Spectrogram generation in SciPy - Time & Space Complexity

Time Complexity: Spectrogram generation

O(n log m)

Understanding Time Complexity

When we create a spectrogram, we split a signal into pieces and analyze each piece.

We want to know how the time to make a spectrogram grows as the signal gets longer.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.signal import spectrogram

fs = 1000  # Sampling frequency
signal = np.random.randn(10000)  # Example signal

frequencies, times, Sxx = spectrogram(signal, fs=fs, nperseg=256)
    

This code computes the spectrogram of a signal by splitting it into overlapping segments and applying Fourier transforms.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Computing the Fourier transform on each segment of the signal.
• How many times: The number of segments, which depends on the signal length and segment size.

How Execution Grows With Input

As the signal length grows, the number of segments grows roughly in proportion.

Input Size (n)	Approx. Operations
10,000	About 40 Fourier transforms (10000/256)
100,000	About 390 Fourier transforms (100000/256)
1,000,000	About 3906 Fourier transforms (1000000/256)

Pattern observation: The total work grows roughly linearly with the signal length.

Final Time Complexity

Time Complexity: O(n log m)

This means the time grows roughly proportional to the signal length times the log of the segment size.

Common Mistake

[X] Wrong: "The spectrogram time grows only with the signal length, ignoring the cost of Fourier transforms."

[OK] Correct: Each segment requires a Fourier transform, which takes extra time depending on segment size, so both length and segment size matter.

Interview Connect

Understanding how spectrogram computation scales helps you explain performance in signal processing tasks clearly and confidently.

Self-Check

"What if we change the segment size to be larger? How would the time complexity change?"

Practice

1. What does the scipy.signal.spectrogram function primarily compute from a signal?

easy

A. The frequency content of the signal over time

B. The total energy of the signal

C. The average amplitude of the signal

D. The phase shift of the signal

Solution

1. Step 1: Understand the purpose of spectrogram

   A spectrogram shows how frequencies in a signal change over time, so it computes frequency content over time.

2. Step 2: Compare with other options

   Total energy, average amplitude, and phase shift are different signal properties, not what spectrogram calculates.

3. Final Answer:

   The frequency content of the signal over time -> Option A

4. Quick Check:

   Spectrogram = frequency over time [OK]

Hint: Spectrogram = frequency changes over time [OK]

Common Mistakes:

• Confusing spectrogram with total energy calculation
• Thinking it returns average amplitude only
• Assuming it gives phase information

2. Which of the following is the correct way to import the spectrogram function from scipy?

easy

A. from scipy import spectrogram

B. import scipy.spectrogram

C. from scipy.signal import spectrogram

D. import spectrogram from scipy.signal

Solution

1. Step 1: Recall correct import syntax for functions

   Functions inside submodules are imported using 'from module.submodule import function'.

2. Step 2: Check each option

   from scipy.signal import spectrogram matches this syntax exactly. Options B, C, and D are invalid Python import statements.

3. Final Answer:

   from scipy.signal import spectrogram -> Option C

4. Quick Check:

   Correct import syntax = from scipy.signal import spectrogram [OK]

Hint: Use 'from scipy.signal import spectrogram' [OK]

Common Mistakes:

• Using 'import scipy.spectrogram' which is invalid
• Trying 'from scipy import spectrogram' ignoring submodule
• Incorrect import order like 'import spectrogram from ...'

3. What will be the shape of the frequency array returned by f, t, Sxx = spectrogram(signal, fs=1000, nperseg=256) if signal length is 1024?

medium

A. (256,)

B. (129,)

C. (512,)

D. (1024,)

Solution

1. Step 1: Understand frequency bins from nperseg

   The frequency array length is nperseg/2 + 1 due to FFT symmetry, so 256/2 + 1 = 129.

2. Step 2: Confirm other options

   256 is the segment length, not frequency bins. 512 and 1024 are unrelated to frequency array size here.

3. Final Answer:

   (129,) -> Option B

4. Quick Check:

   Frequency bins = nperseg/2 + 1 = 129 [OK]

Hint: Frequency bins = nperseg/2 + 1 [OK]

Common Mistakes:

• Assuming frequency array length equals nperseg
• Confusing signal length with frequency bins
• Ignoring FFT symmetry in frequency output

4. Given this code snippet, what is the error?

from scipy.signal import spectrogram
signal = [1, 2, 3, 4, 5]
fs = 100
f, t, Sxx = spectrogram(signal, nperseg=256, fs=fs)

medium

A. nperseg is larger than signal length causing a warning or error

B. fs parameter is missing

C. spectrogram function is not imported correctly

D. signal must be a numpy array, not a list

Solution

1. Step 1: Check parameters and signal length

   Signal length is 5, but nperseg is 256, which is larger than signal length.

2. Step 2: Understand spectrogram behavior

   When nperseg > signal length, scipy issues a warning or error because segment size can't exceed signal length.

3. Final Answer:

   nperseg is larger than signal length causing a warning or error -> Option A

4. Quick Check:

   nperseg > signal length causes error [OK]

Hint: nperseg must be ≤ signal length [OK]

Common Mistakes:

• Ignoring nperseg size relative to signal length
• Thinking fs is missing when it is provided
• Assuming list input causes error (it works but numpy array preferred)

5. You want to generate a spectrogram for a noisy signal sampled at 2000 Hz. To improve time resolution, which parameter should you decrease in scipy.signal.spectrogram?

hard

A. window

B. fs

C. noverlap

D. nperseg

Solution

1. Step 1: Understand time vs frequency resolution trade-off

   Decreasing nperseg (segment length) improves time resolution but reduces frequency resolution.

2. Step 2: Check other parameters

   fs is sample rate, noverlap controls overlap, window shapes segments but doesn't directly improve time resolution.

3. Final Answer:

   nperseg -> Option D

4. Quick Check:

   Decrease nperseg for better time resolution [OK]

Hint: Lower nperseg for better time resolution [OK]

Common Mistakes:

• Trying to change fs which is fixed sample rate
• Confusing noverlap with time resolution
• Changing window type expecting time resolution change