lab1 spectrum analysis

8/6/2019 Lab1 Spectrum Analysis

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Table of Contents

1. Objectives ..........................................................................................................22. Theory ................................................................................................................23. Equipments ........................................................................................................44. Procedures ..........................................................................................................44.1.Part (A) ..............................................................................................................44.2.Part (B) ..............................................................................................................45. Data Sheet and Results.......................................................................................55.1.

Part (a) Results

...................................................................................................

5

5.2.Part (B) Results..................................................................................................76. Discussion Questions .......................................................................................117. Conclusion .......................................................................................................128. References ........................................................................................................12


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1. Objectives To become familiar with the operation of the spectrum analyzer and to study

some of its characteristics.

To study the effect of adding a low pass filter on the spectrum of an inputsignal.

To verify the special components of certain waveforms as Fourier theoremsuggests.

2. TheoryIn mathematics, a Fourier series decomposes any periodic function or periodic signal

into the sum of a (possibly infinite) set of simple oscillating functions, namely sines

and cosines (or complex exponentials). The study of Fourier series is a branch of

Fourier analysis. Fourier series were introduced for the purpose of solving the heat

equation in a metal plate.

The Fourier series has many applications in electrical engineering, vibration analysis,

acoustics, optics, signal processing, image processing, quantum mechanics,

econometrics, etc.

If (x) denotes a function of the real variable x and this function is usually taken to be

periodic, of period 2, which is to say that (x + 2) = (x), for all real numbers x. We

will attempt to write such a function as an infinite sum, or series of simpler 2

periodic functions. We will start by using an infinite sum of sine and cosine functions

on the interval [, ] such that

Since A Fourier series is an expansion of a periodic function in terms of an infinite

sum of sines and cosines. Fourier series make use of the orthogonality relationships of

the sine and cosine functions. The computation and study of Fourier series is known

as harmonic analysis and is extremely useful as a way to break up an arbitrary

periodic function into a set of simple terms that can be plugged in, solved

individually, and then recombined to obtain the solution to the original problem or an

approximation to it to whatever accuracy is desired or practical. An example of

successive approximation to common functions using Fourier series is illustrated inthe figures below.


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Figure (1): Partial Sums of the Fourier Series for the Square-Wave Function.

Talking about the low-pass filter, a low-pass filter is a filter that passes low-frequency

signals but attenuates (reduces the amplitude of) signals with frequencies higher than

the cutoff frequency. The actual amount of attenuation for each frequency varies from

filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in

audio applications. A low-pass filter is the opposite of a high-pass filter, and a band-

pass filter is a combination of a low-pass and a high-pass.

The concept of a low-pass filter exists in many different forms, including electroniccircuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data,

acoustic barriers, blurring of images, and so on. Low-pass filters play the same role in

signal processing that moving averages do in some other fields, such as finance; both

tools provide a smoother form of a signal which removes the short-term oscillations,

leaving only the long-term trend.


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3. Equipments TIMS modules

Audio oscillator. Twin pulse generator. Tunable low pass filter. 60 kHz low pass filter.

Oscilloscope. Spectrum analyzer.

4. Procedures4.1. Part (A)

Use the audio oscillator to obtain a sine wave with frequency of 10 kHz and 3Vp-p. Use the CRO to set the signal generator for the conditions above.

Using spectrum analyzer, plot the spectrum of the above signal. Repeat the above procedure for a square wave of 10 kHz and 3 Vp-p.

4.2. Part (B) Use the pulse generator to obtain a train of pulses with frequency of 10 kHz

and pulse width of 20 s.

Using the spectrum analyzer, plot the spectrum of the above signal and findthe first zero crossing of the spectrum.

Change from 20 s to 10 s, plot the spectrum and find the first zerocrossing of the spectrum.

Adjust the cutoff frequency of the tunable LPF to be 12 kHz and the gain to beone.

Use the signal shown in step (1) as an input to the LPF, Observe the spectrumof the output and sketch it.

Sketch the output waveform. Replace the tunable LPF by the 60 kHz LPF, sketch the output.


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5. Data Sheet and Results5.1. Part (a) Results

Figure (2): A Sine Wave in the Time Domain with 10 KHz Frequency.

Figure (3): The Spectrum of the Sine Wave with Frequency of 10 KHz.


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Figure (4): A Square Wave with Frequency of 10 KHz in the Time Domain.

