Lecture1- Final Released

8/10/2019 Lecture1- Final Released

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Dr. Zulfiqar H [email protected]

Lecture 1
mailto:[email protected]:[email protected]


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Course Objective

To establish the idea of using computing techniques toalter the properties of a signal for desired effects

Understanding of fundamentals of discrete-time, linear,

shift-invariant signals and systems in

Representation: sampling and quantization;

Processing: filtering and transform techniques;

Processing System Design: filter & processing algorithm design.

Efficient computational algorithms and their implementation.

Understanding to real world scenario


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What is DSP?

A-to-D conversion:bandwidth control, sampling and quantization

Computational processing: implemented on computers or ASICs

with finite-precision arithmetic

Basic numerical processing: add, subtract, multiply (scaling,

amplification, attenuation), mute,

Algorithmic numerical processing:convolution or linear

filtering, non-linear filtering (e.g., median filtering), difference

equations, DFT, inverse filtering, MAX/MIN,

D-to-A conversion: re-quantification* and filtering (or

interpolation) for reconstruction


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Course Introduction: What is Signal Processing


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Digital Signal Processing: Historical Context


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Basic Interests & Issues in DSP

Digital Filters: Filter design, noise analysis, structures

Fourier Analysis: Spectrum estimation, FFT, cosine

transform, short-time FT

Signal Modeling and Analysis: Linear prediction, wavelets

Hardware and Software: Minicomputers, array

processors, DSP chips, workstations, PCs, MATLAB, RTOS

Applications:Speech, radar, image, video, data, ...


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Common algorithms Linear filtering: FIR, IIR

FFT, cosine transform, filterbanks

Correlation Matrix calculations

Common features

Lots of multiply/accumulate operations Block processing is often appropriate

Fixed-point arithmetic for economical solutions

DSP Algorithms


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Digital Signal Processing: Advantages


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Digital Signal Processing: Some Disadvantages


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Examples of Signals: Speech Waveform


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Example of 1D Signal


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Digital Speech Signal

Voice frequency range: 20Hz ~ 3.4 KHz Sampling rate: 8 KHz (8000 samples/sec)

Quantization: 8 bits/sample

Bit-rate: 8K samples/sec * 8 bits/sample = 64 Kbps (foruncompressed digital phone)

In current Voice over IP (VOIP) technology, digital speech

signals are usually compressed (compression ratio: 8~10)


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Examples of 1-D Signals: Electrocardiogram


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Examples of 2-D Signals: Image


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Digital Image Signal

One mega-pixel image (1024x1024) Quantization: 24 bits/pixel for the RGB full-color space,

and 12 bits/pixel for a reduced color space (YCbCr)

Bit-rate: 1024x1024 samples/sec * 12 bits/pixel = 12

Mbits = 1.5 Mbytes (for uncompressed digital phone)

How many uncompressed images can be stored in a 2G

SD flash-memory card?


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Examples of 3-D Signals: Video


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Textbook

Discrete-Time Signal processing, A. V. Oppenheim, R. W.

Schafer, and J. R. Buck, Pearson Prentice Hall, US, 1999.: Signal Processing First, J. H. McClellan, R. W. Schafer,and

M. A. Yodar, Pearson Prentice Hall, US, 2003.

Statistical Digital Signal Processing and Modeling byMonson H. Hayes


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Signals

Something that carries information

Speech, audio, image, video, biomedical signals, radar

signals, seismic signals, etc.

Systems

Something that can manipulate, change, record, or

transmit signals

CD, VCD/DVD

Signals and Systems


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Signals: Review of Basic Concepts


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Signal


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Signals Types Most of the signals we will talk about are functions of time.

There are many ways to classify signals. This class is organized

according to whether the signals are continuousin time, or

discrete.

A continuous-time signal has values for all points in time in

some (possibly infinite) interval.

A discrete time signal has values for only discrete points in

time.

Signals can also be a function of space (images) or of space

and time (video), and may be continuous or discrete in each

dimension


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Discrete-Time signal

A sampled version of a continuous signal

What should be the sampling frequency which is

enough for perfectly reconstructing the original

continuous signal?

Nyquist rate (Shannon sampling theorem)

Digital Signal

Sampling + Quantization Quantization: use a number of finite bits (e.g., 8bits) to

represent a sampled value

Discrete-Time Signal vs Digital Signal


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Discrete Time Signals

Fundamentally, a discrete-time signal is sequence of samples,

written,

[]

where is an integer over some (possibly infinite) interval.

Often, at least conceptually, samples of a continuous time

signal

[] = ()

where is an integer, and is the sampling period.

Example


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Signal Characteristics and Models

Operations on the time dependence of a signal

Amplitude Scaling Time scaling

Time reversal

Time shift

Combinations

Even and Odd Symmetry

Discrete Amplitude Signals

Impulse Sequence

Unit Step Sequence Unit Rectangle

Unit Ramp

Unit Triangle Signal

Exponential Signals

Sinusoidal Signals

Complex Signals


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Amplitude Scaling (Continuous-Time)

The scaled signal ax(t) is x(t) multiplied by the constant a


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Amplitude Scaling (Discrete Time)

The scaled signal ax[n] is x[n] multiplied by the constant a


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Time Scaling (Continuous Time)

A signal x(t) is scaled in time by multiplying the time variable

by a positive constant b, to produce x(bt).

A positive factor of b either expands (0 < b < 1) or compresses

(b > 1) the signal in time.


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Time Scaling (Discrete Time)

The discrete-time sequence []is compressed in time by

multiplying the index n by an integer

, to produce the time-

scaled sequence [].

