Foundations — scipy.signal — filtering, convolution, FFT-based analysis

Before you can read the parent note you must be able to see what each squiggle means. This page builds them one at a time, from nothing. No symbol is used until it has a plain-words meaning and a picture.

1. The signal $x[n]$ — a row of dots

Forget curves for a moment. A signal, in the computer, is a finite list of numbers:

$x=[x[0],x[1],x[2],\dots ]$

Why does the topic need this? Because a real-world thing (a sound, a heartbeat, a voltage) is continuous, but a computer can only store a finite number of snapshots. Every tool in scipy.signal operates on these snapshots — so we must agree that a signal = a numbered row of dots.

Look at the figure: each stem (vertical line with a dot on top) is one sample. Its horizontal position is the index $n$; its height is the value $x[n]$. There is nothing between the stems — that empty space is the whole reason we later worry about aliasing.

2. Sampling: turning seconds into slots — $f_s$, $t$, $\Delta t$

The dots did not appear by magic. We took snapshots at a steady rhythm.

So slot number $n$ happens at real time $t=n\Delta t=n/f_s$ seconds. In the parent's code, t = np.arange(0, 1, 1/fs) is exactly this: it builds the list of real times $0,\Delta t,2\Delta t,\dots$ up to $1$ second.

The figure shows the same dots as before, but now the axis is labelled in seconds. The index view and the time view are two rulers under one row of dots.

3. The pure wave — $\sin$, $\cos$, frequency $f$, and $\pi$

Signals are interesting because they are built from waves. The simplest wave is a sine.

Why the topic needs waves: the whole Fourier idea (Section 5) is that any signal is a sum of these pure waves. To measure "how much 50 Hz is in here", you must first know what a 50 Hz wave is.

The figure shows a slow wave (small $f$) and a fast wave (big $f$). Count the peaks in one second — that count is the frequency in Hz.

4. The kernel $h[n]$ and the star $\ast$

The parent's Section 1 hits you with $h[n]$ and $x\ast h$. Here is what they are.

The figure is the flip-and-slide picture that Section 1 assumes you can see: the flipped kernel (mint) sits over part of the signal (lavender); the shaded overlaps are the products being summed into one output dot (coral). Move the kernel one slot right and you get the next output dot. That, repeated, is the entire ${\sum }_{k}x[k]h[n-k]$.

5. The complex probe wave — $e^{-i2\pi kn/N}$, $i$, $|X[k]|$

This is the scariest symbol in the parent note. We disassemble it.

6. Frequency bookkeeping — $f_k$, $\Delta f$, Nyquist $f_{Nyq}$

Once the FFT gives you $X[0],X[1],\dots$, which real frequency does each belong to?

Prerequisite map

Read it top-down: the dot idea feeds everything; sampling gives you time and frequency rulers; waves give you what the FFT looks for; the kernel plus sliding gives convolution and filtering; all rivers meet at the parent topic.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, re-read its section.

Connections

• Fourier Transform — the continuous parent of the DFT sum built here.
• Aliasing and Sampling Theorem — the full story of Section 6's Nyquist limit.
• Linear Time-Invariant Systems — where the kernel $h[n]$ and convolution come from.
• numpy.fft — the tool that computes $X[k]$ from these ingredients.
• Digital Filter Design — what to do with $h[n]$ once you understand it.
• Cross-correlation and Template Matching — convolution's un-flipped cousin.
• Hinglish version →