Compute the 8-point DFT of the sequence
x[n] = 
1 2 3 4 0 0 0 0
↑

Using the radix-2 DIT algorithm. Follow exactly the corresponding signal flow graphs and keep track
of all the intermediate quantities by putting them on the diagram.

To compute the 8-point Discrete Fourier Transform (DFT) of the sequence \( x[n] = \{1, 2, 3, 4, 0, 0, 0, 0\} \) using the radix-2 Decimation-In-Time (DIT) algorithm, we need to follow the steps of the Fast Fourier Transform (FFT) algorithm carefully, which breaks the problem into smaller parts through butterfly computations.

### Step-by-Step Procedure for Radix-2 DIT FFT

1. **Input sequence:**
   \[
   x[n] = \{1, 2, 3, 4, 0, 0, 0, 0\}
   \]

2. **Rearrange input sequence in bit-reversed order:**
   For an 8-point sequence, the bit-reversed indices for the original indices are:

   \[
   \text{Original indices: } \{0, 1, 2, 3, 4, 5, 6, 7\}
   \]
   \[
   \text{Bit-reversed indices: } \{0, 4, 2, 6, 1, 5, 3, 7\}
   \]

   Rearranging the input sequence according to these bit-reversed indices:

   \[
   x_{\text{bit-reversed}} = \{1, 0, 3, 0, 2, 0, 4, 0\}
   \]

3. **First stage of the butterfly computations:**
   This stage involves 4 butterfly computations, where we combine pairs of values.

   \[
   \text{Butterfly computation: } X[k] = x[n] + W_N^k \cdot x[n+N/2]
   \]

   For this stage, \( N = 8 \) and \( W_8^k = e^{-j 2\pi k/8} \) are the twiddle factors. However, for the first stage, we only need to add/subtract the values as the twiddle factors for the first stage are all 1.

   \[
   \begin{aligned}
   \text{Butterfly 1: } & (1, 0) \Rightarrow 1+0 = 1 \text{ and } 1-0 = 1 \\
   \text{Butterfly 2: } & (3, 0) \Rightarrow 3+0 = 3 \text{ and } 3-0 = 3 \\
   \text{Butterfly 3: } & (2, 0) \Rightarrow 2+0 = 2 \text{ and } 2-0 = 2 \\
   \text{Butterfly 4: } & (4, 0) \Rightarrow 4+0 = 4 \text{ and } 4-0 = 4 \\
   \end{aligned}
   \]

   After this stage, the intermediate output is:

   \[
   \{1, 1, 3, 3, 2, 2, 4, 4\}
   \]

4. **Second stage of the butterfly computations:**
   Now, in this stage, we use the twiddle factors for the 8-point FFT. The twiddle factors for \( N = 8 \) are:

   \[
   W_8^0 = 1, \quad W_8^1 = e^{-j\pi/4} = \frac{1}{\sqrt{2}} - j\frac{1}{\sqrt{2}}, \quad W_8^2 = e^{-j\pi/2} = -j, \quad W_8^3 = e^{-3j\pi/4}
   \]

   Perform the butterfly computations using these twiddle factors:

   \[
   \begin{aligned}
   \text{Butterfly 1: } & (1, 3) \Rightarrow 1 + 3 = 4 \text{ and } 1 - 3 = -2 \\
   \text{Butterfly 2: } & (2, 4) \Rightarrow 2 + 4 = 6 \text{ and } 2 - 4 = -2 \\
   \text{Butterfly 3: } & \text{Apply } W_8^1 \text{ to the second half of each pair} \\
   & (1 + j3) \text{ and } (2 + j4)
   \end{aligned}
   \}



Compute the 8-point DFT of the sequence x[n] = {1, 2, 3, 4, 0, 0, 0, 0} using the radix-2 DIT algorithm. Follow exactly the corresponding signal flow graphs and keep track of all the intermediate quantities by putting them on the diagram.

Learn how to compute the 8-point DFT of the sequence {1, 2, 3, 4, 0, 0, 0, 0} using the radix-2 Decimation-In-Time (DIT) algorithm. Follow the Fast Fourier Transform (FFT) process with detailed intermediate steps, butterfly computations, and signal flow graphs. Suitable for university-level students studying Fourier transforms.

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