MAE 315-Lab 2 DSP Writeup

MAE 315 Lab 2:

Digital Signal Processing

Alexander SpiridakisProfessor Glauser

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TABLE OF CONTENTS

Abstract p.3

Introduction

o Digital Signal Processing………………………………………………………………p. 4-9

o Analog/Digital systems and conversion p. 4-6

o Data acquisition p. 6-8

o Resolution p. 8

o Nyquist frequency p. 9

o Signal Conditioning……………………………………………………………………p. 10-13

o Filters p. 10-12

o Amplifiers p. 12-13

o Errors…………………………………………………………………………………………p. 14-17

o Aliasing p. 14-15

o Quantization p. 15-16

o Clipping p. 16-17

o Fourier Analysis………………………………………………………………………..p. 18-28

o Background p. 18-20

o Calculations p. 21-28

Procedure p.29-32

Results and Discussion

o Aliasing and Filtering……………………………………………………………….p. 32-34

o Fourier Analysis: Simulation and Experiment …………………………..p.35-39

o Reconstruction Graphs: Simulation v. Experimental…………………..p.41-42

o Quantization…………………………………………………………………………….p.43-44

o Clipping……………………………………………………………………………………p.45-46

o Noise and Filtering……………………………………………………………………p.47-48

Conclusion p. 49-50

Appendix

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ABSTRACT

Digital signal processing (DSP) and Fourier analysis were used to develop a basic

understanding of how signals are manipulated and analyzed. In this lab, analog to digital

converters (A/D) were used to digitize signals. Through the data acquisition system used,

these digitized signals were fed into the A/D converter and then displayed on a computer.

Using these tools to visually display the signals allowed for analysis and computation of

Fourier coefficients. The lab had two main sections: a simulation and an experimental

section. During the simulation, the signals were displayed and manipulated theoretically.

The experimental section was then conducted, using the simulation results as the ideal

results that should have been obtained. In both experimental and simulation sections,

sine, square, and triangle waves were manipulated by using filters and intentionally

implementing errors, such as aliasing, clipping, and quantization errors.

The parameters, such as fundamental frequency, sampling rate, and bipolar

voltage were changed for the signals from section to section to examine when and how

errors occurred. It was determined that aliasing can be eliminated through the use of low-

pass filters or by increasing the sampling frequency to a value of over twice the signal’s

fundamental frequency. It also became evident that clipping occurred when an incorrect

voltage range was used to record and display the signal. Quantization error related most

closely to the resolution or number of bits used to process a signal. The fewer the bits

processed in the A/D conversion, the more “noise” appeared due to quantization error.

Lastly, it was demonstrated that Fourier analysis could be used to accurately reconstruct

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signals (periodic functions). The Fourier analysis and plots showed that the accuracy of

the reconstructions positively correlated to the number of harmonics used in the analysis.

INTRODUCTION

1. Digital Signal Processing

Background

Digital signal processing is the action of taking one or more input signals and

digitizing them. Input signals can be anything from vocal, audio, or visual samples to

physical properties such as temperature or pressure. Digitizing a signal creates a visual

representation of the inputs that can be mathematically analyzed and applied to real world

situations. Both analog and digital signals can be processed and analyzed accordingly.

Analog systems produce or process continuous signals, while digital systems produce or

directly process signals that vary in discrete steps.1

Figure 11 below shows a visual representation of an analog signal to the left and an example of a digital signal to the right.

DSP and Analog to Digital Conversion

Analog to digital converters (A/D converters) are used to convert continuous,

analog signals into digital, discrete signals. Signals in the real world, such as light, sound,

1 http://lcs3.syr.edu/faculty/glauser/MAE315/index.html

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and most others, are analog.2 In order to analyze them via computer or transfer signals

over wire, they must first be converted to digital. Telephones use A/D converters to relay

audio since the user’s voice is analog and the information must be transferred digitally.

Scanners take analog information given by a picture (light) and convert it into digital to

be displayed on a computer screen.2 Likewise, digital to analog converters (D/A

converters) use the opposite process to generate an analog signal from digital data. Some

examples of various A/D converters and their applications are listed below in Table 1.

2 http://www.hardwaresecrets.com/article/317

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Table 11: A/D converters and there given applications.

One of the most useful advantages in converting analog to digital is the reduction

of noise. Since analog signals can be assumed to have any value, there is no way to

distinctly separate the analog wave from noise without using a filter. Since digital signals

can only maintain two values, zero and one, any other values get discarded.2 For

example, this is why noise is heard on vinyl records and not on CDs. The needle on the

record player reads analog signals and cannot differentiate between the sound coming

directly from the record and the dust lying on the record, so noise from dust is played

through speakers. CDs, on the other hand, don’t experience this because the noise has

already been filtered out in the A/D conversion.

A/D conversion essentially happens by using precise mathematical

approximations, such as integrals and summations. A/D conversion provides a series of

numbers, each of which corresponds to the weighted integral of the analog signal over

some time period, s, called the aperture, which is less than or equal to the sampling

period .1 Equation 1 below demonstrates the relation between an analog signal and its

corresponding digitally sampled signal.

