Description
In this lab, we will consider different types of digital filters and look at their characterization
in time and frequency. This will give you some insight into how digital filters are
implemented and designed and into the properties of different digital filter design algorithms.
1. Filter Implementation and Characterization
Digital filters are usually implemented using a difference equation
y[n]=(1/a0)[b0x[n] + b1x[n-1] + … + bMx[n-M] – a1y[n-1] … – aNy[n-N]]
assuming a causal filter. In MATLAB, this is implemented with the filter command:
y=filter(b,a,x)
where vector a=[ a0 … aN ], b=[ b0 … bM ], and x and y are finite-length input and output
sequences.
You can also implement a filter using the MATLAB convolution function
y=conv(x,h,’same’)
given an impulse response, which you can find by using the filter command with a unit
impulse function, e.g.
impulse = [1 zeros(1,49)];
h=filter(b,a,impulse);
will give you the impulse response up to length 50. The ‘same’ flag will give you an output
that is the same length as x, which will be comparable to what you get with the filter
command. (Without that flag you will get the result with length corresponding to the sum of
the lengths of x and h, which pads x with zeroes at the end.)
Both filter and conv assume zero values for the x signal before its start time.
Like continuous LTI systems, a digital filter can be characterized in the time domain and the
frequency domain. The MATLAB function
freqz(b,a)
will give you log magnitude and unwrapped phase plots for the filter frequency response over
positive normalized frequency [0,0.5]. (Recall that normalized frequency is periodic with
period 1, and [0,0.5] in normalized frequency corresponds to [0, π ] in radians for the DTFT.)
Look at the freqz documentation to find out how you can get values to plot over radians or
Hz.
In order to understand the behavior of different filters in different domains, we will
experiment with a moving average filter and a first difference filter. Download the text file
containing Microsoft stock price over 4 years from the class web site. Load the data using
load command, and plot it. Note that the quick changes correspond to the high frequency
component. People tend to invest based on the long-term trend, which we can find using a
moving average filter. Consider the 30-day moving average filter:
y[n]=1/30( x[n] + x[n-1] + … x[n-29] )
Implement this filter in MATLAB using both the conv and filter functions to confirm that you
get the same answer when applied on the stock data. Note that the big jump at the onset of the
filtered signal is associated with zero-padding – start your plot at sample n=30 to avoid this
artifact. Plot the original and filtered signal together, with the filtered signal plotted in red.
Does the filtered result look smoother?
A simple filter for pulling out the high frequency trends is the first difference:
y[n]=x[n]-x[n-1]
Find the a and b coefficients for this filter, use them to filter the stock signal, and plot the
result.
Plot the frequency response for both filters over the same time range and verify that the
moving average system is a low-pass filter and the first difference system is a high-pass filter.
Note the difference in scale of the two different filtered time signals.
IN-LAB CHECK-OFF: Show your TA your plot of the original and filtered stock data, the
plot of the first-difference filtered signal, and the plot of the log magnitude frequency
responses for each of the two systems. Be sure to label the axes of the plots.
2. Filter Design
MATLAB has numerous built-in functions for generating discrete time filters. In this problem
we are going to use two to look at how FIR vs. IIR systems behave in the frequency domain.
You will not be expected to understand the details of how these algorithms work; you only
need to evaluate their behavior.
Both the filter design functions in the problems below ask you to specify cut-offs in terms
of 0 < W < 1, where W=1 corresponds to half the sampling frequency. So, a cut-off frequency
of 0 < W < 1 corresponds to 0 < v < p in radians for the DTFT, and 0<f<0.5 in normalized
frequency. Thus, a cut-off frequency of W=0.5 corresponds to vc = p/2 in radians and fc=0.25
in normalized.
i. FIR (Finite Impulse Response) Digital Filters
Use the MATLAB function fir1 to create a low pass FIR filter of order 10 with
cutoff frequency of 0.3p.
ii. IIR (Infinite Impulse Response) Digital Filters
Use the MATLAB function butter to create a low pass Butterworth filter of order
10 with cut-off frequency 0.3p.
