Spectrogram visualization in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the shape of the spectrogram matrix?

Given a signal sampled at 1024 Hz with 1 second duration, a window size of 256 samples, and an overlap of 128 samples, what is the shape of the spectrogram matrix produced by the following code?

Signal Processing

import numpy as np
from scipy.signal import spectrogram
fs = 1024
signal = np.sin(2 * np.pi * 50 * np.linspace(0, 1, fs, endpoint=False))
f, t, Sxx = spectrogram(signal, fs=fs, window='hann', nperseg=256, noverlap=128)
print(Sxx.shape)

A (256, 8)

B (128, 8)

C (129, 8)

D (129, 7)

💡 Hint

Remember the number of frequency bins is nperseg//2 + 1, and the number of time segments depends on overlap.

The number of frequency bins is 256/2 + 1 = 129. The number of time segments is calculated as floor((1024 - 256) / (256 - 128)) + 1 = floor(768 / 128) + 1 = 6 + 1 = 7. So the shape is (129, 7).

❓ data_output

intermediate

Identify the dominant frequency from the spectrogram data

Using the spectrogram data from the code below, which frequency has the highest average power over time?

Signal Processing

import numpy as np
from scipy.signal import spectrogram
fs = 500
t = np.linspace(0, 2, 2*fs, endpoint=False)
signal = np.sin(2 * np.pi * 60 * t) + 0.5 * np.sin(2 * np.pi * 120 * t)
f, _, Sxx = spectrogram(signal, fs=fs, nperseg=128)
avg_power = Sxx.mean(axis=1)
max_freq = f[np.argmax(avg_power)]
print(round(max_freq))

A 60

B 120

C 90

D 30

💡 Hint

Look for the frequency with the highest mean power across all time segments.

The signal has two frequencies: 60 Hz and 120 Hz. The 60 Hz component has higher amplitude, so its average power is higher, making 60 Hz the dominant frequency.

❓ visualization

advanced

Which plot shows the correct spectrogram visualization?

You run the following code to plot a spectrogram of a signal. Which option shows the correct plot output?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import spectrogram
fs = 1000
t = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2 * np.pi * 100 * t)
f, t_spec, Sxx = spectrogram(signal, fs=fs, nperseg=256)
plt.pcolormesh(t_spec, f, 10 * np.log10(Sxx), shading='gouraud')
plt.ylabel('Frequency [Hz]')
plt.xlabel('Time [sec]')
plt.title('Spectrogram')
plt.colorbar(label='Intensity [dB]')
plt.show()

A A plot with frequency on y-axis from 0 to 1000 Hz, time on x-axis from 0 to 1 sec, and no visible lines.

B A plot with time on y-axis and frequency on x-axis, showing a vertical line at 100 Hz.

C A plot with frequency on y-axis from 0 to 500 Hz, time on x-axis from 0 to 1 sec, and a bright horizontal line at 100 Hz.

D A plot with frequency on x-axis and time on y-axis, showing a bright horizontal line at 100 Hz.

💡 Hint

Remember the spectrogram frequency axis goes from 0 to Nyquist frequency (fs/2).

The spectrogram frequency axis ranges from 0 to 500 Hz (half the sampling rate). The signal frequency is 100 Hz, so a bright horizontal line appears at 100 Hz over time.

🔧 Debug

advanced

Why does this spectrogram plot show no signal?

The code below produces a spectrogram plot, but the plot is completely dark with no visible signal. What is the most likely cause?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import spectrogram
fs = 1000
t = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2 * np.pi * 50 * t)
f, t_spec, Sxx = spectrogram(signal, fs=fs, nperseg=1024)
plt.pcolormesh(t_spec, f, 10 * np.log10(Sxx), shading='gouraud')
plt.show()

A The sampling frequency fs is too low to capture 50 Hz signal.

B The window size nperseg is larger than the signal length, causing no segments to be computed.

C The signal is zero, so no power is shown.

D The plotting function plt.pcolormesh is used incorrectly without axis labels.

💡 Hint

Check the relationship between signal length and nperseg parameter.

The signal length is 1000 samples, but nperseg=1024 > 1000. This causes the array of start indices np.arange(0, 1000 - 1024 + 1, ...) to be empty (upper limit negative), so no segments are computed and Sxx is empty, resulting in a blank plot.

🚀 Application

expert

How to detect time-varying frequency components using spectrogram?

You have a signal composed of two sine waves: one at 50 Hz active during the first half of the signal, and another at 150 Hz active during the second half. Which approach using spectrogram visualization best reveals these time-varying frequency components?

A Use a short window size (e.g., 64 samples) with overlap to get good time resolution and plot the spectrogram.

B Use a very long window size (e.g., 1024 samples) without overlap to get best frequency resolution.

C Use no windowing and plot the raw FFT of the entire signal.

D Use a window size equal to the entire signal length to maximize frequency resolution.

💡 Hint

Short windows improve time resolution but reduce frequency resolution.

