Digital Signals and Systems - PowerPoint Presentation

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Digital Signals and Systems

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A discrete-time signal

is a function of an integer variable

. In the DS processor, the signal is represented by the discrete encoded values with a finite precision.

Digital Signals

Graphical representation of a discrete-time signal

Mathematically a discrete-time signal can be determined by

Functional representation

Tabular representation

Sequence representation (bold or arrow for origin n=0)

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Common Digital Sequences

Unit-impulse sequence:

Unit-step sequence:

Exponential sequence:

0 < a < 1

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Shifted Sequences

Shifted unit-impulse

Shifted unit-step

Right

shift by

two samples

Left shift by two samples

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Example

1

x(

𝑛) = δ(

𝑛 + 1) + 0.5 δ (

𝑛

− 1) + 2 𝛿(𝑛 − 2).Sketch this following digital signal sequence,Solution:

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Generation of Digital Signals

In order to generate the digital sequence from its analog signal function, the analog

signal

is uniformly sampled at the time interval of

,

T

is the sampling period.

sampling interval :

x(n): digital signal

x(t): analog signal

Also

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Example 2

Solution:

Convert analog signal x(t) into digital signal x(n), when sampling period is 125 microsecond, also plot sample values.

The first five sample values:

Plot of the digital sequence:

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Digital Systems

A digital system

is a device or an algorithm that performs prescribed operations (or transformation) on

a

signal

, called the

input or excitation, to produce another signal

called the

output or response of the system. 

transformation

Digital System

Input signal

(Excitation

)

Output signal

(Response) 

Example

Determine the response of the following

system (mean value of three values)

to the input signal

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Block Diagram

Representation of Discrete-Time Systems

adder

Constant multiplier

signal multiplier

unit delay element

unit advance element9Slide10

Example 3

Sketch

the block diagram representation of the discrete-time system described by the input-output relation.

Solution

The output

is obtained by rearranging the input-output equation:

)

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Causal

and

non-causal systemsA system

is causal

if its output at any time

depends only on present and past inputs [i.e.

, but does not depend on future inputs [i.e.

].

Example 4

Determine whether the systems described below are causal

Solution

Since

for

the output

depends on the current input and its past value the system is causal

.

b. Since for

the output

depends on the current input

and its future value

the system is

non-causal

.

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Linear

System

Continuous system

Time, t

discrete system

Sample number, n

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Linear Systems: Property 1

Homogeneity:

(Deals with amplitude ) If

x[n] 

y[n], then

kx[n]

 ky[n] K is a constant

A digital system

S is linear if and only if it satisfy the superposition principle (Homogeneity and Additivity):

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Linear Systems: Property

2

Additivily

Homogeneity & Additivity

(superposition

principle)

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Example 5 (a)

Let a digital amplifier

,

If the inputs are: Outputs will be:

If we apply combined input to the system:

The output will be:

Individual outputs:

X 10

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Example 5 (b)

System

System

System

If the input is:

4

Then the output is:

= (

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Linear Systems: Property

3

Shift (time) Invariance

A system is called

time-invariant (or shift-invariant) if its input-output characteristics do not change with time (a shifted input signal will produce a shifted output signal, with the same of shifting amount).

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Example 6 (a)

Given the linear system

,

find whether the system is time invariant or not.

Solution:System

Let the shifted input be:

Therefore system output:

Shifting

By

samples leads to

Equal

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Example 6 (b)

Given the linear system

, find whether the system is time invariant or not.

Solution:

System

Let the shifted input be:

Therefore system output:

Shifting

By

samples leads to

Not Equal

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Difference Equation

A causal, linear, and time invariant system can be represented by a difference equation as follows:

Outputs

Inputs

After

rearranging:

Finally:

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Example 7

Identify non zero system coefficients of the following difference equations.

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System Representation Using Impulse Response

A linear time-invariant system (LTI system) can be completely described by its unit-impulse response

due to the impulse input

with zero initial conditions.

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Example 8 (a)

Consider the difference equation with an initial condition

.

a. Determine

the unit-impulse response

.

b. Draw the system block diagram.

c. Write the output using the obtained impulse response.

Solution

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Example 8 (b)

Solution

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Finite Impulse Response (FIR) system:

When the difference equation contains no previous outputs, (

coefficients of

are zero).

Infinite Impulse Response (IIR) system:

When the difference equation contains previous outputs, (

coefficients of

are not all zero).

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BIBO Stability

BIBO: Bounded In and Bounded Out

A stable system is one for which every bounded input produces a bounded output.

Let, in the worst case, every input value reaches to maximum value

M.

Using absolute values of the impulse responses,

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BIBO Stability

To

determine whether a system is stable, we apply the following equation:

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Given a linear system given by:

Which is described by the unit-impulse response:

Determine whether the system is stable or not.

Example 9

Solution

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Digital Convolution

A LTI system can be represented using a digital convolution sum

. The unit-impulse response relates the system input and output. To find the output sequence

for any input sequence

, we use the digital convolution:

The sequences are interchangeable.

Convolution sum requires h(n) to be reversed and shifted.

If h(n) is the given sequence, h(-n) is the reversed sequence. 29Slide30

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Reversed

Sequence Example 10Slide31

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Convolution Using Table

MethodExample 11 (a)Slide32

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Convolution

Using Table Method Example 11 (b) Slide33

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Convolution Properties

the order in which two signals are convolved makes no differenceSlide34

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Examples of ConvolutionSlide35

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Low Pass Filters

(filter's impulse

response)

each

sample in

the output signal being a weighted average of many adjacent points from the input signal.removing high-frequencycomponentsThe Exponential

is the simplest recursive filter.

The rectangular pulse is best at reducing noise while maintaining edgesharpness.

The

sinc

function is

used to separate one band of frequencies from another.Slide36

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High Pass Filters

Typical high-pass filter kernels. These are formed

by subtracting the corresponding low-pass filter kernels from a delta function

.

The distinguishing

characteristic of high-pass filter kernels is a spike surrounded by many adjacent negative samples.Slide37

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Signal-to-noise ratio

 ( SNR) is a measure

that compares the level of a desired 

signal to the level of background noise.

 It is defined as the ratio of signal power to the noise power, often expressed in 

decibels.Signal-to-Noise Ratio (SNR) If the variance of the signal and noise are known, and the signal is zero-mean:

Because many signals have a very wide 

dynamic range, SNRs are often expressed using the logarithmic decibel scale. In decibels, the SNR is defined asA logarithmic scale is a nonlinear scale used when there is a large range of quantities

.Slide38

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Signal-to-Noise Ratio (SNR)

The decibel

 (dB) is a logarithmic unit

 used to express the ratio between two values of a physical quantity, often 

power or 

intensity.Slide39

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Periodicity

Example 12

Consider the following continuous signal for the current

which is sampled at 12.5 ms.

Will the resulting discrete signal be periodic?