ele 374 lab
TRANSCRIPT
Chowdhury Raiyeem Farhan ee08u205
School of Electronic Engineering & Computer Science
ELE 374: Signals & Systems Theory
Formal Report on: Fourier Analysis & Synthesis of Waveforms
Chowdhury Raiyeem Farhan
ID No: 089609202
Email: [email protected]
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Contents
Abstract
Introduction
Background Theory
Description of the Experiment
Discussion & Conclusion
Glossary
Reference
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Abstract:
The Experiment: Fourier Analysis and Synthesis of Waveforms is all about
learning the basic of Signals and Systems Theory, which is Signal. Here we analyze
different signals with the help of Fourier Series Java Applet[1] and explore the
properties of them. This special java applet enables us to characterize a real signal
in the Time domain[2] to its spectrum in Frequency domain[3].
The experiment consists four parts with guide lines for each one. These let us to
learn more about signal spectrum[4], bandwidth[5] , waveforms, effects of
distortion[6] and limitations of signals. In brief the entire experiment allows us to
learn about signals with help of Fourier Series[7] in communication system.
Introduction:
Communication system is a system that includes exchange of information
among entities, this entity might be human, company or computer. Among
human this is done by spoken or written language but in communication system
engineering point of view information is exchanged through signals (analogue or Page | 3
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digital).Now what is signal? Signal, as defined in Signals & Systems-models &
behavior by M.L. Meade and C.R. Dillon, the variation of any measurable quantity
that conveys information concerning the behavior of related system. And as this
ELE 374 Module is concerned about to give participants an understanding of basic
signal and system concepts this entire experiment is to learn about signal, how it
propagates through a network or process, properties & behavior of different
signals using Fourier Series.
The experiment is designed in four different parts with elaborated guidelines:
A. Observing the effects limiting the bandwidth of a real signal
B. Deciding the bandwidth needed to support a binary
representation of an analogue signal[8] using Fourier Analysis
C. Using Fourier Analysis to look at an “Unusual” signal
D. Examining Noise[9] using Fourier Analysis
For these experiments we used Fourier Series Java Applet and theoretical
calculation of Fourier Analysis to compare the results we found.
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For part A: we used Square wave [10], Sawtooth wave [11], phase shift and
rectifier options in Java applet and recorded the behavior as stated in the
guidelines.
In part B: we again used Sawtooth wave but this time quantizer and cosine
option.
Part C: this time option clip and Triangle wave[12] was used.
And in Part D: Noise was observed.
The fundamentals are clarified in the Background Theory section and explanation
for each part of the experiment is provided in the Experiment & Interface section.
And all the data gathered from the parts of the experiment was recorded in the
log book for observation and comparison.( in the applet the white line represents
the real signal, and the red line represents the spectra).
Back Ground Theory:
Signal:
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Basically signal is a wave form of voltage or current which varies with time and
carries data or information.
Figure 1: A random signal
Spectrum (spectra- in plural):
Spectrum is a summation of infinity number of sinusoids having a particular
amplitude and phase. Signals and signal networks are analyzed in terms of
spectral representation. The spectrum of a signal indicates the aspect of the signal
which would not have been obvious when looking at the time domain
representation.
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Figure 2: Audio Spectrum
Classification of signals:
As signal is time dependent it is classified in two categories according to
characteristics and time variable:
1) Continuous –time signal:
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It is a varying quantity (a signal) whose domain, which is often time,
is a continuum.
That is, the
function's
domain is an
uncountable
set. The function itself need not be continuous. The graph
of a continuous-time signal x(t) is thus defined at each and every instant over a
measurement interval
extending from t = t1 and t = t1+TM(where t= time and TM= max time )
Generally it’s called analogue signal. Good example of analogue signal is
speech or music signals.
1. Discrete-time signal:
Is a time-series consisting of a sequence of quantities. In other words,
it is a time series that is a function over a domain of discrete integers.
Each value in the sequence is called a sample. A discrete-time signal
is not a function of a continuous argument; however, it may have
been obtained by sampling from a continuous-time signal.Page | 8
Figure 3: Continuous-time signal
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Perfect example is the signal transmitted
or received by computers.
According to the time period signals can be classified as Periodic signals and
Aperiodic Signals.
