discrete time signals and systems

Discrete-Time signals:

sequencesDiscreet-Time signals are represented

mathematically as sequences of numbers

The sequence is denoted 𝑥[𝑛], and it is

written formally as

𝑥 = 𝑥 𝑛 ; −∞ < 𝑛 < ∞

where n is an integer number

In practice sequences arises from the

periodic sampling of an analog signal

1

Discrete-Time signals:

sequencesIn this case the numeric value of the nth

number in the sequence is equal to the

value of the analog signal, 𝑥𝑎(𝑡), at time

𝑛𝑇𝑥 𝑛 = 𝑥𝑎[𝑛𝑇]

2

Examples of sequences

3

Basic sequences and sequence

operationThe product and sum of two sequences x[n]

and 𝑦[𝑛] are defined as the sample by

sample product and sum

Multiplication of a sequence 𝑥[𝑛] by a

number 𝛼 is defined as the multiplication of

each sample value by 𝛼

A sample 𝑦[𝑛] is said to be delayed or shifted

version of 𝑥[𝑛] if 𝑦 𝑛 = 𝑥[𝑛 −

4

MATLAB exercise

Record a voice signal using the

audiorecorder function for 5 seconds with

the following specifications

sampling frequency of 44100

Number of quantization bits 16

Number of channels = 1 for mono

Try to multiply the recorded samples by a

scaling factor of 𝛼 = 0.1 then by 𝛼 = 2 Play

the signal and hear the voice

5

Special sequences Unit sample

sequenceUnit sample sequence is defined as the

sequence

One of the important aspects of the impulse

sequence is that an arbitrary sequence can

be presented as a sum of scaled, delayed

impulses as shown in the next slide

6

Special sequences Unit sample

sequence

In general any sequence can be written as

𝑥 𝑛 = 𝑘=−∞∞ 𝑥 𝑘 𝛿[𝑛 − 𝑘]

7

Special sequences Unit step

sequenceThe unit step sequence is given by

8

Special sequences Unit step

sequenceThe unit step sequence is given by

9

Special sequences Unit step

sequenceThe unit step sequence in terms of

delayed impulses can be written as 𝑢 𝑛 =𝛿 𝑛 + 𝛿 𝑛 − 1 + 𝛿 𝑛 − 2 +⋯ = 𝑘=0∞ 𝛿 𝑛 − 𝑘

Note that the impulse sequence can be

expressed as the first backward difference

of the unit step sequence

𝛿 𝑛 = 𝑢 𝑛 − 𝑢[𝑛 − 1]

10

Special sequences exponential

sequencesExponential sequence are important in

representing and analyzing linear time

invariant systems

The general form of an exponential sequence

is given by 𝑥 𝑛 = 𝐴𝛼𝑛

If 𝐴 and 𝛼 are real then the sequence is real

If 0 < 𝛼 < 1 and 𝐴 is positive then the

sequence values are positive and decreasing

with increasing 𝑛

11

Special sequences exponential

sequencesGraphical representation of exponential

sequence

12

Special sequences sinusoidal

sequencesThe general form of sinusoidal sequence is

given by 𝑥 𝑛 = 𝐴𝑐𝑜𝑠(𝜔0𝑛 + ∅) as shown

13

Special sequences sinusoidal and

complex exponential sequence

The exponential sequence 𝑥 𝑛 = 𝐴𝛼𝑛 with

complex 𝛼 has a real and imaginary parts that

are exponentially weighted sinusoids

If 𝛼 = 𝛼 𝑒𝑗𝜔0 and 𝐴 = 𝐴 𝑒𝑗∅ then the sequence

can be expressed in either one of the following

forms

14

Notes about sequences

When discussing either complex exponential signals of the form 𝑥 𝑛 = 𝐴𝑒𝑗𝜔0𝑛 or real sinusoidal signal of the form 𝑥 𝑛 = 𝐴𝑐𝑜𝑠 𝜔0𝑛 + ∅ we need only to consider frequencies in an interval of length of 2𝜋only because

