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Hence the region of convergence does not include any pole. Property 3: Consider a finite duration sequence x [n] shown in figure 6. If N1 is negative and N2 is positive, then X z will have both positive and negative powers of z. The negative powers of z becomes unbounded infinity if z " 0. Similarly positive powers of z becomes unbounded infinity if z " 3. Property 4: Consider a right-sided sequence x [n] shown in figure 6.
Let, it is bounded by some value Mx , then equation 6. Property 5: Consider a left-sided signal x [n] Here N 2 can be either positive or negative.
Let it is bounded by some value Mx , then equation 6. The ROC of a left-sided sequence is illustrated in figure 6. Property 6: For any time N 0 , a two-sided sequence can be divided into sum of left-sided and right-sided sequences as shown in figure 6.
The z -transform of x [n] converges for the values of z for which the transform of both xR [n] and xL [n] converges. From property 4, the ROC of a right-sided sequence is a region which is bounded on the inside by a circle and extending outward to infinity i. From property 5, the ROC of a left sided sequence is bounded on the outside by a circle and extending inward to zero i.
The ROC for the right-sided sequence xR [n], the left- sequence xL [n] and their combination which is a two sided sequence x [n] are shown in figure 6. If the z -transform X z of x [n] is rational, then its ROC is bounded by poles or extends to infinity. Property 8: If the z -transform X z of x [n] is rational and x [n] is a right-sided sequence then the ROC is the region in the z -plane outside the outermost pole i.
ROC is the region outside a circle with a radius greater than the magnitude of largest pole of X z. This property can be be proved by taking property 4 and 7 together.
Property 9: If the z -transform X z of x [n] is rational and x [n] is a left-sided sequence then the ROC is the region in the z -plane inside the innermost pole i. ROC is the region inside a circle with a radius equal to the smallest magnitude of poles of X z.
This property can be be proved by taking property 5 and 7 together. TABLE 6.
To obtain the sequence x [n] from its z -transform is called the inverse z -transform. There are other commonly used methods to evaluate the inverse z -transform given as follows 1.
Partial fraction method 2. Power series expansion 6.
The values of coefficients A1 , A2 , A Three cases are given as follows Case I: The coefficients are obtained by equation 6. Refer property 8, section 6. Property 9, section 6. So this terms will corresponds to a left sided equation.
To obtain inverse z -transform i. The proofs of properties are given for bilateral transform only and can be obtained in a similar way for the unilateral transform.
Page The Z-Transform Chapter 6 6. Like Laplace transform, the linearity Proof: Since, the z -transform X1 z is finite within the between x1 [n] and x 2 [n], which may specified ROC, R1. Time reversal property states that time Proof: Rx or z! Rx dz For both unilateral and bilateral z -transforms.
This property states that multiplication Proof: This operation does not affect the ROC. Rx X a z k, with ROC: Multiplication of a time sequence with an Proof: Since the expanded sequence xk [n] is zero everywhere except when n is a multiple of k. As defined in equation 6. For both unilateral and bilateral transform.
Time convolution property states that Proof: The z -transform of signal x [n] is given by equation 6. Thus final value theorem 4 applies. As the Laplace transform is used in solving differential equations which describe continuous LTI systems, the z -transform is used to solve difference equation which describe the discrete LTI systems. Similar to Laplace transform, for CT domain, the z -transform gives transfer function of the LTI discrete systems which is the ratio of the z -transform of the output variable to the z -transform of the input variable.
These applications are discussed as follows 6. The solution or total response y [n] consists of two parts as discussed below. Zero-input Response or Free Response or Natural Response The zero input response yzi [n] is mainly due to initial output in the system. The zero-input response is obtained The zero input response is also called the from system equation 6.
The zero-input response yzi [n] is given by inverse z -transform of Yzi z.
Zero-State Response or Forced Response The zero-state response yzs [n] is the response of the system due to input signal and with zero initial conditions. Leave this field empty. Welcome to EasyEngineering, One of the trusted educational blog. Check your Email after Joining and Confirm your mail id to get updates alerts. Kanodia and Ashish Murolia Books. Analysis and Design By Devendra K.
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