1) Discrete-time complex exponential and sinusoidal signals can be represented as x[n] = Ceβn or x[n] = Acos(ω0n + φ). 2) Discrete-time complex exponentials with frequencies separated by 2π are identical, whereas in continuous time they are distinct. 3) A discrete-time complex exponential is periodic if the frequency ω0/2π is a rational number. The fundamental period is 2π/ω0.

1. EC2610:Fundamentals of Signals and Systems
By
Sadananda Behera
Assistant Professor
Department of Electronics and Communication Engineering
NIT, Rourkela
1
CHAPTER-1 : SIGNALS AND SYSTEMS-Part 3

2. Discrete-Time complex exponential and Sinusoidal
signals
A discrete-time complex exponential signal or sequence is of the form
𝑥 𝑛 = 𝐶𝛼𝑛…………(5) where 𝐶 and 𝛼 are complex numbers
Alternatively, eq(5) can be expressed as 𝑥[𝑛] = 𝐶𝑒𝛽𝑛…..(6), where 𝛼 = 𝑒𝛽
The complex exponential can exhibit different characteristics depending upon its
parameters 𝐶 and 𝛼
2

3. A discrete-time real exponential is a special case of complex exponential
𝑥 𝑛 = 𝐶𝛼𝑛
where 𝐶 and 𝛼 are real numbers
• If 𝛼 > 1, the magnitude of the signal 𝑥[𝑛] grows exponentially with 𝑛 (Growing
Exponential)
• If 𝛼 < 1 , the magnitude of the signal 𝑥[𝑛] decreases exponentially with
𝑛 (Decaying Exponential)
• If 𝛼 = 1, then 𝑥[𝑛] is a constant
• If 𝛼 = −1 , 𝑥[𝑛] alternates between +𝐶 and − 𝐶
Real Exponential signal
3

4. 𝛼 > 1
−1 < 𝛼 < 0 𝛼 < −1
0 < 𝛼 < 1
4

5. Sinusoid Signals
Using eq(6) and constraining 𝛽 as a purely imaginary number (i.e 𝛼 = 1 ) and
𝐶 = 1,we will get
𝑥 𝑛 = 𝑒𝑗𝜔0𝑛 …(7)
As in the continuous time case, the discrete-time sinusoid is represented as
𝑥 𝑛 = Acos(𝜔0𝑛 + 𝜑)…….(8)
𝑛 →dimensionless 𝜔0, 𝜑 → radians
5

6. The signals represented in eq(7) and eq(8) are examples of discrete-time signals with
infinite total energy but finite average power
Since 𝑒𝑗𝜔0𝑛 2
= 1, every sample of the signal in eq(7) contribute 1 to the signal
energy. Thus the total energy for −∞ < 𝑛 < ∞ is infinite, while the average power
per time point is equal to 1.
According to Euler’s relation
𝑒𝑗𝜔0𝑛 =cos 𝜔0𝑛 + 𝑗 sin 𝜔0𝑛 …..(9)
A cos(𝜔0𝑛 + ∅) =
𝐴
2
𝑒𝑗∅𝑒𝑗𝜔0𝑛 +
𝐴
2
𝑒−𝑗∅𝑒−𝑗𝜔0𝑛….(10)
6

7. General Complex Exponential Signal
The general complex exponential signal can be represented as in terms of real
exponential and sinusoidal signals.
General exponential signal
𝑥 𝑛 = 𝐶𝛼𝑛
Let 𝐶 = |𝐶|𝑒𝑗𝜃 and 𝛼 = |𝛼|𝑒𝑗𝜔0
𝑥 𝑛 = 𝐶𝛼𝑛 = |𝐶| 𝛼 𝑛 cos(𝜔0𝑛 + 𝜃) + 𝑗|𝐶||𝛼|𝑛 sin(𝜔0𝑛 + 𝜃)
Re{𝑥[𝑛]} Im{𝑥[𝑛]}
7

8. If 𝛼 = 1, then real(Re{𝑥[𝑛]}) and imaginary (Im{𝑥[𝑛]}) parts of a complex
exponential sequence are sinusoidal.
𝑎 𝛼 = 1
8

9. (𝑏) 𝛼 > 1
(𝑐) 𝛼 < 1
If 𝛼 > 1, corresponds to sinusoidal sequences multiplied by a growing exponential
If 𝛼 < 1, corresponds to sinusoidal sequences multiplied by a decaying exponential
9

