Intuition The ONE core ideaAny wiggle-in-time can be rebuilt by stacking pure spinning waves of different speeds, and the FFT is the machine that tells you exactly how much of each speed is inside your wiggle . Everything below is just the vocabulary you need before that sentence means something — a sample, a frequency, a spinning arrow, a sum.
This page assumes you have seen nothing . We will meet every symbol the parent NumPy FFT — np.fft module uses — x n N k ∑ e − 2 π ik n / N f s Δ t
Definition Sample and the symbolx n not store a smooth curve. It takes a quick measurement, waits a tiny moment, measures again, and so on. Each measurement is called a sample . We write the samples as
x 0 , x 1 , x 2 , … , x N − 1 . x index n x 0 x 3
Picture it. Look at Figure 1: the smooth grey curve is the true sound; the magenta dots sitting on it are the samples. The FFT never sees the curve — only the dots.
Intuition Why start counting at zero?Programmers (and the DFT formula) count 0 , 1 , 2 , … N last index is N − 1 N
Question — what does the subscript in x 5 It is the index: x 5
N N the total number of samples you collected. In the parent's examples N 1000 4 N N
Now, the DFT formula adds up many terms. Writing x 0 + x 1 + ⋯ + x 999
Definition The summation symbol∑ "add up a bunch of similar things."
∑ n = 0 N − 1 x n = x 0 + x 1 + x 2 + ⋯ + x N − 1 . "start the counter n 0 x n n n N − 1
If x = [ 2 , 2 , 2 , 2 ] N = 4 ∑ n = 0 3 x n = 2 + 2 + 2 + 2 = 8. exactly why the parent's Worked Example 3 gave bin 0 the value 8
Question — expand ∑ n = 0 2 x n x 0 + x 1 + x 2
Δ t Δ t tiny amount of time between one sample and the next , in seconds. The triangle Δ Δ t = 0.001
f s f s how many samples you take per second , measured in hertz (Hz) = "per second." It is just the flip of the gap:
f s = Δ t 1 . Δ t = 0.001 f s = 1000
Picture it. In Figure 2 the horizontal spacing between neighbouring dots is Δ t Δ t f s
Intuition Why the topic needsΔ t f s sample indices 0 , 1 , 2 , … no idea what "one second" means. To translate a bin number back into real Hz, we must feed it the timing — that is what Δ t f s Sampling and Nyquist Theorem for why f s
Question — if you sample every 0.002 f s f s = 1/0.002 = 500
Frequency is how many full up-and-down cycles a wave completes each second, in Hz. A low hum = few cycles/second; a high whistle = many. This is the "pitch" your ear actually hears.
Definition The frequency indexk k which frequency slot (called a "bin") we are asking about: k = 0 , 1 , 2 , … , N − 1 k k
The bridge from slot k f k = N k f s .
Intuition Why that formula, in wordsYour whole recording lasts T = N Δ t k k "cycles ÷ seconds" = k / T = k f s / N
Question — with N = 1000 f s = 1000 k = 50 f 50 = 50 ⋅ 1000/1000 = 50
This is the scariest-looking symbol in the parent. We build it in three calm steps.
i i i 2 = − 1 multiplying by i in a flat plane. Don't worry about "imaginary" — treat i quarter-turn operator . Full background: Complex Numbers and Euler's Formula .
Picture it. Figure 3 shows the unit circle: the violet arrow sits at angle θ cos θ sin θ all e i θ
Now read the exponent piece by piece:
2 π N k n how far around you have spun. As the sample counter n 0 → N 0 → k k The minus sign means "spin clockwise instead of counter-clockwise." (The parent explains this minus is the conjugate — the measuring version — of the forward probe.)
Intuition Why a spinning arrow is the perfect measuring stickEach frequency you want to detect is itself a spinning-arrow wave. To ask "how much of speed-k , you spin a matching probe at speed k orthogonality the parent proves. This cancel-unless-matched trick is the heart of the whole FFT.
Question — as n 0 N e − 2 π ik n / N Exactly k
X k X k complex number : it has a size and a direction (like the arrows above), because it records both how much of frequency k what phase (timing offset) it has. Capital X x
∣ X k ∣ length of the arrow (its magnitude). For finding how loud a frequency is, we usually care only about this length, not the direction — that's why the parent writes np.abs(X).
X k on top means "flip the arrow's vertical part," i.e. reflect across the horizontal axis. The parent's symmetry rule X N − k = X k rfft can throw it away.
Question — what does ∣ X 3 ∣ The strength (amplitude-ish size) of frequency bin 3 in the signal.
Every box on the left is a symbol you now own; together they build the single formula the parent lives on:
X k = ∑ n = 0 N − 1 x n e − 2 π ik n / N .
Can you read x 7 "sample number 7" — the eighth measurement (indices start at 0).
Can you expand ∑ n = 0 3 x n x 0 + x 1 + x 2 + x 3
Do you know the relation between Δ t f s f s = 1/Δ t
Can you convert bin k f k = k f s / N
Do you know what e i θ A unit-length arrow at angle θ cos θ + i sin θ
Why does multiplying by i It rotates a point 90° counter-clockwise.
What does ∣ X k ∣ The size/strength of frequency bin k
What does the total window length T = N Δ t The frequency resolution Δ f = 1/ T
