Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

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1 Q.  (i) {1,0,0,…….0} (impulse) ⇔ {1,1,1…..1} (constant)

         (ii) {1,1,1,……1} (constant) ⇔) {N,0,0,&&.0} (impulse)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

(Impulse pair)

Or        Cos (2πnf ) = Cos (wn)

Sol.  x(n)  =  Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

We know that 1⇔ N δ (k )

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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 

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)  

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)?

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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.

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

5 Q.  x(n) = {1, 1, 0, 0, 0, 0, 0, 0}  n = 0 to 7 Find DFT.

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

X(0) = 1+1 = 2

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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  Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

(ii) X(k-1) = {j2, 4, -j2, 0}

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

(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}

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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}

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

The document Questions - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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FAQs on Questions - Discrete Fourier Transform - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is the Discrete Fourier Transform (DFT)?
Ans. The Discrete Fourier Transform (DFT) is a mathematical algorithm used to convert a finite sequence of equally spaced samples of a function into a sequence of complex numbers. It represents the function in terms of its frequency components, allowing analysis of the signal's frequency content.
2. How is the Discrete Fourier Transform calculated?
Ans. The Discrete Fourier Transform (DFT) is calculated using the formula: X(k) = ∑[n=0 to N-1] x(n) * exp(-2πikn/N), where X(k) represents the k-th frequency component, x(n) is the input signal, exp is the exponential function, and N is the number of samples in the input signal.
3. What are the applications of the Discrete Fourier Transform?
Ans. The Discrete Fourier Transform (DFT) has various applications in signal processing and data analysis. It is widely used in fields such as audio and image processing, telecommunications, radar systems, and medical imaging. It allows for frequency analysis, noise reduction, filtering, compression, and much more.
4. What is the difference between the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT)?
Ans. The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) are closely related. The DFT is a mathematical operation, while the FFT is an efficient algorithm to compute the DFT. The FFT significantly reduces the computational complexity of the DFT, making it more practical for real-time applications.
5. Are there any limitations or drawbacks of the Discrete Fourier Transform?
Ans. Yes, the Discrete Fourier Transform (DFT) has some limitations. One limitation is the assumption of periodicity, which can introduce artifacts if the input signal is not periodic. Additionally, the DFT operates on equally spaced samples, limiting its effectiveness for non-uniformly sampled signals. The DFT is also computationally expensive for large input sizes, but this can be mitigated by using the FFT algorithm.
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