{"id":2743,"date":"2024-08-08T18:31:38","date_gmt":"2024-08-08T13:01:38","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2743"},"modified":"2024-08-08T18:31:38","modified_gmt":"2024-08-08T13:01:38","slug":"z-transform","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ec\/signals-systems\/z-transform","title":{"rendered":"z-Transform"},"content":{"rendered":"

The discrete-time counterpart of Laplace transform is z-transform. The frequency domain analysis of discrete time system allows us to represent any arbitrary signal x[n] as a sum of exponential of the form zn<\/sup>.
\nThere are two varieties of z-transform: bilateral and unilateral. The bilateral one, also known as two- sided z-transform.
\ncan handle all causal and non causal signals. It provides insights about system\u2019s characteristics such as stability, causality and frequency response.<\/p>\n

The unilateral one, also known as one-sided z -transform can handle only causal signals and mainly used to solve
\ndifference equations with initial conditions.
\nThe purpose of the z-transform is to map (transform) any point s = \u00b1\u03c3 \u00b1j\u03c9 in the s-plane to a corresponding point z(r \u2220\u2126) in the z-plane by the relationship.<\/p>\n

z = esT<\/sup>, where T is sampling period (seconds)<\/p>\n

Under this mapping, the imaginary axis, \u03c3 = 0 maps on to the unit circle |z| = 1 in the z-plane. Also, the left half –\u00a0 plane, \u03c3 < 0 corresponds to the interior of the unit circle |z| = 1 in the z-plane.<\/p>\n

The mapping of s-plane to z-plane is shown in figure.<\/p>\n

\"S-Plane\"<\/p>\n

The Definition of z-Transform<\/h2>\n

Consider a discrete-time signal x[n]. Its z-transform is defined as<\/p>\n

\"<\/p>\n

Region of Convergence for z-transform<\/h3>\n

The z-transform is guaranteed to converge if x[n] \u22c5 r \u2013n<\/sup> is absolutely summable.<\/p>\n

\"<\/p>\n

Thus, ROC consist of those values of |z| = |rei\u2126<\/sup>| = r for which z-transform converges.<\/p>\n

Properties of ROC<\/h4>\n
    \n
  1. The ROC of (i) X(z) consists of a ring in the z-plane centered about the origin.<\/li>\n
  2. The ROC does not contain any poles.<\/li>\n
  3. If x[n] is of finite duration, then the ROC is entire z-plane, except possibly z = 0 and\/or z = \u221e.<\/li>\n
  4. If x[n] is a right-sided sequence, and if the circle |z | = r0 is in the ROC then all finite values of z for which | z | > r0 will also be in the ROC.<\/li>\n
  5. If x[n] is a left-sided sequence, and if the circle | z |= r0 is in the ROC, then all values of z for which 0 < |z |< r0 will also be in the ROC.<\/li>\n
  6. If x[n] is two sided, and if the circle | z |= r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle | z |= r0.<\/li>\n
  7. If the z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity.<\/li>\n
  8. If the z-transform X(z) of x[n] is rational and if x[n] is right sided, then the ROC is the region in the z-plane outside the outer most pole i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). Furthermore, if x[n] is causal (i.e., if it is right sided and equal to 0 for n < 0), then the ROC also includes z = \u221e.
    \n(ix) If the z-transform X(z) of x[n] is rational, and if x[n] is left sided, then the ROC is the region in the z-plane inside the innermost non-zero pole i.e. magnitude of the poles of X(z) other than any at z = 0 and extending inward to and possibly including z = 0. In particular, if x[n] is anti-causal (i.e., if it is left sided and equal to 0 for n > 0), then the ROC also includes z = 0.<\/li>\n<\/ol>\n

    The z-plane and poles and zeros<\/h4>\n

    The graphical representation of complex number z = rej\u2126<\/sup> in terms of complex plane is called as z-plane\"Z-Plane\"\"Z-Plane\"<\/p>\n

    It is concluded that DTFT corresponds to z-transform evaluated on unit circle.<\/p>\n

    \"Polynomials\"<\/p>\n

     <\/p>\n","protected":false},"excerpt":{"rendered":"

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