DSP First, 2/e. Lecture 11 FIR Filtering Intro

Transcription

1 DSP First, 2/e Lecture FIR Filterig Itro

2 Licese Ifo for DSPFirst Slies This work release uer a Creative Commos Licese with the followig terms: Attributio The licesor permits others to copy, istribute, isplay, a perform the work. I retur, licesees must give the origial authors creit. NoCommercial The licesor permits others to copy, istribute, isplay, a perform the work. I retur, licesees may ot use the work for commercial purposes uless they get the licesor's permissio. Share Alike The licesor permits others to istribute erivative works oly uer a licese ietical to the oe that govers the licesor's work. Full Tet of the Licese This (hie) page shoul be kept with the presetatio Aug , JH McClella & RW Schafer 2

3 READING ASSIGNMENTS This Lecture: Chapter 5, Sects. 5, 52, 53 & 5 (partial) Other Reaig: Net Lecture: Ch. 5, Sects 5, 56, 57 & 58 CONVOLUTION Aug , JH McClella & RW Schafer 3

4 LECTURE OBJECTIVES INTRODUCE FILTERING IDEA Weighte Average Ruig Average FINITE IMPULSE RESPONSE FILTERS FIR Filters Show how to compute the output y from the iput sigal, Aug , JH McClella & RW Schafer

5 DIGITAL FILTERING (t) AtoD COMPUTER y DtoA y(t) Characterize SIGNALS (Fourier series) Coverte to DIGITAL (samplig) Toay: How to PROCESS them (DSP)? CONCENTRATE o the COMPUTER ALGORITHMS, SOFTWARE (MATLAB) a HARDWARE (DSP chips, VLSI) Aug , JH McClella & RW Schafer 5

6 The TMS32, 983 First PC plugi boar from Atlata Sigal Processors Ic. Aug , JH McClella & RW Schafer 6

7 Rockla Digital Filter, 97 Cost was about the same as the price of a small house. Aug , JH McClella & RW Schafer 7

8 Digital Cell Phoe (ca. 2) Now, igital cameras a vieo streamig rely o DSP algorithms Aug , JH McClella & RW Schafer 8

9 DISCRETETIME SYSTEM COMPUTER y OPERATE o to get y WANT a GENERAL CLASS of SYSTEMS ANALYZE the SYSTEM TOOLS: TIMEDOMAIN & FREQUENCYDOMAIN SYNTHESIZE the SYSTEM Aug , JH McClella & RW Schafer 9

10 DT SYSTEM EXAMPLES SYSTEM y EXAMPLES: POINTWISE OPERATORS SQUARING: y = () 2 RUNNING AVERAGE RULE: the output at time is the average of three cosecutive iput values Aug , JH McClella & RW Schafer

11 DISCRETETIME SIGNAL is a LIST of NUMBERS INDEXED by STEM PLOT Aug , JH McClella & RW Schafer

12 3PT AVERAGE SYSTEM ADD 3 CONSECUTIVE NUMBERS Do this for each Make a TABLE y = ( 3 2) = = Aug , JH McClella & RW Schafer 2

13 INPUT SIGNAL y = ( 3 2) OUTPUT SIGNAL Aug , JH McClella & RW Schafer 3

14 PAST, PRESENT, FUTURE SLIDE a WINDOW across is PRESENT TIME Figure 53 Filter calculatio at the preset time (l = ) uses values withi a sliig wiow. Gray shaig iicates the past (l < ); orage shaig, the future (l > ). Here, the sliig wiow ecompasses values from both the future a the past. Aug , JH McClella & RW Schafer

15 ANOTHER 3pt AVERAGER Uses PAST VALUES of IMPORTANT IF represets REAL TIME WHEN & y ARE STREAMS y = ( 3 2) Aug , JH McClella & RW Schafer 5

16 CAUSAL 3pt AVERAGER y = ( 2) 3 Aug , JH McClella & RW Schafer 6

17 Aug , JH McClella & RW Schafer 7 GENERAL CAUSAL FIR FILTER FILTER COEFFICIENTS {b k } DEFINE THE FILTER For eample, å = = M k k k b y = = å = k b y k k,2,} 3, = { b k

