Z transform roc

Definition of ROC : 

The ROC stands for the region of convergence of X(Z) is set for all the values of Z for which X(Z)  attains a finite value. Every time when we find Z transform we must indicate its ROC.

The significance of ROC :
  • ROC will decide whether a system is stable or unstable
  • ROC also determine the type of sequence that means: Causal or noncausal, Finite or infinite
Properties of ROC for Z transform :
  • The ROC is a ring, whose center is at an origin
  • ROC cannot contain any pole
  • If x(n) is causal then ROC is exterior part of a circle of radius say ɖ
  • If x(n) is anticausal then ROC is interior part of a circle of radius say ɖ
  • If x(n) is  two-sided sequence then ROC is the intersection of two circles of radii ɖ and β
  • The ROC must be connected region
  • If ROC of X(Z) include unit circle then and then only the Fourier transform of DT sequence x(n) converges
  • For a finite duration sequence, x(n) the ROC is the entire Z plane except Z = 0 and Z = ∞
Summary of ROC :
  • For Causal system, ROC is entire Z plane except |Z| = 0 
  • For the anti-causal system, ROC is the entire Z plane except |Z| = ∞
  • For two-sided sequence ROC is entire Z plane except  |Z| = 0  and |Z| = ∞

z transform convolution

Statement :

If  x1(n) * x2(n) ↔ X1(Z)  * X2(Z)         


Proof :
  
              ∞
y (n)   = ∑      x(k) h(n-k)
            k=-∞

                 ∞           ∞
y(z)  =       ∑       [   ∑   x(k) h(n-k) ] Z-n
                n=-∞    k=-∞

                  ∞                         ∞
y(z)  =       ∑      x(k) Z-k      ∑  h(n-k)  Z-(n-k)
                k=-∞                n=-∞

n=l and k=0 in second terms;

                  ∞                       ∞
y(z)  =       ∑      x(k) Z-k    ∑  h(l)  Z-l
                k=-∞                l=-∞


Y(z) = X(Z) H(Z)

Application of z transform

The field of signal processing is essentially a field of signal analysis in which they are reduced to their mathematical component and evaluated. One important concept of signal processing is that of the Z transform method which converts unwieldy sequence into a form that can be easily dealt with. Z transform is used for much signal processing application. Here this article gives information about the application of Z transform to know more details about it.
  • Pole-zero description of the discrete-time system
  • Analysis of linear discrete signal
  • Use to analysis digital filter
  • Used to find the frequency response
  • Obtain impulse response estimation
  • Determine the difference equation
  • Analysis of discrete signal
  • Calculation of  a signal to control system
  • Direct computer evolution of frequency response
  • Voice transmission
  • Enhance the electrical as well as mechanical energy to provide a dynamic nature of the system
  • Geometric evolution of frequency response
  • Frequency response estimation via the FFT
  • Used to simulate the continuous systems
  • It also helps in system design, analysis and also checks the system stability
  • For automatic control in telecommunication
  • Used in designing digital filters

Project topic for electronics engineering students

Full forms related to electronics engineering terms

Important definition for electronics / electrical engineering students

Important books to refer for electronics engineering students

Digital logic design books

1. Book name: Digital electronics and logic design
Author: Gate academy team
Publisher: Gate academy publications

2. Book name: Digital  logic and computer design
Author: M Morris Mano
Publisher: Pearson

3. Book name: Fundamental of  Digital  Logic with VHDL design 
Author: Stephen Brown
Publisher: Mcgraw hill education

4. Book name: Digital electronics and logic design
Author: A P Adsul, R H Jagdale, SS  Kulkarni
Publisher: Nirali Prakashan

5. Book name: Digital logic design
Author: Alam mansaf, Alam basis
Publisher: PHI learning Pvt Ltd
6. Book name: Digital logic: application and design
Author: John M Yarbrough
Publisher: Cengage Learning
7. Book name: Digital logic design: principal
Author: Bradley  Carlson norman  balabanian
Publisher: Wiley
8. Book name: Digital circuit and  logic design 
Author: Lee c Samuel
Publisher: PHI learning  

 9.  Book name : Digital principal  logic design 
Author : N manna
Publisher : Laxmi publication 

Advantages of Z transform

The Z transform provides a valuable technique for analysis design of discrete time signals as well as discrete time LTI systems. This article gives information about the advantages and disadvantages of Z transform to know more details about it.

