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# Laboratory

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```									Laboratory
Spectra of Common Signals, Sampling and Aliasing
The magnitude spectrum of a signal is a picture that describes the frequency
content of the signal. At each frequency, the magnitude represents the “strength” of the
signal at that frequency. The Fast Fourier Transform (FFT) is a common way of
producing a magnitude spectrum for a signal. An FFT is an estimate of a signal’s
spectrum that is computed using samples of the signal. The accuracy of the estimate
depends on sampling rate used and the windowing method used to select the time interval
of interest.

The figure below shows spectra for a 1 kHz sine wave. The top picture depicts
the ideal spectrum for the sine wave. The second picture shows the FFT estimate of the
spectrum when a rectangular window is used. The third picture shows the FFT estimate
of the spectrum when a Hanning window is used. Note that the rectangular window gives
a narrower main lobe but a lot more spectral leakage (in the form of side lobes) than the
Hanning window does.

Spectra of Common Signals, Sampling and Aliasing                                        1
A.      Finding a Spectrum
Connect a 3.5 Vpp, 1 kHz sine wave from the signal generator to Ch1 of the scope.
Verify the frequency on the scope.

Press the  key. Under Function 2, choose On. Choose the Function 2 menu. Choose
Ch1 as the source. Select the operation FFT. Use the FFT menu and the Entry knob to
adjust the settings to match those shown below. To set effective sampling time, use the
Time/Div knob and press  to see the current setting.

    Use Cursors and Find Peaks to measure the fundamental frequency of the sinusoid.

    Measure the width of the main lobe where the magnitude is –31 dB.

Change to a Rectangular window, but leave the other settings unchanged.

    Measure the fundamental frequency of the sinusoid.

    Measure the width of the main lobe where the magnitude is –13 dB.

Spectra of Common Signals, Sampling and Aliasing                                          2
B.      Effect of Sampling Rate
Reduce the effective sampling rate to 50 ksamples/sec. Adjust the scope settings as
shown:

    Measure the width of the main lobe at this lower sampling rate, where the magnitude
is –31 dB. The width should be narrower than before.

Change to the Rectangular window, but leave other settings unchanged.

    Measure the width of the main lobe where the magnitude is –13 dB.

Experiment with effective sampling rate until you have found best possible display of the
sinusoidal peak. This should be the sampling rate that gives the narrowest possible main
lobe width (in Hz).

    What effective sampling rate gives the narrowest sinusoidal peak for the 1 kHz sine
wave?

    Which window results in more spectral leakage, Hanning or Rectangular?

Spectra of Common Signals, Sampling and Aliasing                                           3
C.      Spectra of Common Signals
For each spectrum, use 1 kHz, 3.5 Vpp, Hanning window, effective sampling rate 50
ksamples/sec and FFT autoscale:

    Print out the spectrum for a sine wave.

    Print out the spectrum for a triangular wave.

    Print out the spectrum for a square wave.

Use Cursors and Magnitude Difference from First Peak
1
Find Peaks to                            0
(dB)
3 the
measure                             -9.542
5
magnitude                         -13.979
7                            -16.902
differences
9                            -19.085
between the
11                            -20.828
for
peaks13 the                         -22.279
square wave.
 Add a column marked “Frequency” to your table. Use cursors to measure the
List them in a
table for each
frequency location (column spike.
 Predict what the frequency of the 15th harmonic should be. Also predict what the
frequency of the 2nd harmonic would be (if it were present in the signal).
Harmonic
 Which spectrumNumber,   (sine, square, triangular) contains the largest number of harmonics?
Magnitude
D.       Aliasing Difference from
First Peak).
Compare them wave to Ch1 of the scope. Set up the FFT controls to
Connect a 10 kHz, 3.5 Vpp sine
to on the next page.
match the diagramthe theoretical
values listed in
the table below.
On the signal generator, progressively increase the frequency to about 24 kHz, allowing
The peak for the way. You should see the peak lobe of the FFT move
the FFT display to stabilize on the
first harmonic is
to the right as the sinusoidal frequency is increased. At an effective sampling rate of 50
assumed to
ksamples/sec, aliasing should start to occur at 25 kHz. Gradually increase the frequency
from 24 kHz. Notice how aliasing causes the peak to “double back” for frequencies
provide a
beyond 25 kHz. reference. All
Increase the sinusoid frequency to 40 kHz.
other peaks are
 Use Cursors comparedPeaks to measure the peak frequency displayed by the FFT. As
and Find to it.
Note that squareshould erroneously display a peak at about 10 kHz.
a result of aliasing, the FFT
waves do not
have any
Increase the effective sampling rate to 100 ksamples/sec.
frequency
 Measure the peak frequency.
components at Because this sampling rate is more than twice the input
even numbered
frequency, the FFT should be accurate.
harmonics.Harm
onic

Spectra of Common Signals, Sampling and Aliasing                                             4
   For a given sine wave input, how should you choose your effective sampling rate?

Spectra of Common Signals, Sampling and Aliasing                                       5

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