Equation 11: Where u(t) refers to the original signal, un is the digital sample, is

quantization noise, and t is time.

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The reconstruction of the wave in the end depends on how the fundamental period

of a wave, s, relates to the sampling period, .

Figure 21: When period s is less than or equal to sampling period

Data Acquisition

Data acquisition is the process of retrieving data inputs to be later analyzed. In

selecting a system to acquire the data, the most important question to consider is, “what

sort of information is ultimately going to be acquired?”1 Answering this helps the user

choose what instrumentation will be best suited for acquiring data. DAQ systems

conventionally consist of a sensor, a DAQ device, such as an analog to digital converter,

and a computer. The sensor reads an analog signal while the A/D convertor changes the

signal from analog to digital so the computer can interpret it.3

With data acquisition systems, other questions must also be considered such as,

how accurate the instrumentation needs to be to accurately process a signal, how fast

must the wave be sampled at, and how will the data be stored? The speed at which the

wave is sampled is termed acquisition speed, and, as can be seen from Figure 3,

3 http://www.ni.com/data-acquisition/what-is/

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acquisition speed can vary greatly depending on how accurately the data must be

sampled.

Figure 31

1. Low speed: < 1 sample/sec2. Intermediate speed: 100 samples/sec – 1000 samples/sec3. High speed: > 1000 samples/sec

Figure 4 – The original analog signal (top). The wave is then sampled at a given rate,

converted, and reconstructed (bottom).

The more sampling points used in the A/D conversion, the more perfectly the

reconstructed wave will resemble the original. The only downside to this is that the more

samples taken, the more storage space it takes to store them all. Too few samples, and the

data won’t be reconstructed accurately. So, an ideal sample rate should be found used for

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A/D and D/A converting so as to accurately reconstruct a wave and not take up too much

space.

Resolution and Nyquist Frequency

Resolution directly corresponds to the number of bits per each sample. A bit is

simply a binary digit. By varying resolution, the user can vary how much extraneous

noise comes through from the signal, resulting in quantization error, which will be

discussed further later on. Figure 5 shows two waves, each sampled at full resolution, ½

resolution, and then ¼ resolution. As the number of bits per sample decrease, the shape of

the wave becomes less defined. Bits and sampling both play a part in determining

resolution. Whereas the sampling rate changes with respect to time, resolution changes

with respect to the number of samples.

Figure 54: The definition of the waves changes with the resolution of the wave.

4 http://www.iac.es/proyecto/magnetism/pages/activities/instrumentation.php

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The Nyquist frequency is defined as being equal to half of the sampling frequency

of that signal.5 If the signal is continuous, Nyquist frequency also refers to the highest

frequency that the sampled signal can unambiguously represent.5 For example, if a signal

is sampled at accurately at 22000 Hz, the highest frequency that we can expect to be

present in the sampled signal is 11000 Hz.

2. Signal Conditioning

Filters

Filters are used in digital signal processing for two general purposes: separation of

signals that have been combined, and restoration of signals that have been distorted in

some way.6 Various filters affect signals in different ways. For example, a low-pass filter

allows the low-frequency signals to pass and significantly reduces signals at a higher

frequency than the preset cutoff. A high-pass filter filters out low-frequency signals and

lets high frequencies pass. Band-pass filters allow frequencies to pass within a given

range, and band-reject filters are the opposite. Low-pass and high-pass filters work in the

time domain, while band-pass and band reject filters work in the frequency domain.1

Frequency domain filters are used when the information contained relates to amplitude,

frequency and phase components of a wave, and the goal of these filters is to separate one

band of frequencies from another, while time domain filters are used for things such as

5 http://www.fon.hum.uva.nl/praat/manual/Nyquist_frequency.html6 http://www.dspguide.com/ch14.htm

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smoothing and shaping waveformsError! Bookmark not defined.6 High-pass, band-

pass, and band-reject filters are designed by starting with a low-pass filter, and then

converting to obtain the desired response Figure 6 shows a visual a visual representation

of how these filters are combined and created.

Figure 6

There are two main methods for turning low-pass into high-pass filters: spectral

inversion and spectral reversal. Spectral inversion (shown in Figure 6) changes a filter

from low-pass to high-pass, high-pass to low-pass, band-pass to band reject, and band-

reject to band-pass. Spectral inversion works differently in that a high-pass and a low-

pass filter are added to create a band-reject filter. This process is demonstrated below in

Figure 7.

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Filters help with cleaning up signals that are digitally processed. For example, a

low-pass filter can be used to filter out frequencies below the Nyquist frequency of a

signal. This is a preventative measure so that noise doesn’t come up and ensures the

signal won’t experience aliasing. So, filters can remove the useless data from a processed

signal and can prevent error.

