Discrete Fourier transform

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 $$4\pi$$
 * Phase-quadrature (modfase):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 $$ \omega_k=2\pi k f_s/N, k=0,1,2,\ldots,N-1$$