301 EECE Signals & Systems
Prof. Mark Fowler
Note Set #9a
Example: Using D-T Convolution in an Application of Measuring the Speed of a Motor
1/13
Course Flow Diagram
The arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis). New Signal Models
Ch. 1 Intro
C-T Signal Model Functions on Real Line System Properties LTI Causal Etc D-T Signal Model Functions on Integers
Ch. 3: CT Fourier Signal Models
Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals
Ch. 5: CT Fourier System Models
Frequency Response Based on Fourier Transform
Ch. 6 & 8: Laplace Models for CT Signals & Systems
Transfer Function
New System Model
New System Model
Ch. 2 Diff Eqs
C-T System Model Differential Equations D-T Signal Model Difference Equations Zero-State Response Zero-Input Response Characteristic Eq.
Ch. 2 Convolution
C-T System Model Convolution Integral D-T System Model Convolution Sum
New System Model
New Signal Model Powerful Analysis Tool
Ch. 4: DT Fourier Signal Models
DTFT (for "Hand" Analysis) DFT & FFT (for Computer Analysis)
Ch. 5: DT Fourier System Models
Freq. Response for DT Based on DTFT
Ch. 7: Z Trans. Models for DT Signals & Systems
Transfer Function
New System Model
New System Model2/13
Example: D-T Processing for Motor Speed
The m-file motor_speed.m is available for download from the course website. To run it all you need to do is: 1. Download it to a folder on your computer a. Make sure Matlab's current directory is the folder where you have put the m-file (change it if needed) 2. At the Matlab command line type: motor_speed 3. The m-file will run and create several figures
3/13
Physical Set-Up of Motor and Sensor Suppose you have a motor whose shaft is connected to a sensor that can measure the angular position of the shaft... its values will range between - and . Output of Sensor when Motor spins
at constant angular rate of 5 rad/sec
Motor
Assume that the sensor generates 1 volt/radian
Q: How do we measure the speed of the motor?
4/13
Motor and D-T System for Processing
First we could sample the sensor's signal at a fast enough rate to get a D-T signal that accurately conveys the C-T signal coming from the sensor: m[n] m(t) Analog-to-Digital Computer-Based Motor Converter (ADC) D-T System C-T Signal Representing Angular Position D-T Signal Representing Angular Position
The resulting samples could be put into a computer (as binary numbers) and processed using D-T convolution. Now... the trick is to determine how to design the D-T system (i.e., choose the system's impulse response h[n]) to get a measure of the angular rate rm[n] !!! Computer
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
5/13
We can use matlab to explore how to do this.
Simulate the Angular Position's D-T Signal in Matlab
% Create time samples spaced 1 ms apart del_t=0.001; t=0:del_t:3; theta=5*pi*t; %% simulates angular position of motor shaft spinning at a constant rate %% use some matlab "tricks" to simulate the fact that measured angle lies in pi to pi theta_m=angle(exp(j*theta));
Motor
ADC
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
Note: Discontinuities
Figure 1 from motor_speed.m
6/13
1st D-T Convolution Approximates Computing a Derivative
% specify impulse response of D-T system that computes %% the time derivative of the input h_1=[-3 16 -36 48 -25]/(12*del_t);
Comes from Branch of Math known as "Numerical Methods"
%%% Compute D-T system's output using convolution x=theta_m; % just rename position measurements to "typical" input symbol y=conv(x, h_1); Matlab % has a command for convolving two signals
Motor
ADC
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
Figure 2 from motor_speed.m
Correct Values For Rate Large Spurious Values due to Discontinuities
7/13
D-T Tricks to Fix the Effects of the Discontinuities
%% we can do some computer-based "exception processing" to get rid of those %% by throwing away output values whose absolute value is bigger than we would expect %% the rotational rate to be I=find(abs(y)>500); % find the indices of such large values y(I)=0; % replace the values with zeros at those locations
Motor
ADC
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
Correct Values For Rate Zeros Inserted to Replace Spurious Values from Discontinuities "Transient" values "Ramp-Up values due to convolution
8/13
Figure 3 from motor_speed.m
D-T Convolution to Further Fix the Discontinuities
In the previous stage we inserted zeros at the discontinuity points... so what we end up with (ignoring the transient) is many correct values with a few zero values included. So... imagine taking a small block of N such data points and taking the data average (add them up and divide by N)... the small number of zero values will have a small effect and we should end up with the average being close to the true value. If we imagine "sliding the block" used for averaging ahead by one sample each time... then we would get an average for each block that should be a good estimate of the angular rate These indicate the Etc. sliding blocks Figure 3 from motor_speed.m
9/13
With a little thought we can convince ourselves that this sliding average is nothing more that convolving the "zero-corrected" signal with a D-T signal that has N ones and zeros elsewhere and then dividing by N:
y2a=conv(y,ones(1,100)) /100; % do sliding average of 100 pts to reduce effect of inserted zeros y2b=conv(y,ones(1,1000))/1000; % do sliding average of 1000 pts to explore effect of N value
Motor
ADC
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
This indicates the following trade-off for the size of the averaging block:
Blue: N = 100 Red: N = 1000 Figure 4 from motor_speed.m
Pro
Small N Rapid Response Time
Large N Small Fluctuation in Values Slow Response Time
Con Large Fluctuation in Values
10/13
Admission: The way we did it (replace anomalies with zeros, smooth with convolution) is probably not the best way to do this: A better "Trick" would be to replace each anomaly with the non-anomaly value that precedes it. This would reduce the jumps due to the inserted zeros and reduce the need for the second convolution.
11/13
Exploring Impact of Abrupt Rate Change
Motor ADC
m[n]
h1[n]
Some Tricks
rm[n] h2[n]
Blue: N = 100 Red: N = 1000
Rate Change
Figures 5&6 from motor_speed.m
Bigger N gives: More accurate measure... But... slower response to changes
12/13
Things We've Seen Here Using Matlab to simulate D-T signals that will be obtained from an ADC Use D-T convolution to approximately compute the derivative of a signal
The D-T signal represents samples of some "underlying" C-T signal
Use tricks to correct anomalies and spurious results
In practice this kind of thing is pretty common
Use D-T convolution to "smooth out" jumps
Here the jumps were due to physical characteristics of angle measurement
We have seen a fundamental trade-off that often arises in linear systems:
A system that provides more "smoothing" capability typically responds slowly to signal changes So... if we have a noisy signal, we can remove a lot of the noise but only at the expense of smoothing out some of the signal's rapid changes
13/13