Amplifiers

Amplifiers essentially take a signal in one form and translate it into another. Most

amplifiers take an input signal and produce a different, more powerful, output signal.7

Figure 8 below demonstrates the use of a conventional amp in amplifying sound

7 http://electronics.howstuffworks.com/amplifier1.htm

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Figure 87

In digital signal processing, operation amplifiers (op-amps) are used to modify

signals and even produce frequency filtering. Op-amps have two inputs and a single

output. The two main types of op-amps worth considering are inverting and non-

inverting. With an inverting amplifier, the gain delivered to the output signal is

determined by two resistors, as shown in Figure 9 and Equation 2 below. The output

signal of an inverting amplifier is the negative of the input signal.

Figure 98 (left) Equation 2 (right)

8 http://electronics.stackexchange.com/questions/32084/transfer-function-for-inverting-amplifier

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The gain delivered to a signal from a non-inverting op-amp is created by

multiplying the input voltage by a constant greater than one. Figure 10 and Equation 3

show a diagram of how a non-inverting amplifier would be set up in a circuit and how to

calculate the output voltage given a certain input.

Figure 108 (left) Equation 3 (right)

3. Errors

When digitizing or processing digital signals as performed in this experiment, one

can encounter a number of errors based on various parameters used in the processing

method.

Aliasing

Aliasing is a phenomenon that occurs when the sampling rate of a signal being

digitized is not high enough to accurately reconstruct the original analog wave. An

aliased signal may look deceptively correct when being displayed in the time domain, but

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will result in incorrect signal measurements.9 Since aliasing results from not having

sampled a wave at enough points, an aliased signal doesn’t correctly display all of the

information needed for collection, and, therefore, creates error in data. To show an

example, Figure 11 displays a wave (top) that is accurately sampled and the same wave

(below) superimposed on one that is aliased.

Figure 11

(graphed in the time domain)9

The Shannon Sampling Theory (or Nyquist sampling theorem) states that aliasing

can be prevented by setting a sampling frequency greater than or equal to two times the

highest frequency contained in the signal.10 In reality, sampling at twice the bandwidth

signal only preserves frequency information. For accurate amplitude and shape

measurements in the time-domain, the sampling rate must be at least ten times the signal

bandwidth. This will ensure that a sufficient number of points are sampled to accurately

reconstruct the input analog wave. To summarize, the higher the sampling rate, the more

9 http://www.ni.com/white-paper/10669/en/10 http://www.siggraph.org/education/materials/HyperGraph/aliasing/alias1.htm

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processing power and time it takes to recreate a signal, but aliasing is prevented and error

is minimized.

Quantization and Quantization Error

Quantization is the process of mapping a larger set of values onto a smaller set.11

Quantization error results from trying to represent a continuous analog signal with

discrete, stepped, digital data.12 As previously discussed, analog signals can assume

virtually any value, while digital signals can only have finite values in discrete steps.

What happens when an analog value falls between these two digital “steps”? If a signal is

quantized, some round off error occurs as the analog signal assumes the digital value it is

closest to. Figure 12 below shows an ideal analogue wave and the quantization error as it

is converted to digital.

11 http://www.thefreedictionary.com/quantization12 http://www.sweetwater.com/insync/quantization-error/

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Figure 12: To the left, an ideal sample of an original wave and then its quantized

sample as its converted to its closest digital wave.13 To the right, a full sample of an

original wave superimposed on a quantized signal and the quantized error are displayed.

Quantization error is represented in Equation 1 (above) as variable “mu” and acts

as noise introduced by the sampling process. In order to get rid of this, enough gain must

be applied to the input signal to ensure that voltage levels rise above any possible

quantization noise.1 Alternatively, a higher resolution A/D converter can used to reduce

quantization noise to acceptable levels. In other words, if you increase the number of bits

to convert the signal, the reconstruction will not be as severely quantized.

Clipping

Clipping is one of the most analytically simple errors to recognize and

understand. Clipping occurs when a signal is limited to an amplitude value less than its

true amplitude, either because of limits on instrumentation (the fact that an A/D converter

has a finite range) or signal gain. A pure sinusoidal wave that is clipped will look like one

or more of its peaks have been cut off. Instead of having rounded peaks, the wave will

develop flat lines that occur at a cutoff amplitude. A clipped wave will not give the user

sufficient or accurate data. In order to prevent clipping, the instrumentation being used to

collect data must be programmed to be able to capture a wave’s peak-to-peak amplitude.

For example, choosing the right bipolar range for the data is crucial in preventing

clipping.

13http://www.diracdelta.co.uk/science/source/q/u/quantization%20error/source.html#.UmW4C2TwLqs

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Figure 13 – Shows a clean signal (left)

and that same signal clipped (right).14

Fourier Analysis

Fourier analysis is a way to mathematically analyze various waveforms in terms

of series, sequences, and trigonometric functions.15 With Fourier analysis comes the

introduction of the fundamental frequency and harmonics, which are used together to

determine the overall shape of a wave. Fourier, a mathematician, proved that any

continueous function could be produced as an infinite sum of sine and cosine waves.16

14 http://ccs.exl.info/installation/crossovers-installation-tweaking/tweaking/15 http://whatis.techtarget.com/definition/Fourier-analysis16 http://hyperphysics.phy-astr.gsu.edu/hbase/audio/fourier.html-

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As long as the function considered is periodic, its representative Fourier series, or

summation of sinusoids, can be expressed using Equation 4 below, where f is the

fundamental frequency, n is the number of harmonics, and t is time.

