ECE 3100
Project 2
Aliasing
Su 2015
Purpose: To examine the consequence of digitizing a continuous signal and reinforce the idea of prefiltering before A/D conversion. When an analog signal is captured and converted from Analog to Digital
there can be unwanted effects which distort our signal of interest. In this project we will examine two
effects which result from Analog to Digital conversion.
Sample Rate:
In a digital system the clock takes in a sample of the sine wave at specified intervals. Any values in
between samples are lost. Holding the value constant until the next sample is taken is called a Sampleand-Hold and resulting waveform is shown below. This makes choosing the sample rate very important
with we want to accurately capture the signal.
The minimum sample rate which can resolve the oscillation of the sine wave is twice the largest
frequency we need to resolve. This is called the Nyquist sample rate. The image below shows a 120Hz
signal sampled at 240Hz. The digital system takes two samples in a single cycle of the sine wave. While
we have lost almost all the detail of the original signal we can still accurately gage the frequency since
there are two points in every cycle.
If we want to really capture the sinewave in detail the solution appears to be to maximize the sample
rate, but the file size of the digitized waveform can become unmanageably large. For example in single
channel CD quality audio each sample is 16-bits at 44.1 kHz. The file size for 1min is:
44.1𝑘𝑠𝑎𝑚𝑝𝑙𝑒𝑠 60𝑠𝑒𝑐
𝐹𝑖𝑙𝑒𝑆𝑖𝑧𝑒 = 2𝑏𝑦𝑡𝑒𝑠 ∙
∙
= 5.29𝑀𝑏𝑦𝑡𝑒𝑠
𝑠𝑒𝑐
1𝑚𝑖𝑛
To keep data sizes manageable we want a sample rate which adequately represents the frequencies of
interest. There are digital processing techniques to deal with over sampling, which are outside the scope
of this project.
Aliasing:
While we are trying to balance sampling rate and file size there is another problem to consider. When
the sample rate is smaller than the frequency of interest only portions of the high frequency are
captured. The digitized signal has a lower frequency then the original signal. The effect is called Aliasing.
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ECE 3100
Project 2
Aliasing
Su 2015
In the image above the high frequency signal is 2 kHz while the sampling frequency is 1.92 kHz. The
resulting digitized signal appears as a lower frequency signal. The method to determine the new
frequency is shown below.
𝜔𝑑 =
2𝜋 ∙ 𝑓𝑠𝑖𝑔 2𝜋 ∙ 2𝑘
=
= 2.083̅𝜋
𝑓𝑠𝑎𝑚𝑝
1.92𝑘
𝑑 = cos(𝜔𝑑 𝑛) = cos(2.083̅𝜋 ∙ 𝑛)
This is the normalized angular frequency (𝜔𝑑 ) and the new digitized signal is d. If 𝜔𝑑 is larger than 2𝜋,
which is the case here, the continuous signal has been aliased. We look for the nearest even value of 𝜋
to find the new frequency. In this case that is 2𝜋.
𝜔𝑛 = 0.083̅𝜋 = 2.083̅𝜋 − 2𝜋
We want only the difference between the 2𝜋 and 2.083̅𝜋. This is the normalized angular frequency
seen in the digitized signal. To translate back to frequency:
𝑓𝑑 =
𝑓𝑑 =
𝜔𝑛
∙𝑓
2𝜋 𝑠𝑎𝑚𝑝
0.083̅𝜋
∙ 1.92𝑘𝐻𝑧 = 82.3𝐻𝑧
2𝜋
Which is close to the frequency of the digitized sine wave in the picture above. The danger of Aliasing
should be obvious. Are the signals we digitize really the waveforms in the range of interest or are they
high frequency interference? Look at the picture below. The digitized waveform really a 200Hz signal
sampled at 1.92kHz or is it a 19 kHz aliased signal? Without seeing the original continuous wave it is
impossible to tell.
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ECE 3100
Project 2
Aliasing
𝜔𝑑 =
Su 2015
2𝜋 ∙ 19𝑘
= 19.7917𝜋
1.92𝑘
𝜔𝑛 = −0.20833𝜋 = 19.7917𝜋 − 20𝜋
𝑓𝑑 =
0.20833𝜋
∙ 1.92𝑘 = 200𝐻𝑧
2𝜋
The solution to minimizing aliasing is to low pass filter the continuous signal before we sample it. A Low
pass filter will reduce the contribution of his frequency signals while preserving the low frequencies.
Pre-filtering is considered essential before digitizing any signal.
Procedure: Sampling Rate
 Builder the circuit shown in the picture above. The Sample and Hold element is found in the
component library under [Special Functions] -> Sample. It requires a clock which is the square
wave seen in the bottom right (Vclk). We’re going to be cleaver and use variables to set the
clock frequency. The Sample Period (Ts) is set using the .param directive. This will make
changing the sampling period much easier. The input is a Behavioral Voltage Source (bv) and is
a function of the voltage label 1 on source V1.
 Examine the effect the sampling rate has on the digitized circuit by adjusting the rate and view
the effect on a 300Hz sine wave. Make a screen shot at 11.025 kHz and 600Hz. Explain what
happens at 300 Hz. Estimate the file size for 16-bit at the sampling rate for 0.6sec of data.
Sampling Freq
22.05 kHz
11.025 kHz
2.756 kHz
600 Hz
300 Hz
Estimated File Size
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ECE 3100
Project 2
Aliasing
Su 2015
Procedure: Aliasing
 Using the same circuit from previous procedure set the sampling frequency to 3 kHz.
 We will examine the effects of aliasing when the input signal is greater than the sampling rate.
Show printouts for 6.3 kHz and 8.5 kHz. Explain the results at 9 KHz.
Source Freq
Measure Freq
Calculated Freq
5.8 kHz
6.3 kHz
8.5 kHz
9.0 kHz
9.6 kHz
14.5 kHz
Procedure: Pre-Filtering
 Download the audio file data001.wav. The file is corrupted with high frequency noise which you
can hear if you play the file with your PC’s media player.
 We are going to use some of LTSpice’s more advanced capabilities. Build the circuit in the figure
below. The directives shown in the top right let LTSpice save our file as a .wav file. The input file
data001.wav has a sample rate of 44.1 kHz, but the file is going to be saved with a sampling rate
11.025kHz. This will cause the high frequencies to alias into our signal.




Design a 2nd order Sallen-Key pre-filter to reduce the high frequency signals in a separate
LTSpice file. Set the corner frequency to 3 kHz and let C1 = 15nF. Examine the bode plot and
make a printout.
Run the simulation for 0.6sec and set the voltage source V1 to wavefile=“datafile001.wav”.
Examine the pre-filtered signal against resampled output.
Examine the FFT pre-filtered signal against resampled output.
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ECE 3100
Project 2
Aliasing
Su 2015
Deliverables:
Make sure to have a cover page with your name, project title, and semester.
Procedure: Sampling Rate
 Screen shots at 600 Hz and 11.025 kHz.
 Explanation of the 300 Hz sampling rate.
 Table of sampling rates and file size.
Procedure: Aliasing
 Printouts for 6.3 kHz and 8.5 kHz.
 Explain the results at 9 KHz.
 Your table of measured and calculated aliased frequencies.
Procedure: Pre-Filtering
 Your designed pre-filter components and bode plot.
 Printout of your pre-filtered signal and resampled output.
 Printout of the FFT pre-filtered signal and resampled output.
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