Power spectral density in SciPy - Time & Space Complexity

Time Complexity: Power spectral density

O(n)

Understanding Time Complexity

We want to understand how the time to compute power spectral density grows as the input signal gets longer.

How does the computation time change when we analyze bigger data?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.signal import welch

fs = 1000  # Sampling frequency
x = np.random.randn(10000)  # Random signal
frequencies, psd = welch(x, fs=fs, nperseg=256)
    

This code computes the power spectral density of a signal using Welch's method.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Splitting the signal into overlapping segments and computing FFT on each segment.
• How many times: Number of segments depends on signal length and segment size.

How Execution Grows With Input

The total work grows roughly in proportion to the number of segments, which grows as the signal length increases.

Input Size (n)	Approx. Operations
10,000	About 40 FFT computations
100,000	About 400 FFT computations
1,000,000	About 4,000 FFT computations

Pattern observation: Increasing the input size by a factor of 10 roughly increases the number of FFT computations by a factor of 10, so the total work grows linearly with input size.

Final Time Complexity

Time Complexity: O(n)

This means the time grows linearly with input size. Each segment uses FFT on a fixed segment size, which takes constant time (O(m log m) with m fixed), and the number of segments grows linearly with n.

Common Mistake

[X] Wrong: "The computation time is O(n log n) because FFT is used."

[OK] Correct: FFT is on fixed-size segments (nperseg=256), so each takes constant time. Number of segments is O(n), so total time is O(n).

Interview Connect

Understanding how signal length and segment size affect computation time helps you explain performance trade-offs clearly in real data analysis tasks.

Self-Check

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

Practice

1. What does the power spectral density (PSD) represent in signal analysis?

easy

A. The phase shift of a signal at various points

B. The total energy of a signal over time

C. How the power of a signal is distributed over different frequencies

D. The amplitude of a signal at a single point in time

Solution

1. Step 1: Understand the definition of PSD

   Power spectral density shows how the power of a signal is spread across frequencies, not just total energy or amplitude at one time.

2. Step 2: Compare options with PSD meaning

   Only How the power of a signal is distributed over different frequencies correctly describes PSD as power distribution over frequencies.

3. Final Answer:

   How the power of a signal is distributed over different frequencies -> Option C

4. Quick Check:

   PSD = power distribution over frequency [OK]

Hint: PSD shows power spread by frequency, not time or phase [OK]

Common Mistakes:

• Confusing PSD with total signal energy
• Thinking PSD shows amplitude at one time
• Mixing PSD with phase information

2. Which of the following is the correct way to import the Welch method for PSD calculation from scipy?

easy

A. from scipy.signal import welch

B. import scipy.welch

C. from scipy import welch

D. import scipy.signal.welch as welch

Solution

1. Step 1: Recall scipy.signal module structure

   The Welch method is inside scipy.signal, so we import it from there.

2. Step 2: Check import syntax correctness

   Only 'from scipy.signal import welch' is valid Python syntax for this function.

3. Final Answer:

   from scipy.signal import welch -> Option A

4. Quick Check:

   Correct import = from scipy.signal import welch [OK]

Hint: Welch is in scipy.signal, import directly from there [OK]

Common Mistakes:

• Trying to import welch directly from scipy
• Using incorrect import syntax
• Assuming welch is a standalone module

3. What will be the output type of the following code snippet?

from scipy.signal import welch
import numpy as np
fs = 100
x = np.sin(2 * np.pi * 10 * np.arange(1000) / fs)
freqs, psd = welch(x, fs)
print(type(freqs), type(psd))

medium

A. (, )

B. (, )

C. (, )

D. (, )

Solution

1. Step 1: Understand welch output types

   The welch function returns two numpy arrays: frequencies and power spectral density values.

2. Step 2: Check printed types

   Printing type(freqs) and type(psd) will show numpy.ndarray for both.

3. Final Answer:

   (<class 'numpy.ndarray'>, <class 'numpy.ndarray'>) -> Option D

4. Quick Check:

   welch returns arrays = numpy.ndarray [OK]

Hint: Welch returns frequency and PSD as numpy arrays [OK]

Common Mistakes:

• Expecting lists instead of numpy arrays
• Thinking output is a tuple of floats
• Confusing output with single float values

4. Identify the error in this code snippet that calculates PSD:

from scipy.signal import welch
import numpy as np
fs = 50
x = np.random.randn(500)
freqs, psd = welch(x)
print(freqs)

medium

A. The print statement should be print(psd) instead

B. Missing sampling frequency (fs) argument in welch call

C. Random data cannot be used with welch

D. Import statement for welch is incorrect

Solution

1. Step 1: Check welch function call parameters

   The welch function needs the sampling frequency (fs) to correctly compute frequencies.

2. Step 2: Identify missing argument

   Here, fs=50 is defined but not passed to welch, causing incorrect frequency calculation.

3. Final Answer:

   Missing sampling frequency (fs) argument in welch call -> Option B

4. Quick Check:

   Pass fs to welch to get correct frequencies [OK]

Hint: Always pass fs to welch for correct frequency output [OK]

Common Mistakes:

• Forgetting to pass fs argument
• Assuming random data is invalid input
• Confusing print output with error

5. You have a noisy signal sampled at 200 Hz. You want to find the dominant frequency component using scipy's Welch method. Which parameter adjustment helps improve frequency resolution in the PSD estimate?

hard

A. Increase the segment length (nperseg) to use longer data chunks

B. Decrease the sampling frequency (fs) to reduce data size

C. Use a smaller window type to reduce computation

D. Set detrend to False to keep original signal shape

Solution

1. Step 1: Understand frequency resolution in Welch method

   Frequency resolution improves with longer segments (nperseg) because it provides finer frequency bins.

2. Step 2: Evaluate options for improving resolution

   Increasing nperseg increases segment length, improving resolution. Decreasing fs reduces max frequency but not resolution. Window type affects leakage, not resolution. Detrend affects baseline, not resolution.

3. Final Answer:

   Increase the segment length (nperseg) to use longer data chunks -> Option A

4. Quick Check:

   Longer nperseg = better frequency resolution [OK]

Hint: Longer segments (nperseg) give finer frequency detail [OK]

Common Mistakes:

• Reducing sampling frequency to improve resolution
• Confusing window type with resolution effect
• Ignoring segment length impact on frequency bins