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 by the set \(\{20, -4+4j, -4, -4-4j\}\), we need to apply the inverse DFT (IDFT) formula.

The DFT of a sequence \( x(n) \) of length \( N \) is defined as:

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

Given the DFT values \( X(k) = \{20, -4+4j, -4, -4-4j\} \) for \( k = 0, 1, 2, 3 \), and knowing that the length \( N \) of the sequence is 4 (since there are 4 DFT values), the inverse DFT (IDFT) formula is:

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

Let's compute \( x(n) \) step-by-step.

### Step-by-Step Calculation

1. **Determine \( N \):**  
   \( N = 4 \) because there are 4 elements in the DFT array.

2. **Compute \( x(n) \) using the IDFT formula:**

   For \( n = 0, 1, 2, 3 \):

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

3. **Calculate \( x(0) \):**

\[
x(0) = \frac{1}{4} \left( 20 \cdot e^{j0} + (-4+4j) \cdot e^{j0} + (-4) \cdot e^{j0} + (-4-4j) \cdot e^{j0} \right).
\]

Since \( e^{j0} = 1 \):

\[
x(0) = \frac{1}{4} \left( 20 + (-4+4j) + (-4) + (-4-4j) \right).
\]

Combine the real and imaginary parts:

\[
x(0) = \frac{1}{4} (20 - 4 - 4 - 4 + 4j - 4j) = \frac{1}{4} (8) = 2.
\]

4. **Calculate \( x(1) \):**

\[
x(1) = \frac{1}{4} \left( 20 \cdot e^{j0} + (-4+4j) \cdot e^{j\frac{\pi}{2}} + (-4) \cdot e^{j\pi} + (-4-4j) \cdot e^{j\frac{3\pi}{2}} \right).
\]

Substitute the complex exponentials:

\[
e^{j0} = 1, \quad e^{j\frac{\pi}{2}} = j, \quad e^{j\pi} = -1, \quad e^{j\frac{3\pi}{2}} = -j.
\]

Therefore:

\[
x(1) = \frac{1}{4} \left( 20 + (-4+4j)j + (-4)(-1) + (-4-4j)(-j) \right).
\]

Simplify each term:

\[
x(1) = \frac{1}{4} \left( 20 + (-4j - 4) + 4 + (4j - 4) \right).
\]

Combine like terms:

\[
x(1) = \frac{1}{4} (20 - 4 - 4 + 4j - 4j + 4) = \frac{1}{4} (16) = 4.
\]

5. **Calculate \( x(2) \):**

\[
x(2) = \frac{1}{4} \left( 20 \cdot e^{j0} + (-4+4j) \cdot e^{j\pi} + (-4) \cdot e^{j2\pi} + (-4-4j) \cdot e^{j3\pi} \right).
\]

Use \( e^{j2\pi} = 1 \) and \( e^{j\pi} = -1 \):

\[
x(2) = \frac{1}{4} \left( 20 + (-4+4j)(-1) + (-4)(1) + (-4-4j)(-1) \right).
\]

Simplify each term:

\[
x(2) = \frac{1}{4} \left( 20 + (4-4j) - 4 + (4+4j) \right).
\]

Combine like terms:

\[
x(2) = \frac{1}{4} (20 + 4 - 4 - 4j + 4 + 4j) = \frac{1}{4} (24) = 6.
\]

6. **Calculate \( x(3) \):**

\[
x(3) = \frac{1}{4} \left( 20 \cdot e^{j0} + (-4+4j) \cdot e^{j\frac{3\pi}{2}} + (-4) \cdot e^{j3\pi} + (-4-4j) \cdot e^{j\frac{9\pi}{2}} \right).
\]

Use \( e^{j\frac{3\pi}{2}} = -j \) and \( e^{j3\pi} = -1 \):

\[
x(3) = \frac{1}{4} \left( 20 + (-4+4j)(-j) + (-4)(-1) + (-4-4j)(j) \right).
\]

Simplify each term:

\[
x(3) = \frac{1}{4} \left( 20 + (4j + 4) + 4 + (-4j - 4) \right).
\]

Combine like terms:

\[
x(3) = \frac{1}{4} (20 + 4 + 4 - 4j + 4j - 4) = \frac{1}{4} (24) = 6.
\]

### Final Sequence

The computed sequence \( x(n) \) is:

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

Would you like a more detailed explanation, or do you have any questions? Here are some related questions to consider:

1. How is the DFT related to the Fourier series?
2. What are some applications of the DFT in real-world scenarios?
3. How does the IDFT differ from the inverse Laplace transform?
4. Can you compute the DFT for sequences of different lengths?
5. What are the computational complexities of the DFT and FFT?

**Tip:** The DFT is a powerful tool for analyzing frequency components in digital signals, making it essential in fields like audio processing and communications.

Learn how to compute the sequence x(n) from the given Discrete Fourier Transform (DFT) values {20, -4+4j, -4, -4-4j}. Detailed step-by-step solution using the Inverse DFT (IDFT) formula, covering complex numbers, exponential functions, and DFT fundamentals. Suitable for advanced learners.

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