ECET 350 Lab 03 Problems 2015

| October 22, 2018

Generating and
Filtering Digital Signals in Matlab
Introduction to Moving Average Filters
Moving averages
are the simplest of all non-recursive or finite impulse response (FIR),
low-pass filters. The term moving means that the average is taken over a moving
or changing with time set of input samples. Also, in a moving average filter,
the filter coefficients are all the same, meaning they are the reciprocal of
the number of input terms to be averaged.
The difference
equation for a FIR filter is.png”>
For example, if
a moving average of length 5 is to be performed, the difference equation
summation may be expanded into the following form.
y[n]=0.2(x[n]+x[n?1]+x[n?2]+x[n?3]+x[n?4])
All the b k
coefficients are equal to the reciprocal of the length, which in this
case, is equal to one fifth, or 0.2.
As with all FIR
filters, this also introduces the need for storing not only the current input
sample but previous or delayed samples, as well. This is usually implemented in
a buffer referred to as a circular buffer. The circular designation means that
when a new sample input is available, the oldest sample in the buffer is always
overwritten by the newest one.
Again, if the
moving average filter length is 5, that means the average is taken of the
latest five samples of the input signal. To illustrate this, suppose we have
the following input sample sequence shown in Table 1.

Sample
Number

First

Second

Third

Fourth

Fifth

Sixth

Seventh

Eighth

Ninth

Sample
Value

3

-2

5

1

4

-3

-1

6

2

Table 1: Input Sample Sequence
Assuming that
all sampling and antialiasing filtering requirements have been met for the
system and assuming that the sampling frequency equals F s, a new
digital sample is provided as an input to the filter once every sampling
period, which is the reciprocal of F s, which is 1/F s=
T s, where T sis the sampling period. In addition, all
input samples prior to the start of the filtering process have been set to a
value of 0.
Table 2 shows
the moving average filter inputs and the calculated results after each new
sample is received. Please note that until the fifth sample is received, the
input signal buffer is in what is called a boundary condition, meaning that not
enough samples have been received to calculate an accurate average yet.

Sample
Number

Filter
Input

1

2

3

4

5

6

7

8

9

x[n]

3

3

3

3

3

-3

-3

-3

-3

x[n -1]

0

-2

-2

-2

-2

-2

-1

-1

-1

x[n -2]

0

0

5

5

5

5

5

6

6

x[n -3]

0

0

0

1

1

1

1

1

2

x[n -4]

0

0

0

0

4

4

4

4

4

Mov
Avg

0.6

0.2

1.2

1.4

2.2

1.0

1.2

1.4

1.6

Table 2: Moving Average Samples and Filter Calculations
When sample 1
is received, it is stored in the x[n]element of the input buffer while
the remaining elements still contain their initialized values of zero. When
sample 2 is received, it is stored in the x[n-1]element. When sample 3
is received, it is stored in the x[n-2]element, and so on until the
sixth sample is received. At this point, the buffer is completely filled with
the five latest samples when the sixth arrives; the oldest sample in the
buffer, which at this time is the x[n]sample, is overwritten with the
newest sample.
When sample 7
is received, it is used to overwrite the next oldest sample, which is the x[n-1]
element. This process continues with the newest sample overwriting the
oldest until the filter is turned off. The calculated moving average value for
each of the input sequences is shown in the last row of Table 2.
This simplified
version works because all the filter coefficients in a moving average filter
are the same. In most digital filters, the coefficients are not identical and a
more sophisticated scheme is required to match the input sample sequence with
the proper filter coefficient, but that will be covered in a later lecture and
lab.

Generating a discrete or sampled sine
wave in Matlab
a. To generate a sine wave in Matlab,
sampling frequency (or rate), the number of samples, the frequency, amplitude
and phase of the sine wave must be defined.

fs=8000; %sampling rate
Ts=1/fs; %sampling
period
n=0:1:249; %indices
(250 samples will be generated)
frequency=500; %500 Hz
frequency
A=sqrt(2); %Amplitude

x=A*sin(2*pi*500*n*Ts) %Sine wave

stem(n,x); %Plot
xlabel(‘Number of
samples’);ylabel(‘Sample value’);grid;

b. Add noise to the sine wave
v=sqrt(0.1)*randn(1,250); % generate Gaussian random noise
x_noise=x+v; %Add noise
to sine wave
stem(n,x_noise); %Plot
xlabel(‘Number of samples’);ylabel(‘Sample value’);grid;

Compare the clean and noisy versions of the
signal and record your observations. What is the frequency original (clean)
sinusoidal signal?
c. Let’s
look at the spectra (frequency content) of the clean signal.

N=length(x);
f=[0:N/2]*fs/N;
Axk=2*abs(fft(x))/N;
Axk(1)=Axk(1)/2; % calculate single side spectrum
for x(n)
plot(f,Axk(1:N/2+1));
xlabel(‘Frequency
(Hz)’);
ylabel(‘Amplitude |X(f)| ‘);grid;

Using the above code snippet, can you plot the frequency
spectrum of the noisy signal?

What are the differences between the two sets
of spectra? What is the apparent frequency of noise?
d. Now we will use a moving average filter to remove
the noise using the filter function from Matlab.

a=1;

win_size=4; %Window size=4

b=(1/win_size)*ones(1,win_size);

y=filter(b,a,x_noise);

Plot the output signal y using the code snippets from
part a. How does the filtered signal compare to the noisy signal? Label your
graph.

e. Calculate the root mean square error
(rmse):
The filtered signal (y) should be close to the actual
signal (x). Root mean square error allows us to measure how far the error
between the two signals is from zero. Visit.stanford.edu/~susan/courses/s60/split/node60.html”>http://statweb.stanford.edu/~susan/courses/s60/split/node60.html for more details.

Use the following command to find the
rmse=
sqrt(sum((x-y).^2)/length(y))

Repeat the filter function with window sizes of 3, 5 and
7 and calculate the rmse values.

Window size

RMSE

3

4

5

7

Which window size gives you the least rmse?

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