Equation 4

In Fourier analysis, the objective is to calculate coefficients an and bn up to the

largest possible value of n. The more harmonics that are assimilated into the equation, the

more accurate the reconstructed signal becomes to the original signal. The coefficients

are calculated using integral calculus with the equations below, where x(t) is the equation

of the original signal:

Equation 5 Equation 6 Equation 7

Since ∫– T /2

T /2

x (t ) dt is the area beneath function f over the given interval, a0 can be

geometrically interpreted as the average value of function f on that given interval or as

the new center of oscillation.17 The combinations of sine and cosine waves in the Fourier

equation develop a complete orthonormal basis for all continuous sinusoidal functions.

Based on that, the developed Fourier series is equivalent to building x(t) from the basis

sine and cosine vectors, and the coefficients an and bn indicate how many of each sine or

17 http://math.stackexchange.com/questions/340429/intuition-behind-fourier-coefficients

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cosine vectors are needed. Ideally, the Fourier reconstruction would be an infinite

combination of the basis vectors.

If the function at hand is considered odd, then the cosine term in Equation 4 will

go to zero along with the an coefficient. Similarly, if the function is even, the sine term

will go to zero, and so will the bn coefficient. With an odd function, there is a bn

coefficient at a certain value corresponding to the sine term, and an even function has an

an coefficient corresponding to the cosine term.

For a uniform sine wave, a0 and an will always be zero. The bn coefficient will

always have a value equal to the amplitude of the specific sine wave being analyzed. A

sine wave only has one harmonic (n=1) because theoretically, it only takes one sine wave

to make up a sine wave. Therefore, the reconstruction of a uniform sine wave will be:

bn=A

x (t )=A∗sin (2 πn f 0 t)

A = amplitude

f = frequency (1/T)

t = time

n = harmonics

For an odd function square wave, a0 and an coefficients will again be zero, and bn

will have a nonzero value. Also, there will be more than one harmonic for a square wave,

seeing as though it takes multiple superimposed sine waves to create a square-like

waveform shape. The greater number of harmonics, the better the reconstruction looks.

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The bn value for an odd square wave is equal to four times the wave’s amplitude divided

by the product of the number of harmonics with pi.

bn=4 Anπ

x (t )=4 Anπ

sin (2 π nf 0t)

Lastly for an even function triangle wave, a0 and bn will be zero. As discussed

before, with even functions, bn, the coefficient for sine, will be zero, and an will have a

nonzero value. Like the square wave, many harmonics are needed for a good

reconstruction of a triangle wave.

an={ 0 , whenn is even8 An2 π2 ,when n is odd

x (t )= 8 A

n2 π2∗cos (2 πn f 0t )

Fourier Derivations

Sine wave (a0)

A = Amplitude, T= Period, L=T/2, f = Frequency = 1/T f*L = ½

x(t) = Asin (2πft )

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a0=A∗1

T∫−T

2

T2

sin (2 πft ) dt= 12 L

∫−L

L

sin (2 πft ) dt

a0=A∗12 L [−cos (2 πft )

2πf ]−L

L

a0=A

2 L ( 12 πf ) [−cos (2 πfL)−(−cos (0 ))]

a0=A

2 π[−1−(−1)]

a0=0

Sine wave (an)

an=2T∫−T

2

T2

x ( t ) cos (2 πnft )dt = A2 L

∫−L

L

sin (2 πft ) cos (2 πnft )dt

Using identity: sin(u)*cos(v) = sin(u+v) + sin(u-v),

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an=A

2 L∫−L

L

sin (2 πft+2 πnft )+¿ sin (2 πft−2 πnft ) dt ¿

an=A

2 L∫−L

L

sin (t (2 πf +2 πnf ) )+sin (t (2 πf −2 πnf ) ) dt

an=A

2 L [−cos (t (2 πf +2 πnf ) )2 πf +2 πnf

+−cos (t (2 πf−2 πnf ) )

2 πf −2 πnf ]−L

L

an=A

2 L [−cos ( L (2 πf +2πnf ) )2 πf +2 πnf

+−cos ( L (2πf −2πnf ) )

2 πf −2πnf ]− A2 L [−cos (−L (2πf +2 πnf ) )

2 πf +2 πnf+−cos (−L (2 πf −2 πnf ) )

2 πf −2 πnf ]an=

−cos ( L (2πf +2 πnf ) )2 πf+2 πnf

+−cos ( L (2πf −2πnf ) )

2 πf −2 πnf+

cos (−L (2πf +2 πnf ) )2 πf +2πnf

+−cos (−L (2 πf−2 πnf ) )

