Signals and Systems - Electrical Engineering

524 C H A P T E R 9: The Z-Transform For a real valueα=|α|ejω^0 forω 0 = 0 orπ, x[n]=αnu[n] ⇔ X(z)= 1 1 −αz−^1 = z z−α ROC:|z|&g ...

9.4 One-Sided Z-Transform 525 The Z-transform pairs of a cosine and a sine are, respectively, cos(ω 0 n)u[n] ⇔ z(z−cos(ω 0 )) (z ...

526 C H A P T E R 9: The Z-Transform Or the pair nx[n]u[n] ⇔ −z dX(z) dz (9.18) For instance, ifX(z)= 1 /( 1 −αz−^1 )=z/(z−α), w ...

9.4 One-Sided Z-Transform 527 − 2 0 2 − 1 0 1 Real part Imaginary part 0 5 10 15 20 h^2 (n ) − 1 0 1 n − 2 0 2 − 1 0 1 Real part ...

528 C H A P T E R 9: The Z-Transform and X 2 (z)= 1 +b 1 z−^1 Their product is X 1 (z)X 2 (z)= 1 +b 1 z−^1 +a 1 z−^1 +a 1 b 1 z− ...

9.4 One-Sided Z-Transform 529 nExample 9.7 Consider computing the output of an FIR filter, y[n]= 1 2 ( x[n]+x[n−1]+x[n−2] ) for ...

530 C H A P T E R 9: The Z-Transform x[k] h[− 1 −k] h[2−k] h[5−k] x[k] x[k] 045123 k k k − 3 − 2 − 1 − 3 − 2 − 1 − 3 − 2 − 1 0 0 ...

9.4 One-Sided Z-Transform 531 − 2 0 2 n x[ n] h [n ] y[ n] n n 4 6 (a) (b) (c) 8 0 0.5 1 − 2 0 2 4 6 8 0 0.5 1 − 2 0 2 4 6 8 0 0 ...

532 C H A P T E R 9: The Z-Transform Solution Graphical approach:Let us use the formula y[n]= ∑n k= 0 x[k]h[n−k] which keeps the ...

9.4 One-Sided Z-Transform 533 − (^1051015) −0.5 0 0.5 1 1.5 n x[ n] 0 5 10 15 0 0.5 1 n h [n ] − 2 0 2 4 6 8 10 12 14 16 18 − 1 ...

534 C H A P T E R 9: The Z-Transform According to the convolution property the Z-transform of the output of the noncausal filter ...

9.4 One-Sided Z-Transform 535 n It is important to remember the relations H(z)=Z[h[n]]= Y(z) X(z) = Z[y[n]] Z[x[n]] where H(z)is ...

536 C H A P T E R 9: The Z-Transform An equivalent condition is that the poles ofH(z)are inside the unit circle. In this case,h[ ...

9.4 One-Sided Z-Transform 537 has finite length and is given by h[n]=b 0 δ[n]+b 1 δ[n−1]+···+bMδ[n−M] Its transfer function is H ...

538 C H A P T E R 9: The Z-Transform H 1 (z) H 2 (z) = = x[n] y[n] H 1 (z)H 2 (z) x[n] y[n] H 1 (z) x[n] y[n] H 2 (z) x[n] y[n] ...

9.4 One-Sided Z-Transform 539 9.4.5 Initial and Final Value Properties In some control applications and to check a partial fract ...

540 C H A P T E R 9: The Z-Transform FIGURE 9.8 Negative-feedback system with plantG(z). + − x[n] e[n] y[n] K G(z) w[n] Solution ...

9.4 One-Sided Z-Transform 541 Table 9.1 One-Sided Z-Transforms Function of Time Function ofz, ROC δ[n] 1, wholez-plane u[n] 1 1 ...

542 C H A P T E R 9: The Z-Transform 9.5 One-Sided Z-Transform Inverse Different from the inverse Laplace transform, which was d ...

9.5 One-Sided Z-Transform Inverse 543 nExample 9.13 Find the inverse Z-transform of X(z)= 1 1 + 2 z−^2 |z|> √ 2 Solution We c ...