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

Tke tjk 2,...,2,1,0},{ 00

kk

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

Complex sinusoidsBases

Def.

T tjk dtetxT

kX0

0)(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?