15

Periodic sequence

A periodic sequence is a sequence that

satisfies the following equation

𝑥 𝑛 = 𝑥[𝑛 + 𝑁],

Where 𝑁 is an integer number

If this condition is tested for the discrete

time sinusoids, then

Which requires

16

Periodic sequence

Where 𝑘 is an integer

A similar statement holds for the complex

exponential

Where 𝑁 is an integer number

Again

17

Example

Determine if the following sequences are

periodic or not. If the sequence is periodic

find its period

a) 𝑥1 𝑛 = cos𝑛𝜋

4

b) 𝑥2 𝑛 = cos3𝑛𝜋

4

18

solution

a) For the first sequence we have 𝜔0𝑁 =

2𝜋k or 𝜋

4𝑁 = 2𝜋𝑘 → 𝑁 = 8𝑘 since 𝑁 is an

integer value the sequence is periodic

b) For the second sequence 𝜔0𝑁 = 2𝜋𝑘 or 3𝜋

4𝑁 = 2𝜋𝑘 → 𝑁 =

8

3𝑘 since 𝑁 is not an

integer value for 𝑘 = 1 the sequence is

aperiodic if 𝑁 = 8

19

2.2 Discrete time systems

A discrete-time system is a system that

maps an input sequence with an output

sequence 𝑦 𝑛 = 𝑇{𝑥 𝑛 }

20

Discrete time system examples

There are many systems will be

investigated through out this course

Examples of these systems are

1. The ideal delay system which is described

mathematically by 𝑦 𝑛 = 𝑥 𝑛 − 𝑛𝑑 , −∞ <𝑛 < ∞

2. Moving average system which is described

mathematically by 1

𝑀1+𝑀2+1 𝑘=−𝑀1𝑀2 𝑥[𝑛 − 𝑘]

21

Discrete time system

classificationsSystems can be classifieds into one of the

following categories

1. Memoryless Systems. A system is classified

into memoryless system if the output 𝑦 𝑛 at

every value of 𝑛 depends only on the input

of 𝑥[𝑛] at the same value of 𝑛. An example

of a memoryless system is the squarer

system described by 𝑦 𝑛 = 𝑥[𝑛] 2

22

Discrete time system

classifications2. Linear systems. Any system satisfies the

superposition and the scaling property is

classifieds as a linear system. As an

example of a linear system is the

accumulator system described by

𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘]

3. Time-invariant system is a system for which

a time shift or delay of the input sequence

causes a corresponding shift in the output

sequence

23

Discrete time system

classificationsExample show that the accumulator system

𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘] is a time invariant system

solution

Assume that the input to the accumulator is

𝑥1 𝑛 = 𝑥[𝑛 − 𝑛0], then its output is 𝑦1 𝑛 = 𝑘=−∞𝑛 𝑥1[𝑘] = 𝑘=−∞

𝑛 𝑥[𝑘 − 𝑛0]

Let 𝑘1 = 𝑘 − 𝑛0This means that

𝑦1 𝑛 = 𝑘=−∞𝑛−𝑛0 𝑥[𝑘1] = y[n − 𝑛0]

24

Discrete time system

classifications4. Causality, a system is causal if the output

sequence value at the index 𝑛 − 𝑛0 depends

only on the input sequence values for 𝑛 ≤ 𝑛0For example the forward difference system

described by 𝑦 𝑛 = 𝑥 𝑛 + 1 − 𝑥 𝑛 is not causal

because the current value of the output depends on

future value of the input

Another example is the backward difference system

𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 − 1] is a causal system since the

output depends only on the present and past

values of the input

25

Discrete time system

classifications5. Stability, a system is stable if and only if

every bounded input sequence produces a

bounded output sequence

Such a system is called BIBO

in equation form

𝑥 𝑛 ≤ 𝐵𝑥 < ∞ → 𝑦 𝑛 ≤ 𝐵𝑦 < ∞

In general any sequence that has the form

𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘] < ∞ is stable system

26

Linear time-invariant system

The linear time-invariant system is an

important system since many of the system

we deal with in signal processing are of this

type

The output sequence in response to the

input sequence applied to the input of the

linear time-invariant system is given by the

convolutional sum 𝑦 𝑛 = 𝑘=−∞∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘]

27

Linear time-invariant system

In order to compute the convolution we

draw both ℎ 𝑛 − 𝑘 and 𝑥[𝑘] sequences as

shown below

28

Linear time-invariant system

From the Figure, we have 𝑦 𝑛 = 0 𝑓𝑜𝑟 𝑛 <0

The next sequence interval is shown by the

next graph that is 0 ≤ 𝑛 ≤ 𝑁 − 1

29

Linear time-invariant system

The output sequence for this interval is

given by

This equation can be solved analytically by

using the geometric series expansion

𝑘=𝑁1

𝑁2

𝑎𝑘 =𝑎𝑁1 − 𝑎𝑁2 +1

1 − 𝑎

30

Linear time-invariant system

The output sequence for this interval is

given by

This equation can be solved analytically by

using the geometric series expansion

𝑘=𝑁1

𝑁2

𝑎𝑘 =𝑎𝑁1 − 𝑎𝑁2 +1

1 − 𝑎

31

Convolution example

Which yields the following result

𝑦 𝑛 =

𝑘=0

𝑛

𝑎𝑘 =1 − 𝑎𝑛+1

1 − 𝑎𝑓𝑜𝑟 0 ≤ 𝑛 ≤ 𝑁 − 1

We consider the next interval when 0 < 𝑛 −𝑁 + 1

The output sequence is given by

32

Convolution example

Which yields the following result

The final answer for the output sequence for

these three intervals is given by

33

Convolution example

34

Convolution in Matlab

Convolution can be accomplished easily in

matlab by using the function conv(u,v)

