Frequency Component Energy

Implement a function to calculate the energy of a signal from its Discrete Fourier Transform (DFT) magnitudes. This task involves understanding the relationship between a signal's time-domain representation and its frequency-domain representation. The Fourier Transform is a mathematical tool used to decompose a signal into its constituent frequencies, and the DFT is a discrete-time equivalent.

The energy of a signal can be computed using the time-domain representation, but it can also be calculated using the frequency-domain representation, thanks to Parseval's theorem. This theorem states that the energy of a signal is equal to the sum of the squared magnitudes of its frequency components.

To compute the energy, follow these steps:

Square each magnitude of the DFT
Sum the squared magnitudes
Divide the sum by the total number of frequency components, $N$

E = \frac{1}{N}\sum_{k=0}^{N-1} |X[k]|^2

This technique is widely used in image processing to analyze the frequency components of an image.

Example:

Input:

signal_energy([2, 0, 2, 0])

Output:

2.0

Reasoning:

The input list is interpreted as the DFT magnitudes: $X = [2, 0, 2, 0]$ .
Compute squared magnitudes and sum: $|2|^2 + |0|^2 + |2|^2 + |0|^2 = 4 + 0 + 4 + 0 = 8$ .
There are $N = 4$ frequency bins, so energy is $E = \frac{1}{N} \sum_{k=0}^{3} |X[k]|^2 = \frac{1}{4} \cdot 8 = 2.0$ .
Thus, signal_energy([2, 0, 2, 0]) returns $2.0$ .

Constraints:

Return energy rounded to 4 decimal places

Example:

Constraints:

Test Results