Periodic signals:
That signal which repeats itself after a period of time T is called Periodic signal.
This period T can be defined as the time to complete 1 full cycle.
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Figure 4: Discrete-time signal(red line represents the discrete values)
Figure 5: Periodic Signal as a function of t
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Strictly Periodic Signals:
This kind of signal has a property which can be explained mathematically.
If, X(t)=signal (as a function of “t” (time)),
T0= Period of signal,
Then
X(t)= X(t+ T0),for all t.
Such kinds of signals are Square wave signals, Sawtooth signals. For these signals
the range of t is:
|t|< T0 /2 and the function X(t) can be represented as:
X(t)= 2t/T0;
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Figure 6: Strictly Periodic Wave (Sine wave, Square wave, Triangle wave,
Sawtooth wave from top to bottom)
Representation of Wave form:
Signal and wave forms are represented in time and frequency domain.
Time domain:
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The time-amplitude axes on which the sinusoid is shown is called
time-plane. The way of representing the signal as a varying function of time is said
Time-domain representation. In this method the graph is plotted with amplitude
[] taken as the Y-axis and the time in X-axis.
Figure 7: waveform represented in time domain
Frequency domain:
Frequency domain representation is a signal representation in the form of a
function having frequency as the independent variable and amplitude as Page | 12
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dependent variable. The distance along the frequency axis is the frequency of the
sinusoid, which is equal to the inverse of the period of the sinusoid.
Figure 8: Waveform in Frequency Domain
Energy and Power signals:
The following assumes a 1 OHM resistance (if we are considering a voltage signal,
and the convention is that this is the physical signal type we assume).
Total energy in a (continuous) signal is: Etot = ∫t=0
∞
x (t )2dt[volts2 seconds]
The total energy is often infinite in signals.
Power (measured over interval T) in a continuous signal is: Pav = 1/T
∫t=o
T
x (t)2dt [volts 2]
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Signals which have finite total energy are classified as energy signals, e.g. an
isolated rectangular pulse. Signals, e.g. sinusoids, for which the total energy
would be infinite, are classified as power signals. 0Atimeamplitude
Figure 9: An isolated pulse: example of an energy signal
Averaging of discrete and continuous signals
For a discrete signal, the average of the first N samples is:
A = 1/N ∑n=0
N−1
x (n .T )
Where T = the sampling interval, 1/T = the sampling frequency.
For a continuous signal, the average over a period 0 to T is: Page | 14
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A = 1/T ∫t=0
T
x( t)dt
Odd and Even signals
If x(t) = x(-t) the signal is EVEN, e.g. a cosine wave.
If x(t) = -x(-t) the signal is ODD, e.g. a sine wave.
Note the PRODUCT rule:
ODD . ODD = EVEN
EVEN . EVEN = EVEN
EVEN . ODD = ODD
ODD . EVEN = ODD
Note also that:
S = ∫t=−t '
t '
x (t)dt = 0 always if x(t) is ODD
= 0 sometimes if x(t) is EVEN
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Figure 10:Integrating across the origin of an odd signal (a sinewave).
Even Function:
If f(x) is to be a real-valued function of a variable, then f is even if equation ‘
f(x) = f( − x)’ holds for all x in the domain of f.
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Geometrically, an even function is symmetric with respect to the y-axis, which
means that its graph remains unchanged after reflection about the y-axis. The
Cosine wave is an example of even functions
If f(x) is to be a real-valued function of a variable, then f is even if equation ‘
f(x) = f( − x)’ holds for all x in the domain of f.
Geometrically, an even function is symmetric with respect to the y-axis, which
means that its graph remains unchanged after reflection about the y-axis. The
Cosine wave is an example of even functions.
Odd function:
A function is odd if it is symmetric with respect to the origin, meaning that its
graph remains unchanged after rotation of 180 degrees about the origin. For
example if
x(t) = -x (-t) then this function is odd.
The sine wave is an example of an odd function.
A function f (n) is odd if f(n)=-f(-n).
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Figure 11: Even funtion,cosine(left side), Odd Function,sine(right side)
Orthogonality
Orthogonality is fundamental to almost everything that is subsequent in signals
and systems theory. The definitions are:
Discrete signals:
If the product of two signals averages to zero over the period T, then those two
signals are ORTHOGONAL in that interval (T).