10. Periodic Properties of Discrete-time complex
exponentials
Continuous-time counter part: 𝑒𝑗𝜔0𝑡
• The larger is the magnitude of 𝜔0 , the higher is the rate of oscillation in the
signal.
• 𝑒𝑗𝜔0𝑡
is periodic for any value of 𝜔0
• 𝑒𝑗𝜔0𝑡 are all distinct for distinct value of 𝜔0
10

11. Discrete-Time case
• 𝑒𝑗 𝜔0+2𝜋 𝑛 = 𝑒𝑗2𝜋𝑛. 𝑒𝑗𝜔0𝑛 = 𝑒𝑗𝜔0𝑛
• Discrete-time complex exponentials separated by 2𝜋 are same.
• The signal with frequency 𝜔0 is identical to the signals with frequencies
𝜔0 ± 2𝜋, 𝜔0 ± 4𝜋, and so on.
• In discrete-time complex exponentials only a frequency interval of length
2𝜋 is considered. E.g: −𝜋 ≤ 𝜔0≤ 𝜋 or 0 ≤ 𝜔0 ≤ 2𝜋.
• 𝑒𝑗𝜔0𝑛 doesn’t have a continually increasing rate of oscillation as 𝜔0 is
increased in magnitude.
11

12. • As 𝜔0 increases from ‘0’, the signal oscillates more and more rapidly until 𝜔0 = ߨ. After this
the increase in 𝜔0 will decrease in the rate of oscillation until it is reached 𝜔0 =2ߨ, which
produces the same constant sequence as 𝜔0 =0
• So, low frequency (i.e. slowly varying) discrete-time exponentials have values of 𝜔0 near 0, 2ߨ
and any other even multiple of 𝜋, while high frequencies(rapid variations) are located near
𝜔0 = ±𝜋 and other odd multiple of 𝜋.
• In particular for 𝜔0 = 𝜋 or any odd multiple of 𝜋
𝑒𝑗𝜋𝑛 = (𝑒𝑗𝜋)𝑛 = (−1)𝑛
It indicates signal oscillates rapidly (changing sign at each point in time).
12

13. Periodicity
𝑒𝑗𝜔0𝑛
will be periodic with period 𝑁 > 0, if 𝑒𝑗𝜔0(𝑛+𝑁)
= 𝑒𝑗𝜔0𝑛
⇒ 𝑒𝑗𝜔0𝑛. 𝑒𝑗𝜔0𝑁 = 𝑒𝑗𝜔0𝑛
⇒ 𝑒𝑗𝜔0𝑁 = 1
𝜔0𝑁 must be multiple of 2𝜋. There must be an integer 𝑚 such that 𝜔0𝑁 = 2𝜋𝑚
⇒
𝜔0
2𝜋
=
𝑚
𝑁
The signal 𝑒𝑗𝜔0𝑛 is periodic if
𝜔0
2𝜋
is a rational number, and not periodic
otherwise. The same observation is applicable for discrete-time sinusoids
Fundamental frequency of a periodic signal 𝑒𝑗𝜔0𝑛 is
2𝜋
𝑁
=
𝜔0
m
and fundamental
period 𝑁 = 𝑚(
2𝜋
𝜔0
) [𝑁 and 𝑚 has no common factor]
Rational Number
13

14. Comparison of signals 𝒆𝒋𝝎𝟎𝒕
and 𝒆𝒋𝝎𝟎𝒏
𝒆𝒋𝝎𝟎𝒕
Distinct signals for distinct values of
𝜔0
Periodic for any choice of 𝜔0
Fundamental Frequency 𝜔0
Fundamental Period
• 𝜔0 = 0, undefined
• 𝜔0 ≠ 0 ,
2𝜋
𝜔0
𝒆𝒋𝝎𝟎𝒏
Identical signals for values of
𝜔0 separated by 2𝜋
Periodic only if 𝜔0 =
2𝜋𝑚
𝑁
for some
integers 𝑁 > 0 𝑎𝑛𝑑 𝑚
Fundamental frequency
𝜔0
𝑚
(𝑚 and 𝑁 don’t have any factors in
common)
Fundamental period
• 𝜔0 = 0, undefined
• 𝜔0 ≠ 0, 𝑚(
2𝜋
𝜔0
)
(𝑚 and 𝑁 do not have any common
factors) 14

15. Example-1:
Find the fundamental period if the signal is periodic.
• 𝑥(𝑡)=cos
2𝜋
12
𝑡 ⇒ 𝑇 = 12 Periodic
• 𝑥[𝑛]=cos
2𝜋
12
𝑛
⇒
𝜔0
2𝜋
=
𝑚
𝑁
⇒
2𝜋
12
2𝜋
=
𝑚
𝑁
⇒
𝑚
𝑁
=
1
12
⇒ 𝑁 = 12
Periodic
15