2 πf −2 πnf

Terms cancel because cos(x) = -cos(x) and –cos(x)+cos(x) = 0

Therefore, an=0

bn=2T

∫−T /2

T /2

x ( t ) sin (2 πnft ) dt = bn=1L∫−L

L

Asin (2 πft )sin (2 πnft )dt

bn=AL [ cos (2 πft )sin (2 πnft )−nsin (2πft ) cos (2πnft )

n2−1 ]−L

L

bn=AL [ cos (2 πfL) sin (2πnfL )−nsin (2 πfL) cos (2 πnfL)

n2−1 ]− AL [ cos (−2πfL )sin (−2πnfL )−nsin (−2πfL ) cos (−2πnfL )

n2−1 ]

bn=AL [ cos ( π ) sin (πn )−nsin (π )cos (πn )

n2−1 ]− AL [ cos (−π ) sin (−πn )−nsin (−π )cos (−πn )

n2−1 ]For harmonic n=1, bn= 0/0. L’Hospital’s rule applied:

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limn →1

AL [ cos ( π ) sin (πn )−nsin ( π ) cos ( πn )

T2

(n2−1 ) ]+ AL [ cos (−π ) sin (−πn )−nsin (−π ) cos (−πn )

T2

(n2−1 ) ]limn →1

AL [ cos ( π ) cos (πn ) π−nsin ( π )(−sin (πn ))π−cos (πn)sin (π )

T2

(2n) ]+ AL

¿¿

As n goes to 1, the limit becomes: bn=A( 12+ 1

2 )=A

Square Wave (a0)


x (t )={−A ,∧−L≤ t<0A ,∧0≤t <L

a0=1T∫−T

2

0

−A dt + 1T∫0

T2

A dt

a0=1

2 L∫−L

0

−A dt+ 12 L

∫0

L

A dt

a0=1

2 L[−At ]−L

0 + 12 L

[ At ]0L

a0=−12 L

(−A (−L ) )+ 12 L

( A ( L ))

a0=0

Square Wave (an)

an=2T∫−T

2

T2

x (t ) cos (2 πnft )dt

an=1L [∫

−L

0

−A cos (2πnft ) dt+∫0

L

Acos (2 πnft )dt ]24

an=1L [−Asin (2 πnft )

2πnf ]−L

0

+ 1L [ Asin (2 πnft )

2πnf ]0

L

an=1L [ Asin (−2 πnfL)

2πnf+

Asin (2 πnfL)2πnf ]

Since sin(-t) = sin(t) and –sin(t) + sin(t) = 0,

an=0

Square Wave (bn)

bn=2T

∫−T /2

T /2

x ( t ) sin (2 πnft ) dt

bn=1L [∫

−L

0

−Asin (2 πnft )dt +∫0

L

Asin(2 πnft)]bn=

1L [ Acos (2 πnft )

2 πnf ]−L

0

+ 1L [−Acos (2πnft )

2πnf ]0

L

bn=1L [ Acos (0 )

2 πnf−

Acos (2 πnfL )2πnf ]+ 1

L [−Acos (2 πnfL )2 πnf

−−Acos (0 )

2πnf ]bn=

1L [ A

2 πnf−

Acos ( πn )2 πnf ]+ 1

L [−Acos ( πn )2 πnf

+ A2 πnf ]

bn=Aπn

(2−cos (πn )−cos ( πn ) )= Aπn

(2−2 cos ( πn ))

For n odd, bn=4 Aπn

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Triangle wave (ao)

*NOTE: The triangle wave has been analyzed below as an even function whereas

the sine and square waves have been analyzed above as odd functions.


x (t )=A−A( 2 tL )

a0=2∗1

T∫0

T2

x (t ) dt

a0=2∗12 L

∫0

L

(A−A ( 2tL ))dt

a0=1L [ At−

A t2

L ]0

L

a0=1L [ AL− A L2

L ]a0=0

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Triangle wave (an)



an=2∗2

T∫0

T2

x ( t ) cos (2 πnft ) dt=2∗1L

∫0

L

[ A−A ( 2 tL )]cos (2 πnft ) dt

an=2 AL∫

0

L

[cos (2 πnft )−( 2tL )cos (2πnft )]dt

Integration by parts: u = 2t/L , dv = cos(2ft) dt

an=2 AL [ sin (2 πnft )

2 πnf ]0

L

−2 AL [ [ 2t

Lsin (2 πnft )

2 πnf ]0

L

−∫0

L2L (sin (2 πnft )

2 πnf )dt ]an=

2 AL [ sin (2 πnft )

2 πnf ]0

L

−2 AL [ [ 2 t

Lsin (2 πnft )

2 πnf ]0

L

−[ 2 (−cos (2 πnft ) )

L( 12 πnf )

2 ]0

L]an=

2 AL [ sin (2 πnfL)

2 πnf ]−2 AL [ 2 Lsin (2 πnfl )

2 πnfL ]−[ 1Lπnf (−cos (2πnfL )

2 πnf—

−cos (0 )2 πnf )]

an=2 AL [ sin (2 πnfL)

2 πnf ]−[ sin ( πn )πnf

−( 1πn ( (1−cos (πn ))

