# HWa - ECE 493 HW#10 – Version 1.1 February 9 2011 Spring...

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Unformatted text preview: ECE 493 HW #10 – Version 1.1 February 9, 2011 Spring 2011 Univ. of Illinois Due Th, Feb 10 Prof. Allen Topic of this homework: Hilbert transform; Functions classes; Positive definite functions; Deliverable: Show your work. 1 Hilbert transforms Analyze the real impulse response h ( t ) = e- t/τ u ( t ) , with τ = 10 [ms], in terms of its Hilbert transform (integral) relations. 1. Find H ( s ), the Laplace transform of h ( t ). 2. Find the real and imaginary parts of H ( ω ) ≡ H ( s ) | s = iω and plot each. 3. The even and odd parts of a function h ( t ) are defined as h e ( t ) ≡ h ( t ) + h ( − t ) 2 and h o ( t ) ≡ h ( t ) − h ( − t ) 2 . Plot the symmetric h e ( t ) and antisymmetric h o ( t ) functions. 4. Find the Fourier transforms of h e ( t ) ↔ H e ( ω ) and h o ( t ) ↔ H o ( ω ). 5. Determine the Hilbert (integral) relations between H e ( ω ) and H o ( ω ) by use of the Fourier transform relations 1 2 sgn( t ) ↔ 1 iω and/or u ( t ) ↔ πδ ( ω ) + 1 iω . Note: sgn( t ) = t/ | t | . 6. Find the Hilbert (integral) relations between H r ≡ ℜ H ( ω ) (real part) and H i ≡ ℑ H ( ω ) (imag part) of H ( ω ). 7. Discuss the similarities and differences between the Hilbert transform and the Cauchy integral theorem f ( z ) = 1 2 πi contintegraldisplay C f ( ζ ) ζ − z dζ 8. Let ζ = x + iy , ζ = x − iy , and f ( ζ ) = u ( x,y ) + iv ( x,y ).)....
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HWa - ECE 493 HW#10 – Version 1.1 February 9 2011 Spring...

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