18 GENERAL CAUSAL FIR FILTER FILTER COEFFICIENTS {b k } y = M å k= FILTER ORDER is M FILTER LENGTH is L = M NUMBER of FILTER COEFFS is L b k k Aug , JH McClella & RW Schafer 8

19 Aug , JH McClella & RW Schafer 9 SLIDE a WINDOW across M M b b b k b y M M k k = = å =! GENERAL CAUSAL FIR FILTER

20 FILTERED STOCK SIGNAL INPUT OUTPUT 5pt Averager Aug , JH McClella & RW Schafer 2

21 SPECIAL INPUT SIGNALS = SINUSOID has oly oe NONZERO VALUE UNITIMPULSE = FREQUENCY RESPONSE (LATER) ïì í ïî = ¹ Aug , JH McClella & RW Schafer 2

22 UNIT IMPULSE SIGNAL is NONZERO Whe its argumet is equal to ZERO 3 = 3 Aug , JH McClella & RW Schafer 22

23 Sequece Represetatio Eample: = = = 2 = = = = 2 = 2 = 6 = 3 = 3 = 2 =! 6 2 3! Aug , JH McClella & RW Schafer 23

24 UNIT IMPULSE RESPONSE FIR filter escriptio usually give i terms of coefficiets b k y = M å k= b k k Ca we escribe the filter usig a SIGNAL istea? What happes if iput is a uit impulse? Aug , JH McClella & RW Schafer 2

25 Aug , JH McClella & RW Schafer 25 Eample: pt AVERAGER CAUSAL SYSTEM: USE PAST VALUES 3) 2 ( = y INPUT = UNIT IMPULSE SIGNAL = 3 2 = = y OUTPUT is calle IMPULSE RESPONSE Deote h=y whe =

26 Aug , JH McClella & RW Schafer 26 Uit Impulse Respose y 3 2 = y 3 2 h y = = =

27 Aug , JH McClella & RW Schafer 27 SUM of Shifte Impulses 3 2 = h h h h h h h

28 pt Avg Impulse Respose y READS OUT the FILTER COEFFICIENTS h i h eotes Impulse Respose = ( 2 h = {!,,,,,,,,,!} = = = = Aug , JH McClella & RW Schafer 28 = =5 NONZERO Whe wiow overlaps 3)

29 FIR IMPULSE RESPONSE y M = åb k h = å k k= M b k= k k Aug , JH McClella & RW Schafer 29

30 3 Ways to Represet the FIR filter Use SHIFTED IMPULSES to write h h = h 2 2: Plot the values 3: List the values b k = {,, 2,,} True for ay sigal, Aug , JH McClella & RW Schafer 3

31 FILTERING EXAMPLE 7poit AVERAGER Removes cosie By makig its amplitue (A) smaller 6 å k= y = ( ) k 7 7 3poit AVERAGER Chages A slightly 2 å k= y = ( ) k 3 3 Aug , JH McClella & RW Schafer 3

32 3pt AVG EXAMPLE Iput : = (.2) cos(2p /8 p / ) for USE PAST VALUES Aug , JH McClella & RW Schafer 32

33 7pt FIR EXAMPLE (AVG) Iput : = (.2) cos(2p /8 p / ) for CAUSAL: Use Previous Aug , JH McClella & RW Schafer 33 LONGER OUTPUT

34 Aug , JH McClella & RW Schafer = MATH FORMULA for Use SHIFTED IMPULSES to write

35 SUM of SHIFTED IMPULSES This formula ALWAYS works Aug , JH McClella & RW Schafer 35

36 PAST, PRESENT, FUTURE is TIME Aug , JH McClella & RW Schafer 36

37 GENERAL CAUSAL FIR FILTER SLIDE a WINDOW across y = M å k= b k k M Aug , JH McClella & RW Schafer 37