This document provides an introduction to discrete time signals and systems. It defines discrete time signals as functions of integer time and discusses various representations including graphical, functional, tabular, and sequence representations. It then describes common elementary discrete time signals like impulse, step, ramp, and exponential signals. The document also classifies signals as energy/power signals, periodic/aperiodic, even/odd, and discusses basic manipulations. Finally, it defines discrete time systems as those that transform an input signal to an output signal according to a rule and classifies systems as static/dynamic and with finite/infinite memory.

1. Discrete Time Signals and
Systems
-an Introduction

2. Discrete Time Signals
 The discrete time signal is a function of an independent
variable that is an integer.
 It is important to note that discrete time signal is defined for
integer values of time while it does not mean that at non
integer instants the signal is ‘0’. It is just not defined over
those intervals.
 The types of representation of signals are
 Graphical representation
 Functional representation
 Tabular representation
 Sequence representation

3. Contd..
Suppose we have 𝑥 𝑛 = 𝐶𝑜𝑠(𝜋𝑛)
• Then 𝑥 𝑛 𝑎𝑡 𝑛 = 0 𝑚𝑒𝑎𝑛𝑠 𝐶𝑜𝑠 𝜋 × 0 =1
• 𝑥 𝑛 𝑎𝑡 𝑛 = 1𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 1 = −1
• 𝑥 𝑛 𝑎𝑡 𝑛 = 2𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 2 = 1
• 𝑥 𝑛 𝑎𝑡 𝑛 = 3𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 3 = −1
.
.

4. Graphical Representation of Discrete Time Signal

5. Functional Representation
• 𝑥 𝑛 =
1, 𝑓𝑜𝑟 𝑛 = 1,3
4, 𝑓𝑜𝑟 𝑛 = 2
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

6. Tabular Representation

7. Sequence Representation
Also called as four point sequence

8. Elementary Discrete Time Signals
 Unit Impulse Signal
 Unit Step Signal
 Unit Ramp Signal
 Exponential Signal

9. Impulse Signal

10. Unit step signal

11. Unit Ramp Signal

12. Exponential Signal
 𝑥 𝑛 = 𝑎 𝑛 for all n
 We have two conditions here when a is Real, when a is
complex.
 When a is real we have 4 conditions and those are explained in
the graph.
 Let us take for example a is 2 that locates in a>1 then
 𝑥 𝑛 = 2 𝑛 for all n


13. Contd..

14. Contd..
When a is complex valued then
𝑎 = 𝑟𝑒 𝑗𝜃
𝑥 𝑛 = (𝑟𝑒 𝑗𝜃) 𝑛
𝑥 𝑛 = 𝑟 𝑛 𝑒 𝑗𝑛𝜃
Then 𝑒 𝑗𝜃 = 𝐶𝑜𝑠𝜃 + 𝑗𝑆𝑖𝑛𝜃
𝑥 𝑛 = 𝑟 𝑛(𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑆𝑖𝑛𝑛𝜃)
𝑥 𝑛 = 𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃
When you have a+jb there ‘a’ is real and ‘b’ is imaginary.
• 𝑥 𝑅 𝑛 = 𝑅𝑒(𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃)
• 𝑥 𝑅 𝑛 = 𝑟 𝑛
𝐶𝑜𝑠𝑛𝜃 Real value signal
• 𝑥𝐼 𝑛 = 𝐼𝑚(𝑟 𝑛
𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛
𝑆𝑖𝑛𝑛𝜃)
• 𝑥𝐼 𝑛 = 𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃 Imaginary signal

18. Classification of Discrete Time Signals
 In this section we classify discrete-time signals according to a
number of different characteristics.
 The mathematical methods employed in the analysis of
discrete-time signals and systems depend on the
characteristics of the signals.

19. Energy Signals and Power Signals
Energy Signal
 The Energy of a signal 𝑥(𝑛) is defined as
𝐸 =
𝑛=−∞
∞
|𝑥 𝑛 |2
 The magnitude square is considered since 𝑥(𝑛) can be real or
complex.
 This will be useful to know whether the signal is having finite
energy or infinite energy.
 In problems they will ask whether the signal is Energy Signal
or not?
 Then you have to compute E if it is finite (0<E<∞) then it is
Energy Signal.

20. Contd..
Power Signal
Many Signals those posses infinite energy have finite average
power. The average power of discrete time signal is
𝑃 = lim
𝑁→∞
1
2𝑁+1
( 𝑛=−𝑁
𝑁
|𝑥 𝑛 |2)
 Clearly if ‘E’ is finite. ‘P’=0 then Energy Signal.
 On the other hand if ‘E’ is infinite ‘P ‘may be either finite or
infinite.
 If ‘P’ is finite (non zero) it is Power Signal.
 If ‘P’ is infinite it is neither Energy nor Power Signal.

