Euler's formula and phasors
e^{j\theta}=\cos\theta+j\sin\theta Derives Euler's identity from Taylor series and maps rotating complex vectors to sinusoidal waveforms.
Open derivation →KNOWLEDGE / DERIVATIONS
Formula Lab
Twenty structured derivations covering mathematics, electronics, signal processing, imaging, and estimation.
Laplace transform: the nature of the s-domain
X(s)=\int_{0^-}^{\infty}x(t)e^{-st}\,dt Extends Fourier analysis with exponential weighting and explains ROC, poles, and stability.
Open derivation →Z-transform: foundation of discrete systems
X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n} Interactive derivation module for Z-transform: foundation of discrete systems.
Open derivation →Convolution theorem: bridge between time and frequency
y(t)=x(t)*h(t)\Longleftrightarrow Y(\omega)=X(\omega)H(\omega) Interactive derivation module for Convolution theorem: bridge between time and frequency.
Open derivation →Ideal op-amp golden rules
A_v=-\frac{R_f}{R_{in}},\quad A_v=1+\frac{R_f}{R_g} Interactive derivation module for Ideal op-amp golden rules.
Open derivation →Active filter design: Sallen-Key topology
H(s)=\frac{\omega_0^2}{s^2+\frac{\omega_0}{Q}s+\omega_0^2} Interactive derivation module for Active filter design: Sallen-Key topology.
Open derivation →Skin effect and conductor loss
\delta=\sqrt{\frac{2}{\omega\mu\sigma}} Interactive derivation module for Skin effect and conductor loss.
Open derivation →Smith chart and impedance matching
\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0} Interactive derivation module for Smith chart and impedance matching.
Open derivation →Nyquist sampling and aliasing
f_s>2f_{max} Interactive derivation module for Nyquist sampling and aliasing.
Open derivation →DFT matrix and frequency bins
X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N} Interactive derivation module for DFT matrix and frequency bins.
Open derivation →Windowing and spectral leakage
X_w(f)=X(f)*W(f) Interactive derivation module for Windowing and spectral leakage.
Open derivation →FFT butterfly decomposition
X[k]=E[k]+W_N^kO[k] Interactive derivation module for FFT butterfly decomposition.
Open derivation →Gaussian blur kernel
G(x,y)=\frac{1}{2\pi\sigma^2}e^{-\frac{x^2+y^2}{2\sigma^2}} Interactive derivation module for Gaussian blur kernel.
Open derivation →RGB to YUV transform
\begin{bmatrix}Y\\U\\V\end{bmatrix}=M\begin{bmatrix}R\\G\\B\end{bmatrix} Interactive derivation module for RGB to YUV transform.
Open derivation →DCT and JPEG compression
C_{u,v}=\alpha_u\alpha_v\sum x_{m,n}\cos\frac{(2m+1)u\pi}{16}\cos\frac{(2n+1)v\pi}{16} Interactive derivation module for DCT and JPEG compression.
Open derivation →Histogram equalization
s_k=(L-1)\sum_{j=0}^{k}p_r(r_j) Interactive derivation module for Histogram equalization.
Open derivation →PID controller
u(t)=K_pe(t)+K_i\int e(t)dt+K_d\frac{de(t)}{dt} Interactive derivation module for PID controller.
Open derivation →Kalman filter
K_k=P_k^-H^T(HP_k^-H^T+R)^{-1} Interactive derivation module for Kalman filter.
Open derivation →State-space representation
\dot{x}=Ax+Bu,\quad y=Cx+Du Interactive derivation module for State-space representation.
Open derivation →PLL frequency synthesis theory
H(s)=\frac{K_dK_vF(s)/s}{1+K_dK_vF(s)/s} Interactive derivation module for PLL frequency synthesis theory.
Open derivation →No formula matches the current filters.