Power Spectrum

Power Spectrum

The power spectrum shows how the power of a signal is distributed across frequencies. It is computed from the DFT magnitudes:

$P[k] = \frac{|X[k]|^2}{N}$

where $N$ is the number of samples. This normalization makes the power independent of signal length.

Frequency Resolution

For a signal sampled at $f_s$ Hz with $N$ samples, each bin $k$ corresponds to:

$f_k = k \cdot \frac{f_s}{N}$

Bin 0 ($k=0$) is the DC component (zero frequency). The highest meaningful frequency is at $k = N/2$ (the Nyquist frequency).

Dominant Frequency

The dominant frequency is the bin with the highest power, excluding DC (bin 0). For a signal with a strong sinusoidal component, this reveals the fundamental frequency.

Example

A 4-sample signal $[1, 0, 0, 0]$ (impulse) has flat power spectrum:

$P[k] = \frac{1}{4} \quad \text{for all } k$

A DC signal $[1, 1, 1, 1]$ has all power at $k=0$:

$P[0] = \frac{16}{4} = 4, \quad P[k] = 0 \text{ for } k > 0$

Your Task

Implement:

• power_spectrum(x) — returns $[|X[0]|^2/N, \ldots, |X[N-1]|^2/N]$
• dominant_frequency(x, fs) — returns $k^* \cdot f_s / N$ where $k^*$ maximizes $P[k]$ for $k > 0$

import math

def dft(x):
    N = len(x)
    X = []
    for k in range(N):
        val = 0 + 0j
        for n in range(N):
            val += x[n] * math.e ** (-2j * math.pi * k * n / N)
        X.append(val)
    return X

def power_spectrum(x):
    N = len(x)
    X = dft(x)
    return [abs(v)**2 / N for v in X]

def dominant_frequency(x, fs):
    N = len(x)
    ps = power_spectrum(x)
    start = 1 if N > 1 else 0
    best_k = start + ps[start:].index(max(ps[start:]))
    return best_k * fs / N