Motivation for the Laplace transform CT Fourier transform enables us to do a lot of things, e. g Analyze frequency response of lTi systems Sampling Modulation Why do we need yet another transform? One view of Laplace Transform is as an extension of the Fourier
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1.aM with an Arbitrary Periodic Carrier 2. Pulse Train Carrier and Time-Division Multiplexing 3. Sinusoidal Frequency modulation 4. DT Sinusoidal am 5. DT Sampling, Decimation,d Interpolation
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then, assuming we choose wM
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More efficient to transmit e&m signals at higher frequencies Transmitting multiple signals through the same medium using different carriers Transmitting through\channels, with limited passbands
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SAMPLING We live in a continuous-time world most of the signals we encounter are CT signals, e.g. x(). How do we convert them into Dt signals x[n? Sampling, taking snap shots of x(t) every Seconds
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Result: Linear phase e simply a rigid shift in time, no distortion Nonlinear phase e distortion as well as shift DT
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1. DTFT Properties and Examples 2. Duality in fs& ft 3. Magnitude/Phase of Transforms and Frequency Responses
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Convergence Issues Synthesis equation: None. since integrating over a finite interval Analysis Equation: Need conditions analogous to CTFT, e. g
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Fourier series: Periodic signals and lti Systems ()=∑H(k k= ak一→H(ko)ak “g Soak-→|H(jkco)lkl H(7k)=1H(k0e∠B(ko) or powers of signals get modified through filter/system ncludes both amplitude phase akeJhwon
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Convolution Property 0(t)=h(t)*(t)←→Y(j)=H(ju)X( where h(t)←→H(ju) A consequence of the eigenfunction property
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