πnf ))]

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an=2 Asin ( πn )

πn−

4 Asin ( πn )πn

+( 4 A−4 Acos ( πn ) )

π 2n2

For n odd, an=8 A

π2 n2

Triangle wave (bn)



bn=2∗2T

∫0

T2

x ( t ) sin (2 πnft ) dt=2∗1L

∫0

L

[A−A ( 2tL )]sin (2 πnft ) dt

bn=2 AL∫

0

L

[sin (2 πnft )−( 2 tL )sin (2 πnft )]dt

Integration by parts: u = 2t/L , dv = sin(2ft) dt

bn=2 AL [−cos (2πnft )

2πnf ]0

L

−2 AL [[ −2 t

Lcos (2 πnft )

2πnf ]0

L

−∫0

L−2L ( cos (2πnft )

2 πnf )dt ]bn=

2 AL [−cos (2πnft )

2πnf ]0

L

−2 AL [[ −2 t

Lcos (2 πnft )

2πnf ]0

L

−[−2 (sin (2 πnft ) )

L( 12 πnf )

2 ]0

L]Terms cancel because cos(x) = -cos(x) and –cos(x)+cos(x) = 0 and

sin(2) = sin(0) = 0.

bn=0

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PROCEDURE

Simulation

In the simulation, the computer alone digitally generated all signals using LabView. No

A/D converter was used.

2.1 – Digital Oscilloscope Simulation

A sine wave was selected as the input function.

The fundamental frequency was set at 1000 Hz

The amplitude of the wave was set to 5 volts

Sampling frequency was set to 5 kHz

Resolution was set to 12 bits

Bipolar range was set to +/- 10 volts

Low pass filter setting OFF

Press Run

SAVE FILE

2.2 – Fourier Analysis Simulation

Simulated a square and triangle wave in both in time and frequency domains

Square wave was selected as the input function

Waveform frequency was set to 1000 Hz


Resolution, amplitude, and bipolar range were taken from previous section 2.1

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Low pass filter setting ON

Press Run

SAVE FILE (Filtered square wave)

Repeat the above process with the Low pass filter setting OFF

SAVE FILE (Unfiltered square wave)

Repeat above steps with triangle wave

SAVE FILE (Filtered triangle wave)

SAVE FILE (Unfiltered triangle wave)

2.3.1 – Quantization Error and Resolution

Sine wave was selected as input function



Resolution was set to 12 bit

Amplitude and bipolar range were taken from previous sections (2.1 and 2.2)

Low pass filter setting ON

Press Run

Run and SAVE (12 bit sine wave)

Repeat above steps with resolution set to 4 bits

Run and SAVE (4 bit sine wave)

2.3.2 – Clipping

Sine wave was selected as input function

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Amplitude was set to 5 volts


Resolution was set to 12 bit

Bipolar range was set to +/- 1.0 volts

Clipping of the sine wave peaks was noted

SAVE FILE (Clipped sine wave)

Experiment

In the experiment, the function generator was used instead of the computer to create the

waveforms needed. The waveforms were still analyzed through LabView.

3.1– Digital Signal Acquisition

Sine wave was selected on the waveform selector

Frequency was set to 1000 Hz on function generator

Sampling frequency was set to 45 kHz on Labview VI

Press Run

Measured actual frequency of sine wave using cursor

Compare wave to section 2.1 of Simulation

SAVE FILE (Not aliased sine wave)

3.2 – Anti-Aliasing and Filtering

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Repeat process from previous section 3.1 using a sampling frequency of 1500

Hz

SAVE FILE (Aliased sine wave)

Run the function generator through the Krohn-Hite filter

Krohn-Hite filter was set to a cutoff frequency of 750 Hz

Repeat the above procedure

SAVE FILE (Second aliased sine wave)

3.3 – Fourier Analysis Experiment

Square wave was selected on the waveform selector

Frequency was set to 1000 Hz


Measured actual frequency and amplitude of wave using cursor on the

computer

SAVE FILE

Repeat for triangle wave

SAVE FILE

CHANGED to B&K function generator and turn on white noise setting

Sampling frequency to 45 kHz

SAVE FILE (White noise)

Used Low pass filter at cutoff frequency of 7000 Hz

Filter white noise from previous file through low pass filter

SAVE FILE (White noise filtered through low pass filter)

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RESULTS and DISCUSSION

The simulation and the experimental sections were started first by creating a

standard sine wave from which to work off of. The more detailed results and analysis of

the simulation and experiment are below. To examine the sine wave done in simulation

section 2.1, refer to the plots in Appendix 2.1 To look at the sine wave created by the

function generator; refer to the information in Appendix 3.1.

1. Aliasing and Filtering

From experimental section 3.2, the function generator created two sine waves

with the same parameters. Both of these waves experienced aliasing. The first wave

through was sampled at a rate far below its Nyquist frequency. The second wave had the

same frequency as the first and was sampled at the same rate, and it was put through a

low pass filter (Krohn Hite filter). Even though the filter was on, the second wave still

experienced aliasing because the sampling rate was still too low to accurately reconstruct

the wave. The filter prevented some “noise” from coming through at higher frequencies.

