Unit – 2
Transforms
.
Example 1:
Using complex form,find the Fourier series of the function
f(x) = signx =
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 as
n=2k-1,n=
=
Example 2:
Using complex form find the Fourier series of the function f(x) = x2, defined on te interval [-1,1]
Solution:
Here the half-period is L=1.Therefore,the co-efficient c0 is,
For n
Integrating by parts twice,we obtain
=
=
= .
= .
If f, then
Proof .We first claim that
Let By the multiplication formula we get
Since is a good kernel, n the first integral goes to f(0) as tends to 0.Since the
Second integral clearly converges to as our claim is proved .
In general , let F(y)=f(y+x) 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 ,
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:
Where
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 complex-valued 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
Inverse fourier transform:
The fourier transform takes us from f(t) to F(
Now we recall the fourier series of f(t):
Now we transform the sums to integrals from -, and again replace Fm with F.Remembering the fact that we introduced a factor of i(and including a factor of 2 that just crops up),we have
which is a inverse Fourier transform.
Example 1:
Fourier sine integral for even function f(x):
is a Fourier sine transform of f(x)
is a Inverse Fourier sine transformation of FC(x)
Example 2:
Fourier cosine integral for even function f(x):
Fourier cosine transform of f(x)
...Inverse Fourier cosine transform of Fc(x).
Example 3:
Fourier cosine and sine transforms
Consider,
=
Example 4:
Fourier cosine transformation of the exponential function:
F(x)= ex
Solution:
Given,
F(x)= ex
From the know formula we directly apply,
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation. It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
Analysis of continuous time LTI systems can be done using z-transforms. It is a powerful mathematical tool to convert differential equations into algebraic equations.
The bilateral (two sided) z-transform of a discrete time signal x(n) is given as
Z.T[x(n)] = X(Z) =
The unilateral (one sided) z-transform of a discrete time signal x(n) is given as
Z.T[x(n) = X(Z) =
Z-transform may exist for some signals for which Discrete Time Fourier Transform (DTFT) does not exist.
The Z-transform can be defined as either a one-sided or two-sided transform.
Bilateral Z-transform
The bilateral or two-sided Z-transform of a discrete-time 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 Z-transform
Alternatively, in cases where x[n] is defined only for the single-sided or unilateral Z-transform is defined as
X (z) = Z =
In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.
An important example of the unilateral Z-transform is the probability-generating 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 = x-1. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
z-transform properties:
Z-Transform has following properties:
Linearity Property
If x(n)X(Z)
And
y(n)Y(Z)
Then linearity property states that
Time Shifting Property
Then the time shifting property states that
Multiplication by Exponential Sequence Property
If x(n)
Then multiplication by an exponential sequency property is
Time Reversal Property
If x(n)
Then time reversal property states that
Differentiation in Z-Domain OR Multiplication by n Property
If
Then multiplication by n or differentiation in z-domain property states that,
Convolution Property
If
And
Then convolution property states that
Correlation Property
If
And
Then co-relation property states that
Region of Convergence (ROC) of Z-Transform
The range of variation of z for which z-transform converges is called region of convergence of z-transform.
Properties of ROC of Z-Transforms
sequence | z-transformation |
1 | |
Example 1:
Find the z-transformation of the following left-sided sequence
Solution:
=
= 1-
=
If
Example 2:
Solution:
Long division method to obtain
2
Now x(z) can be written as,
X(z) = 2-
Example 1:
Solve the differential equation by z-transformation method.
Solution:
Given,
Let y(z) be the z-transform of
Taking z-transforms 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 Z-transforms, we obtain
Example 2:
Solve using z-transforms
Solution:
Consider,
Taking z-transforms on both sides, we get
=
or
Now,
Using inverse z-transform we obtain,
Reference Books
1. Erwin Kreyszig, “Advanced Engineering Mathematics”, Wiley India,10th Edition.
2. M.D. Greenberg, “Advanced Engineering Mathematics”, Pearson Education, 2 nd Edition.
3. Peter. V and O‟Neil, “Advanced Engineering Mathematics”, Cengage Learning,7th Edition.
4. S.L. Ross, “Differential Equations”, Wiley India, 3rd Edition.
5. S. C. Chapra and R. P. Canale, “Numerical Methods for Engineers”, McGraw-Hill, 7th Edition.
6. J. W. Brown and R. V. Churchill, “Complex Variables and Applications”, McGraw-Hill Inc, 8th Edition.