16. Example-2
 Find the fundamental period if the signal is periodic.
• 𝑥(𝑡)=cos
8𝜋
31
𝑡 Periodic
• 𝑥[𝑛]=cos
8𝜋
31
𝑛 Periodic
𝑥(𝑡)=cos
8𝜋
31
𝑡
𝑇0=
2𝜋
8𝜋
31
=
31
4
𝑥[𝑛]=cos
8𝜋
31
𝑛
⇒
𝜔0
2𝜋
=
𝑚
𝑁
⇒
8𝜋
31
2𝜋
=
𝑚
𝑁
⇒
𝑚
𝑁
=
4
31
⇒ 𝑁 = 31
The discrete time signals is defined only for integer values of independent variable.
16

17. Example-3
 Find the fundamental period if the signal is periodic.
• 𝑥(𝑡)=cos
𝑡
6
• 𝑥[𝑛]=cos
𝑛
6
⇒ 𝑥(𝑡)=cos
𝑡
6
𝑇0 = 12𝜋
𝑥[𝑛]=cos
𝑛
6
⇒
𝜔0
2𝜋
=
1
6 × 2𝜋
=
1
12𝜋
=
𝑚
𝑁
So it is not periodic.
Irrational Number
Rational Number
17

18. Example-4
 𝑥[𝑛]= 𝑒
𝑗
2𝜋
3
𝑛
+ 𝑒
𝑗
3𝜋
4
𝑛
, find the fundamental period if the signal is periodic.
𝑒
𝑗
2𝜋
3
𝑛
is periodic with 3
𝑒
𝑗
3𝜋
4
𝑛
is periodic with 8
𝑥[𝑛] is periodic with 24
• For any two periodic sequences 𝑥1 𝑛 and 𝑥2[𝑛] with fundamental period 𝑁1and 𝑁2,
respectively, then 𝑥1 𝑛 + 𝑥2[𝑛] is periodic with 𝐿𝐶𝑀(𝑁1, 𝑁2)
18

19. Harmonically related periodic exponentials
A set of periodic complex exponentials is said to be harmonically related if all the
signals are periodic with a common period 𝑁
These are the signals of frequencies which are multiples of
2𝜋
𝑁
. That is
∅𝑘[𝑛] = 𝑒𝑗𝑘(
2𝜋
𝑁
)𝑛
, 𝑘 = 0, ±1, ±2, ±3, …
In continuous time case, all of the harmonically related complex exponentials
𝑒
𝑗𝑘
2𝜋
𝑇
𝑡
, 𝑘 = 0, ±1, ±2, ±3, … are distinct
19

20. In discrete time case
∅𝑘+𝑁 𝑛 = 𝑒
𝑗(𝑘+𝑁)
2𝜋
𝑁 𝑛
∅𝑘+𝑁 𝑛 = 𝑒
𝑗𝑘
2𝜋
𝑁
𝑛
. 𝑒𝑗2𝜋𝑛
= ∅𝑘[𝑛] ……..(11)
This implies that there are only 𝑁 distinct periodic exponentials in eq (11)
E.g.:∅0 𝑛 = 1, ∅1 𝑛 = 𝑒𝑗
2𝜋𝑛
𝑁 , 𝑎𝑛𝑑 ∅2[𝑛] = 𝑒𝑗
4𝜋𝑛
𝑁 ,…….∅𝑁−1 𝑛 = 𝑒𝑗
2𝜋(𝑁−1)𝑛
𝑁 are
distinct.
Any other ∅𝑘[𝑛] is identical to one of these
E.g.: ∅𝑁[𝑛] = ∅0 𝑛 𝑎𝑛𝑑 ∅−1[𝑛] = ∅𝑁−1[𝑛]
20

21. Unit Impulse and Unit Step Functions
Discrete-time unit impulse and unit step sequences.
The unit impulse or unit sample sequence is defined as 𝛿[𝑛]=
0, 𝑛 ≠ 0
1, 𝑛 = 0
Discrete-time unit step signal 𝑢[𝑛] is defined as 𝑢[𝑛]=
0, 𝑛 < 0
1, 𝑛 ≥ 0
21

22. Discrete-time unit impulse is the first difference of the discrete-time step
𝛿[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 1]
22

23. 23
The discrete-time unit step is the running sum of the unit sample
𝑢[𝑛] = 𝛿[𝑚]
𝑛
𝑚=−∞ ………(12)
𝑛 < 0 𝑛 > 0

