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 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:
Page 3


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:
Page 4


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:
Read More
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FAQs on Signal & System Formulas for GATE EE Exam - GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

1. What are the key formulas in signal and system analysis for the GATE EE exam in Electrical Engineering?
Ans. Some key formulas in signal and system analysis for the GATE EE exam in Electrical Engineering include: 1. Fourier Transform: F(w) = ∫[x(t)e^(-jwt)]dt 2. Inverse Fourier Transform: x(t) = 1/2π ∫[F(w)e^(jwt)]dw 3. Laplace Transform: X(s) = ∫[x(t)e^(-st)]dt 4. Inverse Laplace Transform: x(t) = 1/2πj ∫[X(s)e^(st)]ds 5. Convolution: y(t) = x(t) * h(t) = ∫[x(τ)h(t-τ)]dτ These formulas are commonly used to analyze and manipulate signals and systems in electrical engineering.
2. How are Fourier Transform and Inverse Fourier Transform related to signal analysis in the GATE EE exam?
Ans. Fourier Transform and Inverse Fourier Transform are essential tools in signal analysis for the GATE EE exam. Fourier Transform is used to decompose a signal into its frequency components. It transforms a signal from the time domain to the frequency domain, allowing us to analyze its frequency content. The formula for the Fourier Transform is F(w) = ∫[x(t)e^(-jwt)]dt, where F(w) represents the frequency domain representation of the signal x(t). Inverse Fourier Transform, on the other hand, is used to reconstruct a signal from its frequency components. It transforms a signal from the frequency domain back to the time domain. The formula for the Inverse Fourier Transform is x(t) = 1/2π ∫[F(w)e^(jwt)]dw, where x(t) represents the time domain representation of the signal F(w). By utilizing Fourier Transform and Inverse Fourier Transform, we can analyze the frequency content of a signal and understand its behavior.
3. How do Laplace Transform and Inverse Laplace Transform aid in system analysis for the GATE EE exam?
Ans. Laplace Transform and Inverse Laplace Transform play a crucial role in system analysis for the GATE EE exam. Laplace Transform is used to analyze the behavior of linear time-invariant (LTI) systems. It transforms a time-domain representation of a system into the complex frequency domain, making it easier to analyze and manipulate. The Laplace Transform formula is X(s) = ∫[x(t)e^(-st)]dt, where X(s) represents the complex frequency domain representation of the system input x(t). Inverse Laplace Transform, on the other hand, is used to obtain the time-domain representation of a system from its complex frequency domain representation. It allows us to understand how a system responds to different inputs in the time domain. The formula for the Inverse Laplace Transform is x(t) = 1/2πj ∫[X(s)e^(st)]ds, where x(t) represents the time-domain representation of the system input X(s). By utilizing Laplace Transform and Inverse Laplace Transform, we can analyze the behavior of systems and predict their response to different inputs.
4. How is convolution used in signal and system analysis for the GATE EE exam?
Ans. Convolution is a fundamental operation used in signal and system analysis for the GATE EE exam. Convolution is used to determine the output of a system when the input signal is passed through it. It involves integrating the product of the input signal and the impulse response of the system over a certain interval. Mathematically, convolution is denoted by the symbol "*", and the formula is y(t) = x(t) * h(t) = ∫[x(τ)h(t-τ)]dτ. By performing convolution, we can analyze how a system modifies the input signal. It helps us understand the output behavior of a system and its response to different inputs. Convolution is commonly used in areas such as signal processing, communication systems, and control systems.
5. How can knowledge of signal and system formulas benefit me in the GATE EE exam for Electrical Engineering?
Ans. Knowledge of signal and system formulas can greatly benefit you in the GATE EE exam for Electrical Engineering. 1. Understanding the formulas allows you to analyze and manipulate signals and systems effectively. You can apply the appropriate mathematical operations to solve problems related to signals and systems. 2. These formulas help you analyze the frequency content of signals and understand their behavior in the frequency domain. This knowledge is crucial in various applications such as communication systems and signal processing. 3. Knowledge of Laplace Transform and Inverse Laplace Transform enables you to analyze the behavior of linear time-invariant systems. You can predict and analyze the system's response to different inputs in the time domain. 4. Convolution is a fundamental operation in signal and system analysis. Understanding and applying convolution allows you to determine the output of a system when the input signal is passed through it. By mastering these formulas, you can confidently approach signal and system analysis questions in the GATE EE exam and solve them accurately. It enhances your overall understanding of electrical engineering concepts and prepares you for real-world applications in the field.
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