NumPy FFT — np.fft module — Hinglish notes

5.4.7 · Coding › Scientific Computing (Python)

YE EXIST KYU KARTA HAI?

DFT KYA HAI? (scratch se derive karo)

$e^{- 2 π ik n / N}$ aata kahan se hai?

YE exact formula kyun? Socho ki tum apne signal ko ek probe wave ke against test kar rahe ho.

Frequency $k$ ki ek pure wave ( $N$ samples ke upar) hai $w_{n} = e^{+ 2 π ik n / N}$ — ye unit circle ke around $k$ full turns karti hai jab $n$ jaata hai $0 \to N$ tak.
Ye poochne ke liye ki "wave $k$ ka kitna $x$ mein hai?", inner product (correlation) lo: har sample ko probe ke conjugate se multiply karo aur sum karo.
$e^{+ 2 π ik n / N}$ ka conjugate hai $e^{- 2 π ik n / N}$ . Isliye minus sign aata hai.

Hum orthogonality prove kar sakte hain (roots of unity ki geometric series): $\sum_{n = 0}^{N - 1} e^{2 π i (k - m) n / N} = {N 0 k = m k \neq = m$ KYU: agar $k = m$ toh har term $1$ hai, sum $= N$ . Agar $k \neq = m$ , toh ye ek geometric series hai ratio $r = e^{2 π i (k - m) / N} \neq = 1$ ke saath aur $r^{N} = 1$ , toh $\frac{1 - r ^{N}}{1 - r} = 0$ .

Inverse DFT

Orthogonality ki wajah se, hum $x_{n}$ recover kar sakte hain: $x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{+ 2 π ik n / N}$ $\frac{1}{N}$ aur $+$ sign exactly wahi hai jo forward transform ko undo karta hai (NumPy ka default convention).

HOW NumPy output organize karta hai

Worked Example 1 — ek tone dhundho

import numpy as np
fs   = 1000.0            # sampling rate, Hz
N    = 1000              # number of samples
t    = np.arange(N) / fs # time axis, seconds
x    = 3*np.sin(2*np.pi*50*t)   # a 50 Hz sine, amplitude 3
 
X    = np.fft.rfft(x)              # complex spectrum
f    = np.fft.rfftfreq(N, d=1/fs) # frequency axis
amp  = np.abs(X) * 2 / N          # convert to physical amplitude
 
print(f[np.argmax(amp)], amp.max())   # ~50.0 Hz, ~3.0

Step	Ye step kyun?
`t = arange(N)/fs`	sahi time stamps banata hai; total window $T = N / f_{s} = 1$ s
`rfft(x)`	real signal → aadha spectrum kaafi hai
`rfftfreq(N,d=1/fs)`	batata hai ki har bin kaun si frequency hai — kabhi hard-code mat karo!
`*2/N`	DFT ek real sine ki energy ko +f aur −f mein split karta hai; 2 se multiply karo aur $N$ se divide karo taaki sahi amplitude 3 mile

Worked Example 2 — reconstruct karo & inverse verify karo

x  = np.array([1.0, 2.0, 0.0, -1.0])
X  = np.fft.fft(x)
xr = np.fft.ifft(X)
print(np.allclose(x, xr.real))   # True

Kyun: ifft(fft(x)) == x bilkul exactly (float error tak) kyunki $1/ N$ aur sign conventions matched hain. Debug karne ke liye hamesha round-trips par trust karo.

Worked Example 3 — DC aur Nyquist

x = np.array([2,2,2,2], dtype=float)
print(np.fft.fft(x))   # [8.+0.j 0 0 0]

Kyun: ek constant signal mein sirf zero frequency (DC) hoti hai. Bin 0 equals $\sum x_{n} = 8$ . Baaki saare bins zero hain — koi oscillation nahi hai. DC bin hamesha samples ka sum hota hai.