Continuous signals:
If the product of two signals integrates to zero over the period T, then those two
signals are ORTHOGONAL in that interval(T)
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Figure 12: Example of orthogonality
Linearity and Time Invariance
Linearity is one of the most important properties defined in engineering. If the
input to a linear system is the weighted sum of several signals, then the output of
that system is the weighted sum of the responses of the system to each of the
signals separately. Formally:
additivity property
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if x1(t) → y1(t) (“input x1(t) produces output y1(t)”)
and x2(t) → y2(t)
then x1(t) + x2(t) → y1(t) + y2(t)
homogeneity (scaling) property
if x1(t) → y1(t)
then a.x1(t) → a.y1(t) (where ‘a’ is any complex constant) If a system has both
these properties, then the system is a linear system. This can be summed up as:
a.x1(t) + b.x2(t) → a.y1(t) + b.y2(t) continuous time
a.x1[n] + b.x2[n] → a.y1[n] + b.y2[n] discrete time
A system is time invariant if the performance of the system does not change over
time, i.e. if x1[n] produces y1[n] then x1[n - n0] produces y1[n - n0]: the same
input at a different time producing the same output.
Typically not all systems are time invariant: circuits using analogue electronics are
notoriously prone to varying performance as they (a) warm up, (b) wear out, and
this is one of the major considerations in favor of using s/w based systems.
Examples:
y(t) = sin(x(t)) is an example of a time invariant system
y[n] = n.x[n] is an example of a time varying system (it has time dependent gain)
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Fourier series:
Fourier series were introduced by Joseph Fourier (1768–1830) 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.
The process of converting a signal into its Fourier series is called Fourier
Transform. Different types of signals have different representation of Fourier
series. An arbitrary signal can be composed of an infinite number of sinusoids,
with frequencies harmonically related to the fundamental frequencies. This
changes the signal form from a time domain graph to a frequency domain graph.
The property of sinusoidal waves enables us to analyze and investigate them
easily. One of the most fundamental quantities is the bandwidth required for
signal transmission. The Fourier series can also be used to find out the bandwidth
and this has many important applications in systems and signals theory.
It states that any signal can be represented by a series of sine waves
The formula for the Fourier series:
x(t) = a0 + Σ an.cos(n.ω.t) + Σ bn.sin(n.ω.t)
The term a0 is the zero frequency term. The coefficients an and bn tell us the
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Amplitudes of cosine and sine and indicate how much they are contributing to the
value of x(t). ‘n’ is the number of terms, ‘w’ is the angular frequency and ‘t’ is the
instantaneous time of the signal. The Fourier series provides frequency domain
representations for periodic signals. We know that real signals are not periodic;
Fourier transform is used for non-periodic signals. The Fourier transform allows
the representation of a non-periodic signal as an uncountable infinite number of
sinusoids.
The formula used for this transform is
The Fourier trigonometric series
Any periodic signal, x(t), whose period is T, can be represented by the appropriate
sum of sine and cosine components:
x (t )=a0+∑n=1
∞
an .cos (n.w .t )+¿∑n=1
∞
bn . sin (n.w .t)¿ (1)
a0 is the mean value, or zero frequency term.
Integrating both sides of eqn (1), between = -T/2 and T/2 :
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∫−T /2
T /2
x (t )= ∫−T /2
T /2
a0+ ∫−T /2
T /2
¿¿ dt
in which all of the a.cos, b.sin terms disappear under integration, as the limits of
the integration represent a whole number of cycles at the lowest frequency (n=1),
and will therefore represent an integer number of cycles at all values of ‘n’.
∫−T /2
T /2
x (t )= ∫−T /2
T /2
a0+ ∫−T /2
T /2
¿¿ dt
∫−T /2
T /2
x (t )= ∫−T /2
T /2
a0=a0T
a0 = 1/T ∫−T /2
T /2
x (t )
To find a formula for an it is necessary to multiply both sides of eqn(1) by
cos(m.ω.t) and then integrate over the same limits:
∫−T /2
T /2
x (t )cos (m¿.ω. t)dt=∫−T2
T2
ao .cos (m .ω.t )+¿∫−T2
T2
¿¿¿¿
From identities we can find out:
cos α . sin β=12
¿
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Where, the odd waveforms disappear under integration.
