s17_ps13_sol

# s17_ps13_sol - Unified Engineering II Spring 2004 Problem...

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Unformatted text preview: Unified Engineering II Spring 2004 Problem S17 Solution To begin, label the signals as shown below: From the problem statement, y ( t ) = [ x ( t ) + A ] cos (2 πf c t + θ c ) Define z ( t ) = x ( t ) + A w ( t ) = cos (2 πf c t + θ c ) The factor w ( t ) can be expanded as w ( t ) = cos (2 πf c t + θ c ) = cos θ c cos 2 πf c t − sin θ c sin 2 πf c t The Fourier transform of w ( t ) is then given by W ( f ) = F [cos (2 πf c t + θ c )] 1 1 = 2 cos θ c [ δ ( f − f c ) + δ ( f + f c )] − 2 sin θ c [ − jδ ( f − f c ) + jδ ( f + f c )] 1 1 = 2 (cos θ c + j sin θ c ) δ ( f − f c ) + 2 (cos θ c − j sin θ c ) δ ( f + f c ) The Fourier transform of z ( t ) = x ( t ) + A is given by Z ( f ) = [ z ( t )] = X ( f ) + Aδ ( f ) F Z ( f ) is bandlimited, because X ( f ) is, and of course the impulse function is bandlim- ited. So the FT of y ( t ) is given by the convolution Y ( w ) = Z ( f ) ∗ W ( f ) 1 = [(cos θ c + j sin θ c ) Z ( f − f c ) + (cos θ c −...
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## This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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s17_ps13_sol - Unified Engineering II Spring 2004 Problem...

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