2.3 Complex numbers and Euler's formula

Complex numbers are essential in bioengineering signals and systems. They allow us to represent and manipulate sinusoidal signals, which are fundamental in biological systems. From brain waves to heart rhythms, these mathematical tools help us analyze and interpret complex physiological data.

Understanding complex numbers in different forms (rectangular, polar, exponential) is crucial for signal processing. These representations enable us to perform arithmetic operations, apply Euler's formula, and use complex exponentials to describe sinusoidal signals in bioengineering applications.

Complex Numbers

Forms of complex number representation

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• Rectangular form represents complex numbers as $a + jb$
  • $a$ denotes the real part and $b$ denotes the imaginary part
  • Imaginary unit $j$ is defined as $\sqrt{-1}$ (electrical engineering convention)
• Polar form expresses complex numbers as $r \angle \theta$ or $r e^{j\theta}$
  • Magnitude (modulus) $r$ represents the distance from the origin on the complex plane (e.g., $\sqrt{2}$)
  • Angle (argument) $\theta$ is measured in radians counterclockwise from the positive real axis (e.g., $\frac{\pi}{4}$)
  • Convert rectangular to polar using $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}(\frac{b}{a})$
• Exponential form writes complex numbers as $r e^{j\theta}$
  • Utilizes Euler's formula $e^{j\theta} = \cos\theta + j\sin\theta$ to represent polar form
  • Convert polar to exponential by substituting $\cos\theta + j\sin\theta$ for $e^{j\theta}$ (e.g., $2e^{j\frac{\pi}{4}} = 2(\cos\frac{\pi}{4} + j\sin\frac{\pi}{4})$)

Arithmetic with complex numbers

• Add and subtract complex numbers by operating on real and imaginary parts separately
  • $(a + jb) \pm (c + jd) = (a \pm c) + j(b \pm d)$ (e.g., $(2 + j3) + (4 - j5) = 6 - j2$)
• Multiply complex numbers using distributive property and $j^2 = -1$
  • $(a + jb)(c + jd) = (ac - bd) + j(ad + bc)$ (e.g., $(2 + j3)(4 - j5) = 23 - j14$)
  • In polar form, multiply magnitudes and add angles: $(r_1 \angle \theta_1)(r_2 \angle \theta_2) = r_1 r_2 \angle (\theta_1 + \theta_2)$
• Divide complex numbers by multiplying numerator and denominator by complex conjugate of denominator
  • $\frac{a + jb}{c + jd} = \frac{(a + jb)(c - jd)}{(c + jd)(c - jd)} = \frac{(ac + bd) + j(bc - ad)}{c^2 + d^2}$ (e.g., $\frac{2 + j3}{4 - j5} = \frac{23 - j14}{41}$)
  • In polar form, divide magnitudes and subtract angles: $\frac{r_1 \angle \theta_1}{r_2 \angle \theta_2} = \frac{r_1}{r_2} \angle (\theta_1 - \theta_2)$

Euler's formula for complex exponentials

• Euler's formula expresses complex exponentials in terms of trigonometric functions
  • $e^{j\theta} = \cos\theta + j\sin\theta$ (e.g., $e^{j\frac{\pi}{4}} = \frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}$)
• Complex exponential $e^{j\omega t}$ represents sinusoidal signals
  • Angular frequency $\omega$ measured in radians per second (e.g., $2\pi f$, where $f$ is frequency in Hz)
  • Time $t$ measured in seconds
  • Real part $\cos(\omega t)$ represents a cosine wave
  • Imaginary part $j\sin(\omega t)$ represents a sine wave

Complex numbers in signal processing

• Fourier transform decomposes time-domain signals into complex exponentials in frequency domain
  • Complex exponentials capture both amplitude and phase information of frequency components
• Frequency response of a system is a complex-valued function
  • Magnitude of complex number represents gain at each frequency
  • Angle of complex number represents phase shift at each frequency
• Complex numbers simplify mathematical operations in frequency domain
  • Enable compact representation of magnitude and phase relationships between signals
• Stability analysis uses complex-valued poles and zeros
  • Poles in right half-plane (positive real part) indicate unstable system
  • Zeros in right half-plane can cancel unstable poles for overall stable system