Wolaita Sodo University

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Wolaita Sodo University B.Sc Department of Electrical and Computer Engineering

Course Outline Course Name DIGITAL SIGNAL PROCESSING

Course Code ECEg 3083

Year & Semester 3rd Year/2nd Semester

Stream Section 1,2 & 3

ECTS 5

Module name Signals and Network Analysis

Module Code ECEg-M

Instructor Name Ramakrishna MM

Course Objective Introduce students to methods of discrete-time signals and systems representation

and analysis Introduce design methods and realization structures of discrete-

time systems. Introduce signal processing applications using signal processing

development kits for a chosen DSP chip

Contents of the syllabus Time allotted

CHAPTER 1 Introduction 1.1 Basic elements of DSP system 1.2 Digital Vs. Analog signal

processing 1.3 Classification of Signals 1.4 Analog-to-Digital and Digital-to-Analog conversion 1.5 weeks

CHAPTER 2 LTI Systems 2.1 Discrete-Time LTI Systems - Interconnections 2.3 Analysis of

DiscreteTime Linear Time Invariant Systems 2.4 Convolution sum – Linear Convolution using

cross table method and matrix method 2.5 Correlation of Discrete Time Signals

2 weeks

CHAPTER 3 The Z-Transform 3.1 Direct and Inverse ZTransform 3.2 Properties of ZTransform

3.3 Poles and Zeros 3.4 Analysis of LTI system in the Z-Transform 2 weeks

CHAPTER 4 Frequency Analysis of Signals and Systems 4.1 Fourier analysis of discrete time

signals 4.2 Properties of Fourier Transform 4.3 Frequency Domain Characteristics of LTI systems 2 weeks

CHAPTER 5 Discrete Time Fourier Transform 5.1 Discrete Domain Sampling 5.2 Properties

of DFT 5.3 Frequency Analysis of signals using DFT 5.4 FFT algorithms 5.5 Applications of FFT

algorithms

2 weeks

CHAPTER 6 Realization of FIR & IIR Filters: Direct form & Cascade form of FIR &

IIR, Design of IIR using Impulse Invariance and Bilinear Transformation Methods

1.5 weeks

Pre-requisites ECEG-2081: Signals and Systems Analysis ECEG-3101: Digital Logic

Teaching & Learning Methods Lecture supported by tutorial, assignment and Lab

Assessment/Evaluation & Grading

System

Assignment (20%), Quiz (10%); Tests (25%); Final Examination (40%)

Attendance (5%)

Attendance Requirements A student must attend at least 85 % of the classes

Text Book References

a) Oppenheim and Schafer, “Discrete-time Signal Processing”,

Prentice Hall

b) Digital Signal Processing by R.A.Barapate,Tech Max Publication

c) Proakis and D. G. Manolakis, “Introduction to Digital Signal

Processing”, Macmilan Publishing Company

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CHAPTER ONE

INTRODUCTION TO DSP

Digital Signal Processing (DSP) refers to processing of signals by digital systems like Personal

computers and systems designed using digital integrated circuits, microprocessors and microcontrollers

Signal:

A signal is defined as a function of one or more variables which conveys information on the

nature of a physical phenomenon. The value of the function can be a real valued scalar

quantity, a complex valued quantity, or perhaps a vector.

System:

A system is defined as an entity that manipulates one or more signals to accomplish a

function, thereby yielding new signals.

Continuos-Time Signal:

A signal x(t) is said to be a continuous time signal if it is defined for all time t.

Discrete-Time Signal:

A discrete time signal x[nT] has values specified only at discrete points in time.

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Basic Elements of a Signal Processing System

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Advantages of DSP

1. A digital programmable system allows flexibility in reconfiguring the digital signal processing

operations by changing the program. In analog redesign of hardware is required.

2. In digital accuracy depends on word length, floating Vs fixed point arithmetic etc. In analog

depends on components.

3. Can be stored on disk.

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4. It is very difficult to perform precise mathematical operations on signals in analog form but these

operations can be routinely implemented on a digital computer using software.

5. Cheaper to implement.

6. Small size.

7. Several filters need several boards in analog, whereas in digital same DSP processor is used for

many filters.

Disadvantages of DSP

1. When analog signal is changing very fast, it is difficult to convert digital form.(beyond 100KHz

range)

2. w=1/2 Sampling rate.

3. Finite word length problems.

4. When the signal is weak, within a few tenths of millivolts, we cannot amplify the signal after it is

digitized.

5. DSP hardware is more expensive than general purpose microprocessors & micro controllers.

Applications of DSP

1. Filtering.

2. Speech synthesis in which white noise (all frequency components present to the same level) is

filtered on a selective frequency basis in order to get an audio signal.

