Power spectral density in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of PSD calculation with scipy.signal.welch

What is the shape of the frequency and power spectral density arrays returned by scipy.signal.welch for a signal of length 1024 with default parameters?

SciPy

import numpy as np
from scipy.signal import welch

fs = 1000  # Sampling frequency
x = np.random.randn(1024)  # Random signal
f, Pxx = welch(x, fs=fs)
print(f.shape, Pxx.shape)

A (1024,) (1024,)

B (512,) (512,)

C (129,) (129,)

D (256,) (256,)

💡 Hint

The default nperseg is 256, which affects the output length.

The default nperseg in scipy.signal.welch is 256. The output frequency and PSD arrays have length nperseg//2 + 1 = 129.

❓ data_output

intermediate

Power spectral density peak frequency detection

Given a signal composed of two sine waves at 50 Hz and 120 Hz sampled at 500 Hz, which frequency corresponds to the highest peak in the PSD computed by scipy.signal.welch?

SciPy

import numpy as np
from scipy.signal import welch

fs = 500
T = 1.0
t = np.linspace(0, T, int(T*fs), endpoint=False)
x = 0.5*np.sin(2*np.pi*50*t) + 0.3*np.sin(2*np.pi*120*t)
f, Pxx = welch(x, fs=fs, nperseg=100)
peak_freq = f[np.argmax(Pxx)]
print(peak_freq)

A 120.0

B 50.0

C 85.0

D 0.0

💡 Hint

The sine wave with larger amplitude usually produces the higher peak in PSD.

The signal has two sine waves: 50 Hz with amplitude 0.5 and 120 Hz with amplitude 0.3. The PSD peak corresponds to the frequency with the largest power, which is 50 Hz.

❓ visualization

advanced

Interpreting PSD plot of noisy signal

You compute the PSD of a noisy sine wave using scipy.signal.welch and plot it. Which feature in the plot indicates the sine wave frequency?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import welch

fs = 1000
T = 2.0
t = np.linspace(0, T, int(T*fs), endpoint=False)
x = np.sin(2*np.pi*60*t) + 0.5*np.random.randn(len(t))
f, Pxx = welch(x, fs=fs)
plt.semilogy(f, Pxx)
plt.xlabel('Frequency (Hz)')
plt.ylabel('PSD (V**2/Hz)')
plt.title('Power Spectral Density')
plt.show()

A A sharp peak near 60 Hz

B A peak near 0 Hz

C A flat line across all frequencies

D Multiple peaks at random frequencies

💡 Hint

A sine wave shows as a peak at its frequency in the PSD plot.

The sine wave at 60 Hz appears as a sharp peak near 60 Hz in the PSD plot. Noise contributes to a broad background.

🧠 Conceptual

advanced

Effect of segment length on PSD resolution

How does increasing the segment length (nperseg) in scipy.signal.welch affect the frequency resolution and variance of the PSD estimate?

A Decreases frequency resolution and decreases variance

B Decreases frequency resolution and increases variance

C Increases frequency resolution and decreases variance

D Increases frequency resolution but increases variance

💡 Hint

Longer segments give finer frequency bins but fewer averages leading to higher variance.

Increasing nperseg improves frequency resolution (df = fs/nperseg decreases) but results in fewer segments for averaging, increasing the variance of the PSD estimate.

🔧 Debug

expert

Identifying error in PSD computation code

What error does the following code raise when computing PSD with scipy.signal.welch? import numpy as np from scipy.signal import welch fs = 1000 x = np.random.randn(500) f, Pxx = welch(x, fs=fs, nperseg=1024) print(f.shape, Pxx.shape)

SciPy

import numpy as np
from scipy.signal import welch

fs = 1000
x = np.random.randn(500)
f, Pxx = welch(x, fs=fs, nperseg=1024)
print(f.shape, Pxx.shape)

A ValueError: nperseg must not be greater than input length

B TypeError: unsupported operand type(s) for /: 'int' and 'str'

C IndexError: index out of range

D No error, runs successfully

💡 Hint

nperseg cannot be larger than the signal length.

The parameter nperseg cannot be larger than the length of the input signal. Here, nperseg=1024 but signal length is 500, so scipy.signal.welch raises a ValueError.