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## Wireless Communications

by: Cassidy Effertz

7

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# Wireless Communications ECE 4606

Cassidy Effertz

GPA 3.64

Staff

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COURSE
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KARMA
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## Popular in ELECTRICAL AND COMPUTER ENGINEERING

This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 4606 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/233906/ece-4606-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15
Matched Filter Instructor MA Ingram ECE4823 g Maximum SNR Property Assume AWGN We want the filter that yields the highest signaltonoise ratio SNR at its output Next show that the optimum filter is the matched filter Set Up Suppose rl Slnl where Mr is WGN with spectral height NC2 and st is a signal with a finite duration T Let rt be the input to a filter with impulse response ht Let yt be the output g Signal and Noise Parts The signal part of the output is t ySt Isuht7 udu 0 The noise part of the output is t ynt jnuht 7udu 0 SI Signaltonoise Ratio SNR yft E yt Jr 110 7 Mam Ejnuht7 Mam g Denominator Eyr nuht 7 udujnvht 7 WW L0 0 HEnunvht 7 uht 7Vdudv g Invoke White Noise Model Eb 0 Jaj5u 7 Vht 7 uht 7Vdudv 7 N0 t 2 7th t7udu SNR So Far To optimize the SNR choose Mu to maximize the numerator Usuht7 Mam SNR 0 Ihz t 7 udu g CauchySchwarz Inequality Say S and Q are two points in a Hilbert space then 5an S ISIZIQIZ with equality when QcS CauchySchwarz for Signals LetSand Q be points in the Hilbert space of squareintegrable functions Then 2 Isuqudu SIs2uduJq2udu Equality is reached when csu qu Apply CauchySchwartz Recall numerator of SNR t 2 Isuht7 Mob 0 Pick ht u to be equal to csu Simplify Optimal SNR Substitute ht ucsu cfs z du js2udu SNRopt 0 t Noiczis2udu 2 0 Optimize t If st has finite duration T then SNR is maximized by setting tT T Is2udu t 0 28 SNR0p N0 N0 3 2 Inner Product Interpretation of Matched Filter yT hurT iudu Essa iurT iudu c surudu Cs Iquot w yo cs T it Max SNR Filter Matched Filter Matched filter Su impulse response is a u flipped in place 0 T version of signal hTu hT7u csu 0 T u or hu Ti u o T Matched Filter Frequency Response Take Fourier Transform of the Matched Filter impulse response H f l hueJW du T cIsT7 ukijz m du 0 g Let rTu Hf c sT 7 ue39Z Wdu cj sre rquot 7 dr T ce j sUkWdr ce lSfgtk Matched in Frequency Domain The magnitude of the matched filter response is just a scaled version of the signal s FT IHfl clsml

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