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# ClassNotes3 - ECE595 Wireless Communications Lecture 3...

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ECE595 Wireless Communications Lecture 3 Spring 2009 [Time-varying Channel Impulse Response] The transmitted signal is s ( t ) = Re { u ( t ) e j 2 πf c t } = Re { u ( t ) } cos(2 πf c t ) - Im { u ( t ) } sin(2 πf c t ) The received signal is the sum of all multipath components: r ( t ) = Re N ( t ) X n =1 α n ( t ) u ( t - τ n ( t )) e j 2 π [ f c f n ( t )] t - j 2 πf c τ n ( t ) [Doppler Shift] f c f c + v c f c cos θ Proof: Due to the movement, The wave will get there Δ d/c sec earlier t + Δ d = t + vt cos θ c cos 2 πf c t + vt cos θ c = cos 2 π f c + v c f c cos θ t

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2 Lecture 3 [Resolvable & Nonresolvable path] Assume τ max - τ min B - 1 u , Narrowband channel The received signal is r ( t ) = Re N ( t ) X n =1 α n ( t ) e n ( t ) u ( t ) e j 2 πf c t φ n ( t ) = 2 π Δ f n ( t ) t - 2 πf c τ n ( t ) The baseband equivalent channel is ch BB eq = N ( t ) X n =1 α n ( t ) e + n ( t ) , α n ( t ) and φ n ( t ) are independent The in-phase and quadrature components are given by ch I = N ( t ) X n =1 α n ( t ) cos( φ n ( t )) ch Q = N ( t ) X n =1 α n ( t ) sin( φ n ( t ))
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