Frequency Response of the Equivalent Filter ¾ Given the original input x t is

Frequency response of the equivalent filter ¾ given

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Frequency Response of the Equivalent Filter ¾ Given the original input x ( t ) is real, is the filtered signal f ( t ) real? Since the spectrum is not conjugate symmetric, the corresponding signal is not real A simple remedy is to extract the real part of the output Equivalently, this corresponds to taking the even part of the spectrum of the complex signal f ( t ). 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 26 Equivalent filter for f ( t ) Equivalent filter for Re{ f ( t )}
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Sinusoidal Carrier 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 27 Multiplication with a sinusoidal signal in the time domain is equivalent to move the spectrum to the right and to the left in the frequency domain.
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Sinusoidal Carrier To extract the original signal, we need to apply a lowpass filter Some simple algebra: 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 28
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Implementation 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 29 The cutoff frequency w of the lowpass filter need to chosen to retain spectrum of the original signal while rejecting the portion that is moved to the high frequency.
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What If There Is a Phase Error? ¾ We have assumed that the demodulator perfectly knows the phase of the modulator How? What if there is a phase difference at the demodulator? Depending on the phase difference, there could be attenuation (fixed phase difference) or time-varying fading (time-varying phase difference) 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 30
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Alternative: Asynchronous Demodulation 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 31
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Envelope Detector ¾ In order for it to function properly, the envelope function must be positive for all time, i.e. A + x ( t ) > 0 for all t. ¾ Can we increase A a lot? There is an energy efficiency problem, the signal A cos Ȧ c t does not carry information Low power efficiency (see Problem 8.27) ¾ So, synchronous or asynchronous detector? There is always design tradeoff considering cost, performance requirement, application, business model, etc. Synchronous demodulation: more efficient, but harder to implement (more expensive); application such as satellite communication Asynchronous detector: easier to implement (low cost), but less power efficient; application such as low cost AM radio 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 32 A counter example
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Modulation for Multiplexing 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 33 ¾ Radio broadcasting, mobile phone networks, analog cable TV systems, all employ mixing with different carrier frequencies to achieve multiplexing – the simultaneous transmission of multiple signal channels
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The Block Diagram for Separating the Signals 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 34
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Double-Sideband (DSB) and Single- Sideband (SSB) AM 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 35
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Single Sideband Modulation 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 36
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Demo – Filtering using Matlab 28-Mar-19 EIE3001 Sig & Sys, Spring 2019 37 % This script demonstrates time domain and frequency domain % characteristics for a discrete-time filter h[n].
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