DSP 24 Summary (Deterministic and Random)web.khu.ac.kr/~tskim/DSP 24 Summary (Deterministic and...

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  • Summary

  • Deterministic Signal Processing

  • Deterministic Signal Processing- If periodic

  • • Given a set of orthogonal functions, • And a real-valued function f(x)• Then the function f(x) can be represented in terms

    of βi(x)

    • A generalized Fourier Series of f(x)

    • αi are called the Fourier constants of f(x)• βi are called a set of “basis” functions• This can be viewed as signal expansions or signal

    decomposition

    baxxi ,)},({

    i

    xxxf

    i

    iii

    ,0 where

    ...)()()( 22111

  • Signal Decomposition and Synthesis

    Lecture No. 6

  • Why Elementary Signals & Systems?Lego Blocks

    Systems Systems

    Bases

  • Fourier Representation of Signals• Fourier basis

    • Orthogonal basis

    frequency lfundamenta,

    ],[

    ,...}cos,sin,...,cos,sin,{

    T

    Rt

    tktktt

    2

    0

    1

    0

    0000

    except when k=l

    < 𝒔𝒊𝒏𝒌𝝎𝟎𝒕, 𝒄𝒐𝒔𝒍𝝎𝟎𝒕>=0

    𝒌𝝎𝟎

    k-th harmonics

    Orthogonal

  • Basic Idea of Signal Representation

    Sum of Fourier basis represents any signal

  • Complex Fourier Series Expansion

    tkjtke

    jetjk

    j

    00 sincos

    sincos

    0

    Tketjk 2,...,2,1,0},{ 00

    kk

    tjk tkjtkkXekXtx )sin](cos[][)( 000

    Complex sinusoidsBases

    Def.

    T tjk dtetx

    TkX

    00)(

    1][

    Continuous-time periodic signals are represented by the FS

    The FS coefficients of the signal x(t)

    FS Pair

  • Deterministic Signal Processing- If aperiodic

  • Fourier Representations

  • Deterministic Signal Processing- Filtering

  • Back to Convolution Properties

    )()()(][*][][

    )()()()(*)()(

    jjj eXeHeYnxnhny

    jXjHjYtxthty

    Filtering

  • Frequency response of ideal continuous- (left panel) and discrete-time (right panel) filters. (a) Low-pass characteristic. (b) High-pass characteristic. (c) Band-pass characteristic.

    Ideal Filters

    Note the difference between and

  • Deterministic Signal Processing- Laplace Transform

  • Laplace Transform

    • Previous basis functions: 1, x, cosx, sinx, exp( jt).• New basis function for the LT => complex exponential functions

    • LT provides a broader characteristics of CT signals and CT LTI systems

    • Two types of LT• Unilateral (one-sided): good for solving differential

    equations with initial conditions.• Bilateral LT (two-sided): good for looking at the system

    characteristics such as stability, causality, and frequency response

  • Laplace Transform

    sine dampledlly exponentia}Im{

    cosine dampedlly exponentia}Re{

    ),sin()cos(

    st

    st

    ttst

    e

    e

    jstjetee

    Real and imaginary parts of the complex exponential est, where s = + j.

  • Random Signal Processing

  • Autocorrelation

    • The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise

  • For White Noise Input, Its PSD?