Chapter 2:DISCRETE-TIME SIGNALS AND
SYSTEMS
DISCRETE-TIME SIGNALS
Representing Discrete-Time Signals Graphical Representation
Functional Representation
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
elsewhere
nfor
nfor
nx
0
24
3,11
)(
DISCRETE-TIME SIGNALS
Representing Discrete-Time Signals Tabular Representation
Sequence Representation
or
n … -2 -1 0 1 2 3 4 …
x(n) … 0 0 0 1 4 1 0 …
,...}0,1,4,1,0,0,0{...)( nx
}1,4,1,0{)( nx
DISCRETE-TIME SIGNALS
Some Fundamental Sequence Unit Sample Sequence [δ(n)]
Unit Step Signal [u(n)]
. Exponential Signal x(n) = an for all n
otherwise
nn
0
01)(
00
01)(
n
nnu
DISCRETE-TIME SIGNALS
Signal Duration Finite-Length Sequence – discrete-time sequence that is
equal to zero for values of n outside a finite interval [N1, N2].
Infinite-Length Sequence – signals that are not finite in length, such as the unit step and exponential sequences.
Right-Sided Sequence – any infinite-length sequence that is equal to zero for all values of n < no for some integer no.
otherwise
NnNxnx
,0
,)( 21
DISCRETE-TIME SIGNALS
Signal Duration Infinite-Length Sequence
Left-sided Sequence – an infinite-length sequence x(n), for some integer no is equal to zero for all n > no. For example, which is a time-reversed and delayed unit step.
Two-Sided Sequence – an infinite-length that is neither right-sided nor left-sided, such as the complex exponential.
o
oo nn
nnnnunx
,0
,1)()(
DISCRETE-TIME SIGNALS
Simple Manipulation of Discrete-time Signal Transformation of the Independent variable (time, n)
Time Shifting. The independent variable, n, is replaced by n – k, where k is an integer.
• If k is a positive integer, the signal is delayed.
• If k is negative integer, the signal is advanced.
Example 2.1
A signal x(n) is graphically illustrated in figure below. Show a graphical representation of the signal x(n – 3) and x(n + 2).
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
Example 2.1
Solution:For x(n-3)n=0x(0-3)=x(-3)Base on graphx(-3)= 1
n=1x(1-3)=x(-2)Base on graphx(-2)= 2
n=2x(2-3)=x(-1)Base on graphx(-1)= 3
n=3x(3-3)=x(0)Base on graphx(0)= 4…
n=-1x(-1-3)=x(-4)Base on graphx(-4)=0
n=-2x(-2-3)=x(-5)Base on graphx(-5)= -1
0 1 2 3 4 5
-5 -4 -3 -2 -1
4
3
2
1n
x(n)
Example 2.1
Graphical representation of x(n-3)
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
3 4 5 6 7 8 -2 -1 0 1 2
4
3
2
1
n
x(n-3)
Example 2.1
Graphical representation of x(n+2)
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
-2 -1 0 1 2 3 -7 -6 -5 -4 -3
4
3
2
1
n
x(n+2)
DISCRETE-TIME SIGNALS
Simple Manipulation of Discrete-time Signal Transformation of the Independent variable (time, n)
Folding or Reflection of the signal about the time origin. The time base is to be replaced n by –n .
Example 2.2
Show the graphical representation of the signal x(–n) where x(n) is the signal illustrated below.
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1n
x(n)
Example 2.2
Solution:Let y(n)=x(-n)n=0y(0)= x(0)Base on graphx(0)= 0
n=1y(1)= x(-1)x(-1)= 2
n=2y(2)= x(-2)x(-2)= 2
n=3y(3)= x(-3)x(-3)= 2
n=4y(4)= x(-4)x(-4)= 0
n=-1y(-1)= x(1)x(1)= 1
n= -2y(-2)= x(2)x(2)= 2
n=-3y(-3)= x(3)x(3)= 3
n= -4y(-4)= x(4)x(4)= 4
n=-5y(-5)= x(5)x(5)= 0
0 1 2 3 4 -5 -4 -3 -2 -1
4
3
2
1 n
x(n)
0 1 2 3 4 5 -4 -3 -2 -1
4
3
2
1
n
x(n)
DISCRETE-TIME SIGNALS
Simple Manipulation of Discrete-time Signal Transformation of the Independent variable (time, n)
Folding or Reflection of the signal about the time origin. Note:
Folding and Time-shifting a signal are not commutative.Let TD = time delay operation
FD = folding operation
TDK[x(n)] = x(n–k), k > 0FD [x(n)] = x(–n)
Now,
TDk {FD[x(n)] }= TD{x(–n)} = x(–n+k)Whereas,
FD {TDk [x(n)]} = FD{x(n–k)} = x(–n–k)
Example 2.3
Using Figure 2.2 show that folding and time-shifting are not commutative. Time-advance the signal x(n) by 2 units in time
then fold. Fold the signal x(n) then time advance it by 2
units in time.
Example 2.3
Time-advance the signal x(n) by 2 units in time then fold.
