Tutorial2_solution - 1 Tutorial 2 Deterministic Signal...

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Principles of Communications 1 Tutorial 2 Deterministic Signal Analysis
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Principles of Communications 2 Problem 1 (Fourier Spectrum) Derive the Fourier spectrum of the following signals: s ( t ) = A cos(2 π f 0 t ) t τ / 2 0 elsewhere ( ) sinc( / ) s t A t τ = 2) Sinc-shaped pulse: 3) Rectangular pulse train: 1) Truncated sinusoidal signal: where f 0 is an integer multiple of 1/ τ . t A τ /2 - τ /2 0 0 T 0 T s ( t )
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Principles of Communications 3 Solution: Spectrum of Truncated Sinusoidal Signal s ( t ) S ( f ) = 1 2 ( δ ( f f 0 ) + δ ( f + f 0 )) A τ sinc( f τ ) = A τ 2 sinc ( f f 0 ) τ ( ) + sinc ( f + f 0 ) τ ( ) ( ) s ( t ) = cos(2 π f 0 t ) x ( t ) According to cos(2 π f 0 t ) 1 2 [ δ ( f f 0 ) + δ ( f + f 0 )] t τ 0 s ( t ) = A cos(2 π f 0 t ) t τ / 2 0 elsewhere s ( t ) can be written as where x ( t ) is a rectangular pulse signal: / 2 ( ) 0 otherwise A t x t τ = x ( t ) A τ sinc( f τ ) and f 0 + 1/ τ f A τ /2 0 f 0 f 0 + 2/ τ f 0 - 1/ τ f 0 - 2/ τ -f 0 -f 0 + 2/ τ -f 0 - 1/ τ -f 0 - 2/ τ -f 0 + 1/ τ
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Principles of Communications 4 Solution: Spectrum of Sinc-shaped Pulse ( ) sinc( / ) s t
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