03_Fourier Series Representation of Periodic Signals

03_Fourier Series Representation of Periodic Signals -...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
1 AKW Fall 2009 Chapter 3 Chapter 3: Fourier series representation of periodic signals Response of LTI systems to complex exponentials FS representation of CT periodic signals Convergence and properties of Fourier series FS representation of DT periodic signals and its properties Fourier series and LTI systems Filtering
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 AKW Fall 2009 Chapter 3 Signal as Linear Combination of Basic Signals In Chapter 2, we have seen how a CT signal can be represented as a linear combination of shift impulse functions x(t) t Δ Δ 3 t Δ Δ Δ + Δ Δ ) ( ) ( t x δ t Δ Δ ) ( ) 0 ( t x t Δ Δ Δ Δ ) ( ) ( t x Δ t ) ( t Δ Δ Δ / 1 Δ Δ Δ = −∞ = Δ Δ k k t k x t x ) ( ) ( lim ) ( 0 ) ( * ) ( ) ( ) ( ) ( t t x d t x t x δτ τδ τ = = t ) ( t
Background image of page 2
3 AKW Fall 2009 Chapter 3 Response of LTI System as Linear Combination of Responses We have gained insight into the input-output relationship of all LTI systems by decomposing a signal into a linear combination of more basic signals - the shifted impulse function Are there other basic signals that we can explore? t ) ( t δ CT system ) ( t h t ) ( t h = ττ δτ d t x t x ) ( ) ( ) ( LTI-CT system = τ d t h x t y ) ( ) ( ) (
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 AKW Fall 2009 Chapter 3 Response of LTI system to complex exponentials Consider the set of complex exponentials - What is the response of an CT LTI system given e st as input? - Again, s can be a complex number: s = r + j ω . LTI system h ( t ) e st y ( t )=? Recall that the complex exponential can be used to represent the real exponential, real sinusoid, and real exponentially growing/decaying sinusoid.
Background image of page 4
5 AKW Fall 2009 Chapter 3 To determine the output, we apply the convolution integral from Chapter 2: st s st t s e s H d e h e d e h d t x h t y ) ( ) ( ) ( ) ( ) ( ) ( ) ( = = = = ττ τ y ( t ) is the same complex exponential e st , except for the multiplication by a constant H ( s ) which is a function of s ! s st t s e e e = ) ( e st is called the eigenfunctions of CT LTI systems. As an input, it is not changed by an LTI system in anyway except for multiplication by H ( s ), the eigenvalue , which is a constant that may depends on s . (Recall the concept of eigenvector and eigenvalue in Linear Algebra.)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 AKW Fall 2009 Chapter 3 What is special about complex exponential Why is e st special? We pointed out in Chapter 1 that differentiating a complex exponential is equivalent to multiplication by a constant. In fact, time shifting a complex exponential is also equivalent to multiplication by a constant : () I f , then ( ) st s t s s xt e e e e e xt ττ τ −− =− = = = Delaying e st by is equivalent to multiplying e st by e -s ! Let y ( t ) be the output of an LTI system to e st . Because of time-invariance of the system, we know that y ( t – ) is the output to e s ( t- ) . But because of linearity, we can also claim that the output is e -s τ y ( t ). Hence: ( ) s y te y t −= The output that satisfies the above must be the same complex exponential e st , although it can be scaled by some constant; i.e., the output is some A e st
Background image of page 6
7 AKW Fall 2009
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/24/2010 for the course ELEC 211 taught by Professor Albertk.wong during the Fall '09 term at HKUST.

Page1 / 92

03_Fourier Series Representation of Periodic Signals -...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online