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.

   \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} }