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PHYS 3152 Frequency Analysis of Electronic Signals

创建日期：2024-05-25 15:52:06

所属分类：物理Physical

标签： PHYS 3152

Frequency Analysis of Electronic Signals

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PHYS 3152 Methods of Experimental Physics I
E5. Frequency Analysis of Electronic Signals
Purpose
This experiment will introduce you to the basic principles of frequency domain analysis of
electronic signals. In particular, you will study the Fast Fourier Transform (FFT), which is a
convenient and powerful tool for performing frequency domain analysis on a variety of
signals.
Equipment and components
Agilent DSO-X 2002A Digital Storage Oscilloscope, signal source (~1.1kHz) and 10kΩ
resistor.
Background
Normally, when a signal is measured with an oscilloscope, it is viewed in the time domain.
That is, the vertical axis is voltage and the horizontal axis is time. For many signals, this is
the most logical and intuitive way to view them. However, when the frequency content of the
signal is of interest, it makes sense to view the signal in the frequency domain. In this case,
the horizontal axis becomes frequency. Using Fourier theory, one can mathematically relate
the time domain with the frequency domain. The Fourier transform is given by:
-
2
+
( ) ( ) i ftV f v t e dtπ
∞
−
∞
= ∫ (1)
where v(t) is a voltage signal represented in the time domain and V(f) is its Fourier transform
in the frequency domain. Some typical signals represented in the time domain and the
frequency domain are shown in the figure below.
Frequency Analysis of Electronic Signals
Revised: 10 January 2023 E5-2/16
The discrete (or digitized) version of the Fourier transform is called the Discrete Fourier
Transform (DFT). This transform takes digitized time domain data and computes the
frequency domain representation. In this case, Equation 1 can be written as:

1
2 ( )
0
0
( ) v(t )
N
i f j
j
V f j e π ττ τ
−
−
=
= +∑ (2)
where t0 is the starting time and τ is the smallest sampling time. The Fourier transform
defined in Equation 1 assumes the life of a signal from -∞ to +∞. In real experiment,
however, one can only obtain a signal with a finite lifetime. In this case, we can define VT (f)
which is the Fourier transform of the signal in a time period T. Another function which is
widely used in the experiment is the frequency power spectrum defined as
21( ) lim | ( ) |TTP f V fT→∞
= (3)
The power spectrum P(f) is a real function and has many important applications in electrical
engineering and physics.
The Agilent DSO-X 2002A Digital Storage Oscilloscope uses a particular algorithm, called
the Fast Fourier Transform (FFT), for computing the DFT. The FFT function in the Agilent
DSO-X 2002A can acquire up to 65,536 data points. When the frequency span is at
maximum, all points are displayed. This display extends in frequency from 0 to fs/2, where
fs is the sampling rate of the time record. Because the Agilent DSO-X 2002A uses the
interleaved mode* when one channel is in use, the frequency resolution fres is given by:
fres = (fs/2) / N where N is the number of points acquired for the FFT record, that can be up to
65,536. In this interleaved mode, the time between two adjacent samples τ is the reciprocal
of fs/2. The sampling rate fs can be set by adjusting the Horizontal scale knob of the scope. As
shown in Equation 2, the unit of the Fourier transform V(f) is Volt⋅Second. Because the
sampling time τ in Equation 2 is a simple constant, it is often normalized to unity in the
DFT. In this case, the unit of V(f) becomes the same as that of the signal v(t). The vertical
axis of the FFT display in the Agilent DSO-X 2002A scope is logarithmic, displayed in dBV
(decibels relative to 1 Volt RMS): dBV = 20 log (V(f)/1Volt RMS). Thus a 1 Volt RMS
sinusoidal wave (2.8 Volts peak-to-peak) will read 0 dBV on the FFT display.
* Please read Appendix 1 Introduction to Fourier Theory for details.
Procedure
The following four laboratory exercises are designed to illustrate the relationship between
frequency resolution, effective signal sampling rate, spectral leakage, windowing, and
aliasing in the FFT (Details about these concepts will be discussed separately in the lecture
class).
Exercise 1 This exercise illustrates the relationship between the sampling rate and the
resulting frequency resolution for the spectral analysis using the FFT. The spectral leakage
properties of the Rectangular and Hanning windows are also demonstrated.
A waveform generator is built into the DSO-X 2002A oscilloscope. Press the [Wave Gen]
key to access the Waveform Generator Menu and enable the waveform generator output on
the front panel Gen Out BNC connector. When waveform generator output is enabled, the
[Wave Gen] key is illuminated. From the Waveform Generator Menu, select a 3.5V (peak-to-
peak), 1 kHz sinusoidal signal and input it to channel 1 of the scope. Use [Auto Scale] to
check the time-domain waveform. Next, press [Math] key to display the Math Waveform
Menu, set Function as f(t), select FFT as Operator and channel 1 as Source. Manually adjust
the settings of the horizontal and vertical scales, Center Frequency, Frequency Span the
oscilloscope so that you obtain a nice peak in the FFT display. (See the Appendix 2 for
assistance in adjusting the FFT Menu settings.)
NOTE: Turn off the time-domain waveform display when observing the signal in frequency-
domain. This will improve the refresh rate.
Frequency Analysis of Electronic Signals
Revised: 10 January 2023 E5-3/16
Use the Horizontal scale knob to set the sampling rate to 50 kSa/s. Use [Cursors] to measure
the fundamental frequency of the peak. Adjust the FFT settings and use [Cursors] to measure
the main lobe width (full width at half maximum) for the Hanning window and the
Rectangular window, respectively. Notice the difference in the spectral leakage for the two
windows. Save and print out the FFT spectra under the two different windows. Repeat the
above measurements using a sampling rate of 10 kSa/s.

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