This document provides an overview of signal and systems topics including: - Complex exponential signals including periodic and non-periodic cases - Common basic signals like unit impulse, unit ramp, unit parabolic, unit signum, and unit sinc functions - Examples using unit impulse and unit step functions - Extra questions about average power of signals, periodicity of signals, representing signals as sums of weighted impulses, sketching functions, and relationships between continuous and discrete time signals

1. Signal and Systems
Chapter 1
Islam K. Sharawneh
islam.Sharawneh@ptuk.edu.ps

2. DT Complex Exponential Signal
• 𝑥 𝑛 = 𝐶𝑒𝛼𝑛 , C 𝑎𝑛𝑑 𝛼 are complex numbers
• 1: if 𝐶 = 1 𝑎𝑛𝑑 𝛼 = 𝑗𝜔0 then 𝑥 𝑛 = 𝑒𝑗𝜔0𝑛
This 𝑒𝑗𝜔0𝑛 is not periodic for all 𝜔0
• 𝑥 𝑛 + 𝑁 = 𝑒𝑗𝜔0(𝑛+𝑁)
= 𝑒𝑗𝜔0𝑛
𝑒𝑗𝜔0𝑁
. If 𝜔0𝑁 = 2𝜋𝑚, this implies
that 𝑥 𝑛 = 𝑥 𝑛 + 𝑁 . 𝑚, 𝑛 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 and 𝜔0 =
2𝜋𝑚
𝑁
or
𝜔0
2𝜋
=
𝑚
𝑁
is
a “rational number”

3. DT Complex Exponential Signal
• 𝑥 𝑛 = 𝑒𝑗𝜔0𝑛
is periodic, then we have
𝜔0
2𝜋
=
𝑚
𝑁
under the assumption that
N and m have no factors in common, 𝑁 =
2𝜋𝑚
𝜔0
, 𝜔0≠ 0.
• The fundamental frequency =
2𝜋
𝑁
=
𝜔0
𝑚
• CT: 𝑥 𝑡 = 𝑒𝑗𝜔0𝑡
are distinct signals for distinct 𝜔
• DT: 𝑥 𝑛 = 𝑒𝑗𝜔0𝑛
= 𝑒𝑗𝜔0(𝑛+𝑁)
= 𝑒
𝑗𝜔0(𝑛+
2𝜋𝑚
𝜔0
)
= 𝑒𝑗𝜔0𝑛
. 𝑒𝑗𝜔02𝜋𝑚
Periodic complex exponential signals with angular frequencies 𝜔0, 𝜔0± 2𝜋,
𝜔0± 4𝜋, … are identical signals. That reason we consider a frequency
interval of length 2𝜋. E.g. −𝜋 ≤ 𝜔0 < 𝜋 ; 0 ≤ 𝜔0 < 2𝜋.

4. Other Signals: Unit Rectangular Pulse

5. Other Signals: Unit Ramp

6. Other Signals: Unit Parabolic

7. Other Signals: Unit Signum

8. Other Signals: Unit Sinc

9. Other Signals: Unit Area Triangular

10. Example on using unit impulse and unit step functions
• 𝑦1 𝑡 = 𝑢 𝑡 − 𝑢 𝑡 − 1
• 𝑦2 𝑡 =
𝑑𝑦(𝑡)
𝑑𝑡
• 𝑦3 𝑡 = 𝑢 𝑡 + 1 − 2𝑢 𝑡 + 𝑢 𝑡 − 1

11. Extra Questions
• Q1: Determine the average power of the following signals
1. 𝑥 𝑡 = 𝐴𝑒𝑗(𝜔0+𝜃)
2. 𝑥 𝑡 = 𝐴𝑐𝑜𝑠(𝜔0 + 𝜃)
Q2: If 𝑥 𝑛 = cos
𝜋𝑛2
8
, show that 𝑥 𝑛 is periodic with fundamental
period 𝑁 = 8.
Q3: Express the following as a sum of weighted delayed impulses
𝑥 𝑛 = {0,1,-2,3,-2,1}
(0 sample starts at n=0)

12. Extra Questions
• Q4: Sketch the following functions
• 1. 𝑥1 𝑡 = ൝
1 −
𝑡
12
, 0 < 𝑡 ≤ 12
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• 2. 𝑥2 𝑡 = 𝑥1 1 − 𝑡 𝑢 𝑡 + 1 − 𝑢 𝑡 − 2
• 3. 𝑥3 𝑡 = 𝑥1 1 − 𝑡 𝑢 𝑡 + 1 − 𝑢 2 − 3𝑡
Q5: 𝑢 𝑡 is a unit impulse function, show that 𝑠𝑛𝑔 𝑡 = 𝑢 𝑡 − 𝑢 −𝑡
Q6: If x t = cos 2𝜋𝑓𝑡 is a continuous time signal, and x[n] = cos 2𝜋𝐹𝑛
is a discrete time signal which is the sampled version of x t at sampling rate
𝐹𝑠 =
1
𝑇𝑠
, show that the sampling rate 𝐹𝑠 =
𝑓
𝐹
.