ElE 374 Lab

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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mailto:[email protected]


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


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


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


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


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


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

is used in multiplexing of various signals, i.e. various signals can be transmitted Page | 62


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


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

http://en.wikipedia.org/wiki/Non-sinusoidal_waveform

http://en.wikipedia.org/wiki/Signal_processing

http://en.wikipedia.org/wiki/Electronics

http://en.wikipedia.org/wiki/Non-sinusoidal_waveform

http://en.wikipedia.org/wiki/Signal_(information_theory)

http://en.wikipedia.org/wiki/Signal_(circuit_theory)

http://en.wikipedia.org/wiki/Continuous_function

http://en.wikipedia.org/wiki/Heat_equation

http://en.wikipedia.org/wiki/Joseph_Fourier

http://en.wikipedia.org/wiki/Joseph_Fourier

http://en.wikipedia.org/wiki/Complex_exponential

http://en.wikipedia.org/wiki/Sine_wave

http://en.wikipedia.org/wiki/Sine_wave

http://en.wikipedia.org/wiki/Periodic_function

http://en.wikipedia.org/wiki/Periodic_function


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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http://en.wikipedia.org/wiki/Roll-off

http://en.wikipedia.org/wiki/Harmonic

http://en.wikipedia.org/wiki/Square_wave

http://en.wikipedia.org/wiki/Waveform

http://en.wikipedia.org/wiki/Non-sinusoidal


[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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