To detect changes in frequency over time, good time resolution is needed. Short windows with overlap provide this, allowing the spectrogram to show when frequencies appear and disappear.

Practice

1. What does a spectrogram primarily show in signal processing?

easy

A. How signal frequencies change over time

B. The total energy of a signal

C. The raw amplitude values of a signal

D. The phase shift of a signal

Solution

1. Step 1: Understand spectrogram purpose

   A spectrogram is a visual tool that shows frequency content of a signal as it changes over time.

2. Step 2: Compare options to definition

   Only How signal frequencies change over time describes frequency changes over time, which matches the spectrogram's function.

3. Final Answer:

   How signal frequencies change over time -> Option A

4. Quick Check:

   Spectrogram = frequency over time [OK]

Hint: Spectrogram = frequency vs time graph [OK]

Common Mistakes:

• Confusing spectrogram with waveform plot
• Thinking it shows only amplitude
• Mixing phase information with frequency

2. Which Python library is commonly used to create spectrograms?

easy

A. matplotlib

B. pandas

C. scikit-learn

D. numpy

Solution

1. Step 1: Identify libraries for plotting

   matplotlib is widely used for plotting graphs including spectrograms.

2. Step 2: Check other libraries' main uses

   pandas is for data frames, scikit-learn for machine learning, numpy for arrays but not direct plotting.

3. Final Answer:

   matplotlib -> Option A

4. Quick Check:

   Plotting spectrogram = matplotlib [OK]

Hint: Plotting graphs? Think matplotlib [OK]

Common Mistakes:

• Choosing numpy for plotting directly
• Confusing pandas with plotting library
• Selecting scikit-learn which is for ML

3. Given this Python code snippet, what is the shape of the output spectrogram array?

import numpy as np
from scipy.signal import spectrogram

fs = 1280  # Sampling frequency
x = np.sin(2 * np.pi * 50 * np.linspace(0, 1, fs))
f, t, Sxx = spectrogram(x, fs)
print(Sxx.shape)

medium

A. (1000, 1000)

B. (129, 9)

C. (50, 1000)

D. (1000, 129)

Solution

1. Step 1: Understand spectrogram output dimensions

   The spectrogram function returns frequency bins and time bins. For default parameters, frequency bins are nfft/2+1 (default nfft=256), so 129 frequency bins.

2. Step 2: Calculate time bins

   With 1 second signal sampled at 1280 Hz and default windowing, time bins are 9 segments.

3. Final Answer:

   (129, 9) -> Option B

4. Quick Check:

   Frequency bins = 129, Time bins = 9 [OK]

Hint: Spectrogram shape = (freq_bins, time_bins) [OK]

Common Mistakes:

• Assuming output shape matches input length
• Confusing frequency bins with sampling rate
• Mixing up rows and columns in output

4. Identify the error in this code snippet for plotting a spectrogram:

import matplotlib.pyplot as plt
from scipy.signal import spectrogram

fs = 500
x = [1, 2, 3, 4, 5]
f, t, Sxx = spectrogram(x, fs)
plt.specgram(Sxx)
plt.show()

medium

A. Using plt.specgram instead of plt.plot

B. Missing import for numpy

C. Sampling frequency fs is too high

D. Passing Sxx instead of x to plt.specgram

Solution

1. Step 1: Understand spectrogram output and plotting

   The spectrogram function returns frequencies, times, and the spectrogram matrix Sxx. plt.specgram expects raw signal, not Sxx.

2. Step 2: Identify misuse of plt.specgram

   Passing Sxx (already spectrogram data) to plt.specgram causes error; should plot Sxx with plt.imshow or pass raw signal to plt.specgram.

3. Final Answer:

   Passing Sxx instead of x to plt.specgram -> Option D

4. Quick Check:

   Plot raw signal with plt.specgram, not spectrogram matrix [OK]

Hint: plt.specgram needs raw signal, not spectrogram data [OK]

Common Mistakes:

• Passing spectrogram matrix to plt.specgram
• Confusing plt.specgram with plt.imshow
• Ignoring correct input types for plotting

5. You want a spectrogram with smoother frequency resolution but less time detail. Which parameter should you increase?

hard

A. Sampling frequency (fs)

B. Overlap between windows (noverlap)

C. Window size (nperseg)

D. FFT length (nfft) smaller than window size

Solution

1. Step 1: Understand window size effect on spectrogram

   Larger window size increases frequency resolution but reduces time resolution (smoother frequency, less time detail).

2. Step 2: Evaluate other parameters

   Increasing overlap smooths time axis but does not improve frequency resolution as much. Sampling frequency affects max frequency, not resolution. nfft smaller than window size is invalid.

3. Final Answer:

   Window size (nperseg) -> Option C

4. Quick Check:

   Increase window size for better frequency resolution [OK]

Hint: Bigger window = better frequency, worse time detail [OK]

Common Mistakes:

• Increasing overlap to improve frequency resolution
• Changing sampling frequency to affect resolution
• Setting nfft smaller than window size