Advantages of Z transform :
  • Z transform is used for the digital signal
  • Both Discrete-time signals and linear time-invariant (LTI) systems can be completely characterized using Z transform
  • The stability of the linear time-invariant (LTI)  system can be determined using the Z transform
  • By calculating Z transform of the given signal, DFT and FT can be determined
  • The entire family of digital filters can be obtained from one proto-type design using Z transform
  • The solution of differential equations can be simplified or obtained by using Z transform theorem
  • Mathematical calculations are reduced using the Z transform. For example, consider the convolution operation is transformed into simple multiplication operation 
Disadvantages of Z transform :
  • Z transform cannot apply in the continuous signal

What is z transform

The Z transform has a real and imaginary part like Fourier transform. A plot of imaginary part versus the real part is called a Z plane or complex Z plane.

The pole-zero plot is the main characteristic of discrete time LTI systems full form. We can also check the stability of the system using a pole-zero plot. Z transform used for many signal processing

There are two types of Z transform :

1. Single sided Z transform
2. Double-sided Z transform 

1. Single sided Z transform :

As we know that a single-sided Z transform of discrete time signal x (n) is defined as,

                ∞
X  (Z) =  ∑     x (n)  Z -n

              n=0

Here "Z" is a complex variable. In this equation limits of summation are from the 0 too ∞. So while expanding the summation we will put only positive values of n (from the range n = 0 ton =  ∞). So this is single-sided or one-sided Z- transform.

2. Double-sided Z transform 

As we know that a double-sided Z transform of discrete time signal x (n) is defined as,

                ∞
X  (Z) =  ∑     x (n)  Z -n

              n= - ∞

While expanding the summation we will put both positive as well as negative values of "n". Thus this is called a double-sided Z transform.

How to denote Z transform?

The relationship between x(n) and X(Z) is indicated as given below :

         Z
x (n) ↔ X (Z)

Note that X (Z) is Z transform of x (n), always when Z transform of the sequence is obtained then it is denoted by capital letter, X (Z) Here arrow is bidirectional. This is because we can also obtain x (n) from X (Z) using the inverse Z transform.

The Z transform of x (n) is also denoted as in this form,

X (Z) = Z  { x(n) }

So here x (n) and  X(Z) are called as Z transform pairs.

In same way if we want to analyze a system, which is already represented in frequency domain, a discrete time signal then we go for inverse Z transformation.

Audio video system books

1. Book name: Audio and video system
Author: Gupta
Publisher: Mcgraw hill education

2. Book name: Audio  video system
Author: SP Bali, Rajeev Bali
Publisher: Khanna book publishing

3. Book name: Principal of multimedia
Author: Ranjan Parekh
Publisher: Mcgraw hill education

4. Book name: Modern recording techniques
Author: David Miles Huber
Publisher: A focal press book

5. Book name: Multi-model processing and integration
Author: Potamianos, Patrick
Publisher: Springer

Time shifting property of z transform

Before we learn about time shifting property first let we learn about what is z transform system in digital signal processing subject.

 Statement : 

              Z
If  x(n) ↔ X (Z)

                      Z
Then x (n-k) ↔ Z -k  X (Z)

Proof :
                                 -∞
So here Z{x(n)} = X  (Z) = ∑     x (n)  Z-n
                                 +∞
Z { x(n-k) }  can be written as,


                    -∞
Z{x(n-k)} = ∑     x (n-k)  Z-n
                     +∞

Now  Z-n can be  written as  Z-n  =   Z-(n-k) Z-k  thus the equation 

                    -∞
Z{x(n-k)} = ∑     x (n-k)  Z-(n-k) Z-k
                     +∞

Since the limits of summation are in terms of  'n'  we can take Z-k outside the summation sign.

                             -∞
so Z{x(n-k)} = Z-k  ∑     x (n-k)  Z-(n-k) 
                             +∞

Now put n-k =m on R.H.S 

At  n = -∞, -∞-k = m → m =  -∞
At n = ∞, ∞-k = m  → m =  ∞

                              ∞
Z{x(n-k)} = Z-k  ∑     x (m)  Z-(m) 
                          m = -∞ 
                  
Comparing Equation : 
                           
Z{x(n-k)} = Z-k   X (Z)


= x (n-k) ↔ Z-k   X (Z)

Similarly, we can write 

= x (n+k) ↔ Z+k   X (Z)

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1. Book name: Wireless communication theory and application 
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2. Book name: Modern Wireless communication
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Publisher: Pearson

3. Book name: Wireless and mobile communication
Author: Upena Dalal, Manoj K Shukla
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4. Book name: Wireless communication principal and practice
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6. Book name: Wireless communication
Author: T l Singal
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7. Book name: Wireless communication
Author: Asrar U H sheik
Publisher: Pearson

8. Book name: Wireless communication network and system
Author: William Stallings, Cory Beard
Publisher: Person

9. Book name: Wireless communication and network 3G and beyond
Author: It Saha Misra
Publisher: Mcgraw hill education

Antenna and wave propagation books

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Author: Sisir K Das
Publisher: Magraw higher Ed

