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References

Vocabulary

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

Complex Numbers

Cartesian form (or rectilinear form)
<math>z = x+iy</math>
Polar form
<math>z = r\cos\theta+ir\sin\theta</math>, using Eulers Identity, we get <math>z = re^{i\theta}</math>
  • <math>r=\sqrt{x^2+y^2}</math>, called modulus or magnitude
  • <math>x^2+y^2</math> 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
<math>z=x+iy\quad\Leftrightarrow\quad\overline z=x-iy</math>
Eulers Identity
<math>e^{i\theta}=\cos\theta+i\sin\theta</math>

Sinusoids

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

Signals

Total energy of a signal x
<math>E_x=x(t)\cdot x(t)=\sum_{n=0}^{N-1}|x_n|^2</math>      (the magnitude of the complex value <math>x_n</math> squared)
Euclidian norm of a signal x
<math>||x||=\sqrt{E_x}</math>
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)
<math>\langle \mathbf{u},\mathbf{v}\rangle=\sum_{n=0}^{N-1}u_n\overline{v_n}</math>
Hilbert space
Intuitively, a Banach space with a defined inner product operator.
Orthogonal projection
projection of <math>y\in\mathbf{C}^N</math> onto <math>x\in\mathbf{C}^N</math> is <math>\mathbf{P}_x(y)=\frac{\langle y,x\rangle}{||x||^2}x</math>. The projection of any vector <math>x\in\mathbf{C}^N</math> onto any orthogonal basis set for <math>\mathbf{C}^N</math> can be summed to reconstruct x exactly.
Orthogonality
<math>x\perp y\Leftrightarrow\langle x,y\rangle=0</math>

Discrete Fourier Transform

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