Matlab Exam:Measurement and Signal Processing

Measurement and Signal Processing
1. Following is a noisy harmonic signal:
Please refer to the given data set “data1.mat”, where “y_noise” is the signal, “t” is the time series
and “dT” is the sampling period.
(a) To eliminate the disturbance, design two moving average filters with window size N = 20 and
N = 150, respectively. Plot the resulting filtered signal (both N = 20 and N = 150) and the given
raw data in the same figure. (You should plot with legend and labels, and the line width of the
filtered data should be 1.5). (5%)
(b) What’s the difference when applying larger window size on MAF? (5%)
(c) Redesign the filter with a 2nd order low pass filter, where the cut-off frequency is 10Hz. Plot
the resulting filtered signal and the given raw data in the same figure. (5%)
(d) Based on (c), please sketch the error between the raw data and fitting result, and calculate the
following requirement of the error response: mean, standard deviation and kurtosis. (5%)
(e) What is the correlation coefficient between the raw data and the filtered signal with LPF? (5%)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time(s)
-5
-4
-3
-2
-1
0
1
2
3
4
5
2. Consider a mathematical model
y ax a x x a x a = + -+ ⋅ ⋅+ ( ) sin . ( ) p 3 2
12 3 4 2 05
where a1 , a2 , a3 and a4 are the undetermined parameters.
(a) To estimate the parameters, please represent the equation in least square form Y AX = for
, , m x xx x = 1 2  , , , m y yy y = 1 2  . (5%)
(b) Based on the given data set “data2.mat”, please find the estimated parameters aaaa ˆ ˆ,,, ˆ ˆ 1234 , and
plot the fitting result and the given data in the same figure (with legend and labels). (5%)
(c) Please plot the error response between the given data and the fitting result. (4%)
(d) Refer to the fitting error, plot the histogram with 50 bars and the corresponding Gaussian
distribution. (4%)
(e) Does the fitting error satisfy Gaussian distribution? Why? (2%)
3. Consider a signal as follows:
y t ft ft ft ( )=+ + sin . sin cos (2 05 2 2 2 p pp 1 23 ) ( ) ( )
f Hz 1 =1 , f Hz 2 = 5 , f Hz 3 = 25
Please recover the 1Hz signal and answer the following questions. Please refer to the given data set
“data3.mat”, where “y” is the measured signal y t( ), “t” is the time series and “dT” is the sampling
period.
(a) If the sampling rate is set to be 20Hz , what are the resulting frequencies after sampling? (2%)
(b) To avoid aliasing, what is the minimum sampling frequency? (2%)
(c) In order to recover the 1Hz signal, please design a 2nd order notch filter without frequency
pre-warping to remove the 5Hz and 25Hz signal, where the bandwidth of NTF is 10Hz. Please
plot the filtered results and the given data in the same figure (with legend and labels). (8%)
(d) Based on (c), plot the filtered results with a NTF with frequency pre-warping. (8%)
(e) What’s the difference between the NTFs with and without frequency pre-warping? (5%)
4. Consider a 1st order fan dynamic system
aw bw gw t (t t tt )++ = ( ) ( ) ( ) 2 
where a,b , g are the parameters of the system; t( )t is the input torque of the system while
w(t) is the output speed.
(a) Please apply bilinear transform to determine the difference equation. (5%)
(b) Based on (a), if the equation can be expressed as:
a k k bk ck k k ( ) w w w w tt () ( ) + – + + -= + – () ( ) () ( ) 2 2 1 11
when a =0 15 . , b = 0 03 . , g = 0 001 . and the sampling rate is 100Hz, determine the values
of a ,b ,c . (5%)
(c) To estimate the parameters, please form the m times measurement samples of data in terms of
Y AX = . (5%)
Hint:
( ) ( )
( )( )
( ) ( )
() ( ) 
opt
kk a
b
km km m m c
tt tt
t t tt
é ùé ù +- + é ùé ù ê úê ú ê úê ú
 = ê úê ú ê úê ú
+ + +- + – ê úê ú ë ûë û ë ûë û
Y Y A X
1 21
1 1
 
  

(d) Consider the given data set “data4.mat”, where “torque” is the input signal, “speed” is the
output signal, “dT” is the sampling time and “time” is the time series.
By the application of least square method, what are the estimated parameters a,b , g ? (5%)
5. Consider the data points which were generated by the following polynomial:
n
n y a ax ax ax =+ + ++ 2
01 2 
To find the parameters of the polynomial, least square method is applied. Please refer to the
given data “data5.mat”, and answer the following questions.
(a) Plot the root mean square elbow trend and find the number of fitting term n. (5%)
(b) Based on the number of fitting term n, please estimate the associated coefficients. (5%)
(c) Plot the fitting result and the given data in the same figure (with legend and labels). (5%)
6. Consider the given data “data6.mat”, where the point clouds are generated by the following
equation.
ax by cz dx y e + + + += 22 2 0
(a) To estimate the parameters a e ~ , please form the m samples of data into the form of
Y AX = . (5%)
(b) Based on (a), please find a e ~ as accurate as possible and draw the original point clouds
and fitting result in the same figure. (10%)