Unit2
Transforms
The Fourier integral of f(x) is
………. (1)
Which can be written as
…………… (2)
Because is an even function of . Also since is an odd function of then we have
……… (3)
Multiply (3) by ‘i’ and add to (2)
We get
Which is the complex form of the Fourier integral.
Example 1:
Using the complex form,find the Fourier series of the function
f(x) = sinx =
Solution:
We calculate the coefficients
=
=
Hence the Fourier series of the function in complex form is
We can transform the series and write it in the real form by renaming it as
n=2k1,n=
=
Example 2:
Using complex form find the Fourier series of the function f(x) = x2, defined on the interval [1,1]
Solution:
Here the halfperiod is L=1.Therefore, the coefficient c0 is,
For n
Integrating by parts twice,we obtain
=
=
= .
= .
Key takeaways
2. The Laplace transform of f(t) exists for s>a, if
2. 2. is finite
3. The inverse of the Laplace transform can be defined as below
4. Multiplication by ‘s’ 
5.
6.
7.
8.
9.
10.
Fourier integral theorem
Fourier integral theorem can be stated as
Proof: we have the Fourier series of a function f(x) in interval (c, c) is
And are given as
Put the values of in equation (1), we get
As we know that cosine functions are even functions, then
Equation (2) reduces to
Here we will assume that c increases indefinitely, therefore we can take,
By this assumption, we get
From equation (3) and (4), we get
So that
Example:1
Find the fourier integral representation of the function
Solution:
The graph of the function is shown in the below figure satisfies the hypothesis of
Theorem 1 . Hence from Eqn,(5) and (6), we have
Substituting these coefficients in Eqn.(4) we obtain
This is the Fourier integral representation of the given function.
Example:2
Find the Fourier integral representation of the function
Solution:
The graph of the given function is shown in the below figure . Clearly, the given function f(x) is an even function. We represent f(x) by the fourier cosine integral . We obtain
And thus ,
Key takeaways
2. complex form of the Fourier integral
3. Fourier integral theorem
The following notations are used to find the Fourier sine and cosine integrals.
If the function f(x) is even
B(w)= 0
If the function f(x) is odd
Example:1
Find the Fourier cosine integral of , where x>0, k>0 hence show that
Solution:
The Fourier cosine integral of f(x) is given by:
Example:2
Find the half –range sine series of the function
Solution: xx
Where
The function
Is called the Fourier transform of the function f(x).
The function f(x) given below is called inverse Fourier transform of F(s)
Properties of Fourier transform
Property1: Linear property If F(s) and G(s) are two Fourier transforms of f(x) and g(x) respectively, then
Where a and b are the constants.
Property2: Change of scale If F(s) is a Fourier transform of f(x), then
Property3: Shifting property If F(s) is a Fourier transform of f(x), then
Property4: Modulation If F(s) is a Fourier transform of f(x), then
Example: Find the Fourier transform of
Hence evaluate
Sol. As we know that the Fourier transform of f(x) will be
So that
For s = 0, we get F(s) = 2
Hence by the inverse formula, we get
Putting x = 0, we get
So
Example: Find the Fourier transform of
Sol. As we know that the Fourier transform of f(x) will be
So that
Now put
So that
Key takeaways
Is called the Fourier transform of the function f(x).
2. Linear property If F(s) and G(s) are two Fourier transforms of f(x) and g(x) respectively, then
3. Change of scale If F(s) is a Fourier transform of f(x), then
4. Shifting property If F(s) is a Fourier transform of f(x), then
5. Modulation If F(s) is a Fourier transform of f(x), then
Then
The function is known as the Fourier sine transform of f(x) in 0<x<∞
And the function f(x) s called the inverse Fourier sine transform of .
And
Then
The function is known as the Fourier cosine transform of f(x) in 0<x<∞
And the function f(x) s called the inverse Fourier cosine transform of .
Finite Fourier sine and cosine transforms
The finite Fourier sine transform of f(x), in 0<x<c, is given by
Where n is an integer.
And the function f(x) is called the inverse finite Fourier sine transform of which is defined by
The finite Fourier cosine transform of f(x), in 0<x<c, is given by
Where n is an integer.
And the function f(x) is called the inverse finite Fourier cosine transform of which is defined by
Inverse Fourier transforms
The f(x) given below is called inverse Fourier transform of F(s)
2. Inverse Fourier cosine transform of F(s)
Example1: Find the Fourier sine transform of
Sol. Here x being positive in the interval (0, ∞)
Fourier sine transform of will be
Example2: Find the Fourier cosine transform of
Sol. We know that the Fourier cosine transform of f(x)
=
=
=
Example3: Find the Fourier sine transform of
Sol. Let
Then the Fourier sine transform will be
Now suppose,
Differentiate both sides with respect to x, we get
……. (1)
On integrating (1), we get
Key takeaways
Where n is an integer.
2. The finite Fourier cosine transform of f(x), in 0<x<c, is given by
Where n is an integer.
3. Inverse Fourier sine transform of F(s)
4. Inverse Fourier cosine transform of F(s)
In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes a function (often a function of time, or a signal) into its constituent frequencies, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.
The Fourier transform of a function of time is a complexvalued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem
The function
Is called the Fourier transform of the function f(x).
The function f(x) given below is called inverse Fourier transform of F(s)
Properties of Fourier transform
Property1: Linear property If F(s) and G(s) are two Fourier transforms of f(x) and g(x) respectively, then
Where a and b are the constants.
Property2: Change of scale If F(s) is a Fourier transform of f(x), then
Property3: Shifting property If F(s) is a Fourier transform of f(x), then
Property4: Modulation If F(s) is a Fourier transform of f(x), then
Example: Find the Fourier transform of
Hence evaluate
Sol. As we know that the Fourier transform of f(x) will be
So that
For s = 0, we get F(s) = 2
Hence by the inverse formula, we get
Putting x = 0, we get
So
Example: Find the Fourier transform of
Sol. As we know that the Fourier transform of f(x) will be
So that
Now put
So that
Finite Fourier sine and cosine transforms
The finite Fourier sine transform of f(x), in 0<x<c, is given by
Where n is an integer.
And the function f(x) is called the inverse finite Fourier sine transform of which is defined by
The finite Fourier cosine transform of f(x), in 0<x<c, is given by
Where n is an integer.
And the function f(x) is called the inverse finite Fourier cosine transform of which is defined by
Example1: Find the Fourier sine transform of
Sol. Here x being positive in the interval (0, ∞)
Fourier sine transform of will be
Example2: Find the Fourier cosine transform of
Sol. We know that the Fourier cosine transform of f(x)
=
=
=
Example3: Find the Fourier sine transform of
Sol. Let
Then the Fourier sine transform will be
Now suppose,
Differentiate both sides with respect to x, we get
……. (1)
On integrating (1), we get
In mathematics and signal processing, the Ztransform converts a discretetime signal, which is a sequence of real or complex numbers, into a complex frequencydomain representation. It can be considered as a discretetime equivalent of the Laplace transform. This similarity is explored in the theory of timescale calculus.
Analysis of continuoustime LTI systems can be done using ztransforms. It is a powerful mathematical tool to convert differential equations into algebraic equations.
The bilateral (twosided) ztransform of a discretetime signal x(n) is given as
Z.T[x(n)] = X(Z) =
The unilateral (one sided) ztransform of a discrete time signal x(n) is given as
Z.T[x(n) = X(Z) =
Ztransform may exist for some signals for which DiscreteTime Fourier Transform (DTFT) does not exist.
Definition
The Ztransform can be defined as either a onesided or twosided transform.
Bilateral Ztransform
The bilateral or twosided Ztransform of a discretetime signal x(n) is the formal power series X(z) defined as
X (z) = Z =
where n is an integer and z is, in general, a complex number:
where A is the magnitude of z,j is the imaginary unit and is the complex argument (also referred to as angle or phase) in radians.
Unilateral Ztransform
Alternatively, in cases where x[n] is defined only for the singlesided or unilateral Ztransform is defined as
X (z) = Z =
In signal processing, this definition can be used to evaluate the Ztransform of the unit impulse response of a discretetime causal system.
An important example of the unilateral Ztransform is the probabilitygenerating function, where the component x[n] is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s) in terms of s = x1. The properties of Ztransforms (below) have useful interpretations in the context of probability theory.
Standard properties
Ztransform properties:
ZTransform has the following properties:
1. Linearity Property
If x(n)X(Z)
And
y(n)Y(Z)
Then linearity property states that
2. Time Shifting Property
Then the timeshifting property states that
Multiplication by Exponential Sequence Property
If x(n)
Then multiplication by an exponential sequence property is
3. Time Reversal Property
If x(n)
Then time reversal property states that
Differentiation in ZDomain OR Multiplication by n Property
If
Then multiplication by n or differentiation in zdomain property states that,
4. Convolution Property
If
And
Then convolution property states that
5. Correlation Property
If
And
Then corelation property states that
Region of Convergence (ROC) of ZTransform
The range of variation of z for which ztransform converges is called the region of convergence of ztransform.
Properties of ROC of ZTransforms
Some standard formulae
1.
2.
3.
4.
5.
6.
Example1: Find Ztransform of the following functions
(i)
(ii)
Sol.(i)
(ii)
Example2: Find Ztransform of the following functions
(i)
(ii)
Sol. (i) As we know that
So that
(ii) we know that
So that
The ZT of important sequences are given in the following table
Sequence k  ZT 