21. Contd..
Examples
• Unit Step is Energy Signal or Power Signal?
Energy 𝐸 = 𝑛=−∞
∞ |𝑥 𝑛 |2
• Unit Step Signal
• 𝑥 𝑛 =
1 𝑓𝑜𝑟 𝑛 ≥ 0
0 𝑒𝑠𝑙𝑒 𝑤ℎ𝑒𝑟𝑒
Energy 𝐸 = 𝑛=0
∞
|𝑥 𝑛 |2
• So 1+1+1…..∞ that is E= ∞
• When E is ∞ we have to verify for power signal.

22. Contd..
𝑃 = lim
𝑁→∞
1
2𝑁+1
( 𝑛=−𝑁
𝑁
|𝑥 𝑛 |2)
 For unit step signal
𝑃 = lim
𝑁→∞
1
2𝑁+1
( 𝑛=0
𝑁
|𝑥 𝑛 |2
)
𝑃 = lim
𝑁→∞
1
2𝑁+1
(N+1)
(if N=3 then 1+1+1+1=4…….)
𝑃 = lim
𝑁→∞
1
2+
1
𝑁
(
1
𝑁
+1)=
1
2
Which is a finite value hence it is Power Signal.

23. Periodic and aperiodic Signals
• 𝑥 𝑛 =𝑥 𝑛 + 𝑁

24. Even Signal
• If the signal satisfies the condition
𝑥 −𝑛 =𝑥 𝑛 then it is an even signal

25. ODD Signal
• If the signal satisfies the condition
𝑥 −𝑛 =−𝑥 𝑛 then it is an odd signal

26. Even or ODD?
𝑥1 𝑛 = 𝐶𝑜𝑠(0.125𝜋𝑛)
To know whether the signal is even or odd? We have to compute 𝑥1 −𝑛 if it is
equal to 𝑥1 𝑛 then it is even signal, if it is equal to −𝑥1 𝑛 then it is odd signal.
So 𝑥1 −𝑛 = 𝐶𝑜𝑠 0.125𝜋 −𝑛
𝑥1 −𝑛 = 𝐶𝑜𝑠(−0.125𝜋(𝑛))
you know Cos(-𝜃)=Cos(𝜃) then
𝑥1 −𝑛 = 𝐶𝑜𝑠 0.125𝜋 𝑛 = 𝑥1 𝑛 So it is even signal
Similarly if you do for 𝑥2 𝑛 you will identify it as odd signal.
𝑥2 𝑛 = 𝑆𝑖𝑛(0.125𝜋𝑛)

27. Simple Manipulations on Discrete Time
Signals
Transformation of the independent variable
 A signal x (n ) may be shifted in time by replacing the
independent variable n by n-k, where k is an integer.
 If k is a positive integer, the time shift results in a delay of the
signal by k units o f time. If k is a negative integer, the time
shift results in an advance of the signal by |k| units in time

29. Folding of a signal
• Another useful modification of the time base is to replace the
independent variable n by -n. The result of this operation is a
folding or a reflection of the signal about the time origin n =0.

30. How to plot 𝑥 −𝑛 + 2 ?
• First do the folding and then shifting.
• Note that because the signs o f n and k in
𝑥 𝑛 − 𝑘 and 𝑥 −𝑛 + 𝑘 are different, the
result is a shift of the signals 𝑥 𝑛 and
𝑥 𝑛 − 𝑘 to the right by k samples,
corresponding to a time delay.

33. Addition, multiplication and scaling of
sequences
Scaling
Addition of two signals
Product of two Signals

34. Discrete Time Systems
A discrete time system is a device or algorithm that performs a
prescribed operation on the discrete time signal, called input or
excitation, according to a well defined rules to produce another
discrete time signal called output or response of the system.
We say that the input signal 𝑥 𝑛 is transformed by the system in
to a signal y 𝑛 and express the general relationship between
𝑥 𝑛 and y 𝑛 as
y 𝑛 ≡ Τ[𝑥 𝑛 ]

35. Input-Output description of the systems
The input-output description of a discrete-time system consists of
a mathe matical expression or a rule, which explicitly defines the
relation between the input and output signals (input– output
relationship).
 The exact internal structure of the system is either unknown or
ignored.
 Examples
𝑦 𝑛 = 𝑥 𝑛 − 2
𝑦 𝑛 = 4𝑥(𝑛)

36. Classification of Discrete Time Systems
Static vs Dynamic
• A discrete-time system is called static or memory less if its
output at any instant ‘n’ depends at most on the input sample at
the same time, but not on past or future samples of the input.
𝑦 𝑛 = 𝑎𝑥(𝑛)
𝑦 𝑛 = 𝑛𝑥 𝑛 + 𝑏𝑥2
(𝑛)
• In any other case, the system is said to be dynamic or to have
memory .
Finite Memory System Infinite Memory System