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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
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?