Now cos .cos terms produce:
cos α .cos β=12¿
Which will not disappear after integration,because:
∫−T /2
T /2
∑n=1
∞
cos (m .ω .t ) . an.cos (n .ω. t)= an . 12¿¿[after integration]
But we are integrating over -T/2 → +T/2 and this represents an integer number of
cycles of the sinusoid, whatever the value of ‘m’ and ‘n’. BUT when m=n, we have
a non-zero term after integration:
∫−T /2
T /2
x (t )cos (m¿.ω. t)dt=∫−T2
T2
ao .cos (m .ω.t )+¿∫−T2
T2
an .12cos (0.ω. t )+¿ ∫
−T /2
T /2
¿¿¿¿¿
∫−T /2
T /2
x ( t ) .cos (m .ω. t )dt=( an2 )|t|−T2
T2 =an .
T2
But m=n, so:
∫−T /2
T /2
x ( t ) .cos (n .ω. t )dt=an2.|t|−T /2
T /2=an .
T2
an= 2T
∫−T /2
T /2
x (t ) .cos (n .ω. t )dt
And by similar reasoning:
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bn= 2T
∫−T /2
T /2
x ( t ) . sin (n .ω. t)dt
The exponential form of the Fourier series
This is a way of reducing the amount of ‘writing out’ the Fourier Series requires:
x (t )=a0+∑n=1
∞
an .cos (n.ω. t )+¿∑n=1
∞
bn . sin (n .ω .t ) ¿
an .cos (n .ω. t )=( an2 ) .[e jnωt+e− jnωt]
bn sin (n .ω. t )=( bn2 ) .[e jnωt−e− jnωt]
So an .cos (n .ω. t )+bn sin (n .ω.t )=( an2 ) .[e jnωt+e− jnωt ]+( bn2 ) .[e jnωt−e− jnωt]
[Where, n≠o¿
So the Original Fourier Series can be written as:
x (t )=∑−∞
∞
Xn .e jnωt[Where, X0= a0]
Description of Experiment:
In this experiment we firstly we used Fourier Series Java Applet and then by
manual mathematical calculation compared the results we got from the applet.
During the four parts of experiment we had to answer guided questions, these
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were jotted down in the lab book, which helped us completing the discussion and
conclusion section.
Java applet interface:
Finding the java applet was easy, you go to Google.co.uk and type Fourier series
java applet and the link comes up with the applet. This looks like the picture given
below:
Figure 13: Java Applet for Fourier Series synthesis
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Here the applet can compare different kind of real signals with Fourier analysis.
The synthesized signal types are shown in the tab place on the right corner
named: sine, cosine …etc. There are also two bars to adjust the number of terms
in the spectra and the frequency. In the graph red line represents the synthesized
signal and white line shows the type of synthesized signal. The real signal is also
represented as function of sine and cosine separately in the applet, this helped us
with the analysis.
Here the applet can compare different kind of real signals with Fourier analysis.
The synthesized signal types are shown in the tab place on the right corner
named: sine, cosine …etc. There are also two bars to adjust the number of terms
in the spectra and the frequency. In the graph red line represents the synthesized
signal and white line shows the type of synthesized signal. The real signal is also
represented as function of sine and cosine separately in the applet, this helped us
with the analysis.
Part A: To use Fourier Analysis to observe the effect of limiting the bandwidth of
a real signal
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At first the default wave was cleared then the square wave was selected. As
the number of terms was reduced to zero the red line became flat. The
reason was noted on the lab book.
Figure 14: cleared field with square wave
In the spectrum of cosine there was a white dot this was also noted. Then
as the guide line showed us to increase the number of terms were
increased accordingly and the questions were answered in the log book.
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Figure 15: Number of terms increased(accordingly)
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Then the number of terms were increased until the signal looked like
original signal
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Then number was observed when it took almost original signal’s shape.
After taking the notes we again refreshed the whole thing by clicking and
then chose squarewave and put the number of terms tab to the half of the
bar. Now we used phase-shift option, firstly one time then one by one to 10
times and noted down the incidents happened.
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Figure 16: Phase shift few times
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Figure 17: Phaseshift after 5 times
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Figure 18: Phase-shift after 10 times
The effects of phase shift was noticed and kept in the log book
Then all the steps from 1-6 in the guide line was done for squarewave but
this time with rectify option.
