unit ii analysis of continuous time...

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NH-67, TRICHY MAIN ROAD, PULIYUR, C.F. – 639 114, KARUR DT.

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

COURSE MATERIAL

Subject Name : Signals and Systems Class/Sem : BE(ECE) / III

Subject Code : EC 2204 Staff name : G.Vijayakumari

Unit II ANALYSIS OF CONTINUOUS TIME SIGNALS

Fourier series analysis - Spectrum of C.T. signals - Fourier Transform

and Laplace Transform in Signal Analysis.

Objective:

The main objective of this unit is that students will be able to

understand, the definition of Fourier series, different ways of Fourier series

representation, calculation of power using parsevals theorem, properties of

Fourier series, the definition of Fourier Transform, the properties of Fourier

Transform The definition of Laplace Transforms, properties of Laplace

transforms, examples of Laplace Transforms, example for deriving Laplace

Transform from waveforms.

Keywords: Continuous Time Signals, Fourier series analysis, Fourier

Transform, parsevals Theorem, Laplace Transform.

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Contents:

2.1 Fourier Series representation of Periodic Signals

2.2 Dirichlet Conditions for convergence of Fourier series.

2.3 Properties of Fourier series

2.4 Fourier Transform

2.5 Dirichlet Conditions for convergence of Fourier Transform.

2.6 Properties of Fourier Transform

2.7 Laplace Transform

2.8 Properties of Laplace Transform

2.9 Sampling

2.1 Fourier Series representation of Periodic Signals

Consider a periodic signal x(t) with fundamental period T, i.e.

Then the fundamental frequency of this signal is defined as the reciprocal of

the fundamental period, so that

Under certain conditions, a periodic signal x(t) with period T can be

expressed as a linear combination of sinusoidal signals of discrete

frequencies, which are multiples of the fundamental frequency of x(t).

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Further, sinusoidal signals are conveniently represented in terms of complex

exponential signals. Hence, we can express the periodic signal in terms of

complex exponentials, i.e.

Such a representation of a periodic signal as a combination of complex

exponentials of discrete frequencies, which are multiples of the fundamental

frequency of the signal, is known as the Fourier Series Representation of the

signal.

Frequency Domain Representation

From the above discussion, we can say that a periodic signal whose

Fourier Series Expansion exists, can be represented uniquely in terms of it's

Fourier co-efficients. These co-efficients correspond to particular multiples

of the fundamental frequency of the signal. Thus, the signal may be

equivalently represented as a discrete signal on the frequency axis.

This is called the Frequency domain representation of the signal.

We next discuss the conditions under which the Fourier Expansion is valid.

2.2 Convergence of Fourier series (Dirichlet’s conditions):

1) x(t) should be absolutely integrable over a period.

2) x(t) should have only a finite number of discontinuities over one

period. Furthermore, each of these discontinuities must be finite.

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3) The signal x(t) should have only a finite number of maxima and

minima in one period.

If the signal satisfies the above conditions, then at all points where the

signal is continuous, the Fourier Series converges to the signal.

Problem 1

Determine the Fourier series representation for the following signal.

Solution

It is periodic with period 2. So, consider segment between .

Let

be the Fourier expansion, where:

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For ,

But, in

Substituting in x(t)

Problem 2


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Solution

It is periodic with period 6. So, consider segment between .

The function in this interval is:

Let


For ,

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Problem 3


Solution

It is periodic with period 3, so take the segment

The function in the interval is:

Let

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For ,


Problem 4

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Solution

It is periodic with period 6. So, take the segment

Here will be

Let


For

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Problem 5


Solution

It is periodic with period 3. So, take the segment

Here will be

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Let


For


Problem 6

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Determine the Fourier series representation for the periodic signal x(t)

with period 2 and for .

Let


Given .

For


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2.3 Properties of Fourier series

1) Time shifting

If x(t) is periodic with period T , x( t - t0) has Fourier series

coefficients

2) Time scaling

If a > 0, x(at) is periodic with period ( T / a ) and now ck becomes Fourier

coefficient corresponding to frequency .

If a < 0, x(at) is periodic with period ( T / -a) and now ck becomes Fourier

coefficient corresponding to frequency .

3) Parseval’s Relation

Let x(t) and y(t) be periodic with a common period T.

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Applying the Convolution theorem equivalent we have just proved on

we get:

Put t=0, to get

If we take x = y, then T becomes the fundamental period of x and:

Note the left-hand side of the above equation is the power of x(t).

