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**The 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 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

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**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.

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**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.

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ROC properties ROC 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|>α A left-sided sequence has a z-transform with a region of convergence that is the interior of a circle: ROC: |z|<β The Fourier Transform of x(n) converges absolutely if and only if ROC of z-transform includes the unit circle

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**Properties of Z-Transform**

Linearity If 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 property If x(n) has a z-transform X(z), Time reversal If 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) is and has a region of convergence , which is denoted by

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**Properties of Z-Transform**

Multiplication by an exponential If a sequence x(n) is multiplied by a complex exponential αn. Convolution theorm If 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. Conjugation If X(z) is the z-transform of x(n), the z-transform of the complex conjugate of x(n) is

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**Properties of Z-Transform**

Derivative If X(z) is the z-transform of x(n), the z-transform of is Initial value theorem If 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:

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Fourier Analysis of Discrete Time Signals

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