Signals and Systems - Electrical Engineering

544 C H A P T E R 9: The Z-Transform 9.5.2 Partial Fraction Expansion The basics of partial fraction expansion remain the same f ...

9.5 One-Sided Z-Transform Inverse 545 where the second term is proper rational as the denominator is of a higher degree in power ...

546 C H A P T E R 9: The Z-Transform TheAandBcoefficients can be found (by analogy with the Laplace transform partial fraction e ...

9.5 One-Sided Z-Transform Inverse 547 9.5.3 Inverse Z-Transform with MATLAB............................................ Symbolic ...

548 C H A P T E R 9: The Z-Transform To find the poles and the zeros ofX(z), given the coefficients{b[k]}and{a[k]}of the numerat ...

9.5 One-Sided Z-Transform Inverse 549 − 1 −0.5^0 0.5^1 −1.5 − 1 −0.5 0 0.5 1 1.5 Real part Imaginary part 0 5 10 (a) (b) 15 20 0 ...

550 C H A P T E R 9: The Z-Transform where the second term is not found in the Z-transforms table. To write it so that each of t ...

9.5 One-Sided Z-Transform Inverse 551 − 1 −0.5 0 (a) (b) 0.5 1 − 2 −1.5 − 1 −0.5 0 0.5 1 1.5 2 22 Real part Imaginary part 0 2 4 ...

552 C H A P T E R 9: The Z-Transform Remarks n If the signal is causal, so that{x[i],−N≤i≤− 1 }are all zero, we then have thatZ( ...

9.5 One-Sided Z-Transform Inverse 553 To obtain a closed-form solution, we use the Z-transform. Taking the one-sided Z-transform ...

554 C H A P T E R 9: The Z-Transform To solve the difference equation witha=0.8,x[n]=u[n]−u[n−11], and zero initial condition y[ ...

9.5 One-Sided Z-Transform Inverse 555 You can then see that the first term in Equation (9.38) is a convolution sum, and the seco ...

556 C H A P T E R 9: The Z-Transform Just as with the Laplace transform, the steady-state response of a difference equation y[n] ...

9.5 One-Sided Z-Transform Inverse 557 rational. That term equals B( 1 −0.5z−^1 )+Cz−^1 ( 1 −0.5z−^1 )^2 = B 1 −0.5z−^1 + Cz−^1 ( ...

558 C H A P T E R 9: The Z-Transform Letting A(z)= 1 +z−^1 − 4 z−^2 − 4 z−^3 =( 1 +z−^1 )( 1 + 2 z−^1 )( 1 − 2 z−^1 ) we can wri ...

9.5 One-Sided Z-Transform Inverse 559 so that y[n]= ( −0.5− 1 6 (− 1 )n+ 8 3 2 n ) u[n] which as expected will grow asnincreases ...

560 C H A P T E R 9: The Z-Transform and the second derivative as d^2 vc(t) dt^2 = ddvdtc(t) dt ≈ d(vc(t)−vc(t−Ts))/Ts) dt ≈ vc( ...

9.5 One-Sided Z-Transform Inverse 561 0 1 2 3 4 5 6 7 8 910 0 0.2 0.4 0.6 0.8 1 1.2 t, nTs y( t), y[ n] FIGURE 9.12 Solution of ...

562 C H A P T E R 9: The Z-Transform a very important role in making this determination. Once this is done, the inverse is found ...

9.5 One-Sided Z-Transform Inverse 563 so thatB=−A=−2. The inverse is then found to be x[n]= 2 u[n] ︸︷︷︸ causal + [ − 2 (n+^1 )u[ ...