2.4 Fourier Transform

A very basic concept in Signal and System analysis is Transformation

of signals. For example, the transformation of a signal from the time domain

into a representation of the frequency components and phases is known as

Fourier analysis.

Need for transformations

We can't analyze all the signals in their existing domain.

Transforming a signal means looking at a signal from a different angle so as

to gain new insight into many properties of the signal that may not be very

evident in their natural domain. Transformation is usually implemented on

an independent variable.

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Every periodic signal can be written as a summation of sinusoidal

functions of frequencies which are multiples of a constant frequency (known

as fundamental frequency). This representation of a periodic signal is called

the Fourier series. An aperiodic signal can always be treated as a periodic

signal with an infinite period. The frequencies of two consecutive terms are

infinitesimally close and summation gets converted to integration. The

resulting pattern of this representation of an aperiodic signal is called the

Fourier Transform.

The Fourier Transform equation

The Fourier Transform of a function x(t) can be shown to be:

This equation is called the Fourier Transform equation.

This equation is called the Inverse Fourier Transform equation, x(t) being

called the Inverse Fourier Transform of X(f).

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Dirichlet Conditions for convergence of Fourier Transform

x(t) is absolutely integrable . i.e.:

x(t) has only a finite number of maxima and minima in any finite

interval.

x(t) has only a finite number of discontinuities in any finite interval.

2.5 Properties of Fourier Transform

1) Differentiation

Hence if

then

Now,

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Hence if,

then,

2) Time scaling

3) Convolution theorem

If two signals x(t) and y(t) are Fourier Transformable, and their

convolution is also Fourier Transformable, then the Fourier Transform of

their convolution is the product of their Fourier Transforms.

4) Parseval's theorem

The Parseval's theorem states that the inner product between

signals is preserved in going from time to the frequency domain. This is

interpreted physically as “Energy calculated in the time domain is same

as the energy calculated in the frequency domain”.

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Proof:

Hence Proved.

2.6 Laplace Transform

Laplace Transform is a powerful tool for analysis and design of

Continuous Time signals and systems. The Laplace Transform differs from

Fourier Transform because it covers a broader class of CT signals and

systems which may or may not be stable.

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Till now, we have seen the importance of Fourier analysis in solving

many problems involving signals. Now, we shall deal with signals which do

not have a Fourier transform.

We note that the Fourier Transform only exists for signals which can

absolutely integrated and have a finite energy. This observation leads to

generalization of continuous-time Fourier transform by considering a

broader class of signals using the powerful tool of "Laplace transform".

With this introduction let us go on to formally defining both Laplace

transform.

Definition of Laplace transforms:

The Laplace transform of a function x(t) can be shown to be:

This equation is called the Bilateral or double sided Laplace transform

equation.

x(t) = dssX est

∫∞

∞−

)(

This equation is called the Inverse Laplace Transform equation, x(t) being

called the Inverse Laplace transform of X(s).

The relationship between x(t) and X(s) is

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Region of Convergence (ROC):

The range of values for which the expression described above is finite

is called as the Region of Convergence (ROC).

Relationship between Laplace Transform and Fourier Transform

The Fourier Transform for Continuous Time signals is infact a special

case of Laplace Transform. This fact and subsequent relation between LT

and FT are explained below.

Now we know that Laplace Transform of a signal 'x'(t)' is given by:

The s-complex variable is given by

But we consider and therefore 's' becomes completely imaginary. Thus

we have . This means that we are only considering the vertical strip at

.

From the above discussion it is clear that the LT reduces to FT when the

complex variable only consists of the imaginary part . Thus LT reduces to

FT along the (Imaginary axis).

Example of Laplace Transform

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(1) Find the Laplace transform and ROC of x(t) = eat−

u(t).

we notice that by multiplying by the term u(t) we are effectively considering

the unilateral Laplace Transform whereby the limits tend from 0 to +∞

Consider the Laplace transform of x(t) as shown below

X(s) = dttx est

∫∞

∞−

−

)(

= dteestat

∫∞

−−

0

= dtetas

∫∞

+−

0

)(

= as +

1; for (s+a) > 0

(2) Find the Laplace transform and ROC of x(t) = eat−

− u(-t).