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
-3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
x(-n+2)
-2 -1 0 1 2 3 -6 -5 -4 -3
4
3
2
1
n
x(n+2)
Example 2.3
Fold the signal x(n) then time advance it by 2 units in time.
0 1 2 3 4 5 -5 -4 -3 -2 -1
4
3
2
1
n
x(n)
-7 -6 -5 -4 -3 -2 -1 0 1 2
4
3
2
1
x(-n-2)
-5 -4 -3 -2 -1 0 1 2 3 4
4
3
2
1 n
x(-n)
DISCRETE-TIME SIGNALS
Simple Manipulation of Discrete-time Signal Transformation of the Independent variable (time, n)
Time-Scaling. The independent variable time, n, is replaced by μn, where μ is an integer.
The signal y(n) = x(μn) is a time-scaled version of x(n).
If |μ| > 1, we are SPEEDING UP or DOWN SAMPLING x(n) by a factor of μ
If |μ| < 1, we are SLOWING DOWN or UP SAMPLING x(n) by a factor of μ.
Example 2.4
Show the graphical representation of the signal y(n) = x(2n) where x(n) is the signal illustrated below.
x(n)
0 1 2 3 4 5 6 7 n -4 -3 -2 -1
-7 -6 -5
43
–3
2
1
–2
–1
Example 2.4
y(n)=x(2n)n=0y(0)=x(0)=4
n=1y(1)=x(2*1)=4
n=2y(2)=x(2*2)=x(4)
=4
x(n)
0 1 2 3 4 5 6 7 n
-4 -3 -2 -1
-7 - 6 -5
43
–3
2
1
–2
–1
n=3y(3)=x(2*3)=x(6)
=4
n=4y(4)=x(2*4)=x(8)
=0
n=-1y(-1)= x(2*(-1))
=x(-2)=2
n=-2y(-2) =x(2*(-2))
=x(-4)=0
n=-3y(-3) =x(2*(-3))
=x(-6)=-2
n=-4y(-4)= x(2*(-4))
=x(-8)=0
Example 2.4
y(n)=x(2n)
0 1 2 3 n -2 -1
4
2
–2
x(n)
0 1 2 3 4 5 6 7 n -4 -3 -2 -1
-7 -6 -5
43
–3
2
1
–2
–1
DISCRETE-TIME SIGNALS
Simple Manipulation of Discrete-time Signal Amplitude Modifications
Amplitude scaling
y(n) = A x(n) Addition of two signals
y(n) = x1(n) + x2(n) Multiplication of two signals
y(n) = x1(n) x2(n)
DISCRETE-TIME SYSTEMS
A device or algorithm that operates on a discrete-time signal, according to some well-defined rules, to produce another discrete-time signal.
y(n) = T[ x(n)]where: T = denotes the transformation
x(n) = input signaly(n) = output signal
Block diagram representation of discrete time signal
Discrete-time system
x(n) y(n)
DISCRETE-TIME SYSTEMS
Input-Output Description of Systems
– consist of a mathematical expression or a rule, which explicitly defines the relation between the input and output signals.
General input-output relationship
T
x(n) y(n)
Example 2-5
Determine the response of the following systems to the input signal
y(n) = x(n) y(n) = x(n–1) y(n) = x(n+1) y(n) = ⅓ [x(n+1) + x(n) + x(n–1) y(n) = max [ x(n+1), x(n), x(n–1)]
...)()()()()(
21 nxnxnxkxnyn
k
otherwise
nfornnx
0
33||)(
Example 2-5
x(n)={3, 2, 1, 0, 1, 2, 3} ↑
a. y(n) = x(n) ={3, 2, 1, 0, 1, 2, 3} ↑
b. y(n) = x(n–1) ={3, 2, 1, 0, 1, 2, 3} ↑
c. y(n) = x(n+1) ={3, 2, 1, 0, 1, 2, 3} ↑
otherwise
nfornnx
0
33||)(
Example 2-5
x(n)={3, 2, 1, 0, 1, 2, 3} ↑
d. y(n) = ⅓ [x(n+1) + x(n) + x(n–1)]x(n-1)={3, 2, 1, 0, 1, 2, 3}
↑ x(n)={3, 2, 1, 0, 1, 2, 3}
↑ x(n+1)={3, 2, 1, 0, 1, 2, 3}
↑
y(n)= ⅓{3,5,6,3,2,3,6,5,3}={1,5/3,2,1,2/3,1,2,5/3,1} ↑ ↑
otherwise
nfornnx
0
33||)(
Example 2-5
x(n)={3, 2, 1, 0, 1, 2, 3} ↑
e. y(n) = max [x(n+1), x(n),x(n–1)]x(n-1)={3, 2, 1, 0, 1, 2, 3}
↑ x(n)={3, 2, 1, 0, 1, 2, 3}
↑ x(n+1)={3, 2, 1, 0, 1, 2, 3}
↑
y(n)= {3,3,3,2,1,2,3,3,3} ↑
otherwise
nfornnx
0
33||)(
Example 2-5
x(n)={3, 2, 1, 0, 1, 2, 3} ↑
f.
y(n) = {3, 5, 6, 6, 7, 9, 12, 12,…} ↑
otherwise
nfornnx
0
33||)(
...)()()()()(
21 nxnxnxkxnyn
k
DISCRETE-TIME SYSTEMS
Accumulator – Computes the running sum of all the past input up to the present time.
y(n) = y(n–1) + x(n) Initial condition – summarizes the effect of
all previous inputs to the system Initially relaxed – had no excitation prior to
the present time instant and the initial condition is zero.
