Kompleks Sonlar va Furye

Kompleks sonlar va Furye tahlili — signal ishlash, boshqarish tizimlari va tebranishlarni tushunish uchun muhim matematik vositalar.

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1. Kompleks Sonlar

1.1 Asosiy Tushunchalar

Kompleks son — real va mavhum qismlardan iborat son:

$$z = a + bi$$

Bu yerda:

• $a = \text{Re}(z)$ — real qism
• $b = \text{Im}(z)$ — mavhum (imaginary) qism
• $i = \sqrt{-1}$ — mavhum birlik ($i^2 = -1$)

Misol: $z = 3 + 4i$ uchun $\text{Re}(z) = 3$, $\text{Im}(z) = 4$

1.2 Kompleks Tekislik

Kompleks sonlarni 2D tekislikda tasvirlash mumkin:

• Gorizontal o'q — real qism
• Vertikal o'q — mavhum qism

Modul (absolyut qiymat):

$$|z| = \sqrt{a^2 + b^2}$$

Argument (faza burchagi):

$$\arg(z) = \theta = \arctan\left(\frac{b}{a}\right)$$

1.3 Algebraik Ko'rinish

Qo'shish: $(a + bi) + (c + di) = (a+c) + (b+d)i$

Ayirish: $(a + bi) - (c + di) = (a-c) + (b-d)i$

Ko'paytirish: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Bo'lish:

$$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

Qo'shma (conjugate):

$$\bar{z} = a - bi$$

Xossalar:

• $z \cdot \bar{z} = |z|^2 = a^2 + b^2$
• $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$
• $\overline{z_1 \cdot z_2} = \bar{z}_1 \cdot \bar{z}_2$

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2. Trigonometrik va Eksponensial Ko'rinish

2.1 Trigonometrik Ko'rinish

$$z = r(\cos\theta + i\sin\theta)$$

Bu yerda:

• $r = |z|$ — modul
• $\theta = \arg(z)$ — argument

O'tkazish:

• $a = r\cos\theta$
• $b = r\sin\theta$

2.2 Euler Formulasi

Matematikaning eng go'zal formulasi:

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Maxsus holatlar:

• $e^{i\pi} = -1$ (Euler identiteti)
• $e^{i\pi/2} = i$
• $e^{i \cdot 2\pi} = 1$

2.3 Eksponensial Ko'rinish

$$z = re^{i\theta}$$

Afzalliklari:

• Ko'paytirish osonlashadi: $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$
• Darajaga ko'tarish: $z^n = r^n e^{in\theta}$
• Ildiz chiqarish: $\sqrt[n]{z} = \sqrt[n]{r} e^{i(\theta + 2\pi k)/n}$, $k = 0, 1, \ldots, n-1$

2.4 De Moivre Formulasi

$$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$

Qo'llanilishi: Trigonometrik identitetlarni chiqarish, $n$-darajali ildizlarni topish.

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3. Kompleks Sonlarning Qo'llanilishi

3.1 Tebranishlarni Tasvirlash

Garmonik tebranish:

$$x(t) = A\cos(\omega t + \phi)$$

Kompleks ko'rinishda:

$$\tilde{x}(t) = Ae^{i(\omega t + \phi)} = Ae^{i\phi}e^{i\omega t}$$

Real qism: $x(t) = \text{Re}(\tilde{x}(t))$

3.2 Fazorlar (Phasors)

Fazor — vaqtga bog'liq bo'lmagan kompleks son:

$$\tilde{X} = Ae^{i\phi}$$

AC zanjirlarida:

• Kuchlanish: $\tilde{V} = V_m e^{i\phi_V}$
• Tok: $\tilde{I} = I_m e^{i\phi_I}$
• Impedans: $\tilde{Z} = R + iX$

