function of independent variable. In 
case of continuous time signal, the independent variable is time. Signals are continuous 
function of time. They can also be termed as Analog Signals. 
Discrete Time Signals  
For discrete time signals, the independent variable is discrete. So, they are defined only 
at certain time instants. These signals have both discrete amplitude and discrete time.  
They are also known as Digital Signals.  
 
Signal Energy and Power  
Another important parameter for a signal is the signal energy and power.  
Energy of a signal ? ?
|x(t)|
2
dt
8
-8
                       for continuous time signals  
                                   ? ? |x[n]|
2 8
?? =-8
                     for discrete time signals  
Power of a signal ? lim
T?8
1
T
?
|x(t)|
2
dt
T
2
-
T
2
             for continuous time signals  
                                   ? lim
N?8
1
2N+1
? |x[n]|
2 N
?=-N
    for discrete time signals  
If we take an example of an electrical circuit given as follows 
 
 
  
 
 
 
Instantaneous power is P(t) = V(t)I(t) =
1
R
V
2
(t) = I
2
(t). R  
Energy dissipated in this circuit, E = ? P(t)dt
8
-8
  
=
1
R
? V
2
(t)dt
8
-8
   
The total energy dissipated in the time interval t
1
= t = t
2
 is  
? P(t)dt =
1
R
? V
2
(t)dt
t
2
t
1
t
2
t
1
   
Average power other this time interval , P
av
=
1
t
2
-t
1
?
1
R
t
2
t
1
V
2
(t)dt  
 
Signal Transformations through Variations of the 
Independent Variable  
A signal can undergo several transformations some of which are:  
1. Time Shifting  
2. Time Scaling  
3. Time Inversion / Time Reversal  
 
Time Shifting  
 
 
  
 
Time shifting is a very basic operation that you never stop to come across if you are 
handling a signals and systems problem. We seek to settle all doubts regarding it for one 
last time.   
Consider that we are given a signal x(t) then how do you implement time shifting and 
scaling to obtain the signal x(–at – ß), x( –at + ß), x(at – ß) or x(at + ß), where a and ß 
are both positive quantities. The first thing we would want to clear forever is that a 
negative time shift implies a right shift and a positive time shift implies a left shift.  
Remember it by the thinking of creating an arrow out of negative sign, - to ? which 
implies right shift for negative time shift. The other (+ time shift) would obviously mean 
a left shift.  
Returning to our original agenda, the next thing to know is that time shifting and scaling 
can be implemented, starting from both the left and right side. Since, we are discussing 
time shifting, we set a = 1 which is responsible for scaling.   
Time Shift - Working from the Right 
This is general method which always works.  
 
Let, x(t) = u(t) – u(t – 1)  
 
Then, to implement x(–t –3), working from the right, we first implement right shift by 3 
(due to -3) and then do time reversal (due to -1 coefficient of t) .  
 
 
  
 
 
 
 
Time Shift - Working from the Left  
When we work from left side, first we take common everything that is coefficient of t. So, 
x(± at ± ß) becomes x[± a(t ± ß/a) ]. For x(-t-3), we get x[– (t+3)]. So, we first implement 
time inversion [due to -1 getting multiplied with (t + 3)] and then do left shift by 3 (due 
to +3).  
 
 
This method works always for continuous time signals but not always for discrete time 
signals. For discrete time signals, when taking common creates a fraction inside the 
bracket, the method fails. The last thing to verify that the final results from both 
approaches is same.  
Origin Shifting  
Suppose we want to shift the origin from (0, 0) and (0, a) (let a be positive), then it is a 
right shift for axis, or left shift for the signal relative to the axis. And hence x(t) change 
to x(t + a) contrary to what we intuitively expect it to be x(t – a) due to right shift for  
axis. Similarly, if we want to shift origin to (0, –a) then this is left shift for axis but a right 
 
 
  
 
shift for signal relative to axis. So, x(t) change to x(t – a). This lack of understanding 
causes us to believe that shifting signals and shifting axes are two different concepts and 
then remember two different sets of rule for them. Now, we see that they are just one 
thing and makes easies to remember.   
Example 1:  
A continuous time signal x(t) is given in the figure. Plot the functions x(t – 2) and x(t + 
3).  
 
 
Time Scaling  
Expansion or compression of a signal with respect to time is known as time scaling. Let 
x(t) be a continuous time signal, and then x(5t) will be the compressed version of x(t) by 
a factor of 5 in time. And x(t/5) will be the expanded version of x(t) by a factor of 5 in 
time.  
In general, if we consider x(at) then for a > 1, the signal will be compressed by the   
factor ‘a’ and for a < 1, the signal will be expanded by the factor (1/a).