Chapter 2

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

)(


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


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


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


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)()(


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)



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)



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)



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


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)


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


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 )()()()(


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)


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


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


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


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


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


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

Chapter 2

Documents

n time

x n u n

values of n

n solution

n1 e n e n

time base

xn example

exponential signal xn