F(z) 
 
1 
 
U(k) 
 
k  z>1  
z>1  
z>a  
z>a  
z<a  
z<a  
a<z<1/a  
Inverse ZT
sequence  ztransformation 
1  
Example 1:
Find the ztransformation of the following leftsided sequence
Solution:
=
= 1
=
If
Example 2:
Solution:
Long division method to obtain
2
Now x(z) can be written as,
X(z) = 2
Working procedure to solve the linear difference equation by ZT
1. Take the ZT of both sides of the differential equations.
2. Transpose all terms without U(z) to the right
3. Divide by the coefficients of U(z)
4. Express this function in terms of the Ztransform of known function and take the inverse ZT of both sides.
This will give as a function of n, is the required solution.
Example 1:
Solve the differential equation by the ztransformation method.
Solution:
Given,
Let y(z) be the ztransform of
Taking ztransforms of both sides of eq(1) we get,
ie.
using the given condition,it reduces to
(z+1)y(z) =
Ie.
Y(z) =
Or Y(Z) =
On taking inverse Ztransforms, we obtain
Example 2:
Solve using ztransforms
Solution:
Consider,
Taking ztransforms on both sides, we get
=
or
Now,
Using inverse ztransform we obtain,
Example3:
Solve the following by using Ztransform
Sol. If then
And
Now taking the Ztransform of both sides, we get
z[
It becomes
So,
Now
On inversion, we get
References
1. Erwin Kreyszig, Advanced Engineering Mathematics, 9thEdition, John Wiley & Sons, 2006.
2. N.P. Bali and Manish Goyal, A textbook of Engineering Mathematics, Laxmi Publications.
3. Higher engineering mathematic, Dr. B.S. Grewal, Khanna publishers