Plot and compare the impulse and frequency responses for the two alternatives. Comment on
the differences in the written report.
Use the MATLAB function filter to implement the two low pass filters you produced in
2a and 2b. Apply the filters to:
• the stock market data; and
• the music data on the class web site.
IN-LAB CHECK-OFF: Play the filtered music signals for your TA. How do the filtered
signals sound compared to the original? Can you hear a difference between the results of the
two different filters?
3. Music Equalizer
In this part, you will design a music equalizer, which breaks the sound file into multiple
frequency bands by filtering, weighting, and summing to reconstruct the signal, as shown in
the figure below for a simple 3-band equalizer. Audio mixing boards do this, though with
many more than the three channels we have here. The term “equalizer” comes from the fact
that people often want to boost the gain of low energy component sounds or reduce the gain
of loud components.
W
For the case where we have 3 filters (3 difference equations), there is one LPF, one BPF, and
one HPF. Together, they cover the full frequency range, as shown below for a signal with a
44k sampling frequency. If there are more channels, the filters become narrower and you
need additional BPFs.
In this lab, you will build a 4-band equalizer:
i. Design four 4th-order filters (an LPF, an HPF and two BPFs) with cut-offs at p/5,
2p/5, and 2p/3. (The high frequency bands are bigger since our perception of
frequency is not on a linear scale.)
ii. Write a function that takes as input gain terms (G1, G2, G3, G4), applies the filters,
multiplies the filter outputs by the gain terms, and sums the results.
Since the way our ears hear a change in amplitude is more like a logarithmic scale than a
linear scale, it is useful to refer to gain in terms of Decibels (dB). Overall dB gain for input
and output sequences a filter is commonly defined as:
.
A gain of ½ corresponds to -6 dB in magnitude, a gain of 2 is an increase of 6dB in
magnitude, and a gain of 1 is a change of 0 dB in magnitude. Normally, an equalizer can drop
or boost the gain in dB from -20 to 20 dB in any one band. What range of values does this
require for the gain terms?
Download the music.wav file from the class web site. The music file is standard CD quality,
sampled at 44.1 KHz with a 16-bit word size for each channel. Test your equalizer by
comparing the output when G1=G2=G3=G4=1 to the original. The sound quality should still
be good.
h1[n]
h2[n]
h3[n]
+ x[n] y[n]
x
G1
x
G2
x
G3
0 0.5 1 1.5 2 2.5
x 104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
2
10 10 2
[ ]
10 log 10log [ ] [ ]
n
n
n
y n
h n
x n =
å
å å
Using either the music.wav file or a 10-20s snippet of any music of your choice, create a
modified version that boosts the frequencies in the vocal range (4-8kHz) by an audible
amount. You may achieve this by boosting some bands and attenuating others.
IN-LAB CHECK-OFF: Play the original and modified sound for your TA. Let your TA
try running your code with a different set of gains of his or her choice with the music.wav
file. Which frequency band was boosted most?
Summary of Assignments and Deadlines
• In-lab group demonstrations (due in lab during the ninth week of class)
o Part 1: Show your TA your plot of the original and filtered stock data, the
plot of the first-difference filtered signal, and the plot of the log magnitude
frequency responses for each of the two systems.
o Part 2: Play the filtered music signals for your TA. How do the filtered
signals sound compared to the original? Can you hear a difference between
the results of the two different filters?
o Part 3: Demonstrate your music equalizer.
• Written report
o Part 2: Turn in your plots of the filter responses and time-domain plots of the
filtered stock market signal comparing the different filters. Comment on the
differences in the frequency response of the two filters (magnitude and phase)
and how this impacts the outputs. In commenting on frequency responses,
consider how close a filter matches an ideal low pass filter.
o Part 3: Describe the filters you implemented for the music equalizer (give
filter coefficients and include frequency response plots) and the mapping of
human specified gains to the gain factor in the implementation. Describe the
gains used to create your file with the boosted vocals.
• Files to be turned in via Canvas (due the day before your lab in the last week of class)
o Written report
o Electronic versions of the original and modified music from part 3
o A zip file with all MATLAB code that you used in this lab