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Figure 19:steps from 1-6 for squarewave with rectify option
Then clearing the screen we’ve selected sawtooth wave form and done
steps 1-6 and then phaseshift just like before and recorded our
observation.
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Figure 20:steps 1-6 for sawtooth
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Figure 21: phaseshift for saw tooth
Calculation for part A:
Sqarewave:
an= 2T
∫−T /2
0
−cos (n .ω.t )dt+ 2T∫0
T /2
cos (n .ω. t )dt
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a1= 1π
[sin (π )+sin (−π ) ]
∴a1=1−1=0
Similarly,1π
¿= 0
bn= 2T
∫−T /2
0
−sin (n .ω. t )dt+ 2T∫0
T /2
sin (n .ω. t )dt
bn= 1nπ
¿
b1= 1π
¿
So,
b2= 12 π
¿
b3= 13 π
¿
b 4=14 π
¿
b5= 15 π
¿
Square wave with rectifier:
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a0= 1T∫0
T /2
x ( t )dt= 1T [T2 −0]=0.5[without rectify this value would be zero]
an= 2T∫0
T2
cos (nωt )dt= 2Tnω
¿¿
∴a1=0
So the 1st term of the cosine is zero. Likewise, as sinπ=0 all the terms of cosine
terms will be zero.
Now,
bn= 2T∫0
T2
sin (nωt )dt= 1nπ
[1−cos (nπ ) ]
So
b1= 1π
[1−cosπ ]= 2π=0.636618
As we’ve seen from the previous calculation even terms of b becomes zero,hence
b3= 13 π
[1−cos (3π ) ]= 23π
=0.212206
And,
b5= 15 π
[1−cos 5π ]= 25 π
=0.1273236
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Part B: To use Fourier Analysis to decide on the bandwidth needed to support a
binary Representation of an analogue signal:
For this part we cleared the field and chose the option sawtooth then
increased the number of terms as far as possible to make the synthesized
signal close to sawtooth.
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Then after clicking the qunatize button 3times the wave formed like
staircase as given in the picture below
After that taking the reading we cleared the screen and selected cosine and
repeated steps 1-4, which were given in the guidelines.
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The applet was refreshed and the field was cleared. Then step 1-2 was
repeated which had sawtooth wave form with 5quantisatoin levels and 5
samples/period. After that all the steps were repeated but the quantize
button was used only twice.
Part C: To use Fourier Analysis to look at an unusual signal.
Here we cleared everything and put on triangle option. Then the slider was
moved until it looked like the original signal.
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Now we used clip button 1-15 times and observed how the spectra
changed.
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Figure 22: After 5 clicks
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Figure 23: After 15 clicks
Later on the slider was moved to the farthest to the left until few
harmonics were left, this had a very poor representation of the desired
signal.
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The effects were written down in the log book copy.
Part D: To use Fourier series to analyse noise.
For this part field was cleared then noise option was introduced. Then the
slider was taken to the right side one click at a time. The change was
observed for the cosine and sine part of the spectra.
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Figure 24: At zero number of terms
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Figure 25: for a large number of terms.
Discussion:
In discussion we’ll talk about the observation and the questions we face during
the whole experiment. Observations are described separately for each section.
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Observation of part A:
Question: Why is the red line flat?
The spectrum is represented by the number of terms. In the 0th term there is no
spectrum so the red line is flat. But we can find a dot in the cosine part because,
cosine0=1, but sine0=0. So there is no dot in sine part.
Moreover, when the frequency spectrum of the periodic wave is set to the
average, a0 of the Fourier series, the red line is flat representing the average of
the square wave, which remains constant throughout the time. The average has a
cosine component with no sine component as the argument of the functions is (n
ωt) =0
The values we got from the java applet and from the calculations were
exact same.
Question: The red line is the bandwidth-limited version of the white line.
Now the question is: ‘At what point does the red Line looks enough like the
white line to be an acceptable compromise’?
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The red line represents the limited amount of the bandwidth while the white line
is the ideal signal. From the experiment, we can say the red line looks like the
white line when the number of terms of the spectrum is quite large. After
observation we decided to keep number of terms: 120 to get the replica of
original waveform.
