Z transform
Upcoming SlideShare
Loading in...5
×
 

Z transform

on

  • 2,413 views

 

Statistics

Views

Total Views
2,413
Views on SlideShare
2,413
Embed Views
0

Actions

Likes
1
Downloads
113
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment
  • 1010

Z transform Z transform Presentation Transcript

  • Z-Transforms
    AakankshaThakre
    AyushAgrawal
    KunalAgrawal
    AkshayPhadnis
    Aakanksha_Kunal_Ayush_Akshay
  • Introduction:
    Just like Laplace transforms are used for evaluation of continuous functions, Z-transforms can be used for evaluating discrete functions.
    Z-Transforms are highly expedient in discrete analysis ,
    Which form the basis of communication technology.
    Definition:
    If a function f(n) is defined for discreet values ( n=0,+1 or -1 , +2 or -2,etc ) & f(n)=0 for n<0,then z-transform of the function is defined as
    Z{f(n)}=


    -n
    f(n) z
    =F(z)
    n=0
    Aakanksha_Kunal_Ayush_Akshay
  • Some standard results & formulae:
    -n

    n=0
    2
    Aakanksha_Kunal_Ayush_Akshay
  • Z{
    }
    n
    =
    a
    Z / (z-a)
    Z{
    n
    }
    =
    -a
    Z / (z+a)
    2
    Z{n}= z /
    (z-1)
    Z{1/n!}=
    e
    1/z
    Z{sin nф}= zSinф / (z -2zcosф +1 )
    2
    2
    2
    Z{Cosnф}= z- zCosф / (z -2zCosф + 1)
    Aakanksha_Kunal_Ayush_Akshay
  • Properties:
    Linearity: -
    Z{a f(n)+b g(n)}=a Z{f(n)}+b Z{g(n)}
    Damping rule:-
    Z{a f(n)} = F(z/a)
    Multiplication by positive integer n :-
    Z{n f(n)}= -z d/dz ( F(z) )
    Aakanksha_Kunal_Ayush_Akshay
  • Initial value theorem:-
    f(0)= lim F(z)
    Z∞
    Final value theorem:-
    f(∞)= lim f(n) = lim (z-1) F(z)
    n∞
    Z1
    Shifting Theorem:-
    Z{ f (n+k) }= z [ F(z) - ∑ f(i) z ]
    K-i
    -i
    K
    i=0
    Aakanksha_Kunal_Ayush_Akshay
  • Division by n property:-


    Z{f(n)/n}=
    F(z)/z
    dz
    Z
    Division by n+k property:-


    k
    Z{f(n) /(n+k)}=
    Z
    K+1
    F(z)/ (Z)
    dz
    z
    Aakanksha_Kunal_Ayush_Akshay
  • Applications of Z-Transforms
    The field of signal processing is essentially a field of signal analysis in which they are reduced to their mathematical components and evaluated. One important concept in signal processing is that of the Z-Transform, which converts unwieldy sequences into forms that can be easily dealt with. Z-Transforms are used in many signal processing systems
    Z-transforms can be used to solve differential equations with constant coefficients.
    Aakanksha_Kunal_Ayush_Akshay
  • Derivation of the z-Transform
    The z-transform is the discrete-time counterpart of the Laplace transform. In this
    section we derive the z-transform from the Laplace transform a discrete-time signal.
    Aakanksha_Kunal_Ayush_Akshay
  • The Laplace transform X(s), of a continuous-time signal x(t), is given by the integral
    ∞ -st
    X(s) = ∫ x(t) e dt
    0-
    where the complex variable s=a +jω, and the lower limit of t=0− allows the possibility
    that the signal x(t) may include an impulse. The inverse Laplace transform is defined
    by:-
    a+j∞
    st
    X(t) = ∫ X(s) e ds
    a-j∞
    Aakanksha_Kunal_Ayush_Akshay
  • where a is selected so that X(s) is analytic (no singularities) for s>a. The ztransform
    can be derived from Eq. by sampling the continuous-time input signal
    x(t). For a sampled signal x(mTs), normally denoted as x(m) assuming the sampling
    period Ts=1, the Laplace transform Eq. becomes

    s
    -sm
    X(e ) = ∑ x(m) e
    m=0
    Aakanksha_Kunal_Ayush_Akshay
  • Substituting the variable e to the power s
    in Eq. with the
    variable z we obtain the one-sided
    ztransform

    -m
    X(z) = ∑ x(m) z
    m = 0
    The two-sided z-transform is defined as:-

    -m
    X(z) = ∑ x(m) z
    m = -∞
    Aakanksha_Kunal_Ayush_Akshay
  • The Relationship Between the Laplace, the Fourier, andthe z-Transforms :-The Laplace transform, the Fourier transform and the z-transform are closely related inthat they all employ complex exponential as their basis function. For right-sidedsignals (zero-valued for negative time index) the Laplace transform is a generalisation of the Fourier transform of a continuous-time signal, and the z-transform is ageneralisation of the Fourier transform of a discrete-time signal.
    Aakanksha_Kunal_Ayush_Akshay