Lecture 4 Outline
• RAT covers Oppenheim/Schafer/Buck Sections 2.5-2.6
• Linear constant-coefficient difference equations
– Impulse response of an LTI system defined by a LCCDE
• DSP in real life example: filtering ocean temperature
• Eigenfunctions and frequency response of DT LTI systems
– Frequency response of moving average filter
Lecture 4 RAT: O/S/B Sections 2.5-2.6
1. Name
2. T/F For DT signals, the high frequencies are those that are close to an
even multiple of π, e.g., 0, 2π, 4π . . .
3. T/F The frequency response of DT LTI systems is periodic with period
2π.
4. T/F The linearity of a system described by a difference equation does
not depend on the auxiliary conditions specified for the system.
5. T/F Complex exponential sequences are not eigenfunctions of DT LTI
systems.
In-class problem 1: LCCDE
A causal DT LTI system is described by the following difference equation
y[n] =
6
X
1
m=0 7
x[n − m]
Determine and sketch the impulse reponse of this system.
DSP in real life: filtering ocean temperature data
Why measure temperature? ⇒ Sound speed depends on temperature
NPAL experiment temperature records
Example:
11
197m
10.5
• Ocean temperature
off California coast
Temperature (degrees C)
10
• Samples every 20
minutes for 1 year
9.5
9
8.5
8
7.5
7
6.5
6
200
250
300
350
400
450
500
550
Yearday
⇒ Need to filter temperature data to remove rapid fluctuations
DSP in real life example: moving average filters
Zoom in on temperature sensor #1 (197m)
9.5
Temperature (degrees C)
• Black line
shows output of
a 7-point causal
moving average
filter
9
8.5
Original
Filtered w/7pt causal moving average
8
210
211
212
213
214
215
Yearday
Note: filtered signal appears somewhat time-shifted compared to original.
Can we implement a non-causal filter? ⇒
DSP in real life example: non-causal moving-average filter
Implement non-causal filtering by running this signal through a causal moving average filter and then time shifting the output!
Zoom in on temperature sensor #1 (197m)
9.5
Temperature (degrees C)
• Blue line shows
output of a 7-pt
non-causal
moving average
9
8.5
Original
Filtered w/7pt non−causal moving average
8
210
211
212
213
214
215
Yearday
What happens when we change the length of the moving average filter?
DSP in real life example: changing length of moving average filter
Zoom in on temperature sensor #1 (197m)
9.5
Temperature (degrees C)
• Blue line for a
7-point
non-causal
moving average
filter
9
8.5
Original
Filtered w/7pt non−causal moving average
Mystery filter
8
210
211
212
213
214
215
Yearday
What was green line processed with?
3-point moving avg? 7-point moving avg? 73-point moving avg?
In-class 2: Frequency-response of 7-pt causal moving average
Magnitude
1
jω
|H(e )|
0.8
0.6
0.4
0.2
0−π
−0.8π −0.6π −0.4π −0.2π
π
0
0.2π
ω
Phase
0.4π
0.6π
0.8π
π
What is output of the
system for the following
inputs?
π
1. x1[n] = ej 5 n
∠ H(ejω)
0.5π
2. x2[n] = e−j
0
−0.5π
−π
−π
−0.8π −0.6π −0.4π −0.2π
0
ω
0.2π
0.4π
0.6π
0.8π
π
4π n
7