NumPy Linspace & FFTs: The Endpoint Best Practice (2024)

If you've ever found yourself staring at a messy FFT plot, wondering why your perfect sine wave looks like a smudged fingerprint, you're not alone. The culprit is often a subtle, one-word parameter hiding in plain sight: `endpoint` in NumPy's `linspace` function. It seems trivial, but getting it wrong can derail your entire signal analysis.

Today, we're diving deep into this exact issue. We'll demystify the `linspace` endpoint and establish the definitive best practice for using it with Fast Fourier Transforms (FFTs) in 2024. Let's clear up the confusion for good.

The Humble `np.linspace`: A Quick Refresher

Before we tackle the main event, let's get reacquainted with our tool. NumPy's `np.linspace` is a workhorse function for creating an array of evenly spaced numbers over a specified interval. Its basic signature is wonderfully simple:

numpy.linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None, axis=0)

Most of us use the first three arguments—`start`, `stop`, and `num`—and call it a day. But that fourth one, `endpoint=True`, is where the magic (and the trouble) happens. It dictates whether or not the `stop` value is included in the resulting array.

The `endpoint` Dilemma: To Include or Not to Include?

This single boolean parameter fundamentally changes the nature of the array you create, specifically by altering the step size between your points. Let's see it in action.

`endpoint=True` (The Default): The Closed Interval

By default, `linspace` includes the `stop` value. This creates a closed interval, mathematically represented as `[start, stop]`. All the points, including the first and the last, are contained within the range.

How does it calculate the step size? It divides the total range by `num - 1` sections.

Step Size Formula: (stop - start) / (num - 1)

Let's create 5 points from 0 to 10:

import numpy as np

# The default behavior
arr_true = np.linspace(0, 10, 5, endpoint=True)
print(arr_true)
# Output: [ 0.   2.5  5.   7.5 10. ]

As you can see, 10 is included. The step size is `(10 - 0) / (5 - 1) = 2.5`.

`endpoint=False`: The Half-Open Interval

When you set `endpoint=False`, you're telling NumPy *not* to include the `stop` value. This creates a half-open interval, `[start, stop)`. The sequence goes up to, but does not include, the final value.

This changes the step size calculation. Now, the range is divided by `num` sections, because the `stop` value itself isn't a point.

Step Size Formula: (stop - start) / num

Let's run the same example:

# Explicitly excluding the endpoint
arr_false = np.linspace(0, 10, 5, endpoint=False)
print(arr_false)
# Output: [0. 2. 4. 6. 8.]

Notice that 10 is missing. The array ends one step *before* it. The step size is now `(10 - 0) / 5 = 2.0`.

`endpoint=True` vs. `endpoint=False`: At a Glance

A quick comparison table can help solidify the difference:

Connecting the Dots: Why This Matters for FFTs

So, why is this distinction so critical for Fast Fourier Transforms?

The answer lies in the core assumption of the Discrete Fourier Transform (DFT), the algorithm that FFTs efficiently compute. The DFT assumes that the finite signal you provide is a single period of an infinitely repeating, periodic signal.

Imagine a sine wave you've sampled over one full cycle, from 0 to 2π. The value at 0 is the same as the value at 2π. If you include both the start point (time=0) and the endpoint (time=2π) in your sample array, you have sampled the *exact same point* in the periodic wave twice.

Using `endpoint=True` for a signal of duration `T` means your time vector includes both `t=0` and `t=T`. For a periodic signal, the value at `t=T` is identical to the value at `t=0`. This redundancy breaks the assumption of a seamlessly repeating signal and introduces a discontinuity, which manifests as **spectral leakage** in your frequency domain plot. The energy that should be concentrated in a single frequency bin gets smeared across its neighbors.

The 2024 Best Practice: `endpoint=False` for Periodic Signals

This leads us to the golden rule:

When generating a time vector for a signal intended for FFT analysis, always use `endpoint=False`.

This ensures that you sample the interval `[0, T)`—you get the beginning of the cycle, but you stop one step *before* the end of the duration. The FFT algorithm implicitly understands that the next point in time would be `t=T`, which is the start of the next cycle (and identical to `t=0`). This creates a perfectly periodic sequence for the FFT to analyze.

A Practical Example: Sine Wave Analysis

Let's prove it with code. We'll generate a 10 Hz sine wave, sample it for 1 second at 100 Hz, and then find its frequency components using `scipy.fft`.

from scipy.fft import fft, fftfreq
import numpy as np
import matplotlib.pyplot as plt

# Signal Parameters
SAMPLING_RATE = 100  # Hz
DURATION = 1.0       # seconds
SIGNAL_FREQ = 10     # Hz

# --- The Correct Way: endpoint=False ---
N = int(SAMPLING_RATE * DURATION)

# Create a time vector over the half-open interval [0, DURATION)
t = np.linspace(0.0, DURATION, N, endpoint=False)
y = np.sin(SIGNAL_FREQ * 2.0 * np.pi * t)

# Compute the FFT and the frequency bins
yf = fft(y)
xf = fftfreq(N, 1 / SAMPLING_RATE) # Note: fftfreq also assumes this interval

# Plotting (we only need the positive frequencies)
plt.plot(xf[:N//2], 2.0/N * np.abs(yf[:N//2]))
plt.grid()
plt.title("FFT with endpoint=False (Correct)")
plt.xlabel("Frequency (Hz)")
plt.ylabel("Amplitude")
plt.show()

When you run this, you'll see a beautiful, sharp peak right at 10 Hz. The analysis is clean because our time vector correctly represented one, non-redundant period of the signal.

If you were to re-run this example using `endpoint=True`, the peak at 10 Hz would be slightly smaller and its energy would have leaked into the adjacent frequency bins. For complex signals, this leakage can completely obscure important frequency components.

Are There Times to Use `endpoint=True`?

Absolutely! The default exists for a reason. `endpoint=True` is the right choice for many common tasks that *don't* involve an assumption of periodicity.

• Plotting a function: If you want to plot `y = x**2` from x=-5 to x=5, you definitely want to include the points at both -5 and 5. `np.linspace(-5, 5, 100, endpoint=True)` is perfect here.
• Numerical integration: Methods like the trapezoidal rule require evaluating the function at both the start and end of the integration interval.
• Defining physical boundaries: If your array represents positions along a physical object, you typically want to include the start and end points.

The key is to think about whether your interval is a self-contained range or one period of a repeating sequence.

Final Takeaways: Your `linspace` Cheat Sheet

Let's boil it all down. When you reach for `np.linspace`, pause for a second and ask yourself what you're doing.

1. What is `np.linspace` for? Creating an array of evenly spaced numbers.
2. What's the difference in endpoints? `endpoint=True` includes the `stop` value (closed interval `[a,b]`), while `endpoint=False` does not (half-open interval `[a,b)`). This changes the step size.
3. THE GOLDEN RULE: For FFTs and periodic signal processing, always use `endpoint=False`. This aligns with the DFT's assumption of a periodic signal and prevents spectral leakage.
4. When to use the default? For most other cases, like plotting a function over a specific range or defining a set of coordinates, the default `endpoint=True` is usually what you want.

This small detail is a classic "gotcha" in scientific computing with Python. By understanding the *why* behind the rule, you can save yourself hours of debugging and produce far more accurate and reliable results. Happy coding!