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The ROC does not contain any poles 3. ROC of z-transform of anu(n) is |z|>|a|. Given x(n)= δ(n-k)=1 at n=k So, the system is BIBO stable. If is a rational z-transform of a right sided function , then the ROC is the region outside the out-most pole. students definitely take this Test: Z Transform exercise for a better result in the exam. ROC of z-transform of bnu(-n-1) is |z|<|b|. The Z-Transform X(z) of a discrete time signal x(n) is defined as: Explanation: The z-transform of a real discrete time sequence x(n) is defined as a power of ‘z’ which is equal to , where ‘z’ is a complex variable. The Region of Convergence has a number of properties that are dependent on the characteristics of the signal, x [ n]. rL ? File \zt", version of … The ROC cannot contain any poles. => X(z)=z-k What is the ROC of a causal infinite length sequence? ROC of z-transform: A. DTFT of x(n) exists only if ROC of X(z) contains unit circle. F. The solved questions answers in this Test: Z Transform quiz give you a good mix of easy questions and tough questions. Property 4 If x[n] is a ﬁnite-length sequence, then the ROC is the entire Z-plane with the possible exception of z = 0 or z = ∞. If the Z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity. Properties of the ROC 11 1. If x (n) is a finite duration causal sequence, the ROC is entire z-plane except at z=0. Answer: a Explanation: Since the value of z-transform tends to infinity, the ROC of the z-transform does not contain poles. Pole: Each of the roots of the denominator polynomial for which is a pole of . What is the ROC of z-transform of finite duration anti-causal sequence? For rational Laplace transforms, the ROC doesn’t contain any poles (since X(s)=∞) 3. Electrical Engineering (EE) The ROC is bounded by the poles or extends to infinity. For a general signal x[n], the ROC will be the intersection of the ROC of its causal and noncausal parts, which is an annulus. The ROC of z-transform of any signal cannot contain poles. We describe any of these shapes as a generalized ring around the origin. An alternative approach is to think of x 1[n] as 1 8 times a version of 1 2 nu[n] that is delayed by 3. What is the set of all values of z for which X(z) attains a finite value? The Fourier transform of xn converges absolutely iff the ROC includes the unit circle. The ROC might extend outward to ¥ or inward to origin depending on the sequence. If x (n) is a finite duration anti-causal sequence or left sided sequence, then the … Given h(n)= an(n) (|a|<1) The z-transform of h(n) is H(z)=z/(z-a),ROC is |z|>|a| If |a|<1, then the ROC contains the unit circle. C. Given x(n) is finite, its ROC is entire z plane except possibly z=0 or z=∞. Explanation: A discrete time LTI is BIBO stable, if and only if its impulse response h(n) is absolutely summable. This extends to cases with … The set of all values of z where X(z) converges to a finite value is called as Radius of Convergence(ROC). What is the z-transform of the finite duration signal. Both other possible ROCs (inside the circle with radius 0.5 and outside the circle with radius 2) do not include the unit circle, and, consequently, do not correspond to impulse responses of stable systems. 2. (4) If x[n] ... DSP (Spring, 2004) The z-Transform 6 Pole Location and Time-Domain Behavior for Causal Signals Reference: Digital Signal Processing by Proakis & Manolakis . ! What is the ROC of the z-transform of the signal x(n)= anu(n)+bnu(-n-1)? Then X 1(z) = X∞ l=0 z−1 2 l+3 = (z−1/2)3 1−(z− 1/2) = 1 8z2(z− 2). The ROC of X(s) consists of strips parallel to the jω-axis in the s-plane 2. B. ROC cannot contain any pole. Definition of ROC of a z-transform should not contain any poles. absolutely if and only if the ROC of the z-transform includes the unit circle 3.ROC cannot contain any poles 4.If x[n] is a finite duration sequence, then the ROC is the entire z-plane except possibly z=0 or z=¥ 5.If x[n] is a right sided sequence, the ROC extends outward from the outermost finite pole in X(z) to z=¥ Is the discrete time LTI system with impulse response h(n)=an(n) (|a| < 1) BIBO stable? ROC of the a-transform of x[n] includes the unit circle. Which of the following series has an ROC as mentioned below? What is the ROC of the signal x(n)=δ(n-k),k>0? The Z transform of 1 2 nu[n] is z z−1 2. The Region of Convergence is a ring or circle in the Z-plane centred about the origin. ROC of the z-transform of x[n] includes the unit circle. The ROC of X(z) consists of a ring in the z−plane centered about the origin 2. Since X (z) must be finite for all z for convergence, there cannot be a pole in the ROC. From the above graph, we can state that the ROC of a two sided sequence will be of the form r2 < |z| < r1. … By definition a pole is a where X (z) is infinite. If x (n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0. If is rational, then its ROC does not contain any poles (by definition dose not exist). What is the z-transform of the signal x(n)= -αnu(-n-1)? Explanation: For a given signal x(n), its z-transform. Determine the pole-zero plot for the signal Im(z) a Re(z) x() ()naun= n 1 1 1 z Xz az z a− == − − The z-transform is One zero at z 1=0 One pole at p 1=a. Explanation: We know that, for a given signal x(n) the z-transform is defined as EduRev is a knowledge-sharing community that depends on everyone being able to pitch in when they know something. Examples 2 & 3 clearly show that the Z-transform X(z) of x[n] is unique when and only when specifying the ROC. If the Z-transform X(z) of x[n] is rational, and x[n] is right-sided, then the ROC is the region in the z-plane outside the outermost pole (outside the circle of radius equal to the largest magnitude of the poles … If the ROC includes the Unit circle, this implies convergence of z-transform for |z| = 1 or, equivalently, the Fourier transform converges. •The Fourier transform of x[n]converges absolutely if and only if the ROC of the z-transform includes …