1n
k
n
k
nxkxkxny )()()()(
DISCRETE-TIME SYSTEMS
Block Diagram Representation of Discrete-Time SystemsMemoryless – a system is said to be memoryless if the output at any
time n = no depends only on the input at time n = no.a. Adder. Performs the addition of two signal sequences to form
another sequence. Memoryless operation.
b. Constant multiplier. Applies a scale factor on the input x(n). Also a memoryless operation.
x2n)
+
x1(n)y(n)= x1(n) + x2(n)
ax(n) y(n) =ax(n)
DISCRETE-TIME SYSTEMS
Block Diagram Representation of Discrete-Time Systems
c. Signal Multiplier Multiplication of two signal sequences to form another sequence. Also a memoryless operation
d. Unit delay Element. A special system that simply delays the signal passing through it by one sample. It requires memory.
e. Unit advance Element. A special system that simply moves the signal passing through it by one sample. It requires memory.
xx1(n)
x2(n)
y(n)= x1(n) x2(n)
z–1x(n) y(n)= x(n–1)
zx(n) y(n)= x(n+1)
Example 2-6
Using the basic building blocks, sketch the block diagram representation of the discrete-time system described by the input-output relation.
y(n)= ¼ y(n–1) + ½ x(n) + ½ x(n–1)
Classification of Discrete-time Systems
1. Static vs. Dynamic Systems
2. Time-Invariant vs. Time Variant Systems
3. Linear vs. Nonlinear Systems
4. Causal vs. Non-Causal Systems
5. Stable vs. Unstable
Classification of Discrete-time Systems
1. Static vs. Dynamic Systems Static – a discrete-time system that is
memoryless Dynamic – a discrete-time system that requires
memory.
Example 2-7
Determine whether the following signals are static or dynamic:
y1(n) = ax(n)
y2(n) = x(n) + 3x(n–1)
y3(n) =
y4(n) = nx(n) b x3(n)
y5(n) = ax(n2)
n
kx )(
Static
Dynamic
Dynamic
Static
Dynamic
Classification of Discrete-time Systems
2. Time-Invariant vs. Time-variant Systems Time-Invariant System – input-output
characteristics do not change with time.
Let y(n) be the response of they system to an arbitrary input x(n). The system is said to be time invariant if, for any delay no, the response to x(n–no) is y(n–no).
Time-Variant System – the input-output characteristics do vary with time.
Example 2-8
Determine if the system equations are time-invariant or time variant. y(n) = x(n) – x(n–1) y(n) = nx(n) y(n) = x(–n) y(n) = x2(n)
Time Invariant
Time Variant
Time Variant
Time Invariant
Classification of Discrete-time Systems
3. Linear vs. Nonlinear Systems Linear System – one that satisfies the superposition principle.
Superposition Principle – requires that the response of the system to a weighted sum of signals be equal to the corresponding weighted sum of the responses of the system to each of the individual input signals.
– The response to the sum of inputs is equal to the sum of the inputs individually.
T[a1x1(n) + a2 x2(n)] = a1T[x1(n)] + a2T [x2(n)]
Nonlinear System – a relaxed system produces a nonzero output with a zero input and does not satisfy the superposition principle.
Example 2-9
Determine if the systems described by the following input-output equations are linear or nonlinear.
y1(n) = nx(n)
y2(n) = x(n2)
y3(n) = x2(n)
y4(n) = Ax(n) + B
y5(n) = ex(n)
Linear
Linear
Nonlinear
Linear
Nonlinear
Classification of Discrete-time Systems
4. Causal vs. Noncausal Systems Causal System – a system whose output at any
time n depends on the present and past inputs but DOES NOT depend on future inputs.
Noncausal System – a system has an output depends not only on present and past inputs but ALSO on future inputs.
Example 2-10
Determine if the system is causal or noncausal.
y(n) = x(n) – x(n-1)y(n) = x(n) + 3x(n+4)y(n) = x(n 2)y(n) = ax(n)y(n) = x(2n)y(n) = x(–n)
Causal
Noncausal
Noncausal
Causal
Noncausal
Noncausal
n
k
kxny )()( Causal
Classification of Discrete-time Systems
5. Stable vs. Unstable Systems
Stable System – it follows BIBO (Bounded Input – Bounded Output). Every bounded input produces a bounded output.
Unstable System – bounded input sequence does not produce a bounded output.
Example 2-11
Consider the following input-output equations y(n) = x(n-1) y(n) = cos [x(n)] y(n) = y2(n-1) + x(n)
As an input sequence x(n) = Cδ(n)
where: C is a constant and the system is initially relaxed
Determine if the system is stable or unstable
Stable
Unstable
Unstable