Multiplexing and more More bandwidth to carry information at a higher carrier

# Multiplexing and more more bandwidth to carry

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Multiplexing, and more… More bandwidth to carry information at a higher carrier frequency Transmit different signals simultaneously (frequency division multiplexing, FDM) Filter design 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 19

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Complex Exponential Carrier 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 20 As we will see later, knowing the phase of the carrier signal is important for demodulation, and that is why we explicitly express it in the notation.
Spectrum of the Modulation Step 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 21

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Demodulation Step 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 22
Implementation using Only Real Signals 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 23

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Design a Bandpass Filter using Exponential Carrier Modulation 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 24 passband ¾ We wish to extract only the portion of the signal fallen within the red window (in frequency domain) ¾ Challenge: Sometimes difficult to design bandpass filter given arbitrary specification Could be difficult to adjust the filter if we change the filter requirement It is often easier to build a fixed filter (fixed center frequency, passband, etc.)
Design a Bandpass Filter using Exponential Carrier Modulation 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 25 passband

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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 ). 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 26 Equivalent filter for f ( t ) Equivalent filter for Re{ f ( t )}
Sinusoidal Carrier 27-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: 27-Mar-19 EIE3001 Sig & Sys, Spring 2019 28
Implementation 27-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?
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