Common mistake Classic galtiyon ko steel-man karo

1. "Bin index = frequency in Hz." Ye sahi kyun lagta hai: Example 1 mein bin 50 ne 50 Hz tone pakda. Haqeeqat: ye sirf isliye kaam kiya kyunki $N / f_{s} = 1$ . Generally $f_{k} = k f_{s} / N$ . Fix: hamesha fftfreq/rfftfreq use karo.

2. "Peak height hi amplitude hai." Ye sahi kyun lagta hai: tum 3 expect karte ho aur... ye zyada bada hai. Haqeeqat: raw $∣ X_{k} ∣$ , $N /2$ se scale hota hai ek real sine ke liye. Fix: $2/ N$ se multiply karo (aur DC bin ke liye $2/ N$ nahi, $1/ N$ use karo).

3. "Window length T bhool jaana." Chhoti window se frequency resolution moti hogi $Δ f = f_{s} / N = 1/ T$ . Lamba recording ⇒ behetar pitch discrimination. Fix: yaad rakho resolution = $1/ T$ .

4. "fft use karo phir arange ke against plot karo." Upar ke bins negative frequencies hain. Fix: fftshift + fftfreq use karo taaki plot 0 ke around symmetric ho.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek music chord — ek saath bajte kai notes, hawa ke ek jhatkay mein smash ho gaye. Tumhara kaam: andar ke alag-alag notes pata karo. FFT ek magical chalni hai: tum har possible note ko us jhatke ke saath gungunate ho, aur jab bhi tumhari awaaz kisi chhupe note se match karti hai, woh "ring" karta hai; jab match nahi karta, chup rahta hai. Har note ke liye ye bahut hoshiyaari se karo (ek-ek nahi balki sab ek saath), aur nikal aata hai ek list: "itna low note, itna high note." Woh list hai spectrum. "Fast" part sirf ek smart shortcut hai taaki kaam lunch se pehle ho jaaye.

Flashcards

np.fft.fft kiske beech transform karta hai?

Time (ya spatial) domain ↔ frequency domain (complex output).

X_{k}

ka DFT formula kya hai?

X_{k} = \sum_{n = 0}^{N - 1} x_{n} e^{- 2 π ik n / N}

FFT "fast" kyun hai — naive DFT ke comparison mein complexity kya hai?

O (N lo g N)

O (N^{2})

Rate

f_{s}

, length

N

diye hue, bin

k

ki frequency Hz mein?

f_{k} = k f_{s} / N

np.fft.fftfreq(N, d) kya return karta hai?

Har bin ki frequency (negative freqs bhi), spacing

d = Δ t

fft aur rfft mein kya fark hai?

rfft real input assume karta hai aur sirf

N /2 + 1

non-redundant bins return karta hai.

DFT bin 0 (DC) ki value kya hai?

Saare samples ka sum

\sum_{n} x_{n}

Amplitude

A

ki real sine ke liye, |rfft| se

A

kaise recover karein?

2/ N

se multiply karo (DC bin ke liye

1/ N

use karo).

Forward DFT exponent mein minus sign kyun hota hai?

Ye probe wave ka conjugate hai jo har frequency ko correlate/measure karne ke liye use hota hai.

FFT ki frequency resolution kya hai?

Δ f = f_{s} / N = 1/ T

, jahan

T

total record length hai.

fftshift kya karta hai aur kyun?

Zero-frequency component ko array ke center mein le jaata hai taaki negative/positive freqs symmetrically plot hon.

Real input ke liye

X_{k}

aur

X_{N - k}

mein kya relation hai?

X_{N - k} = \overline{X_{k}}

(conjugate symmetry).

Connections

Discrete Fourier Transform — woh math jo FFT compute karta hai
Sampling and Nyquist Theorem — kyun $f_{s}$ visible frequencies ko limit karta hai
Convolution Theorem — FFT convolution ko multiplication mein badal deta hai
NumPy arrays — broadcasting — vectorized signal building
Spectral Leakage and Windowing — non-periodic signals ko fix karna
Complex Numbers and Euler's Formula — $e^{i θ} = cos θ + i sin θ$
scipy.signal — higher-level spectral tools jo np.fft par bane hain