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Basic signals

Basic signals. Why use complex exponentials? Because they are useful building blocks which can be used to represent large and useful classes of signals Response of LTI systems to these basic signals is particularly simple and useful. a k H(jk  0 ) e j  0 t. a k e jk  0 t. h(t).

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Basic signals

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  1. Basic signals • Why use complex exponentials? • Because they are useful building blocks which can be used to represent large and useful classes of signals • Response of LTI systems to these basic signals is particularly simple and useful. ak H(jk0) ej0 t akejk0 t h(t) Complex scaling factor

  2. Fourier Series Representation of CT Periodic Signals x(t) = x(t+T) for all t • Smallest such T is the fundamental period • 0 = 2/T is the fundamental frequency ejt periodic with period T   = kw0  • Periodic with period T • {ak} are the Fourier (series) coefficients • k = 0 DC • k = ±1 first harmonic • k = ±2 second harmonic

  3. CT Fourier Series Pair (Synthesis Equation) (Analysis Equation)

  4. Examples • x(t) = cos(4t) + 2sin(8t) • Periodic square wave (from lecture) • Periodic impulse train

  5. Even and Odd Functions • Even functions are defined by the property: f(x) = f(-x) • Odd functions are defined by the property: f(x) = -f(-x)

  6. Some notes on CT Fourier series • For k = 0, a0 = sT x(t) dt  mean value over the period(DC term) • If x(t) is real, ak = a*-k • If x(t) is even, ak = a-k • If x(t) is odd, ak = -a-k • If x(t) is real and even, ak = a*-k = a-k  ak’s will only have real terms • If x(t) is real and odd, ak = a*-k = -a-k  ak’s will only have odd terms

  7. The CT Fourier Transform Pair x(t) ↔ X(j)

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