Compute the sequence x(n) for which the DFT is { 20,
-4+4j, -4, -4-4j}

To compute the sequence \( x(n) \) for which the Discrete Fourier Transform (DFT) is given as \( \{ 20, -4 + 4j, -4, -4 - 4j \} \), we need to use the inverse DFT (IDFT) formula.

The IDFT is defined as:

\[
x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j\frac{2\pi}{N}kn} \quad \text{for } n = 0, 1, 2, \dots, N-1,
\]

where:
- \( X(k) \) is the DFT of the sequence.
- \( N \) is the length of the sequence (in this case, \( N = 4 \)).
- \( j \) is the imaginary unit.

Given \( X(k) = \{ 20, -4 + 4j, -4, -4 - 4j \} \), let's compute \( x(n) \) for \( n = 0, 1, 2, 3 \).

The IDFT formula for \( N = 4 \) becomes:

\[
x(n) = \frac{1}{4} \sum_{k=0}^{3} X(k) e^{j\frac{2\pi}{4}kn}.
\]

Let's calculate each value of \( x(n) \) step-by-step.The sequence \( x(n) \) for which the DFT is \( \{ 20, -4 + 4j, -4, -4 - 4j \} \) is:

\[
x(n) = \{ 2, 4, 6, 8 \}.
\]

Would you like more details on the process or have any questions?

Here are 5 related questions you might find interesting:

1. How is the DFT related to the Fourier series for periodic signals?
2. What is the significance of the imaginary unit \( j \) in the DFT?
3. How would you compute the DFT for a sequence longer than 4 points?
4. What is the effect of zero-padding a sequence before computing the DFT?
5. How does the DFT differ from the continuous Fourier transform?

**Tip:** The DFT assumes periodicity in the input sequence, which means the signal is treated as if it repeats indefinitely.

Learn how to compute the sequence x(n) for a given Discrete Fourier Transform (DFT) of { 20, -4+4j, -4, -4-4j }. Detailed step-by-step solution using the Inverse Discrete Fourier Transform (IDFT) formula. Includes explanation of complex numbers and their roles in DFT computations. Suitable for advanced learners and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation