Presentation on theme: "The z-Transform: Introduction"— Presentation transcript:
1The z-Transform: Introduction 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 proceduresThe 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 –TransformProperties of z- TransformInverse z- Transform
2Definition of the z-Transform Definition:The z-transform of a discrete-time signal x(n) is defined bywhere 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 writeThe 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.
3ROC & z-plane Complex z-plane Zeros and poles of X(z) z = Re(z)+jIm(z) = rejwZeros 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.
4ROC propertiesROC is an annulus or disc in the z-plane centred at the origin. i.e.A finite-length sequence has a z-transform with a region of convergence that includes the entire z-plane except, possibly, z = 0 and z = . The point z = will be included if x(n) = 0 for n < 0, and the point z = 0 will be included if x(n) = 0 for n > 0.A right-sided sequence has a z-transform with a region of convergence that is the exterior of a circle: ROC: |z|>αA left-sided sequence has a z-transform with a region of convergence that is the interior of a circle:ROC: |z|<βThe Fourier Transform of x(n) converges absolutely if and only if ROC of z-transform includes the unit circle
12Properties of Z-Transform LinearityIf x(n) has a z-transform X(z) with a region of convergence Rx, and if y(n) has a z-transform Y(z) with a region of convergence Ry,and the ROC of W(z) will include the intersection of Rx and Ry, that is, Rw contains .Shifting propertyIf x(n) has a z-transform X(z),Time reversalIf x(n) has a z-transform X(z) with a region of convergence Rx that is the annulus , the z-transform of the time-reversed sequence x(-n) isand has a region of convergence , which is denoted by
13Properties of Z-Transform Multiplication by an exponentialIf a sequence x(n) is multiplied by a complex exponential αn.Convolution theormIf x(n) has a z-transform X(z) with a region of convergence Rx, and if h(n) has a z-transform H(z) with a region of convergence Rh,The ROC of Y(z) will include the intersection of Rx and Rh, that is,Ry contains Rx ∩ Rh .With x(n), y(n), and h(n) denoting the input, output, and unit-sample response, respectively, and X(z), Y(x), and H(z) their z-transforms. The z-transform of the unit-sample response is often referred to as the system function.ConjugationIf X(z) is the z-transform of x(n), the z-transform of the complex conjugate of x(n) is
14Properties of Z-Transform DerivativeIf X(z) is the z-transform of x(n), the z-transform of isInitial value theoremIf X(z) is the z-transform of x(n) and x(n) is equal to zero for n<0, the initial value, x(0), maybe be found from X(z) as follows: