Digital signal processing

1. Digital signal processing Done By: IssraaJwadKazim Lec. 7

2. Z TRANSFORM • Basic Definitions • The Z – transform of a discrete time signal or sequence x(n) is defined as: • x(n) and X(z) are said to form an z-transform pair. This is written symbolically as • where z is a complex variable. In polar form . As the angle ‘w’ varies, we get a circle of radius ‘r’ in the z-plane. If r = 1, we get a circle of radius unity . It is called the unit circle.

3. Discrete Time Fourier Transform (DTFT) • Substituting in X(z) , we get, This is called the Discrete Time Fourier Transform (DTFT) of the sequence x(n). Thus Z- transform evaluated on the unit circle gives the DTFT of the sequence .

4. Region of Convergence (ROC) of the Z transform • We see from the definition of the z-transform that it is an infinite power series and hence the summation may not converge (ie., becomes finite) for all values of z. • The set of values of z for which the summation converges is called the Region of Convergence (ROC) of the Z-transform. ROC should always be mentioned along with the z-transform.

5. 5.2 Z – Transform of some time sequences 1) Right side sequences As an example, let us find the z-transform and ROC of the right sided sequence • We see that X(z) becomes infinity at z =0. Except at z = 0, X(z) is finite for all values of z. Therefore we can say that the ROC of this z transform is the entire z-plane except z = 0. ie., ROC : |z| > 0.

6. 2) Left sided sequences: Let us find the z-transform and ROC of the left sided sequence We see that X(z) becomes infinity at z = ∞.. Except at z = ∞, X(z) is finite for all values of z. Therefore we can say that the ROC of this z transform is the entire z-plane except z = ∞ ie., ROC : |z| < ∞

7. 3 ) Double sided sequences: • A sequence x(n) is said to be double sided if x(n) has both right and left sides. For example, x(n) = ( 2, 1, 1, 2 ) is a double sided sequence • ↑ • because x(n) exists in the range -2 ≤ n ≤ 1. Z transform of this sequence is given by • We see that X(z) becomes infinity both at z =0 and z = ∞, and X(z) is finite for all other values of z. Therefore we can say that the ROC of this z transform is the entire z-plane except z = 0 and z = ∞. ie., ROC : 0 < |z| < ∞ .

8. 4) Positive time exponential sequences: Positive time exponential sequence

9. Poles and zeros of the Z transform Values of z for which X(z) = 0 are called the zeros of X(z). A zero is indicated by a ‘O’ in the z plane. Values of z for which X(z) = ∞ are called the poles of X(z). A pole is indicated by a ‘X’ in the z plane. For example, in the previous example, where X(z) = z / (z-a) The z transform has one zero at z=0 and one pole at z=a.

11. Negative time exponential sequences: This is a left sided sequence of infinite duration. Its z transform is obtained as ie., X(z) = z / (z-b) , with |z| < |b| which defines the ROC of the z transform. The ROC in this case, is the interior of a circle of radius ‘b’. This is true, in general, for any left sided sequence.

13. Note: If b = a, then the z-transforms of the two sequences are identical, even though the sequences are different. Hence we cannot determine the time domain sequence just from its z transform alone. However we may note that the ROC of the z transforms are unique (ie., in the first case, it is the exterior of a circle and in the second case, it is the interior of a circle , both of finite radius).

14. Double sided exponential sequences A double sided exponential sequence can be obtained as a sum of positive and negative time exponential sequences. Thus summing the sequences in (1) and (2), we get a double sided exponential sequence . The ROC of the z transform is the intersection of ROCs of the two individual ROCs. |z| > |a| and |z| < |b| ie., ROC : { |z| > |a| } ∩ {|z| < |b|}. There can be two cases: (1) |b| < |a| (2) |b|>|a|

15. Case (2): |b| > |a| In this case, the z transform exists and converges in the annular region {|a| <|z| < |b|}