The above example can be solved easily in

matalb by using the following code in matlab

n=1:10;

h=ones(1,5);

x=0.4.^n;

Y=conv(x,h);

stem(y);

35

2.4 Properties of linear time

invariant systemThe output sequence 𝑦[𝑛] of all LTI are

described by the convolution sum

𝑦 𝑛 =

𝑘=−∞

∞

𝑥 𝑘 ℎ[𝑛 − 𝑘]

Where ℎ[𝑛] is the impulse response of the LTI

system

This means that ℎ[𝑛] is a complete

characterization of the properties of a specific

LTI system

36

Properties of the convolution

sumcommutative

𝑥 𝑛 ∗ 𝑦 𝑛 = 𝑦 𝑛 ∗ 𝑥 𝑛

Distribution over addition𝑥 𝑛 ∗ ℎ1 𝑛 + ℎ2 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 + 𝑥 𝑛 ∗ ℎ2 𝑛

Associative 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 ∗ ℎ2 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 ∗ ℎ2 𝑛

37

Graphical representation of

combined LTI systems

38

Cascaded systems can be presented

by a single system whose impulse

response is given by ℎ 𝑛 = ℎ1[𝑛] ∗ℎ2[𝑛]. Cascaded systems satisfy the

convolution commutative property

Systems connected in parallel

can be replaced by a single

system whose ℎ 𝑛 = ℎ1 𝑛 +ℎ2[𝑛].

Stability and causality in terms

of ℎ[𝑛]LTI are stable if and only if there impulse

response is absolutely summable i.e.

𝑆 =

𝑘=−∞

∞

ℎ[𝑘] < ∞

LTI is causal if ℎ 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0

Causality means that the difference

equations describing the system can be

solved recursively

39

FIR systems – reflected in the

h[n]Ideal delay

𝑦 𝑛 = 𝑥 𝑛 − 𝑛𝑑 , −∞ < 𝑛 < ∞ℎ 𝑛 = 𝛿 𝑛 − 𝑛𝑑 , 𝑛𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

Forward difference

𝑦 𝑛 = 𝑥 𝑛 + 1 − 𝑥 𝑛ℎ 𝑛 = 𝛿 𝑛 + 1 − 𝛿 𝑛

Backward difference

𝑦 𝑛 = 𝑥 𝑛 − 𝑥 𝑛 − 1ℎ 𝑛 = 𝛿 𝑛 − 𝛿[𝑛 − 1]

Finite-duration impulse response (FIR) system are

characterized by an impulse response has that has only a

finite number of nonzero samples

40

IIR systems – reflected in the

ℎ[𝑛]Accumulator

𝑦 𝑛 =

𝑘=−∞

𝑛

𝑥[𝑘]

ℎ 𝑛 =

𝑘=−∞

𝑛

𝛿[𝑘] = 𝑢[𝑛]

Infinite duration impulse response (IIR) system has ℎ[𝑛]whose duration extends to infinity

Stability S = 𝑘=−∞∞ ℎ[𝑘] <? ∞

FIR systems always are stable, if each value of ℎ[𝑛] values is

finite in magnitude

IIR systems can be stable, e.g. ℎ 𝑛 = 𝑎𝑛𝑢 𝑛 , 𝑎 < 1 →

𝑛=0∞ 𝑎 𝑛 =

1

1− 𝑎< ∞

41

Cascading system examples

Determine if the following system is causal or not

Solution

Since the impulse response of the cascaded

system satisfy ℎ 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0 the resulting

