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

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

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

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

Even functions | \(f(x) = f(-x)\)

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

Imaginary Number Definition | \(j = \pm \sqrt{-1}\)

Odd functions | \(-f(x) = f(-x)\)

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

Cartesian Form of a Complex Number | \(Z = a + b j\)

Definition of a Derivative | \(f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)

Product Rule | \((u \cdot v)' = u' \cdot v + v' \cdot u\)

Polar Form of a Complex Number | \(Z = r \cdot \cos(\theta) + r \cdot j \cdot \sin(\theta)\)

Derivative of exp(x) | \((e^{x})' = e^{x}\)

Derivative of Sine | \((\sin(x))' = \cos(x)\)

Derivative of Cosine | \((\cos(x))' = -\sin(x)\)

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

Angle addition for cosine | \(\cos(\theta + \phi) = \cos(\theta) \cdot \cos(\phi) - \sin(\theta) \cdot \sin(\phi)\)

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

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

Angle addition for sine | \(\sin(\theta + \phi) = \sin(\theta) \cdot \cos(\phi) + \sin(\phi) \cdot \cos(\theta)\)

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

Pythagorean theorem | \(a^{2} + b^{2} = c^{2}\)

Definition of Exponentiation | \(x^{y} = \prod_{i = 1}^{y} x\)

Exponent Addition | \(x^{y + z} = x^{y} \cdot x^{z}\)

Angle subtraction for sine | \(\sin(\theta - \phi) = \sin(\theta) \cdot \cos(\phi) - \sin(\phi) \cdot \cos(\theta)\)

Angle subtraction for cosine | \(\cos(\theta - \phi) = \cos(\theta) \cdot \cos(\phi) + \sin(\theta) \cdot \sin(\phi)\)

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

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

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

Definition of a Quadratic Polynomial | \(f(x) = a \cdot x^{2} + b \cdot x + c\)

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

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

Radian Frequency | \(\omega = 2 \pi f\)

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

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

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

General Form for the Equation of a Straight Line | \(y = m x + C\)

Gradient of a Straight Line | \(m = \frac{\Delta y}{\Delta x}\)

Definition of a Definite Integral | \(\int_{a}^{b} f(x) \:d x = \lim_{n \to \infty} \sum_{i = 1}^{n} f(x_{i}) \Delta x_{i}\)

Area of a Rectangle | \(A = W \cdot H\)

Power Rule for Differentiation | \(\frac{\:d}{\:d x}(a x^{k}) = k a x^{k - 1}\)

Power Rule for Integration | \(\int (a x^{k}) \:d x = \frac{a}{k + 1}x^ {k + 1} + C\)

Integral of a Derivative | \(\int f'(x) \:d x = f(x) + C\)

Derivative of an Integral | \(\frac{\:d}{\:d x} (\int f(x) \:d x) = f(x)\)

Integral of exp(x) | \(\int e^{x} \:d x = e^{x} + C\)

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

Integration by Parts | \(\int u \:d v = u v - \int v \:d u\)

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

Kullback-Leibler Divergence | \(KL(P, \hat{P}) = \sum_{\mathbf{s} \in S} P(\mathbf{s}) \frac{\log(P(\mathbf{s}))}{\log(\hat{P}(\mathbf{s}))}\)

Division of Exponents | \(\frac{a^{x}}{a^{y}} = a^{x - y}\)

Mathematics

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Associated Equations

Associated Symbols