Module 2: Discrete-time Signals and Systems

2.1 Discrete-Time Signals

Outline
• Introduction
• Discrete-time signals
Basic operations
Basic sequences
Discrete-Time Signals
• Discrete-time signals are usually represented mathematically
as sequences of numbers

x = {x[n]}, −  n  

n is an integer!!!
Discrete-Time Signals: Sampling
• In practice, discrete-time signals may arise from periodic
sampling of analog signals
Discrete-Time Signals: Sampling
• Mathematical representation
x[n] = xa (nT ), −  n  
n is an integer and T is the sampling period

Can we apply any T?

Sampling theory
Discrete-Time Signals: Generating
Discrete-Time Signals
⚫ Graphical Representation
y[n] = x[mn]
x[0]

x[-1] 1
x[1] m=
x[2] 2
-5 -4 -3 3 4 5
m =2
-2 -1 0 1 2 6

x[3] m =-1
Basic Operations of Sequences
◼ Product and Sum: sample-by-sample
x[0]
x[1]
x[2]

3 4
x[n] x[n] y[n] 2 3
0 1 2 0 1 4

y[1]
y[0]
2
y[n] x[n]+y[n]
0 1 3 4 0 1 2 3 4

y[2]
Basic Operations of Sequences
◼ Multiplication a sequence by a number 
multiplication of each sample value by 

x[0] x[1]
x[2]
3 4 3 4
x[n] 2x[n]
0 1 2 0 1 2
Basic Operations of Sequences
◼ Delay or shift

y[n] = x[n − n0 ] n0  0
n0  0

x[0] x[0]
x[1] x[1]
x[2] x[2]

3 4
x[n] x[n-2]
0 1 2 0 1 2 3 4
Basic Sequences
◼ Unit sample sequence 1

0, n  0,
 [ n] = 
1, n = 0. 0

◼ Often referred to as discrete-time impulse or impulse

◼ A delayed impulse 0, n  k ,
 [n − k ] = 
1, n = k .
◼ Representing an arbitrary sequence using a sum of scaled,
delayed impulses.

x[n] =  x[k ] [n − k ]
k = −

p[n] = a− 2 [n + 2] + a1 [n − 1] + a3 [n − 3]

Basic Sequences
◼ Unit step sequence
1

1, n  0,
u[n] = 
0, n  0. u[n]
0
◼ Represent a unit step sequence using impulses
u[n] =  [n] +  [n − 1] +  [n − 2] + . . .


u[n] =   [n − k ]
k =0
n
=   [ m]
m = −

 [ n] = ?
Basic Sequences
◼ Sinusoidal sequence

x[n] = A cos( 0 n +  ), for all n

with parameters real constants 2

1.5

0.5

-0.5

-1

-1.5

-2
-5 0 5 10 15 20 25
Basic Sequences
◼ Exponential sequence (real)

x[n] = A n , for all n

If the parameters are real constants, the sequence is real.

1

0.8

0.6
 1
0.4

0.2 ... ...

-0.2

-0.4

-0.6
◼ Exponential sequence (complex)

x[n] = A n , for complex A and 

= Ae j
 e j
n 0n

= A  e j ( 0 n + )
n

= A cos( 0 n +  ) + j A  sin( 0 n +  )
n n

 =1 frequency phase
End of Module 2.1