The aliasing could’ve been predicted using the Shannon or Nyquist Sampling Theorem.

Aliasing of the wave without the low pass filter can be seen in Figure 1.1, and aliasing of

the wave with the filter can be seen in Figure 1.2. Additional information and plots

associated wit experimental section 3.2 can be found in Appendix 3.2.

Amplitude 3.3817 volts ±0.0012 volts

Frequency 1000 Hz± 500 HzCutoff filter frequency 750 Hz ± 375 HzSampling frequency 1500 Hz ± 750 Hz

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Number of Samples 8192

Figure 1.1

Figure 1.2

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2. Fourier Analysis: Simulation and Experiment

From simulation section 2.2, square and triangle waves were created with the

parameters listed below. The waves were both run a second time through a low pass

filter.

Amplitude Frequency Sampling frequency Number of samples

5 volts ± 0.0012 v 1000 Hz± 500 Hz 45 kHz± 22.5 Hz 8192

The first square wave that was simulated without the filter was close to a perfect

square. The square wave with the low pass filter on developed peaking where the wave

was originally flat. The slopes of the filtered wave were more noticeable slanted instead

of clearly resembling straight vertical lines. The differences occurred as a result of the

low pass filter being turned on. The low pass filter blocked higher frequencies from

coming through that were needed to make a more precise square waveform. The

unfiltered square wave can be seen in Figure 2.1 while the filtered can be seen in Figure

2.2.

The triangle wave experienced the same thing as the square wave. The unfiltered

triangle wave had peaks that were more pointed compared to the filtered wave that had

peaks that were more rounded. Again, this was a result of the filter cutting out

frequencies in the higher range. These high frequencies are needed in triangle and square

waveforms to create the visible pointed edges. The unfiltered triangle wave and filtered

triangle wave can be seen in Figures 2.3 and 2.4, respectively. The Nyquist plots

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corresponding to simulation 2.2 can be found in Appendix 2.2, while the Fourier

reconstruction graphs can be found on pages 38-40 in this section.

Figure 2.1

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Figure 2.2

Figure 2.3

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Figure 2.4

In the first two steps of experimental section 3.3, both square and triangle

waveforms were created through the function generator.

Amplitude 3.6

Frequency 1111.11 Hz ± 555

Sampling frequency 45 kHz ± 22.5

Number of Samples 8192

Harmonics n=1,3,5,…,63,65,67

The data showed that the square and triangle waveforms in the experimental

section resembled the corresponding waveforms in the simulation very closely. The data

collected in the time domain showed that the results were accurate. In the frequency

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domain, the experimental graphs had noticeably more peaks than the frequency domain

graphs of the simulation. The frequency peaks were a result of needing more frequencies

to accurately reconstruct the waves. Square and triangle waves have pointed edges

whereas sine waves do not, and therefore, they need multiple frequencies to increase the

definition around the pointed edges. The experimental square and triangle waves can be

seen in Figures 2.5 and 2.6, respectively. Additional information on experiment 3.3 can

be found in Appendix 3.3.

Figure 2.5

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Figure 2.6

3. Reconstruction Graphs

The square and triangle waves from simulation section 2.2 and experimental

section 3.3 were reconstructed using harmonics. Below are the results displaying each

reconstruction up to 75 harmonics. The results showing progressive reconstruction using

a few harmonics at a time and Nyquist plots are in Appendix 2.2 for simulation 2.2 and

Appendix 3.3 for experimental section 3.3. The results show that enough harmonics were

used to reconstruct the graphs, and the reconstructed graphs from the simulation closely

resemble the reconstructed graphs from the experiment.

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Reconstruction: Unfiltered Square wave Simulation 2.2

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Reconstruction: Unfiltered Square wave Experiment 3.3

Reconstruction: Unfiltered Triangle wave Simulation 2.2

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Reconstruction: Unfiltered Triangle wave Experiment 3.3

4. Quantization Error

From simulation 2.3.1, a sine wave was simulated once with 12 bit resolution and

the second time with 4 bit resolution. The sine wave with 12 bits appeared smooth and

uniform. The sine wave taken with 4 bit resolution appeared more jagged and had edges

as opposed to curves. The low pass filter was activated to prevent any aliasing from

occurring.

Amplitude 2 volts

Frequency 1000 Hz± 500 Hz

Sampling Frequency 45 kHz± 22.5 Hz

Bi-polar range +/- 10 volts

Low pass filter cutoff 27.5 kHz± 13.75 Hz

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The low resolution caused quantization error to appear for the second wave. The

number of bits used for each wave dictated how many data points could be recorded. The

wave generated with 4 bits had significantly fewer data points and much larger spaces in

between each point. The quantized wave taken at 4 bits experienced significantly more

error than the reference sine wave. This created the quantization error seen in figure 4.2.

The reference sine wave can be seen in Figure 4.1. The remaining data for these two

waves can be found in Appendix 2.3.1.

