DSP First, 2/e. This Lecture: Lecture 13 Frequency Response of FIR Filters SINUSOIDAL INPUT SIGNAL. Frequency Domain (sum of sinusoids)

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1 DSP First, 2/ Lctur 3 Frquc Rspos of FIR Filtrs READING ASSIGNENTS Tis Lctur: Captr 6, Sctios 6- to 6-5 Otr Radig: Rcitatio FREQUENCY RESPONSE EXAPLES Nt Lctur: Cap. 6, Scts. 6-6, 6-7 & 6-8 Aug , J ccllla & RW Scafr 3 LECTURE OBJECTIVES SINUSOIDAL INPUT SIGNAL DETERINE t FIR FILTER OUTPUT FREQUENCY RESPONSE of FIR AG PLOTTING vs. Frquc AGNITUDE vs. Frq PASE PASE vs. Frq DOAINS: Tim & Frquc Tim-Domai: =tim discrt-tim sigal t cotiuous-tim sigal Frquc Domai sum of siusoids Spctrum vs. f z ANALOG vs. DIGITAL Spctrum vs. omga-at ov ac ad fort QUICKLY Aug , J ccllla & RW Scafr 4 Aug , J ccllla & RW Scafr 5

2 DIGITAL FILTERING FILTERING EXAPLE t t A-to-D FILTER D-to-A CONCENTRATE o t SPECTRU SINUSOIDAL INPUT INPUT = SU of SINUSOIDS T, OUTPUT = SU of SINUSOIDS 7-poit AVERAGER 7 Rmovs cosi B maig its amplitud A smallr 3-poit AVERAGER Cags A sligtl Aug , J ccllla & RW Scafr 8 Aug , J ccllla & RW Scafr 9 7-pt FIR EXAPLE AVG Iput :.2 CAUSAL: Us Prvious cos2 /8 / 4 for 4 SINUSOIDAL RESPONSE INPUT: = SINUSOID OUTPUT: will also a SINUSOID Diffrt Amplitud ad Pas SAE Frquc APLITUDE & PASE CANGE Calld t FREQUENCY RESPONSE Aug , J ccllla & RW Scafr 2 LONGER OUTPUT Aug , J ccllla & RW Scafr 3

3 DCONVDEO: ATLAB GUI DCONVDEO: ATLAB GUI Aug , J ccllla & RW Scafr 4 COPLEX EXPONENTIAL COPLEX EXPONENTIAL A A is t iput sigal a compl potial FIR DIFFERENCE EQUATION Aug , J ccllla & RW Scafr 5 COPLEX EXP OUTPUT COPLEX EXP OUTPUT Us t FIR Diffrc Equatio Us t FIR Diffrc Equatio A A A A Aug , J ccllla & RW Scafr 6 A FREQUENCY RESPONSE FREQUENCY RESPONSE At f DEFINE FREQUENCY At ac frquc, w ca DEFINE FREQUENCY RESPONSE Compl-valud formula as AGNITUDE vs. frquc q Ad PASE vs. frquc Notatio: i plac of Aug , J ccllla & RW Scafr 7 Notatio: i plac of

4 EXAPLE 6. { } {, 2 2, } cos EXPLOIT SYETRY PLOT of FREQ RESPONSE { } {,2,} 2 2cos RESPONSE at Sic 2 2cos agitud is ad Pas is 2 2cos radias Aug , J ccllla & RW Scafr 8 Aug , J ccllla & RW Scafr 9 EXAPLE 6.2 Fid id ad w 2 / 4 is 2 2cos ow EXAPLE 6.2 aswr Fid w 2 Evaluat at 2 2cos 3 / 3 / 4 2 /4 6 /2 Aug , J ccllla & RW Scafr 2 Aug , J ccllla & RW Scafr 2

5 EXAPLE: COSINE INPUT Fid w is ow ad 2cos cos Aug , J ccllla & RW Scafr 22 EX: COSINE INPUT as-2 2cos Fid w 3 4 / 4 3 / cos 2 2cos 3 3 /2 /2 3 2 / 4 / 4 Aug , J ccllla & RW Scafr 24 EX: COSINE INPUT 2cos Fid w 3 4 2cos Us Liarit / 4 / / 4 / 4 Aug , J ccllla & RW Scafr 23 ATLAB: FREQUENCY RESPONSE = frqz,,ww VECTOR cotais Filtr Cofficits SP-First: = frz,,ww FILTER COEFFICIENTS { } Aug , J ccllla & RW Scafr 25

6 LTI SYSTES LTI: Liar & Tim-Ivariat i COPLETELY CARACTERIZED : FREQUENCY RESPONSE, or IPULSE RESPONSE Siusoid IN -----> Siusoid OUT At t SAE Frquc Tim & Frquc Rlatio Gt Frquc Rspos from r is t FIR cas: IPULSE RESPONSE Aug , J ccllla & RW Scafr 26 Aug , J ccllla & RW Scafr 27 BLOCK DIAGRAS UNIT-DELAY SYSTE Equivalt Rprstatios Fid ad for { } {, } Aug , J ccllla & RW Scafr 28 Aug , J ccllla & RW Scafr 29

7 FIRST DIFFERENCE SYSTE DLTI Dmo wit Siusoids Fid ad Equatio : for t Diffrc FILTER = Aug , J ccllla & RW Scafr 3 CASCADE SYSTES Aug , J ccllla & RW Scafr 3 CASCADE EQUIVALENT Dos t ordr of S &S 2 mattr? NO, LTI SYSTES ca rarragd!!! WAT ARE TE FILTER COEFFS? { } WAT is t ovrall FREQUENCY RESPONSE? 2 S S 2 2 Aug , J ccllla & RW Scafr 32 Aug , J ccllla & RW Scafr 2 33