cascaded system is stable

Any FIR system can be made causal by

cascading it with a sufficiently long delay

42

Cascading system examples

Determine the impulse response of the

following cascaded systems

An inverse system is given by

43

Linear constant-coefficient

difference equationsThe Nth order linear constant coefficient

equations are a subclass of linear time

invariant systems

The general form of these equations is

44

Example of difference equations

Write the accumulator system in terms of

difference equations

Solution

The accumulator equation is given by

The output for 𝑛 − 1 can be written as

45

Example of difference equations

Now the output sequence can be written as

Or alternatively it can be written as

If we compare the last equation with 𝑘=0𝑁 𝑎𝑘[𝑛 − 𝑘] =

𝑘=0𝑀 𝑏𝑚[𝑚 − 𝑘] we find that 𝑁 = 1, 𝑎0 = 1, 𝑎1 =− 1,𝑀 = 0, 𝑏0 = 1

46

Example of difference equations

The difference equations gives a better

understanding of how we can be

implement the accumulator system in this

example

47

Solving the Linear constant

coefficient difference equationsDifference equations are similar to differential

equations in continuous systems

The solution for the difference equations is

composed from the homogeneous and

particular solutions as described

mathematically by

𝑦 𝑛 = 𝑦𝑝 𝑛 + 𝑦ℎ[𝑛]

48

Solving the Linear constant

coefficient difference equationsThe homogeneous solution 𝑦ℎ 𝑛 is obtained

with 𝑥 𝑛 = 0

This means that the difference equation reduces

to

𝑘=0

𝑁

𝑎𝑘𝑦ℎ[𝑛 − 𝑘] = 0

Since 𝑦ℎ 𝑛 has 𝑁 undetermined coefficients, a

set of 𝑁 auxiliary conditions is required for the

unique specification of 𝑦[𝑛] for a given 𝑥 𝑛

49

Solving the Linear constant

coefficient difference equationsThese auxiliary conditions might consist of

specifying fixed values of 𝑦[𝑛] at specific values

of 𝑛, such as 𝑦[−1], 𝑦[−2], … , 𝑦[−𝑁]

The above step results in a set of 𝑁 linear

equations for the 𝑁 undetermined coefficients,

which can be solved to produce the required

coefficients

50

Recursive solution of the

difference equationsThe output samples for 𝑛 ≥ 0 can be computed

recursively by rearranging the difference

equation as shown below

𝑦 𝑛 = −

𝑘=1

𝑁𝑎𝑘𝑎0𝑦 𝑛 − 𝑘 +

𝑘=0

𝑀𝑏𝑘𝑎0𝑥[𝑛 − 𝑘]

If the input 𝑥[𝑛], together with a set of auxiliary

values 𝑦 −1 , 𝑦 −2 ,… , 𝑦[−𝑁] is specified then

the output 𝑦[0] can be computed

51

Recursive solution of the

difference equationsWith 𝑦 0 , 𝑦 −1 ,… , 𝑦[−𝑁 + 1] available 𝑦[1] can

be computed

To generate values of 𝑦[𝑛] for 𝑛 < −𝑁, we can

rearrange the linear constant coefficient

difference equation as shown below

𝑦 𝑛 − 𝑁 = −

𝑘=0

𝑁−1𝑎𝑘𝑎𝑁𝑦 𝑛 − 𝑘 +

𝑘=0

𝑀𝑏𝑘𝑎𝑁𝑥[𝑛 − 𝑘]

52

Recursive computation example

Example: solve the following difference

equation recursively

𝑦 𝑛 = 𝑎𝑦 𝑛 − 1 + 𝑥 𝑛

Assume that the input is 𝑥 𝑛 = 𝐾𝛿 𝑛 and

𝑦 −1 = 𝑐

53

Recursive computation example

When 𝑛 > −1, we can use recursive

computation as follows

Let 𝑛 = 0 then

𝑦 0 = 𝑎𝑦 0 − 1 + 𝑥 0𝑦 0 = 𝑎𝑦 −1 + 𝐾𝛿 0

Since 𝑦 −1 = 𝑐, then

𝑦 0 = 𝑎𝑐 + 𝐾

54

Recursive computation example

Next we do the same procedure when 𝑛 = 1

• 𝑦 1 = 𝑎𝑦 1 − 1 + 𝑥 1

• 𝑦 1 = 𝑎𝑦 0 + 0 = 𝑎 𝑎𝑐 + 𝐾 = 𝑎2𝑐 + 𝑎𝐾

• 𝑦 2 = 𝑎𝑦 1 + 0 = 𝑎 𝑎2𝑐 + 𝑎𝐾 = 𝑎3𝑐 + 𝑎2𝐾

• 𝑦 3 = 𝑎𝑦 2 + 0 = 𝑎 𝑎3𝑐 + 𝑎2𝐾 = 𝑎4𝑐 + 𝑎3𝐾

• 𝑦 𝑛 = 𝑎𝑛+1𝑐 + 𝑎𝑛𝐾

To determine the output for 𝑛 < 0, we express the

difference equations in the form

𝑦 𝑛 − 1 = 𝑎−1(𝑦 𝑛 − 𝑥 𝑛 )