2. Book name: Wave propagation and Antenna engineering
Author:  Kumar Sanjay, Shukla Saurabh
Publisher: PHI Learning Pvt Ltd

3. Book name: Antenna and wave propagation
Author: K D Prasad
Publisher: Satya Prakashan

4. Book name: Antenna and wave propagation
Author: Kamal Kishore
Publisher: I K international publishing house Pvt Ltd

5. Book name: Antenna and wave propagation
Author: A K Gautam
Publisher: S K Kataria and sons

6. Book name: Antenna and wave propagation
Author: U A Bakshi
Publisher: Technical publication 

7. Book name: Antenna and wave propagation
Author: Raju
Publisher: Pearson

8. Book name: Antenna and wave propagation
Author: Ahmad s khan, Ronald J Marhefka, John D Kraus
Publisher: Magraw higher Ed

9. Book name: Antenna and wave propagation
Author: A R Harish, M Satchidananda
Publisher: Oxford

Digital image processing books

1. Book name: Digital image processing
Author: Rafael c Gonzalez, Richard E. Wood
Publisher: Pearson

2. Book name: Principal of digital image processing

Author: Burger Burge, Wilhelm mark j
Publisher: Springer Verlag London Ltd

3.  Book name: Digital image processing
Author: T Veerkumar, S Jayaraman, S esakkirajan
Publisher: Mcgraw higher Ed

4.  Book name: Digital image processing
Author: Abhishek Yadav
Publisher: Laxmi publications new Delhi

5.  Book name: Digital image processing using MATLAB
Author:  Rafael c Gonzalez
Publisher: Mcgraw higher Ed

6.  Book name: Fundamental of digital image processing 
Author:  Anil k jain
Publisher: PHI learning

7.  Book name: Introductory digital image processing 
Author: John r Jensen
Publisher: Pearson India

8. Book name: Fundamental of digital image processing 
Author: Annadhurai
Publisher: Pearson 

9. Book name: Digital image processing  method
Author: Dougherty
Publisher: Taylor and Francis

10. Book name: Digital signal and image processing 
Author: Soman k.p.Et.al
Publisher: Cengage Learning

Properties of even and odd signals

  • The sum of two even signals is even signal
  • The sum of two odd signals are odd
  • The sum of an even signal and an odd signal is neither even nor odd signal
  • The product of two even signals is even
  • The product of two odd signals is  even
  • The product of two odd signals and even the signal is odd
Show in  :

Addition / Subtraction :
In addition :
Even + Even = Even
Odd + Odd = Odd
Even + Odd = we can't say anything

In Subtraction :
Even - Even = Even
Odd - Odd = Odd
Even - Odd = we can't say anything

In Multiplication :
Odd * Odd = Even
Even * Odd = Odd
Even * Even = Even

Full form of DSP

What is the full form of DSP?


Answer :
  • Digital Signal Processing

What does DSP mean?

DSP stands for digital signal processing, which refers to manipulating analog information, such as sound that has been converted into a digital form.

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Advantages and disadvantages of step index fiber

This article gives information about the advantages and disadvantages of step index fiber to know more details about it.

Advantages of step index fiber :
  • It is easy to manufacture
  • It is cheaper compared to the graded index
  • Light is propagated through the graded index
  • Larger bandwidth 
  • High information rate
  • Less attenuation
  • Can run over a longer distance without repeaters
Disadvantages of step index fiber :
  • Pulse distortion is more
  • The information carrying capacity is less
  • Losses are more
  • Difficult to couple light in and also out of the tiny core
  • A high directive light source is required
  • Interfacing module is more expensive
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dwdm applications

DWDM stand for dense wavelength division multiplexing. Hence dense means the wavelength channel is very narrow and close to each other. Here this article gives the application of DWDM to know more details about it.
  • It can be used for signal transmission
  • It is ready to make for long distance, Telecommunication operator that use either point to point or ring topology 
  • Utilize the existing thin fiber
  • Building or expanding network
  • The transparency of the DWDM system to  various bit-rate and protocols
  • Network wholesalers can lease capacity rather than entire fiber cable

Advantages and disadvantages of DWDM

DWDM stands for dense wavelength division multiplexing, It has to increases bandwidth by allowing different data streams to be sent simultaneously over a single optical fiber network. Here this article gives information about DWDM  to know more details about it.

DWDM advantages :
  • Improve scalability
  • Provide extra resilience
  • New services to be provide
  • Maximum capacity system available
  • Can conserve switched bandwidth
  • Permits multiple logical topological over single physical MAN
  • Greater fiber capacity
  • DWDM system capable of longer span length
  • Easier network expansion
DWDM disadvantages :
  • Not cost effective for low channel numbers
  • Complex technology requires
  • More space
  • More power
  • Startup cost more than equivalent CWDM
  • High accuracy laser and wave filter
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