Lecture 1,2 Introduction to DSP

8/8/2019 Lecture 1,2 Introduction to DSP

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INTRODUCTION T0 DSP

1Lecture 1

Class time table:

Section I:

Mon & Tue 9:00 to 11:00 PST

Section II:

Wed & Thu 9:00 to 11:00 PST

1: 911

2: 911


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THE TEACHING PLAN

OTHER DETAILS

INTRODUCTION TO DSP

ANALOG TO DIGITAL CONVERSION

SAMPLING SAMPLING THEOREM, ALIASING

QUANTIZATION QUANTIZATION NOISE

In todays class


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Teaching plan

Introduction

Preliminaries

Review of Signals & Systems

Analog to Digital Conversion

Convolution

Correlation

Z-domain transformation

Fourier transformation

Discrete Fourier Transform (Review)

Fast Fourier Transform

2 lectures

1 lecture

1 lecture

4 lectures

7 lectures


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Contd

Digital Filters

Finite Impulse Response (FIR) filters

Infinite Impulse Response (IIR) filters

Realization of Digital Filters

Multi-rate Signal Processing

Adaptive Signal Processing

Spectrum Estimation & Analysis

11 lectures

10 lectures

5 lectures

4 lectures

2 lectures

5 lectures

Pre-requisite : Signals & Systems (Theory)


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Reading material

Text book

Digital Signal Processing:

Principles, Algorithms and

Applicationsby

John Proakis,

Dimitris Manolakis


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Reference books

Discrete-time Signal Processing by Alan Oppenheim

and Ronald Schaffer

By Emmanuel Ifeacher

Web based demos


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Submitting Assignments!

Assignments will have to be submitted in the form of

groups

Each group can have 6 members max.

Every group musthave at leastone person in the

Top10 students of your class (section)

Group leader will be the student among Top10

Submit the name of your group members tomorrow!Mention roll nos

Email address of group leader


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Digital Signal Processing: What is it?

What is digital signal processing (DSP) anyway?

The term DSP generally refers to the use of digital

computers to process signals. Normally, these signalscan be handled by analog processors but, for a

variety of reasons, we may prefer to handle them

digitally


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Why Digital Signal Processing?

Why Digital Signals & Digital Systems?

Analog signals cant be stored properly

Processing analog signals is not efficient

e.g. An analogue filter with sharp cutoff has non-linearphase response

Analog systems are usually implemented usingresistors, capacitors, inductors, op-amps, transistors etc

Potentially consume a lot of power

Analog systems can get very complex

An analog system made for one task/purpose may notwork for another task/purpose


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Why Digital Signals & Digital Systems?

Digital signals can be easily stored and processed Digital Signal Processing is implemented in processors

Processors in your cell phones, mp3 players etc

Same processor can be used for many applications

Multi-purpose PC can be used for education, gaming, watching

movies, listening to songs what else?

Digital systems will always consume less power than their

analogue counterparts!


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DSP for Telecom engineers: Why?

DSP has made deep inroads in to Telecommunications

All modern communication systems are based on

digital communication principles, which makes them

robust against noise, that is always present in thechannel

Robustness is due to the signal processing algorithms

that run in the digital system Regenerative repeaters

Equalizers

Error-correction codes and so on


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Contd

Modern communication systems are based on Field

Programmable Gate Array (FPGAs), Digital Signal

Processors (DSPs) or Application Specific Integrated

Circuits (ASICs)

All these reallyimplement DSP algorithms


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DSP Applications

Medical

Diagnostic imaging (MRI, CT, ultrasound, etc.)

Electrocardiogram (ECG) analysis

Electroencephalogram (EEG) analysisMedical image storage and retrieval

Scientific

Data acquisitionData extraction (DIP)

Simulation & modeling

Spectral analysis


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Contd

Commercial

Sound Processing

Its no fluke that suddenly every singer has a good voice!!

MP3, WMA, RM etc Image Processing

Adobe Photoshop

JPEG, BMP, GIF etc

Djvu (new compression format for scanned documents) Video Processing

Better video formats that occupy less space

Video stabilization


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Contd

Military

More secure information transfer (better encryption)

Jammers

RADAR

GPS guided missiles?

So many more applications!


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Need for A/D conversion

We know by now the benefits of digital signals and

systems

But most signals of practical interest are still analog

Voice, Video

RADAR signals

Biological signals etc

So in order to utilize those benefits, we need toconvert our analog signals into digital

This process is called A/D conversion


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Three step process

Analog to Digital conversion is really a three stepprocess involving

Sampling

Conversion from continuous-time, continuous valuedsignal to discrete-time, continuous-valued signal

Quantization

Conversion from discrete-time, continuous valued signal

to discrete-time, discrete-valued signal Coding

Conversion from a discrete-time, discrete-valued signalto an efficient digital data format

Represent as bit?


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SAMPLING QUANTIZATIO

N

CODING

CT-CV DT-CV DT-DV DT-DV

Analog signal Binary bits

2 4 6 8 10

-1

-0.5

0

0.5

1

2 4 6 8 10

-1

-0.5

0

0.5

1

2 4 6 8 10

-1

-0.5

0

0.5

1

1 2 3 4 5 6 7 8 9 104.5

5

5.5

6

6.5

7

7.5

Arbitrarily, Ive chosen Differential

PCM. Can you re-create these graphs?