3. Speech compression and expansion for use in radio voice communication.

4. Speech recognition.

5. Signal analysis.

6. Image processing: filtering, edge effects, enhancement.

7. PCM used in telephone communication.

8. High speed MODEM data communication using pulse modulation systems such as FSK, QAM etc.

MODEM transmits high speed (1200-19200 bits per second) over a band limited (3-4 KHz) analog

telephone wire line.

9. Wave form generation.

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Signals Based on Representation:

Signals Based On Duration

Basic Signals

1. Even and Odd

2. Periodic and Non-Periodic

3. Energy and Power

4. Deterministic and Random etc.

Elementary Signals

1. Unit Step

2. Unit Ramp

3. Unit Impulse

4. Sinusoidal

Etc…

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Sampling

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Quantization

Coding

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CHAPTER TWO

LTI SYSTEMS

Convolution Sum


Linear Convolution

1. Plot both x(k) and h(k)

2. Reflect h(k) about k=0 to obtain h(-k)

3. Shift h(-k) by n (towards left for -n and towards right for +n)

4. Let the initial value of n be negative

5. Multiply each element of x(k) with h(n-k) and add all the product terms to obtain y(n)

6. Shift h(n-k) by incrementing the value of n by one, and do step 5

7. Do step 6 until the product of x(k) and h(n-k) reduces to zero


Practice Problem

Practice Problem


Linear Convolution using Cross Table Method

Problem

Perform the Convolution of x(n) and h(n) , where x(n) ={1,2,3,4} and h(n) = {1,1,1,1}


Practice Problem

Linear Convolution using Matrix Method


Introduction to Correlation


Cross Correlation

Problem

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CHAPTER THREE

Z TRANSFORM

The Z-transform is a powerful method for solving difference equations and, in general, to

represent discrete systems. Although applications of Z-transforms are relatively new, the

essential features of this mathematical technique date back to the early 1730s when DeMoivre

introduced the concept of a generating function that is identical with that for the Z-transform.

Recently, the development and extensive applications of the Z-transform are much enhanced

as a result of the use of digital computers.

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which

is a sequence of real or complex numbers, into a complex frequency-domain representation.

It can be considered as a discrete-time equivalent of the Laplace transform

The Defination of Z Transform

The Z-transform can be defined as either a one-sided or two-sided transform.

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal is the formal power

series defined as

X(z) = z{x(n)} = ∑ 𝑥(𝑛) 𝑧−𝑛∞𝑛=−∞

where is an integer and z is, in general, a complex number:

One sided or Single sided Z Transform

X(z) = z{x(n)} = ∑ 𝑥(𝑛) 𝑧−𝑛∞𝑛=0

Region of Convergence

The Region of Convergence (ROC) of x(z) is all set of values of z for which x(z) attains a finite

value.

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Signal_processing

https://en.wikipedia.org/wiki/Discrete-time_signal

https://en.wikipedia.org/wiki/Sequence

https://en.wikipedia.org/wiki/Real_number

https://en.wikipedia.org/wiki/Complex_number

https://en.wikipedia.org/wiki/Frequency-domain

https://en.wikipedia.org/wiki/Laplace_transform

https://en.wikipedia.org/wiki/Formal_power_series

https://en.wikipedia.org/wiki/Formal_power_series

https://en.wikipedia.org/wiki/Complex_number

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Problem

Obtain the Z Transform of the following finite sequence

X(n) = {1,2,4,5,0,7}

Solution:

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Problem

Determine the Z Transform and ROC of the signal

X(n) = [ 3(4n) – 5(3n)] u(n)

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Practice Problems

Problem one

1.

Problem Two

Problem Three

Problem Four

Problem Five


CHAPTER FOUR

FREQUENCY ANALYSIS

The Frequency analysis of signal involves the resolution of signal in to its frequency

components. These frequency components are basically sinusoidal components. If any signal

is decomposed in to sinusoidal components, the sum of all these sinusoidal components results

in original waveform. The basic aim for performing the frequency analysis is to provide the

mathematical representation of frequency components contained in a given signal.

One of the important technique to do the frequency analysis is the Fourier representation of

signals. There are four distinct Fourier representation.

1. Fourier series (FS) – Applicable to analyses of Continuous time periodic signals

2. Fourier Transform (FT) – Applicable to analyses of Continuous time non periodic

signals

3. Discrete time Fourier series (DTFS) - Applicable to analyses of Discrete time periodic

signals

4. Discrete time Fourier Transform (DTFT) - Applicable to analyses of Discrete time Non

periodic signals


Practice Problems

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CHAPTER FIVE

DISCRETE TIME FOURIER TRANSFORM


Practice Problems


FAST FOURIER TRANSFORM


Practice Problems


CHAPTER SIX

RELAIZATION OF FIR & IIR FILTERS

Problems


IIR FILTERS


Practice Problem


Practice Problem

Practice Problem


FILTER DEISGN


Pratice Problem


Solved Problem

Practice Problem

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