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Test: Fourier Transforms Properties


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10 Questions MCQ Test Signals and Systems | Test: Fourier Transforms Properties

Test: Fourier Transforms Properties for Electrical Engineering (EE) 2022 is part of Signals and Systems preparation. The Test: Fourier Transforms Properties questions and answers have been prepared according to the Electrical Engineering (EE) exam syllabus.The Test: Fourier Transforms Properties MCQs are made for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Fourier Transforms Properties below.
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Test: Fourier Transforms Properties - Question 1

 What is the energy density spectrum of the signal x(n)=anu(n), |a|<1? 

Detailed Solution for Test: Fourier Transforms Properties - Question 1

Explanation: Given x(n)= anu(n), |a|<1
The auto correlation of the above signal is
rxx(l)=1/(1-a2 ) a|l|, -∞< l <∞
According to Wiener-Khintchine Theorem,
Sxx(ω)=F{ rxx(l)}= [1/(1-a2)].F{a|l|} = 1/(1-2acosω+a2 )

Test: Fourier Transforms Properties - Question 2

 What is the convolution of the sequences of x1(n)=x2(n)={1,1,1}?

Detailed Solution for Test: Fourier Transforms Properties - Question 2

Explanation: Given x1(n)=x2(n)={1,1,1}
By calculating the Fourier transform of the above two signals, we get
X1(ω)= X2(ω)=1+ ejω + e -jω = 1+2cosω
From the convolution property of Fourier transform we have,
X(ω)= X1(ω). X2(ω)=(1+2cosω)2=3+4cosω+2cos2ω
By applying the inverse Fourier transform of the above signal, we get
x1(n)*x2(n)={1,2,3,2,1}

Test: Fourier Transforms Properties - Question 3

 What is the Fourier transform of the signal x(n)=a|n|, |a|<1? 

Detailed Solution for Test: Fourier Transforms Properties - Question 3

Explanation: First we observe x(n) can be expressed as
x(n)=x1(n)+x2(n)
where x1(n)= an, n>0
=0, elsewhere

 

x2(n)=a-n, n<0 =0, elsewhere Now applying Fourier transform for the above two signals, we get X1(ω)= 1/(1-ae)/((1-ae(-jω) )(1-ae)) = (1-acosω-jasinω)/(1-2acosω+a2 )

Now, X(ω)= X1(ω)+ X2(ω)= 1/(1-ae^(-jω) )+(ae^jω)/(1-ae^jω ) = (1-a2)/(1-2acosω+a2).

Test: Fourier Transforms Properties - Question 4

 If x(n)=A, -M<n<M, then what is the Fourier transform of the signal?
=0, elsewhere

Detailed Solution for Test: Fourier Transforms Properties - Question 4

Explanation: Clearly, x(n)=x(-n). Thus the signal x(n) is real and even signal. So, we know that

Test: Fourier Transforms Properties - Question 5

What is the value of |X(ω)| given X(ω)=1/(1-ae-jω ) ,|a|<1? 

Detailed Solution for Test: Fourier Transforms Properties - Question 5

Explanation: For the given X(ω)=1/(1-ae-jω ) ,|a|<1 we obtain
XI(ω)= (-asinω)/(1-2acosω+a2 ) and XR(ω)= (1-acosω)/(1-2acosω+a2 )
We know that |X(ω)|=√(〖X_R (ω)〗2+〖X_I (ω)〗2 )
Thus on calculating, we obtain
|X(ω)|= 1/√(1-2acosω+a2 )

Test: Fourier Transforms Properties - Question 6

What is the value of XI(ω) given X(ω)=1/(1-ae-jω ) ,|a|<1? 

Detailed Solution for Test: Fourier Transforms Properties - Question 6

Explanation: Given, X(ω)= 1/(1-ae-jω ) ,|a|<1
By multiplying both the numerator and denominator of the above equation by the complex conjugate of the denominator, we obtain
X(ω)= (1-ae)/((1-ae(-jω) )(1-ae)) = (1-acosω-jasinω)/(1-2acosω+a2 )
This expression can be subdivided into real and imaginary parts, thus we obtain
XI(ω)= (-asinω)/(1-2acosω+a2 ).

Test: Fourier Transforms Properties - Question 7

What is the value of XR(ω) given X(ω)=1/(1-ae-jω ) ,|a|<1? 

Detailed Solution for Test: Fourier Transforms Properties - Question 7

Explanation: Given, X(ω)= 1/(1-ae-jω ) ,|a|<1
By multiplying both the numerator and denominator of the above equation by the complex conjugate of the denominator, we obtain
X(ω)= (1-ae)/((1-ae(-jω) )(1-ae)) = (1-acosω-jasinω)/(1-2acosω+a2 )
This expression can be subdivided into real and imaginary parts, thus we obtain
XR(ω)= (1-acosω)/(1-2acosω+a2 ).

Test: Fourier Transforms Properties - Question 8

 If X(ω) is the Fourier transform of the signal x(n), then what is the Fourier transform of the signal x(n-k)? 

Detailed Solution for Test: Fourier Transforms Properties - Question 8

Explanation: Given

Test: Fourier Transforms Properties - Question 9

 If x(n) is a real signal, then 

Detailed Solution for Test: Fourier Transforms Properties - Question 9

We know that if x(n) is a real signal, then xI(n)=0 and xR(n)=x(n)
We know that,

Test: Fourier Transforms Properties - Question 10

Which of the following relations are true if x(n) is real?

Detailed Solution for Test: Fourier Transforms Properties - Question 10

Explanation: We know that, if x(n) is a real sequence

If we combine the above two equations, we get
X*(ω)=X(-ω)

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