![Example for the inspection method
• Consider a causal LTI system specified by its
system function H(z). Compute its unit impulse
response h[n].
6
€
H (z) =
1− z−1
1+
3
4
z−1](https://image.slidesharecdn.com/2c-231016095243-cae42f76/85/The-inverse-Z-Transform-pdf-6-320.jpg)
![Inverse ZT via power series
expansion
• We start from the definition of X(z)
• We notice that x[n] is the coefficient of n-th power of z-1
• If we have the Z transform expressed as a series of powers
of z-1, then we can retrieve x[n] by direct identification
• Main idea
• For rational ZT: long division
11
€
X(z) = x[n]z−n
n=−∞
+∞
∑
X(z) =
bkz−k
k=0
M
∑
akz−k
k=0
N
∑
power series expansion
⎯ →
⎯⎯⎯⎯⎯⎯ X(z) = x[n]z−n
n=−∞
+∞
∑](https://image.slidesharecdn.com/2c-231016095243-cae42f76/85/The-inverse-Z-Transform-pdf-11-320.jpg)
![Example 1 for power series expansion
• Determine the sequence x[n] corresponding to
the following ZT:
12
€
X(z) = (1+ 2z)(1+ 3z−1
)](https://image.slidesharecdn.com/2c-231016095243-cae42f76/85/The-inverse-Z-Transform-pdf-12-320.jpg)
![Example 2 for power series expansion
• Determine the sequence x[n] that
corresponds to the following Z Transform,
knowing that this sequence is right-sided
13
€
X(z) =
1
1+
1
2
z−1](https://image.slidesharecdn.com/2c-231016095243-cae42f76/85/The-inverse-Z-Transform-pdf-13-320.jpg)
The inverse Z-Transform.pdf
- 1. 1 The Z transform (3)
- 2. 2 Today • The inverse Z Transform • Section 3.3 (read class notes first) • Examples 3.9, 3.11
- 3. 3 The inverse Z-transform • by expressing the Z-transform as the Fourier Transform of an exponentially weighted sequence, we obtain • The formal expression of the inverse Z-transform requires the use of contour integrals in the complex plane.
- 4. Computational methods for the inverse Z-Transform • For rational Z-transforms we can compute the inverse Z-transforms using alternative procedures: – Inspection (Z Transform pairs) – Partial Fraction Expansion – Power Series Expansion 4
- 5. Inspection method • Makes use of common Z-Transform pairs in Table 3.1 and of the properties of the Z- Transform (Table 3.2), which we will discuss in the next lecture. – Most useful Z-Transform pairs: 1, 5, 6 – Most useful property: time shifting • The inspection method can be used by itself when determining the inverse ZT of simple sequences • Most often, it represents the final step in the expansion-based methods 5
- 6. Example for the inspection method • Consider a causal LTI system specified by its system function H(z). Compute its unit impulse response h[n]. 6 € H (z) = 1− z−1 1+ 3 4 z−1
- 7. Table 3.1 SOME COMMON z-TRANSFORM PAIRS
- 8. Table 3.2 SOME z-TRANSFORM PROPERTIES
- 9. Inverse ZT via partial fraction expansion • We will study only the case of first-order poles (all poles are distinct) and M<N. • Equations 3.45, 3.46, 3.47 are not required • General idea: • The partial fraction expansion process computes all coefficients Ak 9 € X(z) = bkz−k k=0 M ∑ akz−k k=0 N ∑ partial fraction expansion ⎯ → ⎯⎯⎯⎯⎯⎯ ⎯ X(z) = Ak 1− dkz−1 k=1 N ∑
- 10. 10 Example for partial fraction expansion • Compute the inverse Z-transform for:
- 11. Inverse ZT via power series expansion • We start from the definition of X(z) • We notice that x[n] is the coefficient of n-th power of z-1 • If we have the Z transform expressed as a series of powers of z-1, then we can retrieve x[n] by direct identification • Main idea • For rational ZT: long division 11 € X(z) = x[n]z−n n=−∞ +∞ ∑ X(z) = bkz−k k=0 M ∑ akz−k k=0 N ∑ power series expansion ⎯ → ⎯⎯⎯⎯⎯⎯ X(z) = x[n]z−n n=−∞ +∞ ∑
- 12. Example 1 for power series expansion • Determine the sequence x[n] corresponding to the following ZT: 12 € X(z) = (1+ 2z)(1+ 3z−1 )
- 13. Example 2 for power series expansion • Determine the sequence x[n] that corresponds to the following Z Transform, knowing that this sequence is right-sided 13 € X(z) = 1 1+ 1 2 z−1
- 14. Summary • The inverse ZT is in general a contour integral. • For rational ZTs, it is not necessary to explicitly compute this integral • 3 methods: – Inspection (ZT pairs and properties of the ZT) – Partial fraction expansion – Power series expansion • Which method should we choose? – Power series expansion is an excellent tool when the power series is finite – For infinite length sequences, we will work mostly with inspection and partial fraction expansion, since long division is computationally expensive 14