Spfirst l10 Ev

Copyright © Monash University 2009

Signal

Processing

First

Lecture 10

FIR Filtering Introduction

1


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

2


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]

3


DIGITAL FILTERING

• CONCENTRATE on the COMPUTER

– PROCESSING ALGORITHMS

– SOFTWARE (MATLAB)

– HARDWARE: DSP chips, VLSI

• DSP: DIGITAL SIGNAL PROCESSING

4

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


The TMS32010, 1983

5

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


Rockland Digital Filter, 1971

6

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


Digital Cell Phone (ca. 2000)

7

Now it takes and plays video


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

8

COMPUTER y[n] x[n]


D-T SYSTEM EXAMPLES

• EXAMPLES:

– POINTWISE OPERATORS

• SQUARING: y[n] = (x[n])2

– RUNNING AVERAGE

• RULE: Eg “the output at time n is the average of three consecutive input values”

9

SYSTEM y[n] x[n]


DISCRETE-TIME SIGNAL

• x[n] is a LIST of NUMBERS

– INDEXED by “n” : …, x[-1]=0, x[0]=2, x[1]=4, …

10

STEM PLOT


SIMPLEST FILTER -3-PT AVERAGE • ADD 3 CONSECUTIVE NUMBERS

– Do this for each “n”

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Make a TABLE

n=0

n=1

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


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12

INPUT SIGNAL

OUTPUT SIGNAL

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


ANOTHER 3-pt AVERAGER

• Uses “PAST” VALUES of x[n]

– IMPORTANT IF “n” represents REAL TIME

• WHEN x[n] & y[n] ARE STREAMS

13

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


PAST, PRESENT, FUTURE

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“n” is TIME

Causal filtering

Noncausal filtering


GENERAL FIR FILTER

• FILTER COEFFICIENTS {bk}

– DEFINE THE FILTER

– For example,

15

M

k

k knxbny0

][][

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

][][3

0

nxnxnxnx

knxbnyk

k

}1,2,1,3{ kb


GENERAL FIR FILTER

• FILTER COEFFICIENTS {bk}

• FILTER ORDER is M

• FILTER LENGTH is L = M+1

– NUMBER of FILTER COEFFS is L

16

M

k

k knxbny0

][][


GENERAL FIR FILTER

• SLIDE a WINDOW across x[n]

17

x[n] x[n-M]

M

k

k knxbny0

][][


FILTERED STOCK SIGNAL

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OUTPUT

INPUT

50-pt Averager


SPECIAL INPUT SIGNALS

• x[n] = SINUSOID

• x[n] has only one NON-ZERO VALUE

19

00

01][

n

nn

1

n

UNIT-IMPULSE

FREQUENCY RESPONSE (LATER)


SHIFTED UNIT IMPULSE [n-i]

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20

[n-i] is NON-ZERO

Only when n-i = 0 !!!

]3[ n 3n


MATH FORMULA for x[n]

• Use SHIFTED IMPULSES to write x[n]

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

NOTE: {[n-k]}

for - < k < is an

orthonormal basis

for D-T signals

[ ] [ ] [ ]k

x n x k n k


General formula for x[n]

• Use SHIFTED IMPULSES to write x[n]

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NOTE:

k[n] = [n-k] for - < k <

is an orthonormal basis for

Discrete-Time signals

[ ] [ ] [ ] [ ] [ ]k

k k

x n x k n k x k n


UNIT STEP SIGNAL

24

00

01][

n

nnu

[ ]

[ ] [ 1]

n

u n u n

u[n]

u[n-1]

[n]


4-pt averager with [n] input

• CAUSAL SYSTEM: USE PAST VALUES

25

])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


4-pt Avg Impulse Response

[n] “READS OUT” the FILTER COEFFICIENTS

26

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

41

41

41 nh

“h” in h[n] denotes

Impulse Response

1

n

n = 0

n=4

n=5

n = –1

NON-ZERO

When window

overlaps [n]

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

n = 1


FIR IMPULSE RESPONSE

• Convolution = Filter Definition

– Filter Coeffs = Impulse Response

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M

k

knxkhny0

][][][

CONVOLUTION

M

k

k knxbny0

][][


FILTERING EXAMPLE

• 7-point AVERAGER

– Removes cosine

• By making its amplitude (A) smaller

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3-point AVERAGER

Changes A slightly

6

071

7 ][][k

knxny

2

031

3 ][][k

knxny

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


3-pt AVG EXAMPLE

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USE 3 PAST VALUES

Phase shift

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