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.
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.
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
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
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
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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