Questions
1 Q. (i) {1,0,0,…….0} (impulse) ⇔ {1,1,1…..1} (constant)
(ii) {1,1,1,……1} (constant) ⇔) {N,0,0,&&.0} (impulse)
(Impulse pair)
Or Cos (2πnf ) = Cos (wn)
Sol. x(n) =
We know that 1⇔ N δ (k )
I. Inverse DFT of a constant is a unit sample.
II. DFT of a constant is a unit sample.
2 Q. Find 10 point IDFT of
3 Q. Suppose that we are given a program to find the DFT of a complex-valued sequence x(n). How can this program be used to find the inverse DFT of X(k)?
∴
1. Conjugate the DFT coefficients X(k) to produce the sequence X*(k).
2. Use the program to fing DFT of a sequence X*(k).
3. Conjugate the result obtained in step 2 and divide by N.
5 Q. x(n) = {1, 1, 0, 0, 0, 0, 0, 0} n = 0 to 7 Find DFT.
X(0) = 1+1 = 2
By conjugate symmetry X(k) = X*(N-k) = X*(8-k)
∴X(5) = X*(3) = 0.293 + j 0.707
X(6) = X*(2) = 1+j
X(7) = X*(1) = 1.707 + j 0.707
X(k) = { 2 , 1.707 - j 0.707, 0.293 - j 0.707, 1-j, 0, 1+j, 0.293 + j 0.707, 1.707 + j 0.707 }
6 Q. x(n) = {1, 2, 1, 0} N=4
X(k) = {4, -j2, 0, j2}
(i) y(n) = x(n-2) = {1, 0, 1, 2}
Y(k) = X(k) e
(ii) X(k-1) = {j2, 4, -j2, 0}
(iii) g(n) = x(-n) = 1, 0, 1, 2
G(k) = X(-k) = X*(k) = {4, j2, 0, -j2}
(iv) p(n) = x*(n) = {1, 2, 1, 0}
P(k) = X*(-k) = {4, j2, 0, -j2}* = {4, -j2, 0, j2}
(v) h(n) = x(n) x(n)
= {1, 4, 1, 0}
S(k) = X(k) X(k) = {16, -2.35- j 10.28, -2.18 + j 1.05, 0.02 + j 0.03, 0.02 - j 0.03, -2.18 -j 1.05, -2.35 + j 10.28}
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1. What is the Discrete Fourier Transform (DFT)? |
2. How is the Discrete Fourier Transform calculated? |
3. What are the applications of the Discrete Fourier Transform? |
4. What is the difference between the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT)? |
5. Are there any limitations or drawbacks of the Discrete Fourier Transform? |
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