Processing math: 1%

---
## 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

Notes for current slide

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1 / 29

Euler's Identity2 / 29

Euler's Identity

$e^{j\theta} = \cos(\theta) + j\sin(\theta) \qquad\qquad \mbox{(Euler's Identity)}$

2 / 29

Euler's Identity

$e^{j\theta} = \cos(\theta) + j\sin(\theta) \qquad\qquad \mbox{(Euler's Identity)}$

Properties of Exponents

2 / 29

Euler's Identity

$e^{j\theta} = \cos(\theta) + j\sin(\theta) \qquad\qquad \mbox{(Euler's Identity)}$

Properties of Exponents

$a^{n_1} a^{n_2} = a^{n_1 + n_2}$

2 / 29

Euler's Identity

$e^{j\theta} = \cos(\theta) + j\sin(\theta) \qquad\qquad \mbox{(Euler's Identity)}$

Properties of Exponents

$a^{n_1} a^{n_2} = a^{n_1 + n_2}$

$\left(a^{n_1}\right)^{n_2} = a^{n_1 n_2}$

2 / 29

The Exponent Zero3 / 29

The Exponent Zero

$a^0 a = a^0 a^1 = a^{0+1} = a^1 = a$

3 / 29

The Exponent Zero

$a^0 a = a^0 a^1 = a^{0+1} = a^1 = a$

$a^0 a = a$

3 / 29

The Exponent Zero

$a^0 a = a^0 a^1 = a^{0+1} = a^1 = a$

$a^0 a = a$

${a^0 = 1.}$

3 / 29

Negative Exponents4 / 29

Negative Exponents

$a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1.$

4 / 29

Negative Exponents

$a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1.$

${a^{-1} = \frac{1}{a}.}$

4 / 29

Negative Exponents

$a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1.$

${a^{-1} = \frac{1}{a}.}$

${a^{-M} = \frac{1}{a^M}}$

4 / 29

Rationale Exponents5 / 29

Rationale Exponents

A rational number is a real number that can be expressed as a ratio of two finite integers:

5 / 29

Rationale Exponents

A rational number is a real number that can be expressed as a ratio of two finite integers:

$\displaystyle x = \frac{L}{M}, \quad L\in\mathbb{Z},\quad M\in\mathbb{Z}$

5 / 29

Applying property (2) of exponents, we have6 / 29

Applying property (2) of exponents, we have

$\displaystyle a^x = a^{L/M} = \left(a^{\frac{1}{M}}\right)^L.$

6 / 29

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

6 / 29

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

$\displaystyle \left(a^{\frac{1}{M}}\right)^M = a^{\frac{M}{M}} = a$

6 / 29

we see that $a^{1/M}$ is the M-th root of a. This is sometimes written

7 / 29

we see that $a^{1/M}$ is the M-th root of a. This is sometimes written

${a^{\frac{1}{M}} = \sqrt[M]{a}.}$

7 / 29

e^(j theta)

$f(x)=e^x$ is one of the simplest imaginable infinite series:

8 / 29

e^(j theta)

$f(x)=e^x$ is one of the simplest imaginable infinite series: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots$

8 / 29

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 $n$

8 / 29

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 $n$

$e^{j\theta} = \sum_{n=0}^\infty \frac{(j\theta)^n}{n!} = 1 + j\theta - \frac{\theta^2}{2} - j\frac{\theta^3}{3!} + \cdots \,.$

8 / 29

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 $n$

$e^{j\theta} = \sum_{n=0}^\infty \frac{(j\theta)^n}{n!} = 1 + j\theta - \frac{\theta^2}{2} - j\frac{\theta^3}{3!} + \cdots \,.$

$\text{Re} \left(e^{j\theta}\right) = 1 - \frac{\theta^2}{2} + \frac{\theta^4}{4!} - \cdots$

8 / 29

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\$

8 / 29

Polar form9 / 29

Polar form

$\displaystyle z = r e^{j\theta}$

9 / 29

In-Phase & Quadrature Sinusoidal Components10 / 29

In-Phase & Quadrature Sinusoidal Components

From the trig identity $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ , we have

10 / 29

In-Phase & Quadrature Sinusoidal Components

From the trig identity $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ , we have

Derive

10 / 29

every sinusoid can be expressed:as the sum of a sine function (phase zero) 
a cosine function (phase  \pi/2 ). 

