to ECE645
Introduction Digital Signal
Processing
Project Report
Submitted By
Damanjit Singh
Nirupama Zambre
Objective: To change the sampling frequency of a given digital signal
Steps:
1. We use a music file sampled at 44.1 KHz in .wav format. Matlab has functions like
wavread to read the .wav file and convert it to data file for further processing and
wavwrite to write the processed output to .wav file.
2. Up sampling- The operation of up sampling by a factor L N describes insertion
of L-1 zeros between every sample of the input signal. i.e. padding data with intermediate
zero valued samples.
3. Low pass filtering- We are using IIR filters for low pass filtering. We choose an
Elliptic filter (LP IIR). Forward and backward processing is done to eliminate the effects
of non-linear phase characteristic of the filter, at each stage of low pass filtering.
4. Down Sampling- The operation of down- sampling by a factor of D N describes the
process of keeping every D th sample and discarding the rest i.e. retaining only one out of
the N samples of the filtered output.
For our project, we want to convert sampling frequency of 44.1 KHz to 48 KHz i.e. we
need to re-sample our signal by the following ratio:
48
480 10 8 6
=
=
44.1 441 9 7 7
Here, the first step would be up- sampling by L=10, then low pass filtering followed by
down- sampling by D=9. Then repeat this for L=8 and D=7, and then with L=6 and D=7.
Description and Details:
We read 100*1024 (100k) samples of music file Chellame S1.wav uploaded on the
website using wavread and store data in orgData, the sampling frequency in Fs and
the number of bits in bits. Full size of the given file could not be taken because its size
was too large to fit into MATLABs memory.
The above read samples are written to input.wav file using wavwrite for comparison
purposes.
We now plot FFT of the sample music file (input.wav or orgData) to observe the
spectrum of the given signal (data):
1
FFT of original/input data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
Frequency -->
8
10
12
4
x 10
In the FFT of the sample music file, we observe two colors blue and green. This is
because of the presence of two channels. The sample music file gets stored in orgData in
form of two columns, one for each channel. The sampling frequency is Fs = 44.1 KHz
per channel.
In up- sampling the spectrum gets compressed by a factor of L. We plot the FFT after
every up- sampling to see if the processing is proceeding as desired.
Up- sampling for L=10, we should be able to see 10 peaks.
2
FFT after upsampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
Frequency -->
8
10
12
5
x 10
Design of Elliptic filter (for low pass filtering) - We use fifth order Elliptic filter (N=5),
passband ripple Ap=1.25 dB, the stop band attenuation for As=40.27, =1 and the filter
ratio (ratio of stop band and cut off frequency) s / c =1.2062. We find the coefficients
for normalized 5 th order elliptic filter using the table given in the class notes. That is, we
take the following values
0 = 0.35693; 1 = 0.20145; 2 = 0.04529;
1 = 0.73787; 2 = 0.9946; 3 = 1.2402; 4 = 1.7368;
And substitute them in the following given equation of fifth order elliptic filter:
T ( s) = K
( s 2 + 32 )( s 2 + 42 )
( s + 0 )( s + 1 j1 )( s + 2 j 2 )
We find the value of K by using the fact that the dc response of an odd order Elliptic
filter is unity and thus obtain the analog normalized low pass filter.
We then convert it to a filter digital using the following transformation from analog
normalized low pass filter.
3
That is, we perform pre-warping and bilinear transformation in a single step.
Then forward and backward processing is performed to eliminate the effects of non-linear
phase characteristic of the filter, at each stage of low pass filtering. That is, a symmetric
low pass filter is used with the same magnitude characteristic but phase characteristic
negative to that of the original in series with the original filter. Thus, converting it to a
zero phase filter. Note that as one of these filters (forward or backward processing filters)
will be non-causal, this processing is possible only on the stored data and not the data
which is coming in real time.
After the low pass filtering, we plot the FFT to verify the design of our elliptical low pass
filter:
FFT after low pass filtering of data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
Frequency -->
8
10
12
5
x 10
4
The last step is down- sampling. Here the spectrum expands by a factor of D. In order to
avoid aliasing, all the frequencies should be within 1/D of cut-off frequency Fc,
otherwise the spectrums will overlap. We plot the FFT after every down- sampling to see
if the processing is proceeding as desired.
FFT after down sampling D=9,
FFT after down sampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
Frequency -->
8
10
12
4
x 10
We repeat the above process of up-sampling, low pass filtering and down sampling two
more times as mentioned in the beginning (L=8 & D=7, L=6 & D=7). The FFTs obtained
during these two repetitions are given below:
Up- sampling for L=8, we should be able to see 8 peaks.
5
FFT after upsampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
1
2
3
4
5
6
Frequency -->
7
8
9
10
5
x 10
FFT after low pass filtering
FFT after low pass filtering of data
350
300
Magnitude -->
250
200
150
100
50
0
0
1
2
3
4
5
6
Frequency -->
7
8
9
10
5
x 10
6
FFT after down sampling D=7,
FFT after down sampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
8
Frequency -->
10
12
14
4
x 10
Up- sampling for L=6, we should be able to see 6 peaks.
FFT after upsampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
1
2
3
4
5
Frequency -->
6
7
8
5
x 10
7
FFT after low pass filtering
FFT after low pass filtering of data
350
300
Magnitude -->
250
200
150
100
50
0
0
1
2
3
4
5
Frequency -->
6
7
8
5
x 10
FFT after down sampling D=7,
FFT after down sampling of data
350
300
Magnitude -->
250
200
150
100
50
0
0
2
4
6
Frequency -->
8
10
12
4
x 10
8
This is the FFT of the processed signal. We write the processed signal using wavwrite
to an output file called output.wav.
Observations and Conclusions:
We can now compare the size of files input.wav and output.wav. We find that the size
of the output file has increased by the re-sampling ratio of (48 kHz/44.1 kHz). This is a
check to ensure that the sampling frequency has been converted from 44.1 KHz to 48
KHz.
On changing the sampling frequency from 44.1 KHz to 48 KHz, we listen to the Original
data sample (at 44.1 KHz) and the processed data sample (at 48 KHz), we find that both
of the music samples play the same. Hence, music on CDs (digitized at 44.1 KHz) and
same music on DVD (digitized at 48 KHz) use different sampling frequencies to
represent the signal.
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