Amplitude Uncertainty for 12-bit +/- 0.001 volts

Amplitude Uncertainty for 4-bit +/- 0.333 volts

Figure 4.1

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Figure 4.2

5. Clipping

In simulation 2.3.2, a sine wave was taken with the parameters listed below with a

bipolar range of +/-10 volts. That same sine wave was then simulated with a bipolar

range of +/-1 volt.

Amplitude 5 volts ± 1.22(1 0−4) volts

Frequency 4000 Hz± 2000 Hz

Sampling Frequency 80 kHz± 40 Hz

Resolution 12 bit

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Figure 5.1

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Figure 5.2

The clipping in above Figure 5.2 is a direct result of decreasing the bipolar range.

The flat peaks of the sine wave are notably where the bipolar range cuts off. The slopes

leading up to the peaks are slightly more curved when the wave is clipped. Figure 5.1 can

be used as a reference to see what the wave would look like if it had not been clipped. It

should be noted that the frequency domain plots are slightly different. The clipped wave

is no longer a perfect sine wave and, therefore, needs more frequencies to accurately

reconstruct its shape. More information can be found on the clipped wave in Appendix

2.3.2.

6. Noise and Filtering

In the last part of experiment section 3.3, white noise was created using the

function generator. The noise was produced with frequencies varying 2 to 20,000 Hz. The

data taken for the noise can be seen in Figure 6.1 below.

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Figure 6.1

The noise coming out of the generator was then run through a low pass filter with

a cutoff frequency of 7000 Hz. The results can be seen in Figure 6.2 below. As expected,

the filter cleared most of the frequencies above the cutoff. However, the filter used was

not perfect. There was a negative slope in the frequency domain plot of Figure 6.2

indicating that the filter did not cutoff exactly at 7000 Hz and still let some of the higher

frequencies pass through. This is general error associated with low pass filters. As shown,

the instrumentation is not perfect. The data shows that the filter significantly helped in

reducing the original noise and stopped most noise frequencies from passing above 7000

Hz.

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Figure 6.2

CONCLUSION

Through analysis and comparison of the results, multiple conclusions can be

drawn regarding error, Fourier analysis, and applications regarding digital signal

processing. Comparison of the experimental results to the results of the simulation

showed how close the actual experimental results were to what would have happened in

ideal conditions with minimal error. As expected, the experimental results had a larger

margin of error than the results taken from the simulation.

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The Fourier analysis conducted in digital signal processing tied together

mathematical theory with real world application. Fourier analysis allowed for the

reconstruction of the experimental and simulated waves using harmonics. Looking at the

square and triangle wave reconstruction graphs, it became evident that the more

harmonics incorporated, the more the reconstructed plot resembled the original. The

reconstructed graphs were close but never exactly as precise as the original because a

finite number of harmonics was used in each case. To have a reconstructed graph using

harmonics identical to the original, theoretically an infinite number of harmonics must be

incorporated. Essentially, an infinite number of sine waves must be used to create a

perfect square or perfect triangle waveform. Any other number of sine waves will

produce a waveform that is only close to a perfect square or triangle wave. The Fourier

coefficients that were found in the analysis were used to create an accurate

reconstruction. Overall, the simulations, the experimental graphs, and the Fourier

reconstruction were similar, meaning the experimental results behaved as predicted, and

the Fourier reconstructions were done accurately.

The exercises both in the simulated and experimental tests showed how various

waveforms behaved given certain parameters. The types of waveforms that were

examined were pure sine, square, and triangle, and the three prominent errors that

occurred were aliasing, quantization error, and clipping. The errors occurred in the

process of converting an analog signal into its digital counterpart. By simulating these

errors, a lot was learned on why these errors happen circumstantially and how to prevent

them from happening in future signal processing. Aliasing is the most difficult to analyze

because, as discussed, a wave that is aliased can still give certain correct information.

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Aliasing occurred when the sampling rate was not twice the frequency of the wave.

Sampling at a good rate and filtering a signal through low-pass filters can prevent

aliasing. Quantization error occurred in this experiment when the resolution was set

incorrectly. Altering the resolution for a signal to increase the number of bits would

prevent this error. Clipping occurred simply when a wave was processed using an

incorrect voltage range. Clipping is obvious when a visual representation is created of the

signal in the time-domain and can be corrected by simply changing the bipolar voltage.

Digital signal processing is indisputably significant in today’s technological

world. This experiment in particular provided a lot of information on how different

waveforms are manipulated and used, the errors that can occur when processing them,

and how to recognize and fix them when they do occur. Along with Fourier analysis, this

knowledge can be applied to anything involving vibrations and waves, such as speakers

and telephones used in everyday life or waves traveling through solid objects.

Uncertainty Equations

ut=±12

t

u f=±12

f

uA=±12 ( Range

2N−1 )ubn

=√ f 2+u f2+ A2+ua

2

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MAE 315-Lab 2 DSP Writeup

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Transcript of MAE 315-Lab 2 DSP Writeup