3. WHY Z -TRANSFORM?
• Many of signals (such as x(n)=u(n), x(n) = (0.5)nu(-n), x(n) =
• sin(nω) etc. ) do not have a DTFT.
• Advantages like Fourier transform provided:
• Solution process reduces to a simple algebraic procedures
• The temporal domain sequence output y(n) = x(n)*h(n) can be
• represent as Y(z)= X(z)H(z)
• Properties of systems can easily be studied and characterized in z
• – domain (such as stability..)
• Topics:
• Definition of z –Transform
Properties of z- Transform Inverse
z- Transform
4. DEFINITION OF THE Z-TRANSFORM
Definition:
The z-transform of a discrete-time signal x(n) is defined by
where z = rejw is a complex variable. The values of z for which the sum
converges define a region in the z-plane referred to as the region of
convergence (ROC).
Notationally, if x(n) has a z-transform X(z), we write
The z-transform may be viewed as the DTFT or an exponentially
weighted sequence. Specifically, note that with z = rejw, X(z) can be
looked as the DTFT of the sequence r--nx(n) and ROC is determined by
the range of values of r of the following right inequation.
5. ROC & Z-PLANE COMPLEX Z-PLANE
ZEROS AND POLES OF X(Z)
z = Re(z)+jIm(z) = rejw
Zeros and poles of X(z)
Many signals have z-transforms that are rational function of z:
Factorizing it will give:
The roots of the numerator polynomial, βk,are referred to as the zeros (o)
and αk are referred to as poles (x). ROC of X(z) will not contain poles.
6.
7. PROPERTIES AND THEOREMS
1.Multiplication by a Constant
2.Linearity of the z Transform
3.Multiplication by ak
4.Shifting Theorem
5.Complex Translation Theorem
6.Initial Value Theorem Final Value Theorem
1.Multiplication by a constant
If X(z) is the z transform of x(n), then
Z [ax(n)] = a Z[x(n)] = a X(z)
where a is a constant...
Linearity of the z transform
The z transform possesses an important property: linearity. This
means
that, if f(n) and g(n)...
8. 2.LINEARITY OF THE Z TRANSFORM
Linearity of the z transform The z transform possesses an important
property: linearity. This means that, if f(n) and g(n) are z-transformable
and α and β are scalars, then x(n) formed by a linear combination
x(n) = αf(n) + βg(n) has the z transform X(z) = αF(z) + βG(z) where F(z) and
G(z) are the z transforms of f(n) and g(n), respectively.
3. Multiplication By Ak
If X(z) is the z transform of x(k), then the z transform of ak x(n) can be
given by X(a-1 z): Z[ak x(n)] = X(a-1 z) This can be proved as follows: Z[an
x(n)] = = X(a-1 z) ∑∑ ∞ = −− ∞ = − = 0 1 0 ))(()( n n n nn zanxznxa
4.Shifting Theorem
Also call real translation theorem. If x(t) = 0 for t < 0 and x(t) has the z
transform X(z), then Z[x(t-nT)] = z-n X(z) and Z[x(t+nT)] = n = zero or a
positive integer − ∑ − = − 1 0 )()( n k kn zkTxzXz