11 / 29

every sinusoid can be expressed:as the sum of a sine function (phase zero) 
a cosine function (phase  \pi/2 ). 

sine part is called the in-phase component, 
11 / 29

every sinusoid can be expressed:as the sum of a sine function (phase zero) 
a cosine function (phase  \pi/2 ). 

sine part is called the in-phase component, 
the cosine part called the phase-quadrature component. 
11 / 29

every sinusoid can be expressed:as the sum of a sine function (phase zero) 
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.
11 / 29

Complex Sinusoids in Frequency Domain12 / 29

Complex Sinusoids in Frequency Domain

$\displaystyle x(t) = A_x e^{j\omega_x t}$

12 / 29

13 / 29

Fourier Analysis14 / 29

Fourier AnalysisFourier analysis is the process of decomposing a function into simple sinusoids known as basis functions.
14 / 29

Fourier AnalysisFourier 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}  
14 / 29

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$th frequency sample (rad/sec)}$ $T = 1/f_s = \mbox{time sampling interval (sec)}$ $\Omega = 2\pi f_s/N = \mbox{frequency sampling interval (rad/sec)}$

14 / 29

The transform kernel:15 / 29

The transform kernel:

$\displaystyle e^{-j\omega_k t_n} = \cos(\omega_k t_n) - j \sin(\omega_k t_n)$

15 / 29

Inner Product = Projection16 / 29

Inner Product = Projection

$X(\omega_k)$ , the DFT at frequency $\omega_k$ , is a measure of the amplitude and phase.

16 / 29

Signals: Vector View17 / 29

Signals: Vector ViewA signal consists of N  samples, is equivalent to a two-dimensional vector. 
17 / 29

Signals: Vector ViewA signal consists of N  samples, is equivalent to a two-dimensional vector. 
Each sample is a corrdinate in the N-dimensional space.
17 / 29

Signals: Vector ViewA 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.
17 / 29

DFT not decomposing into sinusoids18 / 29

DFT not decomposing into sinusoids

DFT decompose singals up into complex exponentials.

18 / 29

DFT not decomposing into sinusoids

DFT decompose singals up into complex exponentials.

DFT Matrix

18 / 29

DFT not decomposing into sinusoids

DFT decompose singals up into complex exponentials.

DFT Matrix

The transformation matrix W can be defined as $W = \left( \frac{\omega^{jk}}{ \sqrt{N} } \right )_{\left(j,k=0,\ldots ,N-1\right)}$ ,

18 / 29

or equivalently:19 / 29

or equivalently:

19 / 29

or equivalently:

where $\omega =e^{-2\pi i/N}$

19 / 29

20 / 29

Examples21 / 29

ExamplesTwo-pointThe two-point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference).
21 / 29

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).

21 / 29

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).
The first row performs the sum, and the second row performs the difference.

21 / 29

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).
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

21 / 29

The four-point DFT matrix is as follows:22 / 29

The four-point DFT matrix is as follows:

22 / 29

The four-point DFT matrix is as follows:

where $\left(\displaystyle \omega =e^{-{\frac {\pi i}{2}}}=-i\right)$ .

22 / 29

Eight-point23 / 29

Eight-pointThe first non-trivial integer power of two case is for eight points:
23 / 29

24 / 29

where

$\omega = e^{-{\frac{2\pi i}{8}}} = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$

(Note that $\omega^{8 + n} = \omega^{n}$ .)

24 / 29

New Task: Writing Python/MATLAB Wrappers for C/C++25 / 29

New Task: Writing Python/MATLAB Wrappers for C/C++Hints25 / 29

New Task: Writing Python/MATLAB Wrappers for C/C++

Hints

25 / 29

References26 / 29

ReferencesDigital Signal Processing, 2nd Edition, Fundamentals and Applications. Authors: Li Tan Jean Jiang.
Signals and Systems using MATLAB. by Luis Chaparro Ph.D. University of California Berkeley.
Mitra Digital Signale Processing, Computer Based Approach.
Oppenheim Discrete-Time Signal Processing.
26 / 29

Readings27 / 29

Readings

27 / 29

Demo file

$ git clone https://github.com/sbme-tutorials/sbe309-week3-demo.git

28 / 29

ExtraGit
Markdown
Latex
29 / 29

Euler's Identity2 / 29

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