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Sampling

A continuous-time signal has some value defined at every time instant

So it has infinite number of sample points

2 4 6 8 10

-1

-0.5

0

0.5

1

2 4 6 8 10

-1

-0.5

0

0.5

1

2 4 6 8 10

-1

-0.5

0

0.5

1

2 4 6 8 10

-1

-0.5

0

0.5

1

sample

every

1 sec

sample

every

0.1 sec

sample

every

1 sec


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It is impossible to digitize an infinite number of points

because infinite points would require infinite amount

of memory and infinite amount of processing power So we have to take some finite number of points

Sampling can solve such a problem by taking samples

at the fixed time interval

If an analog signal is not appropriately

sampled, aliasing will occur, where a discrete-time

signal may be a representation (alias) of multiple

continuous-time signals

Aliasing:


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Shannons sampling theorem

The sampling theorem guarantees that an analogue signal can be in theory perfectly

recovered as long as the sampling rate is at least twice as large as the highest-frequency

component of the analogue signal to be sampled

max2FFs

A signal with no frequency component above a certain maximum frequency is known as

a band-limited signal (in our case we want to have a signal band-limited to Fs)

Some times higher frequency components are added to the analog signal (practical signalsare not band-limited)

In order to keep analog signal band-limited, we need a filter, usually a low pass that stops

all frequencies above Fs. This is called an Anti-Aliasing filter


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In order to sample a voice signal containing

frequencies up to 4 KHz, we need a sampling rateof 2*4000 = 8000 samples/second

Similarly for sampling of sound with frequencies up

to 20 KHz, we need a sampling frequency of2*20000 = 40000 samples/second

What is the sampling rate for CDs?

Isnt it more than the one we just calculated?

E l 1 F th f ll i l i l fi d th N i t li


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Example 1: For the following analog signal, find the Nyquist sampling

rate, also determine the digital signal frequency and the digital signal

t)70cos(3)( tx

The maximum frequency component is x(t) is

Therefore according to Nyquist, we need a sampling rate of

The digital signal would have a frequency

The digital signal can be represented as

HzF 352

70max

HzFFs 702 max

70

352w

)cos(3][ nnx


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Anti-aliasing filters

Anti-aliasing filters are analog filters as they process the signal

before it is sampled. In most cases, they are also low-pass filters

unless band-pass sampling techniques are used

The ideal filter has a flat pass-band and the cut-off is verysharp, since the cut-off frequency of this filter is half of that of the

sampling frequency, the resulting replicated spectrum of the

sampled signal do not overlap each other. Thus no aliasing occurs


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Practical low-pass filters cannot achieve the ideal characteristics.

What can be the implications?

Firstly, this would mean that we have to sample the filtered signals at

a rate that is higher than the Nyquist rate to compensate for the

transition band of the filter

Thats why the sampling rate of a CD is 44.1 KHz, much higher than

the 40 KHz we calculated

Go through the assignment it has some reading task along with

some problems


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Example 2: Find the Nyquists rate for the following signal

t)100cos(-t)300sin(10t)50cos(3)( tx

This composite signal comprises three frequencies

f1 = 25 Hz, f2 = 150 Hz, f3 = 50 Hz

So, according to Nyquist we need to sample at 300 Hz

However, for the sine term, the sampled signal has values

sin(n), meaning the samples are taken at the zero crossings, so the

sine term is not counted in the process

Therefore, a solution is to sample at higher than twice the max. freq

component


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Quantization

Now that we have converted the continuous-

time, continuous-valued signal into a discrete-

time, continuous-valued signal, we STILL need to make

it discrete valued This is where Quantization comes into picture

The process of converting analog voltage with

infinite precision to finite precision

For e.g. if a digital processor has a 3-bit word, the

amplitudes of the signal can be segmented into 8 levels


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Quanitization


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General rules for Quantization

Important properties

of the quantizer

include

Number ofquantization levels

Quantization resolution

Note the minimum &

maximum amplitude of

the input signal

Ymin & Ymax

0 1 2 3 4 5 6 7 8 9 10

-1

-0.5

0

0.5

1

Ymax = 1

Ymin = -1


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Note the word-length of DSP

m-bits

Number of levels of quantizer is equal toL = 2m

The resolution of the quantizer is given as

Resolution of a quantizer is the distance between two

successive quantization levels More quantization levels, better resolution!

Whats the downside of more quantization levels?

)(1

)( minmax voltsL

yy


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0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00

09.0][

n

nnx

n

m = 4, L = 16Ymin = 0

Ymax = 1

= 1/15 = 0.0667


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Quantization error

The error caused by representing a continuous-valued

signal(infinite set) by a finite set of discrete-valued

levels

Suppose a quantizer operation given by Q(.) is

performed on continuous-valued samples x[n] is given

by Q(x[n]), then the quantization error is given by

][][][ nxnxne qq


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Lets consider the signal , which is to be

quantized.

In the figure (previous slide), we saw that there was a

difference between the original signal and the quantized

signal. This is the error produced while quantization

It can be reduced, however, if the number of quantization

levels is increased as illustrated on next slide

00

09.0][

n

nnx

n


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0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

0.02

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0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 250

0.1

0.2

0.3

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0.5

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0.8

0.9

1

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

3-bit ADC

8-bit ADC

Quant. error

Quant. error


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Signal-to-Quantization-noise ratio

Provides the ratio of the signal power to the

quantization noise (or quanitization error)

Mathematically,

where

Px = Power of the signal x (before quantization)

Pq = Power of the error signal xq

q

x

P

P

dBSQNR 10log10

1

0

21

0

2 11N

n

q

N

n

q nxnxN

neN

Pq

Lecture 1,2 Introduction to DSP

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