Sine-Cosine equivalence | \(\cos(\theta) = \sin(\theta + \pi/2)\)

Cosine Periodicity | \(\cos(\theta) = \cos(\theta + 2\pi k), k \in \mathbb{Z}\)

Cosine is an even function | \(\cos(\theta) = \cos(-\theta)\)

Sine is an odd function | \(\sin(-\theta) = -\sin(\theta)\)

Full cycle cosine idenity | \(\cos(2 \pi k) = 1, k \in \mathbb{Z}\)

Cosine quarter cycle zeros identity | \(\cos(\pi k + \frac{\pi}{2}) = 0, k \in \mathbb{Z}\)

Pythagorean identity | \(\cos^{2}(\theta) + \sin^{2}(\theta) = 1\)

Product to sum sine cosine | \(\sin(\theta) \cdot \cos(\phi) = \frac{1}{2}(\sin(\theta - \phi) + \sin(\theta + \phi))\)

Product to sum sine sine | \(\sin(\theta) \cdot \sin(\phi) = \frac{1}{2}(\cos(\theta - \phi) - \cos(\theta + \phi))\)

Product to sum cosine cosine | \(\cos(\theta) \cdot \cos(\phi) = \frac{1}{2}(\cos(\theta + \phi) + \cos(\theta - \phi))\)

Imaginary Number Relation to Real Numbers | \(j = \sqrt{-1}\)

Complex Conjugate Definition | \((a + bj)^{*} = a - bj\)

Eulers Formula | \(e^{j \cdot \theta} = \cos(\theta) + j \sin(\theta)\)

Euler Definition of Sine | \(\sin(\theta) = \frac{e^{j \cdot \theta} - e^{-j \cdot \theta}}{2 \cdot j}\)

Euler Definition of Cosine | \(\cos(\theta) = \frac{e^{j \cdot \theta} + e^{-j \cdot \theta}}{2}\)

Integral of exp(ax) wrt x | \(\int e^{a x} \:d x = \frac{e^{a x}}{a} + C, a \neq 0\)

Definition of Product Symbol | \(\prod_{a = i}^{b} x_{i} = a_{i} a_{i + 1} ... a_{b}, a < b\)

Integral of (a exp(ax)) wrt x | \(\int x e^{a x} \:d x = \frac{a x - 1}{a^{2}} e^{a x} + C\)

Quadratic Formula | \(x_{1, 2} = \frac{-b \pm \sqrt{b^{2} - 4 a c}}{2 a}\)

Definition of Sampling Frequency | \(f_{s} = \frac{1}{T_{s}}\)

Normalized Radian Frequency | \(\hat \omega = \omega T_{s}\)

Discrete signal | \(x[n] = x(nT_{s}) = a \cos(\omega n T_{s} + \phi) = a \cos(\hat \omega n + \phi)\)

Definition of a Finite Impulse Response | \(h[n] = \sum_{k = 0}^{M} b_{k} \delta[n - k]\)

General form of a Finite Impulse Response Filter | \(h[n] = \sum_{k = 0}^{M} b_{k} x[n - k]\)

Convolution of a Signal and a Filter | \(h[n] \ast x[n]= \sum_{k = 0}^{M} h[k] x[n - k]\)

Sinusoidal Response of Finite Impulse Response Filter | \(H(e^{j \hat \omega}) = \sum_{k = 0}^{M} b_{k} e^{-j \hat \omega k} = \sum_{k = 0}^{M} = h[n] e^{je \hat \omega}\)

Fourier Series (Exponential Form) | \(x(t) = \sum_{k = -\infty}^{\infty} a_{k} e^{j(2 \pi / T_{0} ) k t}\)

Signals and Systems

Equations List

Symbols List