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Phase shift:
When the phase shift button was pressed, the wave form changed. At time
of 5th time the wave shifted 90⁰ and after 10th time the wave shifted 180⁰.
So each click of phase-shift shifts 18⁰ in the wave form.
Question: Has the requirement for bandwidth been altered in terms of the
magnitudes of the Components as the spectra moves to the right? What is
the possible importance of this shift for a communications system?
From the observation, the more the number of harmonics used to represent the
square wave, or a discrete signal, the better is the interpretation of the signal. So
the number of harmonic sinusoids used is important in any transmission of signal.
Therefore implying better understanding of the information transmitted at the
receiver end, i.e. better quality of transmission. But for more harmonics we need
higher bandwidth. Higher bandwidth costs more. So as an engineer’s point of
view we would check the resources and try to use all of them to get the optimum
bandwidth for the best quality of service for the communication system.
The term phase-shift has a great importance in communication system. Because it
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through the same medium at the same frequency at same time by varying their
phase angles. But phase shifting can also be an undesired matter, if it occurs
naturally. This might phase shift the transmitted signal during transmission hence
making demodulation at the receiver end. Phase shifting can also cause time
delay in communication systems. This can be a real problem in full duplex
communications.
For sawtooth wave form after great deal of observation we decided to keep
the number of terms as 50 and the frequency as 540Hz. When we used the
phase-shift button same thing happened for the sawtooth as happed for
squarewave.
Observation of part B:
Quantization: there were 5quantizaton on level L, so n per sawtooth period
was 2.32 and the sampling frequency 0.2Hz. when the cosine was used we
found
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f=1Hz
T = 1 sec
1 sample = 3 bits
10 samples = 30 bit
= 30Hz
So the sampling frequency = 30Hz.
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Problem: a digital communications system which transmits this quantized
sawtooth as a stream of bits. There are 5 levels, so how many bits k will you
need per sample?
Ans: Here the number of levels= 5,
Then the minimum number of bits =k
We know, 2k=¿number of combinations
So, 2k ≥5 , this means, 23≥5∧22≤5.
So k= 3.
The signal of 5 levels shown below is created in the java applet manually.
We can represent our signal using approximately 35 harmonics.
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A cosine wave of frequency 1Hz and amplitude 1volt can be represented in
two discrete levels, where an amplitude 1volt is represented by ‘1’ and an
amplitude 0volt can be represented by ‘0’. The fundamental frequency
required to do this would be 1Hz, same as that of the cosine wave. A
rectified square could be used to analyze the number of harmonics
required to describe this quantized cosine wave, as this square wave is the
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worst case representation of the quantized cosine wave which would
require the most bandwidth.
If the cosine wave is only quantized twice the amount of harmonics
required to represent the bit stream would be lesser, but compromising the
quality of transmission, i.e. a less precise interpretation at the receiver end.
Observation of Part C:
Question: why the signal requires so many spectra to represent it
accurately?
Ans: the reasons are
I. Triangle wave has only cosine components. The slope of the wave
shows the cosine component of the 1st harmonic. Then the number
of line spectra required to represent is very few.
II. When the wave is clipped 15 times and when only few harmonics are
used to represent the resultant wave, the red line would be a very
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poor representation. The reason for this is slope of the resultant
wave is either zero or very closer to infinite.
III. When the wave is clipped the amplitude of the wave is increased
keeping the frequency the same and also representing any amplitude
greater that of the original wave by that of the original wave. Hence
if we keep clipping the wave, it starts to look more like a square
wave.
Observation of part D:
When the slider is moved to right the sine and cosine waves changes and the
change is unpredictable. Because noise is unpredictable as we slide it to the right
side the red signal becomes like the white line which means it represents noise.
Conclusion:
The whole experiment gives us broader picture of signals and how it responses to
different conditions. Fourier series helps us to break down the signals to sine and Page | 69
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cosine parts. The java applet helps us more to analyze the waveforms in details as
in point to point. Moreover Fourier analysis converts the signal from time domain
to frequency domain.
The four parts of the analysis stresses about the effect of rectification, phase-shift,
clipping, quantization of different wave forms like cosine, sawtooth, square and
triangle wave form.