X(s) = dttx est

∫∞

∞−

−

)(

= dteestat

∫ −∞−

−−

0

= dtetas

∫∞−

+−

0)(

= as +

1; for (s+a) < 0

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If we consider the signals e-at

u(t) and -e-at

u(-t), we note that although

the signals are differing, their Laplace Transforms are identical which is

1/( s+a). Thus we conclude that to distinguish L.T's uniquely their ROC's

must be specified.

2.7 Properties of Laplace Transform

1) Linearity

If with ROC R1 and with ROC

R2, then with ROC containing

.

The ROC of X(s) is at least the intersection of R1 and R2, which

could be empty, in which case x(t) has no Laplace transform.

2) Differentiation in the time domain

If with ROC = R then with ROC =

R.

This property follows by integration-by-parts. Specifically let

Then,

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And Hence

The ROC of sX(s) includes the ROC of X(s) and may be larger.

3) Time Shift

If with ROC = R then

with ROC = R

4) Time Scaling

If with ROC=R, then

Let

Inverse Laplace transform

Problem 1

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Determine the function of time x (t), for the following Laplace Transform

and its associated regions of convergence:

Solution

Problem 2



Solution

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Problem 3



Solution

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Note:

Relationship between Laplace and Fourier Transform

Laplace transform becomes Fourier transform

if σ=0 and s=jω.

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2.8 SAMPLING

Band-limited signals:

A Band-limited signal is one whose Fourier Transform is non-zero on

only a finite interval of the frequency axis.

Consider a signal x(t) having bandwidth less than B.

The signal x(t) and the signal ( ) obtained by multiplying the signal by a

periodic train of impulses, separated by , having strength 1 are shown

below.

Our goal of achieving a sampled signal is possible by the multiplication of

the original C.T. signal with the generated train of pulses. Now these two

signals are multiplied practically with the help of a multiplier as shown in

the schematic below. In our analysis so far, this is how we imagined

sampling of a signal.

Now the Fourier Series Representation of 'p(t)' is given as:

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Where the Fourier Coefficients of the series are defined as:

We now see what happens to the spectrum of continuous time signal on

multiplication with the train of pulses. Having obtained the Fourier Series

Expansion for the train of periodic pulses the expression for the sampled

signal can be written as:

Taking Fourier transform on both sides and using the property of the Fourier

transform with respect to translations in the frequency domain we get:

This is essentially the sum of displaced copies of the original

spectrum modulated by the Fourier series coefficients of the pulse train. If

'x(t)' is Band-limited so long as the the displaced copies in the spectrum do

not overlap. For this the condition that 'fs' is greater than twice the bandwidth

of the signal must be satisfied. The reconstruction is possible theoretically,

using an Ideal low-pass filter as shown below:

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Thus the condition for faithful reconstruction of the original

continuous time signal is fs ≥2fm: where is the bandwidth of the original

band-limited signal.

Test your Understanding:

Determine the Fourier transform of unit impulse signal.

Determine the Fourier transform of signum function.

Find the Laplace transform of hyperbolic sine function.

Summary:

Any function can be decomposed into sin and cos functions.

If the function is continuous Fourier series is infinite.

A function should satisfy Dirichlet conditions for the Fourier series to

be valid.

The plot of Fourier coefficient as function of frequency is called

Fourier spectrum.

The Fourier transform possesses a wide variety of important

properties that describe how different characteristics of signals are

reflected in their transforms.

Convolution property leads to the description of an LTI system in

terms of its frequency response.

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Two sided or bilateral Laplace Transform pair is defined as

The unilateral or single sided transform is defined as

We denote the transform from the time domain to frequency domain

and vice versa as

Review Questions:

What is Fourier series representation of a signal?

State Dirichlet’s conditions for convergence of Fourier series.

Write discrete time Fourier series pair.

List out the properties of Fourier series.

What is the need for transformation of signal?

Define Fourier Transform.

State Dirichlet’s conditions for convergence of Fourier Transform.

Define Laplace Transform.

Mention the properties of Laplace Transform.

What do you mean by sampling?

What is a band limited signal?

What are the criteria for sampling frequency to recover the original

signal?

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State Parseval’s theorem for discrete time signal.

Write the condition for the LTI system to be causal and stable.

Write synthesis and analysis equation of continuous time Fourier

series.

How do we fine Fourier series coefficient for a given signal.

State time shifting property of CT Fourier series.

State time shifting property of DT Fourier series.

Determine the Fourier series coefficient of sinωot.

State conjugate symmetry of CT Fourier series.

Determine the Fourier series coefficient of cosωon.

unit ii analysis of continuous time...

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