Chapter 1 Introduction

Digital Signal ProcessingDigital Signal ProcessingDigital Signal ProcessingDigital Signal Processing

www.ptcdb.edu.ps

PALESTINE PALESTINE TECHNICAL COLLEGETECHNICAL COLLEGE

Eng. Akram Abu Garad

Digital Signal ProcessingCourse at a glance

Discrete-TimeSignals &Systems

Fourier DomainRepresentation

Sampling &Reconstruction

SystemStructure

SystemAnalysis

System

Z-Transform DFT

Filter

Filter Structure Filter Design

Chapter 1- Introduction

Digital Signal Processing

Signals, Systems and Signal Processing.

Classification of Signals.

Concept of Frequency in Continuous-

Time & Discrete-Time Signals.

Analog to Digital & Digital to Analog

Conversion.

Introduction

1.1. Signals, Systems and Signal Processing

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Signal is defined as any physical quantity that varies with independent

variables. For Example, the functions

S1(t) = 5t or S2(t) = 20t2 one variable

S(x,y) = 3x+4xy+6x2 two variables x and y

Speech signal


N

iiii ttFtA

1

))()(2sin()(

Amplitude Frequency Phase

Introduction

System, is defined as a physical device that performs an operation on a

signal.

Basic elements of a digital signal processing system:


A/D ConverterDigital SignalProcessing

D/A Converter

Analog input signal

Analog outputsignal

Digital input signal

Digital output signal


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Introduction



Advantages of DSP

Flexibility (software change)

Accuracy

Reliable Storage

Complex process realized by simple code

Cost, Cheaper than analog

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Introduction

tAts 3sin)(1 tjtAAets tj 3sin3cos)( 3

2

1.2. Classification of Signals

1.2.1. Multi-channel & Multidimensional Signals:

A signal is described by a function of one or more independent variables. The

value of function can be REAL-VALUED Scalar, a COMPLEX-VALUED, or a

VECTOR.


Real-Valued SignalComplex-Valued Signal

)(

)(

)(

)(

3

2

1

3

ts

ts

ts

tS

Vector Signal

A signal can be generated by a single source (1-channel) or multiple source (M-

channel). Vector of signals as a multi-channel. ECG (Electrocardiogram) are often

used 3-channel and 12-channel.

Multi-channel Signals

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Introduction


1.2.1. Multi-channel & Multidimensional Signals:

Multi-dimensional Signals• If the signal is a function of a single independent variable, the signal called a one-dimensional signal.• On the other hand , a signal called M-dimensional if its value is a function of M independent variables.


• The gray picture is an example of a 2-dimensional signal, the brightness or the intensity I(x,y) at each point is a function of 2 independent variables.

• The black & white TV picture represented as I(x,y,t) “3-Dimensional” since the brightness is a function of time.

• The color TV picture has 3 intensity functions Ir(x,y,t), Ig(x,y,t) and Ib(x,y,t).

),,(

),,(

),,(

),,(

tyxI

tyxI

tyxI

tyxI

b

g

r

1

Introduction

ttx cos)(1 tetx )(2


1.2.2. Continuous-Time versus Discrete-Time Signals:

Continuous-Time or analog signal are defined for every value of time.


are examples of analog signals

x(t)

t0 Analog Signal• Continuous in time. • Amplitude may take on any value in the continuous range of (-∞, ∞).

Analog Processing• Differentiation, Integration, Filtering, Amplification.• Implemented via passive or active electronic circuitry.

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Introduction


1.2.2. Continuous-Time versus Discrete-Time Signals:


Discrete-Time signals are defined only at certain specific value of time.

• Continuous Amplitude.• Only defined for certain time instances.• Can be obtained from analog signals via sampling.

The function provide an example of a discrete-time signal.

x(n)

n0 1 2 3 4 5 6 7-1

Undefined

Defined

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Introduction


1.2.3. Continuous-Valued versus Discrete-Valued Signals:The values of a CT or DT Signal can be continuous or discrete.If a signal takes on all possible values of a finite or an infinite range, it is CONTINUOUS-VALUED Signal.If the signal takes on values from a finite set of possible values, it is DISCRETE-VALUED Signal. Also called Digital SignalDigital Signal because of the discrete values.


x(n)

n0 1 2 3 4 5 6 7-1 8

Digital Signal with 4 different amplitude values

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Introduction


1.2.4. Deterministic versus Random Signals:


Random SignalA signal in which cannot be approximated by a formula to a

reasonable degree of accuracy (i.e. noise).

Deterministic SignalAny signal whose past, present and future values are

precisely known without any uncertainty

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Introduction

1.3. Concept of Frequency in CT & DT Signals

The concept of frequency is directly related to the concept of time. It has the dimension of inverse time.

1.3.1. Continuous-Time Sinusoidal Signals: A simple harmonic oscillation is mathematically described by the following CT sinusoidal signal:


AnalogSignal

Amplitude

Ω is frequency in rad/s

θ phase in rad

Instead of Ω the frequency F in Hz is usedΩ = 2πF

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Introduction



Analog Sinusoidal Signal Properties :

For every fixed value of the frequency F, xa(t) is periodic.xa(t+TP) = xa(t) where TP = 1/F is the fundamental period of

the sinusoidal signal. CT sinusoidal signal with different frequencies are themselves different.

Increasing the frequency F results in an increase in the rate of oscillation of the signal.

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Introduction



Analog Sinusoidal Signal Periodicity:

TP is the smallest value to satisfy the above property.