Part A: explains about bandwidth required to transmit periodic signal. The fourier
analysis java applet and the manual calculation made clear how to calculate
bandwidth and the importance of phase-shift.
Part B: This part shows the importance of functions of quantization and the
transformation of the analogue signal into a discrete signal. And it also emphasis
on the concept of number of bits needed for transmitting a quantized signal.
Part C: This part says more about the function of clip altering the shape of a
triangular waveform into a square wave signal
Part D: This experiment made the fundamentals of white noise clear and its
properties, also the effect of noise.
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To sum up, the experiment tells about signal, time domain, frequency domain,
Fourier series & analysis and how we can use the knowledge in communication
system as an engineer.
Glossary:
[1] Fourier Series Java Applet: a special program written in JAVA language to
express an arbitrary periodic function as a sum of cosine terms. In other words,
Fourier series can be used to express a function in terms of the frequencies
(harmonics) it is composed of.
[2] Time domain: is a term used to describe the analysis of mathematical
functions, or physical signals, with respect to time. In the time domain, the signal
or function's value is known for all real numbers, for the case of continuous time,
or at various separate instants in the case of discrete time. An oscilloscope is a
tool commonly used to visualize real-world signals in the time domain. A time
domain graph shows how a signal changes over time, whereas a frequency
domain graph shows how much of the signal lies within each given frequency
band over a range of frequencies.
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[3] Frequency domain: is a term used to describe the analysis of mathematical
functions or signals with respect to frequency. Frequency-domain graph shows
how much of the signal lies within each given frequency band over a range of
frequencies. A frequency-domain representation can also include information on
the phase shift that must be applied to each sinusoid in order to be able to
recombine the frequency components to recover the original time signal.
[4] Spectrum: an array of entities, as light waves or particles, ordered in
accordance with the magnitudes of a common physical property, as wavelength
or mass: often the band of colors produced when sunlight is passed through a
prism, comprising red, orange, yellow, green, blue, indigo, and violet.
[5] Bandwidth: Measurement of the capacity of a communications signal. For
digital signals, the bandwidth is the data speed or rate, measured in bits per
second (bps). For analog signals, it is the difference between the highest and
lowest frequency components, measured in hertz (cycles per second)
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[6] Distortion: is the alteration of the original shape (or other characteristic) of an
object, image, sound, waveform or other form of information or representation.
Distortion is usually unwanted.
[7] Fourier series: a Fourier series decomposes a periodic function or periodic
signal into a sum of simple oscillating functions, namely sines and
cosines (or complex exponentials). Fourier series were introduced by Joseph
Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.
[8]Analogue signal: is any continuous signal for which the time varying feature
(variable) of the signal is a representation of some other time varying quantity, i.e
analogous to another time varying signal
[9] Noise: is fluctuations in and the addition of external factors to the stream of
target information (signal) being received at a detector.
[10] Square wave: is a kind of non-sinusoidal waveform, most typically
encountered in electronics and signal processing. An ideal square wave alternates
regularly and instantaneously between two levels
[11] Sawtooth wave: is a kind of non-sinusoidal waveform. It is named a sawtooth
based on its resemblance to the teeth on the blade of a saw.Page | 73
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The convention is that a sawtooth wave ramps upward and then sharply drops.
[12] Triangular wave: is a non-sinusoidal waveform named for its triangular
shape. Like a square wave, the triangle wave contains only odd harmonics.
However, the higher harmonics roll off much faster than in a square wave
(proportional to the inverse square of the harmonic number as opposed to just
the inverse.
Reference:
[1] Lecture Notes, Signals and Systems Theory background notes for
academic year 2006-2007 by John Schormans, Dept of Electronic
Engineering, Queen Mary, University of London
[2] Lab Sheet, Fourier analysis, Signals and Systems, Dept of Electronic
Engineering, Queen Mary, University of London
[3] Signal and Systems by Meade and Dillon (ISBN 041240110)
[4] http://en.wikipedia.org/wiki/Fourier_transform
[5] http://en.wikipedia.org/wiki/Signal
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[6] http://en.wikipedia.org/wiki/Even_and_odd_functions
[7] http://www.falstad.com/fourier/
[8] http://www.see.ed.ac.uk/~mjj/dspDemos/EE4/tutFT.html
[9] http://mathworld.wolfram.com/FourierTransform.html
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