Digital Signal Processing

1. Properties of Discrete Time Z- transform
Presented By:
Md. Kamal Hossain
Assistant Professor
Department of Electronics & Communication Engineering
Hajee Mohammad Danesh Science & Technology University, Dinajpur-5200

2. Linearity Property of Discrete Time z-
Transform:
Statement:
if 𝑥1(𝑛) ↔ 𝑋1(𝑧) and 𝑥2 𝑛 ↔ 𝑋2 𝑧
then 𝑎1 𝑥1 𝑛 + 𝑎2 𝑥2 𝑛 ↔ 𝑎1 𝑋1 𝑧 + 𝑎2 𝑋2(𝑧)
for any constants 𝑎1 and 𝑎2.

3. Linearity Property of Discrete Time z-
Transform:
Proof: We know z- transform of signal
𝑥(𝑛) is 𝑛=−∞
∞ 𝑥(𝑛)𝑧−𝑛
so, z- transform of the signal 𝑎1 𝑥1 𝑛 is
𝑍 𝑎1 𝑥1 𝑛 =
𝑛=−∞
∞
𝑎1 𝑥1 𝑛 𝑧−𝑛
= 𝑎1
𝑛=−∞
∞
𝑥1 𝑛 𝑧−𝑛
= 𝑎1 𝑋1(𝑧)

4. Linearity Property of Discrete Time z-
Transform:
Again, z- transform of the signal 𝑎2 𝑥2 𝑛 is
𝑍 𝑎2 𝑥2 𝑛 = 𝑛=−∞
∞
𝑎2 𝑥2 𝑛 𝑧−𝑛
= 𝑎2
𝑛=−∞
∞
𝑥2 𝑛 𝑧−𝑛
= 𝑎2 𝑋2(𝑧)
∴ 𝑎1 𝑥1 𝑛 + 𝑎2 𝑥2 𝑛 ↔ 𝑎1 𝑋1 𝑧 + 𝑎2 𝑋2(𝑧) (Proved)

5. Time Shifting of Discrete Time z-Transform:
Statement: if 𝑥(𝑛) ↔ 𝑋(𝑧)
then 𝑥(𝑛 − 𝑘) ↔ 𝑧−𝑘 𝑋(𝑧) if k is positive
Proof: We know z- transform of signal 𝑥(𝑛) is
𝑛=−∞
∞
𝑥(𝑛)𝑧−𝑛
so, z- transform of the signal 𝑥(𝑛 − 𝑘) is
𝑍 𝑥(𝑛 − 𝑘) =
𝑛=−∞
∞
𝑥(𝑛 − 𝑘)𝑧−𝑛

6. Time Shifting of Discrete Time z-Transform:
let 𝑛 − 𝑘 = 𝑚
∴ 𝑛 = 𝑚 + 𝑘
=
𝑛=−∞
∞
𝑥(𝑚)𝑧−(𝑚+𝑘)
=
𝑛=−∞
∞
𝑥(𝑚)𝑧−𝑚
𝑧−𝑘
= 𝑧−𝑘
𝑋(𝑧)
∴ 𝑥(𝑛 − 𝑘) ↔ 𝑧−𝑘
𝑋(𝑧) (Proved)

7. Time Shifting of Discrete Time z-Transform:
Statement: if 𝑥(𝑛) ↔ 𝑋(𝑧)
then 𝑥(𝑛 + 𝑘) ↔ 𝑧 𝑘 𝑋(𝑧) if k is negative
Proof: We know z- transform of signal 𝑥(𝑛) is
𝑛=−∞
∞
𝑥(𝑛)𝑧−𝑛
so, z- transform of the signal 𝑥(𝑛 + 𝑘) is
𝑍 𝑥(𝑛 + 𝑘) =
𝑛=−∞
∞
𝑥(𝑛 + 𝑘)𝑧−𝑛

8. Time Shifting of Discrete Time z-Transform:
let 𝑛 + 𝑘 = 𝑚
∴ 𝑛 = 𝑚 − 𝑘
=
𝑛=−∞
∞
𝑥(𝑚)𝑧−(𝑚−𝑘)
=
𝑛=−∞
∞
𝑥(𝑚)𝑧−𝑚
𝑧 𝑘
= 𝑧 𝑘
𝑋(𝑧)
∴ 𝑥(𝑛 − 𝑘) ↔ 𝑧 𝑘
𝑋(𝑧) (Proved)