Proof:

Fundamental Period:

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Introduction



Complex Exponential Signal:

Euler Manipulations:

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Introduction


1.3.2. Discrete-Time Sinusoidal Signals:


A discrete-time sinusoidal signal may be expressed as:

Where n is integer variable, called the sample number.

Amplitude

ω is frequency in rad/sample

θ phase in rad

Instead of the ω frequency f in cycle per sample is used

ω = 2πf

DiscreteSignal

Example of a discrete-time sinusoidalsignal (ω = π/6 (f = 1/12) and θ = π/3)

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Introduction



Discrete-Time Sinusoidal Signal Properties: A discrete-time sinusoid signal is periodic only if its frequency f is a rational number.

The period N MUST be an integer > 0.

Discrete Signals whose frequencies are separated by a multiple of 2πk are identical. (k = integer)Proof:

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Introduction



Discrete-Time Sinusoidal Signal Periodicity:

Because k and N are integers, f0 is rational.

Proof:

for all n

Unique Frequencies: If sinusoids with frequencies ω1 and ω2 both exist within the interval [- π , π ] then ω1 ≠ ω2 . (the frequencies are different).

Therefore discrete frequencies have a “unique range”:

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Introduction



Example:

Is the signal periodic, If periodic, what is fundamental period (N)?

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Introduction



Example:ω0 = 0, π/8, π/4, π/2, π corresponding to f = 0, 1/16, 1/8, 1/4, 1/2 which results Periodic sequences having periods N = ∞, 16, 8, 4, 2 .

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Introduction



Example:Identical SinusoidsIdentical Sinusoids::

If sinusoidal frequencies exist outside of the unique range, identical sinusoids can be found within the unique range.

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Introduction


1.3.3. Harmonically Related Complex Exponentials:


CT Exponentials:

In some cases we deal with sets of harmonically related complex exponentials or sinusoid.

The basic signals for CT harmonically related exponentials are:

For each value of k , sk(t) is periodic with fundamental period 1/(kF0) = TP/k or fundamental frequency kF0 .

From the basic signal we can construct a harmonically related complex exponentials by,

where Ck is arbitrary complex constant

The signal xa(t) is periodic with fundamental period TP=1/F0 and its representation in terms of Fourier Series Expansion.

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Introduction


1.3.3. Harmonically Related Complex Exponentials:


DT Exponentials:

A discrete-time complex exponential is periodic if its relative frequency is a rational number, f0 = 1/N , then

In Contrast to the CT case,

There are only N distinct periodic complex exponential in the set sk(n)

This is Fourier Series Representation

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Introduction

1.4. A/D & D/A Conversion


Sampler Quantizer Coder

Analogsignal

Digitalsignal

Discrete-Time signal

Quantized signal

x(n)

xa(t)

xq (n)

0101101…..

A/D Converter

1.4.1. Analog to Digital Converter (A/D):

Conceptually, the A/D comprise 3 step process as in the following figure.

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1.4.1.1. Sampling:

Introduction




It is the conversion of a CT signal into DT signal obtained by taking “Samples” of the CT signal at DT instants.

Periodic or Uniform Sampling:This type of sampling is used most often in practice, describe by the relation:

where x(n) is the DT signal obtained by taking samples of the analog signal xa(t) every T seconds.

The rate at which the signal is sampled is Fs: Fs = 1/T

Fs is called the SAMPLING RATE or SAMPLING FREQUENCY (Hz)

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1.4.1.1. Sampling:

Introduction




Consider an analog sinusoidal signal of the form:

Sampling Frequency:

Normalized frequency:

Sampled Signal:

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1.4.1.1. Sampling:

Introduction




Relation among frequency variable:

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1.4.1.1. Sampling:

Introduction1.4. A/D & D/A Conversion


4.1. Analog to Digital Converter (A/D):

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We observe that the fundamental difference between CT and DT signals in their range of values of the frequency variables F and f or Ω and ω.

Means Sampling from infinite frequency range for F (or Ω) into a finite frequency range for f (or ω).

Since the highest frequency in a DT signal is ω = π or f = 1/2.

With sampling rate Fs the corresponding highest values of F and Ω are:

1.4.1.1. Sampling:

Introduction




Examples:

I. Two analog sinusoidal signals:

Which are sampled at a rate Fs = 40 Hz.

Discrete-time signals:

This mean

However,

The frequency F2 = 50 Hz is an alias of the frequency F1 = 10 Hz at the sampling rate of 40 samples per second.

F2 is not the only alias of F1

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1.4.1. Analog to Digital Converter (A/D):II. Two analog sinusoidal signals, F1 = 1 Hz & F2 = 5 Hz are sampled at a rate Fs = 4 Hz.

F2 is the alias of F1

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Introduction




Aliasing

• Aliasing occurs when input frequencies (again greater than half the sampling rate) are folded and superimposed onto other existing frequencies.

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In order to prevent alias

where Fmax is the highest input frequency

Nyquist Rate:

Minimum sampling rate to prevent alias.

1.4.1.1. Sampling:




Given Band Limited (Frequency Limited Signal) with highest frequency Fmax:The signal can be exactly reconstructed provided the following is satisfied: Sampling Frequency: The samples are not

quantized (analog amplitudes)

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Sampling Theorem:




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Reconstruction Formula:

The signal:

The samples:

Formula:

Interpolation Function:




1.4.1.2. Quantization:

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The process of converting a DT continuous amplitude signal into

digital signal by expressing each sample value as a finite number of

digits is called QUANTIZATION.





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Fs = 1 Hz





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Numerical illustration of quantization with one significant digit using truncation or rounding

Introduction




1.4.1.3. Coding:

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Introduction



1.4.2. Digital to Analog Converter (A/D):

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Chapter 1 Introduction