lecture 1,2 introduction to dsp
TRANSCRIPT
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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
0.04
0.06
0.08
0.1
0.12
0.14
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
0.5
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