mugup | revision rigorously

Question 1

Marks : +2 | -2

Pass Ratio : 100%

If x(n) and X(k) are an N-point DFT pair, then x(n+N)=x(n).

True

False

Explanation:

We know that the expression for an DFT is given as

Question 2

Marks : +2 | -2

Pass Ratio : 100%

What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}?

{14,14,16,16}

{16,16,14,14}

{2,3,6,4}

{14,16,14,16}

Explanation:

We know that the circular convolution of two sequences is given by the expression

Question 3

Marks : +2 | -2

Pass Ratio : 100%

If x(n) is real and even, then what is the DFT of x(n)?

\\(\\sum_{n=0}^{N-1} x(n) sinâ¡\\frac{2Ï€kn}{N}\\)

\\(\\sum_{n=0}^{N-1} x(n) cosâ¡\\frac{2Ï€kn}{N}\\)

-j\\(\\sum_{n=0}^{N-1} x(n) sinâ¡\\frac{2Ï€kn}{N}\\)

None of the mentioned

Explanation:

Given x(n) is real and even, that is x(n)=x(N-n)

Question 4

Marks : +2 | -2

Pass Ratio : 100%

If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), X2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then what is the expression for x3(m)?

\\(\\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)\\)

\\(\\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)\\)

\\(\\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \\)

\\(\\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)_N \\)

Explanation:

If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), x2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then according to the multiplication property of DFT we have x3(m) is the circular convolution of X1(n) and x2(n).

Question 5

Marks : +2 | -2

Pass Ratio : 100%

What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}, find using the DFT and IDFT concepts?

{16,16,14,14}

{14,16,14,16}

{14,14,16,16}

None of the mentioned

Explanation:

Given X1(n)={2,1,2,1}=>X1(k)=[6,0,2,0]

Question 6

Marks : +2 | -2

Pass Ratio : 100%

If x(n) and X(k) are an N-point DFT pair, then X(k+N)=?

X(-k)

-X(k)

X(k)

None of the mentioned

Explanation:

We know that

Question 7

Marks : +2 | -2

Pass Ratio : 100%

If x(n) is a real sequence and X(k) is its N-point DFT, then which of the following is true?

X(N-k)=X(-k)

X(N-k)=X*(k)

X(-k)=X*(k)

All of the mentioned

Explanation:

We know that

Question 8

Marks : +2 | -2

Pass Ratio : 100%

If x(n) is a complex valued sequence given by x(n)=xR(n)+jxI(n), then what is the DFT of xR(n)?

\\(\\sum_{n=0}^N x_R (n) cosâ¡\\frac{2Ï€kn}{N}+x_I (n) sinâ¡\\frac{2Ï€kn}{N}\\)

\\(\\sum_{n=0}^N x_R (n) cosâ¡\\frac{2Ï€kn}{N}-x_I (n) sinâ¡\\frac{2Ï€kn}{N}\\)

\\(\\sum_{n=0}^{N-1} x_R (n) cosâ¡\\frac{2Ï€kn}{N}-x_I (n) sinâ¡\\frac{2Ï€kn}{N}\\)

\\(\\sum_{n=0}^{N-1} x_R (n) cosâ¡\\frac{2Ï€kn}{N}+x_I (n) sinâ¡\\frac{2Ï€kn}{N}\\)

Explanation:

Given x(n)=xR(n)+jxI(n)=>xR(n)=1/2(x(n)+x*(n))

Question 9

Marks : +2 | -2

Pass Ratio : 100%

If X1(k) and X2(k) are the N-point DFTs of X1(n) and x2(n) respectively, then what is the N-point DFT of x(n)=ax1(n)+bx2(n)?

X1(ak)+X2(bk)

aX1(k)+bX2(k)

eakX1(k)+ebkX2(k)

None of the mentioned

Explanation:

We know that, the DFT of a signal x(n) is given by the expression

Question 10

Marks : +2 | -2

Pass Ratio : 100%

If x(n) is real and odd, then what is the IDFT of the given sequence?

\\(j \\frac{1}{N} \\sum_{k=0}^{N-1} x(k) sinâ¡\\frac{2Ï€kn}{N}\\)

\\(\\frac{1}{N} \\sum_{k=0}^{N-1} x(k) cosâ¡\\frac{2Ï€kn}{N}\\)

\\(-j \\frac{1}{N} \\sum_{k=0}^{N-1} x(k) sinâ¡\\frac{2Ï€kn}{N}\\)

None of the mentioned

Explanation:

If x(n) is real and odd, that is x(n)=-x(N-n), then XR(k)=0. Hence X(k) is purely imaginary and odd. Since XR(k) reduces to zero, the IDFT reduces to

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