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# INTRO TO DIG SIG PRO E C E 467

Clemson

GPA 3.84

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This 9 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 467 at Clemson University taught by John Gowdy in Fall. Since its upload, it has received 64 views. For similar materials see /class/214305/e-c-e-467-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.

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Date Created: 09/26/15

ECE T467667 Introduction to Digital Signal Processing LECTURE 3 Frequency domain representath of LSI systems Recall time domain case xn hn WI Yn xn hn Let xn elmquot cos on j sin um complex exponential signal with frequency 2 0 Then yn Z hkxn k ZR hkequot quot k k elm Emmet k k Hequot frequency response of system Note that Hej DTFThn discrete time Fourier Transform gt hn IDTFTHei E1 73Hequot equot dm ECE quot 467667 Introduction to Digital Signal Processing Note that He quotquot2 Z hkequot 2 39 k Zhkequotj 1k 32 v4 k 1 Hequotquot1 gt Hel is periodic with period 2n Also recall He can be written as He1 z Heiwelarg Hlel Consider a real sinusoidal input with frequency no and phase 4 xn A cos coon 1 2 A m iwon A i iwon e e e e 2 2 Using previous results the resulting output is W 9 Hequot eImon el Helmo elmon l I l V I l l COMPLEX CONJUGATES ECE quot 467 667 Introduction to Digital Signal Processing Note that a jb a jb 2a Z W 2Re39 el Hetwo elwon A Reej elmon Helmo 191mg He A HequotDo lcos coon q arg Hej 0 magmod phase mod Likewise it xn A sinmon 1 then yn A Hejmo sin won q arg Hkequot 0 Examgle LSI system with hn aquotuna1 lt1 Corresponds to system with lO relation yn xn ayn 1 Heim Zhne jmn Zane Icon n n0 iae39iwquot 1 ae jm lt1 quot0 1 eequot ao e39lmid nw J 1 N 39ECE 467667 Introduction to Digital Signal Processing 1 acos c39o j sin 40 Hequot 1 1 1 acos n ja sin co 1 H 1 e A J1 acosco2asinco2 1 x1 2acosua2 cos2 n32 sin2 0 1 f1 2acossma2 quot usable frequency rangequot for filter ECE 467 667 Introduction to Digital Signal Processing Phase Resoon se arg Hej arg 1 arg1 a cos 0 ja sin 03 Otan1 asino 1 acos a tan1 asinm 1 a case arg Hej 15 I It 713 a In radians 15 ECE 39 467 667 Introd uction to Digital Signal Processing Examgle LSI system with hn Sn 5n 11 6n 2 hn 11 4 FIR System 0 1 2 n Corresponds to 10 relation to yn xn xn 1 xn 2 NI l Frequencv Response Hequot Zhnequot quot 1 9quot 1e 2 n 2 2 1 e quot 2e k1 Ie equot 20081 e 1 cos 0 Mel 1 cos 0 2 Low Pass Filter Ta 1 UsableFrequency Hengequot 39ECE T 467 667 Introduction to Digital Signal Processing arg He m 2 03 lt example of linear phase Km with K 1 arg lle39 03 a r 3 A 8 51 J i Artificial Discontinultles to assist plotting Why linear phase is good Consider xn A cos won o as input to LSl system from previous results yn A Hej 00 cos non q arg He vJ Kmo if phase is linear gt yn Metro A cos mom k 1 xar39k pure delay of input note that delay K is independent 9 no gt If xn consists of a sum of sinusoids all are delayed by an egual time by a linear phase LSl system 39 EOE H67667 introduction to mng Signal Processing gt time stgnat re is preserved Important 99 for ECG signals Another regugngy response gxgmpjg LSI System with IO relation N1 yn Zxn k xn xn 1 xn N1 to J 39wlndow39 which Include most recent N Inputs Find frequency response of system Step 1 Find hn 2 Find Hequot To find hn let xn 8Vn Then hn yn 5 n 8 n1 6 nN1 ie hO 1 h1 1 And hn 0 for all other n hIi11 N 1 N1 gt Hequot Zhkequotquot Zequot k kO kO 1 equot N 1 6quot 39 ECE quot34676637 Introduction to Digital Signal Processing Simglim 19N 33E e 2 e 2 e 2 1 94 N J 10 n a 1 8 g39 e equot CON 12N4 2Slh 2 m 2 sm 1 2 0N Sin lw gt He 0 sun 2 mN 39 m SKI 392 and arg He N 1 arg sin 2 gv 00m

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