--- ## Euler's Identity -- $$e^{j\theta} = \cos(\theta) + j\sin(\theta) \qquad\qquad \mbox{(Euler's Identity)}$$ -- ### Properties of Exponents -- $$ a^{n_1} a^{n_2} = a^{n_1 + n_2} $$ -- $$ \left(a^{n_1}\right)^{n_2} = a^{n_1 n_2} $$ --- ### The Exponent Zero -- $$ a^0 a = a^0 a^1 = a^{0+1} = a^1 = a $$ -- $$ a^0 a = a $$ -- $$ {a^0 = 1.} $$ --- ### Negative Exponents -- $$ a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1. $$ -- $$ {a^{-1} = \frac{1}{a}.} $$ -- $$ {a^{-M} = \frac{1}{a^M}} $$ --- ### Rationale Exponents -- A *rational* number is a real number that can be expressed as a ratio of two finite integers: -- $$ \displaystyle x = \frac{L}{M}, \quad L\in\mathbb{Z},\quad M\in\mathbb{Z} $$ --- #### Applying property (2) of exponents, we have -- $$ \displaystyle a^x = a^{L/M} = \left(a^{\frac{1}{M}}\right)^L. $$ -- Thus, the only thing new is $$ a^{1/M} $$ . Since -- $$ \displaystyle \left(a^{\frac{1}{M}}\right)^M = a^{\frac{M}{M}} = a $$ --- we see that $$ a^{1/M}$$ is the M-th root of a. This is sometimes written -- $$ {a^{\frac{1}{M}} = \sqrt[M]{a}.} $$ --- ### e^(j theta) $ f(x)=e^x$ is one of the simplest imaginable infinite series: -- $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots $$ -- The simplicity comes about because $ f^{(n)}(0)=1$ for all $ n$ -- $ e^{j\theta} = \sum_{n=0}^\infty \frac{(j\theta)^n}{n!} = 1 + j\theta - \frac{\theta^2}{2} - j\frac{\theta^3}{3!} + \cdots \,. $ -- $$ \text{Re} \left(e^{j\theta}\right) = 1 - \frac{\theta^2}{2} + \frac{\theta^4}{4!} - \cdots $$ -- $$ \text{Im} \left(e^{j\theta}\right) = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\ $$ --- ### Polar form -- $\displaystyle z = r e^{j\theta} $ --- ## In-Phase & Quadrature Sinusoidal Components -- From the trig identity $ \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ , we have -- ### Derive --- * every sinusoid can be expressed: 1. as the sum of a sine function (phase zero) 2. a cosine function (phase $ \pi/2$ ). -- * sine part is called the **in-phase** component, -- * the cosine part called the **phase-quadrature** component. -- *In general, **phase quadrature** means **90 degrees out of phase. --- ## Complex Sinusoids in Frequency Domain -- $\displaystyle x(t) = A_x e^{j\omega_x t} $ --- <img src="/gallery/img474.png" style="width:70%"> --- ## Fourier Analysis -- * Fourier analysis is the process of decomposing a function into simple sinusoids known as basis functions. -- $$ X_k = \sum_0^{N-1} x_n\cdot e^{-i 2 \pi k n / N} $$ -- $$ t_n = nT = \mbox{$n$th sampling instant (sec)} $$ $$ \omega_k = k\Omega = \mbox{$k$th frequency sample (rad/sec)} $$ $$ T = 1/f_s = \mbox{time sampling interval (sec)} $$ $$ \Omega = 2\pi f_s/N = \mbox{frequency sampling interval (rad/sec)} $$ --- #### The transform *kernel*: -- $\displaystyle e^{-j\omega_k t_n} = \cos(\omega_k t_n) - j \sin(\omega_k t_n) $ --- #### **Inner Product** = **Projection** -- $ X(\omega_k)$ , the DFT at frequency $ \omega_k$ , is a measure of the amplitude and phase. --- ### Signals: Vector View -- * A signal consists of $N $ samples, is equivalent to a two-dimensional vector. -- * Each sample is a corrdinate in the N-dimensional space. -- * We already have that intuition when carrying signals in `Numpy` arrays, just as vectors. --- ### DFT not decomposing into sinusoids -- **DFT** decompose singals up into complex exponentials. -- ## DFT Matrix -- The transformation matrix W can be defined as $$ W = \left( \frac{\omega^{jk}}{ \sqrt{N} } \right )_{\left(j,k=0,\ldots ,N-1\right)} $$, --- #### or equivalently: -- <img src="/gallery/dftmat.svg" style="width:70%"> -- where $${\displaystyle \omega =e^{-2\pi i/N}}$$ --- <img src="/gallery/Fourierop_rows_only.svg" style="width:70%"> --- ### Examples -- #### Two-point * The two-point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference). -- <img src="/gallery/22mat.svg" style="width:20%"> -- * The first row performs the sum, and the second row performs the difference. -- * The factor of $ \frac{1}{\sqrt{2}} $ is to make the transform unitary --- #### The four-point DFT matrix is as follows: -- <img src="/gallery/44mat.svg" style="width:50%"> -- * where $ \left(\displaystyle \omega =e^{-{\frac {\pi i}{2}}}=-i\right) $. --- #### Eight-point -- * The first non-trivial integer power of two case is for eight points: --- <img src="/gallery/88mat.svg" style="width:90%"> -- * where $ \omega = e^{-{\frac{2\pi i}{8}}} = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} $ (Note that $\omega^{8 + n} = \omega^{n}$.) --- ## New Task: Writing Python/MATLAB Wrappers for C/C++ -- ### Hints -- * [{Boost.Python}](http://www.boost.org/doc/libs/1_66_0/libs/python/doc/html/index.html) * [{Wrapping C/C++ for Python}](http://intermediate-and-advanced-software-carpentry.readthedocs.io/en/latest/c++-wrapping.html) * [{How to use C library in python? (Generating Python wrappers for C library)}](http://karuppuswamy.com/wordpress/2012/01/28/how-to-use-c-library-in-python-generating-python-wrappers-for-c-library/) * [{Build MEX function from C/C++ or Fortran source code}](https://www.mathworks.com/help/matlab/ref/mex.html?requestedDomain=true) --- ## References -- 1. Digital Signal Processing, **2nd Edition**, *Fundamentals and Applications*. Authors: Li Tan Jean Jiang. 2. Signals and Systems using MATLAB. by Luis Chaparro Ph.D. University of California Berkeley. 3. Mitra Digital Signale Processing, Computer Based Approach. 4. Oppenheim Discrete-Time Signal Processing. --- ## Readings -- * [{Complex Numbers}](https://ccrma.stanford.edu/~jos/mdft/Complex_Numbers.html) * [{Proof of Euler's Identity}](https://ccrma.stanford.edu/~jos/mdft/Proof_Euler_s_Identity.html) * [{In-Phase & Quadrature Sinusoidal Components}](https://ccrma.stanford.edu/~jos/mdft/In_Phase_Quadrature_Sinusoidal.html) * [{The DFT}](https://ccrma.stanford.edu/~jos/mdft/DFT.html) * [{DFT matrix}](https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/DFT_matrix.html) --- ## Demo file ```bash $ git clone https://github.com/sbme-tutorials/sbe309-week3-demo.git ``` --- ## Extra * Git * Markdown * Latex