24. Put 𝑘 = 𝑛 − 𝑚 in eq(12)
𝑢[𝑛] = 𝛿[𝑛 − 𝑘]
0
𝑘=∞
𝑢[𝑛] = 𝛿[𝑛 − 𝑘]
∞
𝑘=0 …….(13)
𝑛 < 0
𝑛 > 0
24

25. Eq(13) can be interpreted as superposition of delayed impulses.
25
𝑢[𝑛] = 𝛿[𝑛 − 𝑘]
∞
𝑘=0 )

26. 26
The unit impulse sequence can be used to sample the value of a signal.
𝑥 𝑛 . 𝛿 𝑛 = 𝑥 0 . 𝛿[𝑛]
𝑥 𝑛 . 𝛿 𝑛 − 𝑛0 = 𝑥 𝑛0 . 𝛿[𝑛 − 𝑛0]

27. Continuous time unit-step and unit impulse function
The continuous time unit step function 𝑢(𝑡) is defined as
𝑢(𝑡)=
0, 𝑡 < 0
1, 𝑡 > 0
27

28. The continuous time unit impulse function 𝛿(𝑡) is related to the unit step function
as
𝑢 𝑡 = 𝛿 𝜏 𝑑𝜏 … … … … … … … … (14)
𝑡
−∞
𝛿 𝑡 =
𝑑𝑢(𝑡)
𝑑𝑡
𝑢(𝑡) is discontinuous at 𝑡 = 0
𝑢(𝑡) = lim
∆→0
𝑢∆(𝑡)
𝛿∆ 𝑡 =
𝑑𝑢∆(𝑡)
𝑑𝑡 28

29. 𝛿∆ 𝑡 is a short pulse of duration ∆ and with unit area for any value of ∆
𝛿(𝑡) = lim
∆→0
𝛿∆(𝑡)
The arrow at 𝑡 = 0 indicates the area of the pulse is concentrated at 𝑡 = 0 and the
height of the arrow and ‘1’ next to arrow are used to represent area of the impulse
Continuous Time Unit Impulse Scaled Impulse
29

30. Scaled impulse 𝑘𝛿 𝑡 will have an area 𝑘, so
𝑘𝛿 𝜏 𝑑𝜏
𝑡
−∞
= 𝑘𝑢(𝑡)
𝑢 𝑡 = 𝛿 𝜏 𝑑𝜏
𝑡
−∞
Put 𝜎 = 𝑡 − 𝜏
⇒ 𝑢 𝑡 = 𝛿 𝑡 − 𝜎 (−𝑑𝜎)
𝑡
−∞
⇒ 𝑢 𝑡 = 𝛿 𝑡 − 𝜎 (𝑑𝜎) … … … … … … … … … (15)
∞
0
30

31. 𝑡 < 0 𝑡 < 0
𝑡 > 0
𝑡 > 0
𝑢 𝑡 = 𝛿 𝜏 𝑑𝜏
𝑡
−∞
𝑢 𝑡 = 𝛿 𝑡 − 𝜎 (𝑑𝜎)
∞
0 31

32. Properties of Impulse Function
𝛿 𝑡 =
0, 𝑡 ≠ 0
𝛿 𝑡 𝑑𝑡 = 1
∞
−∞
𝑥(𝑡). 𝛿(𝑡) = 𝑥(0). 𝛿(𝑡)
𝑥(𝑡). 𝛿(𝑡 − 𝑡0) = 𝑥(𝑡0). 𝛿(𝑡 − 𝑡0)
 𝑥(𝑡). 𝛿 𝑡 − 𝑡0 𝑑𝑡
∞
−∞
= 𝑥(𝑡0)
𝛿 𝑎𝑡 =
1
𝑎
𝛿(𝑡)
𝛿 −𝑡 = 𝛿(𝑡)
𝛿[𝑛]=
0, 𝑛 ≠ 0
1, 𝑛 = 0
𝑥 𝑛 . 𝛿 𝑛 = 𝑥 0 . 𝛿[𝑛]
𝑥 𝑛 . 𝛿 𝑛 − 𝑛0 = 𝑥 𝑛0 . 𝛿 𝑛 − 𝑛0
 𝑥 𝑛 . 𝛿 𝑛 − 𝑛0
∞
𝑛=−∞ =𝑥 𝑛0
𝛿 𝑎𝑛 = 𝛿[𝑛]
𝛿 −𝑛 = 𝛿[𝑛]
Continuous-time Discrete-time
32
Definition Definition

33. THANK YOU
33