𝑦 𝑛 = 𝑎−1(𝑦 𝑛 + 1 − 𝑥 𝑛 + 1 )

55

Recursive computation example

If we use the auxiliary conditions 𝑦[−1] = 𝑐, we

can compute 𝑦[𝑛] for 𝑛 < −1 as follows

• 𝑦 −2 = 𝑎−1 𝑦 −1 − 𝑥 −1 = 𝑎−1𝑐

• 𝑦 −3 = 𝑎−1 𝑦 −2 − 𝑥 −2 = 𝑎−1 𝑎−1𝑐 = 𝑎−2𝑐

• 𝑦 −4 = 𝑎−1 𝑦 −3 − 𝑥 −3 = 𝑎−1𝑎−2𝑐 = 𝑎−3𝑐

𝑦 𝑛 = 𝑎𝑛+1𝑐 𝑓𝑜𝑟 𝑛 ≤ −1

By combining the solutions for 𝑛 > −1 and 𝑛 ≤− 1, we got the following solution

𝑦 𝑛 = 𝑎𝑛+1𝑐

56

2.6 Frequency-domain representation

of discrete time signals and systems

The frequency response of a given system

with impulse response of ℎ[𝑛] is defined

by

𝐻 𝑒𝑗𝜔 =

𝑘=−∞

∞

ℎ 𝑘 𝑒−𝑗𝜔𝑘

The output of any system characterized by

its frequency response is given by

𝑦 𝑛 = 𝐻 𝑒𝑗𝜔 𝑒𝑗𝜔𝑛

57

Frequency response of the ideal

delay system

Example determine the frequency response

of an ideal delay system described by the

following equation

𝑦 𝑛 = 𝑥 𝑛 − 𝑛𝑑Solution

To find the frequency response we first find

the impulse response of the system which

can be found by substituting 𝑥 𝑛 = 𝛿 𝑛

58

Frequency response of the ideal

delay system This means that

𝐻 𝑛 = 𝛿 𝑛 − 𝑛𝑑Now the frequency response is given by

𝐻 𝑒𝑗𝜔 =

𝑛=−∞

∞

𝛿 𝑛 − 𝑛𝑑 𝑒−𝑗𝜔𝑛 = 𝑒−𝑗𝜔𝑛𝑑

𝐻 𝑒𝑗𝜔 can be written in rectangular form as

illustrated below

𝐻 𝑒𝑗𝜔 = 𝐻𝑅 𝑒𝑗𝜔 +𝐻𝐼 𝑒

𝑗𝜔

𝐻𝑅 𝑒𝑗𝜔 = cos(𝜔𝑛𝑑) , 𝐻𝐼 𝑒

𝑗𝜔 = −sin 𝜔𝑛𝑑 from

Euler identity

59

2.7 Representation of sequences

by Fourier transforms

In order to represent a given sequence by its

Fourier transform we can use the following

equation

𝑋 𝑒𝑗𝜔

𝑛=−∞

∞

𝑥 𝑛 𝑒−𝑗𝜔𝑛

However the inverse Fourier transform is given

by

𝑥 𝑛 =1

2𝜋 −𝜋

𝜋

𝑋 𝑒𝑗𝜔 𝑒𝑗𝜔𝑛𝑑𝜔

60

Representation of sequences by

Fourier transformsFor the discrete time signals, the value of

𝜔 is restricted to an interval of 2𝜋

The low frequency component of discrete

time signals are located around 𝜔 = 0

The high frequency component are

located around 𝜔 = ±𝜋

61

Convergence of the Fourier

transformIn general not all the signals have Fourier

transform

Only the absolutely summable signals

have their Fourier transform exits

Absolutely summable signals are signals

satisfying the following condition

𝑛=−∞

∞

𝑥 𝑛

62

Example

Determine if 𝑥 𝑛 = 𝑎𝑛𝑢[𝑛] has a Fourier

transform or not. If the Fourier transform exist,

find the value of 𝑋 𝑒𝑗𝜔

Solution

The summation

𝑛=−∞

∞

𝑥[𝑛] =

𝑛=0

∞

𝑎 𝑛 =1

1 − 𝑎< ∞

If and only if 𝑎 < 1 this means that the discrete

Fourier transform exists only for 𝑎 < 1

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Example

The summation

𝑋 𝑒𝑗𝜔 =

𝑛=0

∞

𝑎𝑛𝑒−𝑗𝜔𝑛 =

𝑛=0

∞

(𝑎𝑒−𝑗𝜔)𝑛 =1

1 − 𝑎𝑒−𝑗𝜔

If and only if 𝑎 < 1 this means that the discrete

Fourier transform exists only for 𝑎 < 1

64