03_Fourier Series Representation of Periodic Signals

# 03_Fourier Series Representation of Periodic Signals -...

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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

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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
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 ) ( ) ( ) (

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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.
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.)

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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
7 AKW Fall 2009

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03_Fourier Series Representation of Periodic Signals -...

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