Vocabulary

Polar Coordinates
• $\theta=\tan^{-1}(y/x)$, called angle or phase

Complex Numbers

Cartesian form (or rectilinear form)
$z = x+iy$
Polar form
$z = r\cos\theta+ir\sin\theta$, using Eulers Identity, we get $z = re^{i\theta}$
• $r=\sqrt{x^2+y^2}$, called modulus or magnitude
• $x^2+y^2$ is called the norm
Norm
a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector.
Complex Conjugate
$z=x+iy\quad\Leftrightarrow\quad\overline z=x-iy$
Eulers Identity
$e^{i\theta}=\cos\theta+i\sin\theta$

Sinusoids

Amplitude
value of signal at a given time t
Peak amplitude
if peak amplitude is A, then $|x(t)|\leq A$ for all t
Sinusoid
$x(t)=A\sin(\omega t+\phi)\quad$ (A is peak amplitude, $\omega$ is radian frequency rad/s, $\phi$ is initial phase)
Instantaneous phase
$\omega t+\phi$. E.g. if the initial phase is 0, and the frequency is 1 Hz, the instantaneous phase at time t=2s is $4\pi$
Two sinusoids are in phase-quadrature when their phases differ by $\frac{\pi}{2}$
Complex sinusoid
$s(t)=Ae^{i(\omega t+\phi)}=A\cos(\omega t+\phi)+iA\sin(\omega t+\phi)$. Note that the real part is in phase-quadrature with the imaginary part.
Orthogonality
Sinusoids with different frequencies are orthogonal. For sampled signals, orthogonality holds only for the harmonics of the sample rate divided by N $f_k=k\frac{f_s}{N},k=0,1,2,...,N-1$.
DFT Sinusoids
The complex sinusoids corresponding to the frequencies $f_k$ are $s_k(n)=e^{i\omega_knT},\quad\omega_k=k\frac{2\pi}{N}f_s,\quad k=0,1,2,...,N-1$. They can also be written $s_k(n)=e^{i2\pi kn/N},\quad n=0,1,2,...,N-1$. All DFT sinusoids are orthogonal.

Signals

Total energy of a signal x
$E_x=x(t)\cdot x(t)=\sum_{n=0}^{N-1}|x_n|^2$      (the magnitude of the complex value $x_n$ squared)
Euclidian norm of a signal x
$||x||=\sqrt{E_x}$
Banach space
the set of signal vectors is a linear vector space, in that it is closed under addition and scalar multiplication (linear combinations). It is also a Banach space, which intuitively means that we can define a norm of any signal.
Inner product (dot/scalar product)
$\langle \mathbf{u},\mathbf{v}\rangle=\sum_{n=0}^{N-1}u_n\overline{v_n}$
Hilbert space
Intuitively, a Banach space with a defined inner product operator.
Orthogonal projection
projection of $y\in\mathbf{C}^N$ onto $x\in\mathbf{C}^N$ is $\mathbf{P}_x(y)=\frac{\langle y,x\rangle}{||x||^2}x$. The projection of any vector $x\in\mathbf{C}^N$ onto any orthogonal basis set for $\mathbf{C}^N$ can be summed to reconstruct x exactly.
Orthogonality
$x\perp y\Leftrightarrow\langle x,y\rangle=0$

Discrete Fourier Transform

The DFT can be viewed as a change of coordinates from coordinates relative to the natural basis in $\textbf{C}^N$, $\{e_n\}_{n=0}^{N-1}$, to coordinates relative to the sinusoidal basis for $\textbf{C}^N$, $\{s_k\}_{k=0}^{N-1}$, where $s_k(n)= e^{j\omega_k t_n}$ The sinusoidal basis set for $\textbf{C}^N$ consists of length N sampled complex sinusoids at frequencies $\omega_k=2\pi k f_s/N, k=0,1,2,\ldots,N-1$