signal spectra signal processing

2.  Sometimes called as the direct z-transform.
 Defined as the power series
𝑋 𝑧 =
𝑛=−∞
∞
𝑥 𝑛 𝑧−𝑛
where z is a complex variable.
 For convenience, the z-transform of a signal x(n) is denoted by
𝑋 𝑧 = 𝑍 𝑥 𝑛
whereas the relationship between x(n) and X(z) is indicated by

3.  Due to the fact that the z-transform is an infinite power series, it exists only for
the values of z for which the series converges
 The region of convergence (ROC) of X(z) is the set of all values of z for which X(z)
reaches a finite value.
 As a result, whenever we mention a z-transform, we also need to mention its ROC.

4. 𝑋1 𝑧 = 1 + 2𝑧−1
+ 5𝑧−2
+ 7𝑧−3
+ 𝑧−5
𝑅𝑂𝐶 = 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = 0

5. 𝑥2 𝑛 = 3,1,0,4,0,7
𝑋2 𝑧 = 3𝑧1 + 1 + 4𝑧−2 + 7𝑧−4
𝑅𝑂𝐶 = 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = 0 𝑎𝑛𝑑 𝑧 = ∞

6. 𝑥3 𝑛 = {0,0,3,8,2,4,6}
𝑋3 𝑧 = 3𝑧−2
+ 7𝑧−3
+ 2𝑧−4
+ 4𝑧−5
+ 6𝑧−6
𝑅𝑂𝐶 = 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = 0

7. 𝑥4 𝑛 = 3,4,1,2,0
𝑋4 𝑧 = 3𝑧4 + 4𝑧3 + 𝑧2 + 2𝑧
𝑅𝑂𝐶 = 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = ∞

8. 𝑥5 𝑛 = 𝛿 𝑛
𝑋5 𝑧 = 1
𝑅𝑂𝐶 = 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒

9. 𝑥6 𝑛 = 𝛿 𝑛 − 𝑘
𝑋6 𝑧 = 𝑧−𝑘
𝑅𝑂𝐶: 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = 0

10. 𝑥7 𝑛 = 𝛿 𝑛 + 𝑘
𝑋7 𝑧 = 𝑧𝑘
𝑅𝑂𝐶: 𝑒𝑛𝑡𝑖𝑟𝑒 𝑧 − 𝑝𝑙𝑎𝑛𝑒 𝑒𝑥𝑐𝑒𝑝𝑡 𝑧 = ∞

12. 𝑥 𝑛 =
1
2
𝑛
𝑢 𝑛
𝑋 𝑧 =
𝑛=−∞
∞
1
2
𝑛
𝑢(𝑛)𝑧−𝑛
=
𝑛=0
∞
1
2
𝑛
𝑧−𝑛
𝑋 𝑧 =
1
1 −
1
2
𝑧−1
𝑅𝑂𝐶: 𝑧 <
1
2

13.  An important family of z-transforms are those of which X(z) is a rational function,
a ratio of polynomials in 𝑧−1 or (z)

14.  The Zeros of a z-transform X(z) are the values of z for which X(z)=0
 The Poles of a z-transform X(z) are the values of z for which 𝑋 𝑧 = ∞
𝑋 𝑧 =
𝐵(𝑧)
𝐴(𝑧)

15. 𝑥 𝑛 = 𝑎𝑛𝑢 𝑛
𝑋 𝑧 =
1
1 − 𝑎𝑧−1
=
1
1 −
𝑎
𝑧
=
1
𝑧 − 𝑎
𝑧
=
𝑧
𝑧 − 𝑎
Zero=0
Pole=a