1. Signal Processing First
Lecture 10
Filtering Intro
5/27/2018
© 2003, JH McClellan & RW Schafer

2. © 2003, JH McClellan & RW Schafer
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
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© 2003, JH McClellan & RW Schafer

3. © 2003, JH McClellan & RW Schafer
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]
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© 2003, JH McClellan & RW Schafer

4. © 2003, JH McClellan & RW Schafer
DIGITAL FILTERING
x(t)
A-to-D
x[n]
COMPUTER
y[n]
D-to-A
y(t)
CONCENTRATE on the COMPUTER
PROCESSING ALGORITHMS
SOFTWARE (MATLAB)
HARDWARE: DSP chips, VLSI
DSP: DIGITAL SIGNAL PROCESSING
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© 2003, JH McClellan & RW Schafer

5. © 2003, JH McClellan & RW Schafer
The TMS32010, 1983
First PC plug-in board from Atlanta Signal Processors Inc.
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© 2003, JH McClellan & RW Schafer

6. Rockland Digital Filter, 1971
For the price of a small house, you could have one of these.
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© 2003, JH McClellan & RW Schafer

7. Digital Cell Phone (ca. 2000)
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© 2003, JH McClellan & RW Schafer
Now it plays video

8. © 2003, JH McClellan & RW Schafer
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
x[n]
COMPUTER
y[n]
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© 2003, JH McClellan & RW Schafer

9. © 2003, JH McClellan & RW Schafer
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”
x[n]
SYSTEM
y[n]
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© 2003, JH McClellan & RW Schafer

10. © 2003, JH McClellan & RW Schafer
DISCRETE-TIME SIGNAL
x[n] is a LIST of NUMBERS
INDEXED by “n”
STEM PLOT
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© 2003, JH McClellan & RW Schafer

11. © 2003, JH McClellan & RW Schafer
3-PT AVERAGE SYSTEM
ADD 3 CONSECUTIVE NUMBERS
Do this for each “n”
Make a TABLE
n=0
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© 2003, JH McClellan & RW Schafer
n=1

12. © 2003, JH McClellan & RW Schafer
INPUT SIGNAL
OUTPUT SIGNAL
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© 2003, JH McClellan & RW Schafer

13. © 2003, JH McClellan & RW Schafer
PAST, PRESENT, FUTURE
“n” is TIME
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© 2003, JH McClellan & RW Schafer

14. © 2003, JH McClellan & RW Schafer
ANOTHER 3-pt AVERAGER
Uses “PAST” VALUES of x[n]
IMPORTANT IF “n” represents REAL TIME
WHEN x[n] & y[n] ARE STREAMS
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© 2003, JH McClellan & RW Schafer

15. © 2003, JH McClellan & RW Schafer
GENERAL FIR FILTER
FILTER COEFFICIENTS {bk}
DEFINE THE FILTER
For example,
5/27/2018
© 2003, JH McClellan & RW Schafer

16. © 2003, JH McClellan & RW Schafer
GENERAL FIR FILTER
FILTER COEFFICIENTS {bk}
FILTER ORDER is M
FILTER LENGTH is L = M+1
NUMBER of FILTER COEFFS is L
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© 2003, JH McClellan & RW Schafer

17. © 2003, JH McClellan & RW Schafer
GENERAL FIR FILTER
SLIDE a WINDOW across x[n]
x[n-M]
x[n]
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© 2003, JH McClellan & RW Schafer

18. © 2003, JH McClellan & RW Schafer
FILTERED STOCK SIGNAL
INPUT
OUTPUT
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© 2003, JH McClellan & RW Schafer
50-pt Averager

19. © 2003, JH McClellan & RW Schafer
SPECIAL INPUT SIGNALS
x[n] = SINUSOID
x[n] has only one NON-ZERO VALUE
FREQUENCY RESPONSE (LATER)
UNIT-IMPULSE 1 n
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© 2003, JH McClellan & RW Schafer

20. UNIT IMPULSE SIGNAL d[n]
d[n] is NON-ZERO
When its argument
is equal to ZERO
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© 2003, JH McClellan & RW Schafer

21. © 2003, JH McClellan & RW Schafer
MATH FORMULA for x[n]
Use SHIFTED IMPULSES to write x[n]
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© 2003, JH McClellan & RW Schafer

22. SUM of SHIFTED IMPULSES
This formula ALWAYS works
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© 2003, JH McClellan & RW Schafer

23. © 2003, JH McClellan & RW Schafer
4-pt AVERAGER
CAUSAL SYSTEM: USE PAST VALUES
INPUT = UNIT IMPULSE SIGNAL = d[n]
OUTPUT is called “IMPULSE RESPONSE”
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© 2003, JH McClellan & RW Schafer

24. 4-pt Avg Impulse Response
d[n] “READS OUT” the FILTER COEFFICIENTS
n=0
“h” in h[n] denotes
Impulse Response
NON-ZERO When window overlaps d[n] 1 n
n=–1
n=0
n=1
n=4
n=5
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© 2003, JH McClellan & RW Schafer

25. © 2003, JH McClellan & RW Schafer
FIR IMPULSE RESPONSE
Convolution = Filter Definition
Filter Coeffs = Impulse Response
CONVOLUTION
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© 2003, JH McClellan & RW Schafer

26. © 2003, JH McClellan & RW Schafer
FILTERING EXAMPLE
7-point AVERAGER
Smooth compared to 3-point Averager
By making its amplitude (A) smaller
3-point AVERAGER
Changes A slightly
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© 2003, JH McClellan & RW Schafer

27. © 2003, JH McClellan & RW Schafer
3-pt AVG EXAMPLE
USE PAST VALUES
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© 2003, JH McClellan & RW Schafer

28. © 2003, JH McClellan & RW Schafer
7-pt FIR EXAMPLE (AVG)
CAUSAL: Use Previous
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© 2003, JH McClellan & RW Schafer
LONGER OUTPUT