6_003(I)

Introduction

Abstraction: Describing a system; input, output
Mathematics:
1.Solving differiential equations.

Ex1:

2.Geometric sum

Ex2: Use Taylor or division method.

3.Partial fraction
4.Multiplying polynomials (e.g. tabular representation)
5.Complex Numbers: $e^{j\theta} = \cos\theta+j\sin\theta$

CT transformation: odd/even, shift, scaling…


DT Systems

Unit sample:

Operator: $\mathcal{R}$, right-shift operator
Fundamental modes: e.g. $y[n]=(0.5)^n$

Multiple representations:

  • Verbal description
  • Difference equation: precise and concise
  • Block diagram: (start at rest) trace the flow of information
    • step-by-step
    • operator approach, polynomials
Lumping: from samples to signals
Declarative vs. Imperative (equation; diagram)
Recipe vs. Constraint
Acyclic vs. Cyclic
Finite vs. Infinite response

Standard form (DT)

Geometric growth: pole (unit-sample response)
1.png

converge: |p|<1;   diverge: |p|>1

Second-Order System: poles (still unit-sample)
  see the Ex3.
  two methods: multipying polynomial; partial fraction

Complex poles:
  Thinking about the difference equations, if the coefficients are real, even though the roots may be complex, the output is still real.
Note: visualize the complex plane.

Summary:
Systems composed of adders, gains, and delays can be characterized by their poles.
The poles of a system determine its fundamental modes.
The unit-sample response of a system can be expressed as a weighted sum of fundamental modes.

Z tranform

Replace $\mathcal{R}$ with $z^{-1}$.
Def: (bilateral)

Region of Convergence (ROC)
Note: For z transform, always consider both the functional form and the region of convergence.
Properties:
  1.Linearity
  2.Delay property   $x[n-1]\leftrightarrow z^{-1}X(z) $ for $z$ in ROC.
IMPORTANT :
  right-sided signal $\rightarrow$ outside region
  IF inside region, solve the difference equation by iterating backwards in time.
截屏2020-02-16下午10.20.42.png
Inverse transform:
  Formally, $x[n]=\frac{1}{2\pi j}\int_CX(z)^{n-1}dz $
  Better ways: partial fractions; using $\mathcal{R}$,….


CT Systems

Unit impulse:
  Unit area but zero width. (represented by an arrow with the number 1)
Unit step:
  $u(t)=\int_{-\infty}^{t}\delta(t)dt = 1 \ (\text{only when }t\geq 0) $
Operator: $\mathcal{A}$, accumulator, $Y=\mathcal{A}X$ means $y(t)=\int_{-\infty}^{t}x(t)dt$

Fundamental operation:  Delays in DT are replaced by integrators in CT.

Standard form (CT)

Methods for solving CT Systems:
  1.differential equation $\rightarrow$ solve it;
  2.use operators (Using poles; or Taylor series)
2020-02-15.48.25.png

converge: p<0; diverge: p>0;

Laplace tranform (CT)

Def: (bilateral)

ROC: Thinking about convergence, only thinking about the real part.
      $e^{pt}=e^{(\sigma+jw)t}= e^{\sigma t}(\cos wt+j\sin wt) $
      right-sided signal $\rightarrow$ right-sided region

截屏2020-02-16下午11.19.45.png

Solving Differential equations with Laplace transform:
  Laplace transform of the derivative $\rightarrow$ s times the L-transform of the original function.


Summary

截屏2020-02-16下午6.05.59.png
Connections:

截屏2020-02-16下午10.48.55.png


Examples

Ex3:
2.png

Note: 可以将原来的diagram化成两个simpler system,cascaded system

Method 1: Multiplying polynomials $$ \begin{aligned} \frac{Y}{X} &=(1+0.7\mathcal{R}+0.7^2\mathcal{R^2}+\cdots)(1+0.9\mathcal{R}+0.9^2\mathcal{R^2}+\cdots)\\ &=1+(0.7+0.9)\mathcal{R}+(0.7^2+0.7\times0.9+0.9^2)\mathcal{R^2}+\dots\end{aligned} $$   Note: 可以用tabular
Method 2:
Partial Fraction $$ \begin{aligned} \frac{Y}{X}&=\frac{4.5}{1-0.9\mathcal{R}}-\frac{3.5}{1-0.7\mathcal{R}}\\ \text{if } x[n] &= \delta[n],\ \text{then }\ y[n] = 4.5(0.9)^n-3.5(0.7)^n \text{ for $n\geq0.$} \end{aligned} $$ Check:

Ex4: Find the poles of Fibonacci system.
Key: difference equation: $y[n] = y[n-1]+y[n-2]+x[n]$
  poles: 1.618… and -0.618…

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