3.3 Boshqarish Tizimlarida

Uzatish funksiyasi $H(s)$ da $s = \sigma + i\omega$:

$$H(i\omega) = |H(i\omega)|e^{i\angle H(i\omega)}$$

• Amplituda: $|H(i\omega)|$
• Faza: $\angle H(i\omega)$

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4. Furye Qatorlari

4.1 Davriy Signallar

Agar $f(t)$ davri $T$ bo'lsa: $f(t + T) = f(t)$

Asosiy chastota: $\omega_0 = \frac{2\pi}{T}$ yoki $f_0 = \frac{1}{T}$

4.2 Furye Qatori Ta'rifi

Har qanday davriy funksiyani sinusoidalar yig'indisi sifatida ifodalash mumkin:

$$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t) \right]$$

4.3 Furye Koeffitsientlari

$$a_0 = \frac{2}{T}\int_0^T f(t)\,dt$$ $$a_n = \frac{2}{T}\int_0^T f(t)\cos(n\omega_0 t)\,dt$$ $$b_n = \frac{2}{T}\int_0^T f(t)\sin(n\omega_0 t)\,dt$$

4.4 Kompleks Furye Qatori

$$f(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t}$$

Bu yerda:

$$c_n = \frac{1}{T}\int_0^T f(t)e^{-in\omega_0 t}\,dt$$

Bog'lanish:

• $c_0 = \frac{a_0}{2}$
• $c_n = \frac{a_n - ib_n}{2}$ ($n > 0$ uchun)
• $c_{-n} = \bar{c}_n$

4.5 Spektral Tahlil

Amplituda spektri: $|c_n|$ vs $n$

Faza spektri: $\arg(c_n)$ vs $n$

Parseval teoremasi (energiya saqlanishi):

$$\frac{1}{T}\int_0^T |f(t)|^2\,dt = \sum_{n=-\infty}^{\infty} |c_n|^2$$

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5. Furye Transformatsiyasi

5.1 Davriy Bo'lmagan Signallar

Davr $T \to \infty$ bo'lganda, diskret spektr uzluksiz spektrga aylanadi.

5.2 Furye Transformatsiyasi Ta'rifi

To'g'ri transformatsiya:

$$F(\omega) = \mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$$

Teskari transformatsiya:

$$f(t) = \mathcal{F}^{-1}\{F(\omega)\} = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}\,d\omega$$

5.3 Asosiy Xossalar

Xossa	Vaqt sohasi	Chastota sohasi
Chiziqlilik	$af(t) + bg(t)$	$aF(\omega) + bG(\omega)$
Vaqt siljishi	$f(t - t_0)$	$e^{-i\omega t_0}F(\omega)$
Chastota siljishi	$e^{i\omega_0 t}f(t)$	$F(\omega - \omega_0)$
Masshtablash	$f(at)$	$\frac{1}{a}F(\frac{\omega}{a})$
Hosila	$\frac{df}{dt}$	$i\omega F(\omega)$
Konvolyutsiya	$f * g$	$F \cdot G$

5.4 Muhim Transformatsiya Juftlari

• $\delta(t) \leftrightarrow 1$
• $1 \leftrightarrow 2\pi\delta(\omega)$
• $e^{-at}u(t)$ (a > 0) $\leftrightarrow \frac{1}{a + i\omega}$
• $e^{-a|t|} \leftrightarrow \frac{2a}{a^2 + \omega^2}$
• $\cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$

5.5 Konvolyutsiya

Ta'rif:

$$(f * g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t - \tau)\,d\tau$$

Konvolyutsiya teoremasi:

$$\mathcal{F}\{f * g\} = F(\omega) \cdot G(\omega)$$

Vaqt sohasida konvolyutsiya = chastota sohasida ko'paytirish.

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6. Diskret Furye Transformatsiyasi (DFT)

6.1 Diskret Signallar

Uzluksiz signal namunalanganda (sampling):

$$x[n] = x(nT_s)$$

Namunalash chastotasi: $f_s = 1/T_s$

6.2 Nyquist Teoremasi

Signal to'liq tiklanishi uchun:

$$f_s \geq 2f_{max}$$

Aliasing: Agar bu shart bajarilmasa, yuqori chastotalar past chastotalarga "yopilib" ketadi.

6.3 DFT Ta'rifi

$N$ ta namuna uchun:

To'g'ri DFT:

$$X[k] = \sum_{n=0}^{N-1} x[n]e^{-i2\pi kn/N}, \quad k = 0, 1, \ldots, N-1$$

Teskari DFT:

$$x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]e^{i2\pi kn/N}$$

6.4 Tez Furye Transformatsiyasi (FFT)

FFT — DFT ni samarali hisoblash algoritmi.