Figure (5): The Spectrum of the Square Wave with Frequency of 10 KHz in the

Frequency Domain.

kHz

0 10 20 30 40 50 60 70 80 900.0

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3.0

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5.2. Part (B) Results

Figure (6): A Train of Pulses with Frequency of 10 KHz and Pulse Width of 20

s in the Time Domain.

Figure (7): The Spectrum of the Train of Pulses with a First Zero Crossing of the

Spectrum of 50 KHz.

s

-50 0 50 100 150 200 250 300 350 400 450

V

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0

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10

ch A: High pulse width(us) 20.00

ch A: Frequency(kHz) 9.87627Jun2010 15:22

kHz

0 20 40 60 80 100 120 140 160 1800.0

0.5

1.0

1.5

2.0

2.5

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Figure (8): A Train of Pulses with Frequency of 10 KHz and Pulse Width of 10

s in the Time Domain.

Figure (9): The Spectrum of the Train of Pulses with a First Zero Crossing of the

Spectrum of 100 KHz.

s

-50 0 50 100 150 200 250 300 350 400 450

V

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-8

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-4

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0

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ch A: High pulse width(us) 10.08ch A: Frequency(kHz) 9.876

27Jun2010 15:30

o

o=98.47kHz, A=0.01V

kHz

0 20 40 60 80 100 120 140 160 1800.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

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27Jun2010 15:32


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Figure (10): The Sine Wave in the Time Domain after Entering a Filter with a 12

KHz Cutoff Frequency.

Figure (11): The Spectrum of the Sine Wave with a 12 KHz Filter.

Comment We can notice that the pulses with frequency less than the cutoff

frequency of the filter which is 12 KHz here passed through the filter but

the frequencies above the cutoff frequency of the filter were eliminated.


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(a)

(b)

Figure (12), a and b: Spectrum of the Train of Pulses after Entering the 60 KHz

Low Pass Filter where all the Frequencies Less than 60 KHz Pass through It.


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6. Discussion Questions In part b, how does the result obtained in step 2 relate to the Fourier series

of pulse train in fig.3?

Answer:

The train of pulses shown in figure 3 in the manual has a high pulse width and

a low pulse width. Also, each train of pulses has a zero crossing frequency of

the spectrum where this zero is determined by the pulse width in which the

first zero crossing the spectrum equals:

First Zero Crossing =

What is the effect of changing the pulse width on the spectrum?Answer:

Since the first zero crossing of the spectrum is related to the pulse by the

relation:

First Zero Crossing =

Then, changing the pulse width affects the first zero crossing of the spectrum

in the frequency domain in such increasing the pulse width will decrease the

value of the frequency of the first crossing of the spectrum and vice versa. As

we can observe from the results above, when the pulse width was 20 s, the

first zero frequency crossing the spectrum was 1/ 20 = 50 KHz. But when the

pulse width was decreased to 10 s, the first zero crossing frequency increased

to 100 KHz which ensure the relation mentioned above.

Comment on the results of steps 5 and 6.Answer:

We can notice from step 5 and 6 the following:

After entering the sine wave in a low-pass filter with a cutofffrequency of 12 KHz, the pulses that have frequencies less than the

cutoff frequency of the filter will pass while the pulses that have

frequencies higher than the cutoff frequency of the filter will not pass,

therefore, it will be eliminated.


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The sine wave in the time domain after adding a low-pass filter will beshifted up by a specific DC gain component.

7. ConclusionFrom this experiment, we can conclude the following points:

Fourier theorem decomposes the periodic functions into sum of sinesand cosines that can be easier in treatment more than the function

itself.

In addition to analyzing the sounds of conventional musicalinstruments, Fourier series enable us to synthesize sounds.

The low pass filter is a device that allows the signals frequencies thatare less than its cutoff frequency to pass through it in which the

frequencies higher than the filter cutoff frequency will be eliminated.

For a train of pulses, the value of the first zero crossing frequency ofthe spectrum is proportional to the inverted value the pulse width.

8.

References

http://en.wikipedia.org/wiki/Low-pass_filter http://en.wikipedia.org/wiki/Fourier_series#Definition http://mathworld.wolfram.com/FourierSeries.html
http://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Fourier_series#Definitionhttp://en.wikipedia.org/wiki/Fourier_series#Definitionhttp://mathworld.wolfram.com/FourierSeries.htmlhttp://mathworld.wolfram.com/FourierSeries.htmlhttp://mathworld.wolfram.com/FourierSeries.htmlhttp://en.wikipedia.org/wiki/Fourier_series#Definitionhttp://en.wikipedia.org/wiki/Low-pass_filter

lab1 spectrum analysis

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