This extracts every sample of [].

Intermediate samples are lost.

The sequence is shorter.

Called downsampling, or decimation


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Time Scaling (Discrete Time)

The discrete-time sequence [] is expanded in time by

dividing the index n by an integer m, to produce the time-

scaled sequence [ = ].

This specifies every sample.

The intermediate samples must be synthesized (set to zero, or

interpolated).

The sequence is longer.

Called upsampling, or interpolation


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Time Reversal

Continuous time: replace with , time reversed signal is

()

Discrete time: replace n with , time reversed signal is [].

Same as time scaling, but with b = -1.


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Time Shift: Continuous Time

For a continuous-time signal (), and a time 1 > 0,

Replacing t with gives a delayed signal

Replacing t with + gives an advanced signal x( + )

May seem counterintuitive. Think about where is zero


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Time Shift: Discrete Time

For a discrete time signal [], and an integer 1 > 0

[ ] is a delayed signal. [ ] is an advanced signal.

The delay or advance is an integer number of sample times.

Again, where is zero?


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Combination of Operations

Time scaling, shifting, and reversal can all be combined.

Operation can be performed in any order, but care is required. This will cause confusion.

Example: (2( 1))

Scale first, then shift

Compress by 2, shift by 1


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Combination of Operations

Example (2( 1)), continued

Shift first, then scale Shift by 1, compress by 2

Shift first, then scale

Rewrite (2( 1)) = (2 2)

Shift by 2, scale by 2


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Combination of Operations: HW


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Event and Odd Symmetry

An even signal is symmetric about the origin

An odd signal is antisymmetric about the origin

d dd


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Any signal can be decomposed into even and odd components

and that

d dd


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Example

Same type of decomposition applies for discrete-time signals.

E d Odd S HW


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Even and Odd Symmetry: HW

The decomposition into even and odd components depends on

the location of the origin. Shifting the signal changes the

decomposition.

Plot the even and odd components of the previous example,

after shifting ()by 1/2to the right.

Di A li d Si l


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Discrete Amplitude Signals

Discrete amplitude signals take on only a countable set of

values.

Example: Quantized signal (binary, fixed point, floating point).

A digital signal is a quantized discrete-time signal.

Requires treatment as random process, not part of this

course.


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Impulse Sequence (Discrete-time)


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Expressing Sequence as Sum of Unit ImulseSequences


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The unit step function u(t) is defined as

Alternate definitions of value exactly at zero, such as 1/2.. Also known as the Heaviside step function.

Unit Step Sequence (Continuous Time)


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f h ( )


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Extracting part of another signal. For example, the piecewise-

defined signal

Can be written as

Uses for the Unit Step (Continuous Time)

( )


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Unit Step Sequence (Discrete-Time)

i l


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Unit Rectangle

U i R


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Unit Ramp


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Exponential Signals


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Exponential Signals

An exponential signal is given by

If < 0, this is exponential decay. If > 0 this is exponential growth.

Damped Exponential Signals


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Damped Exponential Signals

Exponential growth ( > 0)or decay( < 0), modulated bya sinusoid.

Di t E ti l S


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Discrete Exponential Sequence

Sinusoidal Signals


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Sinusoidal Signals

A sinusoidal signal is of the form

where f is the frequency in Hertz.

We will often refer to as the frequency, but it must be kept in

mind that it is really the , and the

frequency is actually .

where the radian frequency is , which has the units of

radians/s.

Also very commonly written as

Sinusoidal Signals


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Sinusoidal Signals

The period of the sinusoid is

With the units of seconds.

The phase or phase angle of the signal is , given in radians.


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C l E ti l S


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Complex Exponential Sequence


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Discrete Time Periodicity


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Discrete Time Periodicity

More Complex Signals


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Many more interesting signals can be made up by

combining these elements.

Example: Pulsed Doppler RF Waveform


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System: Review of Basic Concepts

Systems: Intuitive Description


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Systems: Intuitive Description

Systems: Definition


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Systems: Definition

Systems: Block Diagram


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Systems: Block Diagram

System: Examples


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System: Examples

System Examples


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System Examples

System Examples


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System Examples

System Examples: Multiple Inputs


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System Examples: Multiple Inputs

System Interconnection


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System: Linearity


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System: Linearity


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System: Linearity


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System: Linearity

System Memory


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System Memory

System: Time Invariance


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System: Stability


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System: Stability

System Stability


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System Stability

System Inevitibility


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System Inevitibility

System Invertibility


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System: Described by Differential Equation


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y y q

System: Examples


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y p

System Examples: 2ndorder Circut


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y p

System Examples: Mechanical Systems


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y p y

System Examples: Mechanical Systems


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y p y

System Types


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y ypSystems are classified according to the types of input and output signals

Continuous-time system has continuous-time inputs and outputs.

AM or FM radio

Conventional (all mechanical) car

Discrete-time system has discrete-time inputs and outputs.

PC computer game

Matlab

Hybrid systems are also very important (A/D, D/A converters).

You playing a game on a PC

Modern cars with ECU (electronic control units)

Most commercial and military aircraft

Discrete Type Systems


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yp y

Discrete Time Systems


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y

The system T computes the outputs as a function of the

inputs. In general, the output at time n may depend on

several sa0mples of the input sequence x.

We typically denote the input of a discrete-time system

as x[n] and the output as y[n].

Example: Moving average

Causal Discrete-Time Systems


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y

Stable Discrete-Time Systems


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y

Input-Output Description: Capabilities & Limitations


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Discrete-Time Convolution


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Lecture1- Final Released

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