• DFT murakkabligi: $O(N^2)$
• FFT murakkabligi: $O(N\log N)$

Misol: $N = 1024$ uchun FFT ~100 marta tezroq.

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7. Signal Filtrlash

7.1 Filtr Turlari

Filtr turi	O'tkazadi	To'sadi
Past chastotali (LPF)	$\omega < \omega_c$	$\omega > \omega_c$
Yuqori chastotali (HPF)	$\omega > \omega_c$	$\omega < \omega_c$
O'tkazuvchan (BPF)	$\omega_1 < \omega < \omega_2$	Tashqaridagilar
To'suvchi (BSF)	Tashqaridagilar	$\omega_1 < \omega < \omega_2$

7.2 Ideal Past Chastotali Filtr

$$H(\omega) = \begin{cases} 1, & |\omega| \leq \omega_c \\ 0, & |\omega| > \omega_c \end{cases}$$

Impuls javobi:

$$h(t) = \frac{\omega_c}{\pi}\text{sinc}(\omega_c t)$$

7.3 Filtrlash Jarayoni

Chastota sohasida:

$$Y(\omega) = H(\omega) \cdot X(\omega)$$

Vaqt sohasida:

$$y(t) = h(t) * x(t)$$

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8. Qo'llanilishi

8.1 Robotika va Dronlarda

IMU signal filtrlash:

• Akselerometr va giroskopdan keluvchi shovqinli signallarni tozalash
• Past chastotali filtr — yuqori chastotali shovqinni yo'qotish
• Kalman filtri — sensor ma'lumotlarini birlashtirish

Motor tebranishlari:

• Furye tahlili orqali rezonans chastotalarini aniqlash
• Tebranishlarni kamaytirish uchun filtrlar loyihalash

8.2 Raketada

Aerodinamik kuchlar tahlili:

• Davriy kuchlanishlarni Furye qatori bilan ifodalash
• Flutter (tebranish nobarqarorligi) chastotalarini aniqlash

Navigatsiya:

• GPS va IMU signallarini filtrlash
• Shovqinni kamaytirish

8.3 Kommunikatsiya

Modulyatsiya:

• AM, FM, PM — tashuvchi to'lqin chastotasini o'zgartirish
• Kompleks sonlar orqali ifodalash

Spektral samaradorlik:

• Furye tahlili orqali chastota band kengligini baholash

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9. Python Misollari

Furye Qatori Koeffitsientlari

import numpy as np
import matplotlib.pyplot as plt

def fourier_coefficients(f, T, n_terms):
    """Furye qatori koeffitsientlarini hisoblash."""
    omega0 = 2 * np.pi / T
    a0 = (2/T) * np.trapz(f(np.linspace(0, T, 1000)), 
                          np.linspace(0, T, 1000))
    
    a_n, b_n = [], []
    t = np.linspace(0, T, 1000)
    for n in range(1, n_terms + 1):
        a_n.append((2/T) * np.trapz(f(t) * np.cos(n*omega0*t), t))
        b_n.append((2/T) * np.trapz(f(t) * np.sin(n*omega0*t), t))
    
    return a0, np.array(a_n), np.array(b_n)

FFT Misoli

# Signal yaratish
fs = 1000  # Hz
t = np.linspace(0, 1, fs)
signal = np.sin(2*np.pi*50*t) + 0.5*np.sin(2*np.pi*120*t)

# FFT
fft_result = np.fft.fft(signal)
frequencies = np.fft.fftfreq(len(signal), 1/fs)

# Faqat musbat chastotalar
pos_mask = frequencies >= 0
plt.plot(frequencies[pos_mask], np.abs(fft_result[pos_mask]))
plt.xlabel('Chastota (Hz)')
plt.ylabel('Amplituda')
plt.title('FFT Spektri')
plt.show()

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Xulosa

Mavzu	Asosiy g'oya
Kompleks sonlar	$z = a + bi = re^{i\theta}$
Euler formulasi	$e^{i\theta} = \cos\theta + i\sin\theta$
Furye qatori	Davriy signal = sinusoidalar yig'indisi
Furye transformatsiyasi	Vaqt $\leftrightarrow$ Chastota
FFT	Tez hisoblash algoritmi
Filtrlash	Kerakli chastotalarni ajratish

Keyingi qadam: Masalalar va Python amaliyoti orqali mustahkamlash.