Short Notes: Signals and Systems | Short Notes for Electronics and Communication - Electronics and Communication Engineering (ECE) PDF Download

 Page 2


3. Time scaling: if x(t) is a signal (continuous/discrete) then x(at) is the time scaled
version of x(t) where a is a constant.
4. Periodic signal: A periodic signal has a property that there is a positive value of T
such that x(t) = x(t+nT) for n = 0, 1, 2 … and T is a constant known as period.
5. Even or odd signal: From time reversal if the mirror image of the signal is same
as that of the image. That is if x(-t) = x(t) then the signal is even else if the signal
x(-t) = -x(t) then the signal is odd. Note that any signal can be broken to sum of
signal one which is even and other is odd as given below: -
Even {x(t)}=1/2[x(t)+x(-t)]
Odd {x(t)}=1/2[x(t)-x(-t)]
Exponential and sinusoidal signals:-
Continuous time complex exponential signal is of the form
x(t) = Ce
at
where C and a are generally complex number.
1. Real and exponential signal that is if ‘C’ and ‘a’ are real then the signal is as
follows:
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3. Time scaling: if x(t) is a signal (continuous/discrete) then x(at) is the time scaled
version of x(t) where a is a constant.
4. Periodic signal: A periodic signal has a property that there is a positive value of T
such that x(t) = x(t+nT) for n = 0, 1, 2 … and T is a constant known as period.
5. Even or odd signal: From time reversal if the mirror image of the signal is same
as that of the image. That is if x(-t) = x(t) then the signal is even else if the signal
x(-t) = -x(t) then the signal is odd. Note that any signal can be broken to sum of
signal one which is even and other is odd as given below: -
Even {x(t)}=1/2[x(t)+x(-t)]
Odd {x(t)}=1/2[x(t)-x(-t)]
Exponential and sinusoidal signals:-
Continuous time complex exponential signal is of the form
x(t) = Ce
at
where C and a are generally complex number.
1. Real and exponential signal that is if ‘C’ and ‘a’ are real then the signal is as
follows:
2. Periodic complex exponential and sinusoidal signal: That is ‘a’ is imaginary
x(t) = Ce
jwot
x(t) = Ce
jwo(t+T)
=C. e
jwot
.e
jwoT
but e
jwoT
=1
i.e. if w
0
=0 then x(t)=1 and if w
0
?0 then T=2p/|w
0
|
Signal closely related is x(t)= a cos(w
0
t+f)
Euler’s relation: e
jwot
=cosw
0
t+jsinw
0
t
Acos(w
0
t+f)=A.Re{e
j(wot+f)
} and Asin(w
0
t+f)=A.Im{e
j(wot+f)
}
3. Growing and decaying sinusoidal signal:
x(t)=Ce
rt
cos(w
0
t+f) if r>0 then growing signal and if r<0 then decaying signal
Sinusoidal signal multiplied by decaying exponential is referred as damped
exponential. Similarly for the discrete time characteristic where t becomes n.
Unit impulse and unit step function:
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3. Time scaling: if x(t) is a signal (continuous/discrete) then x(at) is the time scaled
version of x(t) where a is a constant.
4. Periodic signal: A periodic signal has a property that there is a positive value of T
such that x(t) = x(t+nT) for n = 0, 1, 2 … and T is a constant known as period.
5. Even or odd signal: From time reversal if the mirror image of the signal is same
as that of the image. That is if x(-t) = x(t) then the signal is even else if the signal
x(-t) = -x(t) then the signal is odd. Note that any signal can be broken to sum of
signal one which is even and other is odd as given below: -
Even {x(t)}=1/2[x(t)+x(-t)]
Odd {x(t)}=1/2[x(t)-x(-t)]
Exponential and sinusoidal signals:-
Continuous time complex exponential signal is of the form
x(t) = Ce
at
where C and a are generally complex number.
1. Real and exponential signal that is if ‘C’ and ‘a’ are real then the signal is as
follows:
2. Periodic complex exponential and sinusoidal signal: That is ‘a’ is imaginary
x(t) = Ce
jwot
x(t) = Ce
jwo(t+T)
=C. e
jwot
.e
jwoT
but e
jwoT
=1
i.e. if w
0
=0 then x(t)=1 and if w
0
?0 then T=2p/|w
0
|
Signal closely related is x(t)= a cos(w
0
t+f)
Euler’s relation: e
jwot
=cosw
0
t+jsinw
0
t
Acos(w
0
t+f)=A.Re{e
j(wot+f)
} and Asin(w
0
t+f)=A.Im{e
j(wot+f)
}
3. Growing and decaying sinusoidal signal:
x(t)=Ce
rt
cos(w
0
t+f) if r>0 then growing signal and if r<0 then decaying signal
Sinusoidal signal multiplied by decaying exponential is referred as damped
exponential. Similarly for the discrete time characteristic where t becomes n.
Unit impulse and unit step function:
Important Properties of Signals:
Page 5


3. Time scaling: if x(t) is a signal (continuous/discrete) then x(at) is the time scaled
version of x(t) where a is a constant.
4. Periodic signal: A periodic signal has a property that there is a positive value of T
such that x(t) = x(t+nT) for n = 0, 1, 2 … and T is a constant known as period.
5. Even or odd signal: From time reversal if the mirror image of the signal is same
as that of the image. That is if x(-t) = x(t) then the signal is even else if the signal
x(-t) = -x(t) then the signal is odd. Note that any signal can be broken to sum of
signal one which is even and other is odd as given below: -
Even {x(t)}=1/2[x(t)+x(-t)]
Odd {x(t)}=1/2[x(t)-x(-t)]
Exponential and sinusoidal signals:-
Continuous time complex exponential signal is of the form
x(t) = Ce
at
where C and a are generally complex number.
1. Real and exponential signal that is if ‘C’ and ‘a’ are real then the signal is as
follows:
2. Periodic complex exponential and sinusoidal signal: That is ‘a’ is imaginary
x(t) = Ce
jwot
x(t) = Ce
jwo(t+T)
=C. e
jwot
.e
jwoT
but e
jwoT
=1
i.e. if w
0
=0 then x(t)=1 and if w
0
?0 then T=2p/|w
0
|
Signal closely related is x(t)= a cos(w
0
t+f)
Euler’s relation: e
jwot
=cosw
0
t+jsinw
0
t
Acos(w
0
t+f)=A.Re{e
j(wot+f)
} and Asin(w
0
t+f)=A.Im{e
j(wot+f)
}
3. Growing and decaying sinusoidal signal:
x(t)=Ce
rt
cos(w
0
t+f) if r>0 then growing signal and if r<0 then decaying signal
Sinusoidal signal multiplied by decaying exponential is referred as damped
exponential. Similarly for the discrete time characteristic where t becomes n.
Unit impulse and unit step function:
Important Properties of Signals: