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Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS...

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2/15/2017 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 10 Filtering Intro
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Page 1: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 1

Signal Processing First

Lecture 10

Filtering Intro

Page 2: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 2

READING ASSIGNMENTS

This Lecture:

Chapter 5, Sects. 5-1, 5-2 and 5-3 (partial)

Other Reading:

Recitation: Ch. 5, Sects 5-4, 5-6, 5-7 and 5-8

CONVOLUTION

Next Lecture: Ch 5, Sects. 5-3, 5-5 and 5-6

Page 3: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 3

LECTURE OBJECTIVES

INTRODUCE FILTERING IDEA

Weighted Average

Running Average

FINITE IMPULSE RESPONSE FILTERS

FIR Filters

Show how to compute the output y[n] from

the input signal, x[n]

Page 4: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 4

DIGITAL FILTERING

CONCENTRATE on the COMPUTER

PROCESSING ALGORITHMS

SOFTWARE (MATLAB)

HARDWARE: DSP chips, VLSI

DSP: DIGITAL SIGNAL PROCESSING

COMPUTER D-to-A A-to-D x(t) y(t) y[n] x[n]

Page 5: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 5

The TMS32010, 1983

First PC plug-in board from Atlanta Signal Processors Inc.

Page 6: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 6

Rockland Digital Filter, 1971

For the price of a small house, you could have one of these.

Page 7: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 7

Digital Cell Phone (ca. 2000)

Now it plays video

Page 8: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 8

DISCRETE-TIME SYSTEM

OPERATE on x[n] to get y[n]

WANT a GENERAL CLASS of SYSTEMS

ANALYZE the SYSTEM

TOOLS: TIME-DOMAIN & FREQUENCY-

DOMAIN

SYNTHESIZE the SYSTEM

COMPUTER y[n] x[n]

Page 9: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 9

D-T SYSTEM EXAMPLES

EXAMPLES:

POINTWISE OPERATORS

SQUARING: y[n] = (x[n])2

RUNNING AVERAGE

RULE: “the output at time n is the average of three

consecutive input values”

SYSTEM y[n] x[n]

Page 10: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 10

DISCRETE-TIME SIGNAL

x[n] is a LIST of NUMBERS

INDEXED by “n”

STEM PLOT

Page 11: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 11

3-PT AVERAGE SYSTEM

ADD 3 CONSECUTIVE NUMBERS

Do this for each “n”

Make a TABLE

n=0

n=1

])2[]1[][(][31 nxnxnxny

Page 12: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 12

INPUT SIGNAL

OUTPUT SIGNAL

])2[]1[][(][31 nxnxnxny

Page 13: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 13

PAST, PRESENT, FUTURE

“n” is TIME

Page 14: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 14

ANOTHER 3-pt AVERAGER

Uses “PAST” VALUES of x[n]

IMPORTANT IF “n” represents REAL TIME

WHEN x[n] & y[n] ARE STREAMS

])2[]1[][(][31 nxnxnxny

Page 15: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 15

GENERAL FIR FILTER

FILTER COEFFICIENTS {bk}

DEFINE THE FILTER

For example,

M

k

k knxbny0

][][

]3[]2[2]1[][3

][][3

0

nxnxnxnx

knxbnyk

k

}1,2,1,3{ kb

Page 16: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 16

GENERAL FIR FILTER

FILTER COEFFICIENTS {bk}

FILTER ORDER is M

FILTER LENGTH is L = M+1

NUMBER of FILTER COEFFS is L

M

k

k knxbny0

][][

Page 17: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 17

GENERAL FIR FILTER

SLIDE a WINDOW across x[n]

x[n] x[n-M]

M

k

k knxbny0

][][

Page 18: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 18

FILTERED STOCK SIGNAL

OUTPUT

INPUT

50-pt Averager

Page 19: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 19

00

01][

n

nn

SPECIAL INPUT SIGNALS

x[n] = SINUSOID

x[n] has only one NON-ZERO VALUE

1

n

UNIT-IMPULSE

FREQUENCY RESPONSE (LATER)

Page 20: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

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UNIT IMPULSE SIGNAL [n]

[n] is NON-ZERO

When its argument

is equal to ZERO

]3[ n3n

Page 21: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 21

]4[2]3[4]2[6]1[4][2][ nnnnnnx

MATH FORMULA for x[n]

Use SHIFTED IMPULSES to write x[n]

Page 22: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

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SUM of SHIFTED IMPULSES

This formula ALWAYS works

Page 23: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 23

4-pt AVERAGER

CAUSAL SYSTEM: USE PAST VALUES

])3[]2[]1[][(][41 nxnxnxnxny

INPUT = UNIT IMPULSE SIGNAL = [n]

]3[]2[]1[][][

][][

41

41

41

41

nnnnny

nnx

OUTPUT is called “IMPULSE RESPONSE”

},0,0,,,,,0,0,{][41

41

41

41 nh

Page 24: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 24

},0,0,,,,,0,0,{][41

41

41

41 nh

4-pt Avg Impulse Response

[n] “READS OUT” the FILTER COEFFICIENTS

n=0 “h” in h[n] denotes

Impulse Response

1

n

n=0

n=1

n=4

n=5

n=–1

NON-ZERO

When window

overlaps [n]

])3[]2[]1[][(][41 nxnxnxnxny

Page 25: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

2/15/2017 © 2003, JH McClellan & RW Schafer 25

FIR IMPULSE RESPONSE

Convolution = Filter Definition

Filter Coeffs = Impulse Response

M

k

knxkhny0

][][][

CONVOLUTION

M

k

k knxbny0

][][

Page 26: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

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FILTERING EXAMPLE

7-point AVERAGER

Smooth compared to 3-point Averager

By making its amplitude (A) smaller

3-point AVERAGER

Changes A slightly

6

071

7 ][][k

knxny

2

031

3 ][][k

knxny

Page 27: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

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3-pt AVG EXAMPLE

USE PAST VALUES

400for)4/8/2cos()02.1(][ :Input nnnx n

Page 28: Signal Processing First - CmpE WEB 362... · Signal Processing First ... PROCESSING ALGORITHMS SOFTWARE ... First PC plug-in board from Atlanta Signal Processors Inc.

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7-pt FIR EXAMPLE (AVG)

CAUSAL: Use Previous

LONGER OUTPUT

400for)4/8/2cos()02.1(][ :Input nnnx n


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