To compute the Inverse Discrete Fourier Transform (IDFT) using the provided parameters, we'll use the following IDFT equation:

$x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) \cdot e^{j\frac{2\pi}{N}kn}$

Where:

• $N = 4$ (number of points)
• $n = 3$ (the index of the output we want to calculate)
• $X(k)$ are the DFT coefficients: $X(0) = 50$, $X(1) = 2 - 2j$, $X(2) = -2$, $X(3) = 2 + 2j$.

Step-by-Step Calculation:

We need to compute $x(3)$ as follows:

$x(3) = \frac{1}{4} \left[ X(0) \cdot e^{j\frac{2\pi}{4} \cdot 0 \cdot 3} + X(1) \cdot e^{j\frac{2\pi}{4} \cdot 1 \cdot 3} + X(2) \cdot e^{j\frac{2\pi}{4} \cdot 2 \cdot 3} + X(3) \cdot e^{j\frac{2\pi}{4} \cdot 3 \cdot 3} \right]$

Breaking it down:

• For $k = 0$: $X(0) \cdot e^{j\frac{2\pi}{4} \cdot 0 \cdot 3} = 50 \cdot e^{j \cdot 0} = 50$

• For $k = 1$: $X(1) \cdot e^{j\frac{2\pi}{4} \cdot 1 \cdot 3} = (2 - 2j) \cdot e^{j \cdot \frac{3\pi}{2}} = (2 - 2j) \cdot (-j) = 2j + 2$

• For $k = 2$: $X(2) \cdot e^{j\frac{2\pi}{4} \cdot 2 \cdot 3} = -2 \cdot e^{j \cdot 3\pi} = -2 \cdot (-1) = 2$

• For $k = 3$: $X(3) \cdot e^{j\frac{2\pi}{4} \cdot 3 \cdot 3} = (2 + 2j) \cdot e^{j \cdot \frac{9\pi}{2}} = (2 + 2j) \cdot j = -2 + 2j$

Summing the terms:

$x(3) = \frac{1}{4} \left[ 50 + (2 + 2j) + 2 + (-2 + 2j) \right]$

Simplifying:

[ x(3) = \frac{1}{4} \left[ 50 + 2 + 2 + 2j + 2j \right] = \frac{1}{4} \left[ 54 + 4j \right] = 13.5 + j ]

Final Answer:

$x(3) = 13.5 + j$

Would you like any further details or have any questions?

Here are some related questions to explore further:

1. What is the general form of the IDFT equation?
2. How does the IDFT relate to the DFT?
3. What are the properties of the Fourier Transform?
4. How do you compute the IDFT for different values of $n$?
5. Can you explain the role of the complex exponential in the IDFT?
6. How does changing $N$ affect the IDFT result?
7. What are the practical applications of IDFT in signal processing?
8. How do you interpret the real and imaginary parts of the IDFT result?

Tip: When calculating the IDFT, make sure to account for both the real and imaginary parts, especially when multiplying by complex exponentials.