CMPE2322 Signals and Systems Page 1 of 2
Exam 2 Project Due Tuesday April 7th in class (no late submission) Spring 2009
In this project you are going to do a comparative design of a low pass filter. The low pass
filter will filter out or attenuate signals with frequencies higher than your designated
cutoff frequency. In the design effort you will compare a total of seven filters. The filters
are a running average low pass filter, three truncated sinc filters with different window
functions, the FIR filters generated by the matlab functions fir1 and fir2, and a IIR filter
generated by the matlab function ellip. You will be asked to evaluate certain
characteristics of the filter based on your simulation results.
The sampling frequency of the system is 40 kHz. The cutoff frequencies for your low
pass filters are as specified. The order M of all your FIR filters is assigned and will be the
same except for the running average filter. The IIR filter order is also assigned but
remember that the order of the IIR filter is the order N of the feedback coefficients.
The Mth order running average filter impulse response function h[n] has M+1 ones and is
expressed as:
1....11
1
1
M
nh
The frequency response
2/
ˆ
exp
2/
ˆ
sin)1(
2/1
ˆ
sin
ˆ
exp Mj
M
M
jH
is linear in angle
and has magnitude
2/
ˆ
sin)1(
2/1
ˆ
sin
M
M
given by the Dirichlet function (diric in matlab)
which is equal to the sinc
2/
ˆ
)1(
2/1
ˆ
sin
M
M
at small ^ where
2/1
ˆ
2/1
ˆ
sin MM
but is larger as ^ approaches the maximum possible
^ without aliasing. For ^ less than the numerator determines the zero crossings of
the magnitude function and they are the same as the sinc function. The magnitude
function is equal to 1 at ^ = 0 and decreases to the
2
at the normalized cutoff just
before the first zero crossing
1
2
ˆ
M
. This indicates for the running average FIR filter
being used as a low pass filter M itself determines the desired cutoff frequency and the
highest cutoff frequency possible is ^= or 1/2 the sampling frequency. Assuming the
cutoff frequency is approximately one half the zero point crossing
1
2
2
1
ˆ
M
c
can be
used to find M for your required c. M should be rounded up to the largest integer to
make sure the cutoff frequency is lower than your specified cutoff frequency. For your
order M find hn the impulse response. For FIR filters the hn are also the bk values and for
the running average filter is a vector with M+1 values of 1/(M+1) Do a frequency plot
freqz(b,1,wh) where wh = -pi:pi/300:pi; defines ^ from -pi to +pi. If M is odd such that
M+1 is even to get a good graph on the graph click Edit, Axis Properties. In the Axis
Properties Editor pop up select Y axis and change the limit number on the left to -10dB
CMPE2322 Signals and Systems Page 2 of 2
