lec1-intro fall 2014

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    Dr. Shoab Khan

    Adv Digital Signal Processing

    Lecture 1

    Introduction: DSP is Everywhere

    Source: Internet CARE/EME Projects

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    Outline

    Introduction of DSP Applications

    Projects

    DSP in Biomedical Engineering DSP in Radars

    DSP in Communication Systems

    Course Outline, Text Book, Grading

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    INTRODUCTION TO DSP

    3

    Adv DSP:Focus on DSP Software Desi n

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    DSP

    Application of mathematical operations (linearand non-linear) to digitally represented signals

    IN OUT

    A/D D/ADSP

    -3 -2 -1 0 1 2 3 4

    x[0]x[1]

    n

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    General IntroductionDiscrete Time Signal

    sequence x[n]

    - as opposed to continuous-timesignals x(t)

    - time = independent variable

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    Example- 1D Signal

    Sampled continuous-time (analog) signals

    - Speech

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    DSP

    Roots in 17-th and 18-th centurymathematics

    An important modern tool in a

    multitude of fields of science andtechnology

    Techniques and Applications of DSP

    As old as Newton and Gauss As new as digital computers and integrated

    circuits

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    Purpose

    To estimate characteristic parametersof signals

    Statistical Signal Processing

    To transform a signal into a moredesirable form

    Fourier, Wavelet etc

    Classical numerical analysis formulasare also DSP

    interpolation, integration, anddifferentiation

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    2-D Array: A Digital Image

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    10

    Typical Scenario

    Step 1: Analog sensor picking analog signal (e.g., microphone picking sound)

    Step 2: Analog to Digital Converter

    Step 3: DSP processes the digital signals (e.g., compression, noise suppression)

    Step 4: Digital to analog converter to recover the analog signal

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    Real-Time DSP

    Example: Processor clocked at 120 MHz and can

    perform 120MIPS Sampling rate = 48KHz (Digital Audio Tape - DAT)

    number of instructions per sample = (120 x106)/(48 x 103) = 2500.

    Sampling rate = 8KHz (voice-band, telephony)number of instructions per sample = 15000.

    Sampling rate = 75MHz (CIF 360x288 Video at 30frames per second) number of instructions persample = 1.6.

    Real-TimeDigital Processing

    Digital Signal in Digital Signal out

    Time-constrained Operation or Transformationperformed on digitalsignals within a required period

    of time to maintain synchronization with occurring events.

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    DSP Targets: Cell Phone

    -Speech Coders

    -Speech Recognition

    - Equalizers

    - Antenna noise cancellation

    -Image enhancement techniques

    DSP

    Chip

    RF

    Codec

    Voice

    Codec

    RF

    Receiver

    Microprocessor

    Chip

    Cell

    Peripherals

    Controlled by Power Management Unit

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    DSP Targets: Cell Phone

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    Purpose

    DSP SW Design: Availability of high-speed digital computers has fostereddevelopment of increasingly complexand sophisticated signal processingalgorithms

    Digital Design of DSP: Advances in VLSItechnology have made possible

    economical implementations of verycomplex digital signal processingalgorithms

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    DSP APPLICATIONS

    15Adv DSP:Focus on DSP Software Desi n

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    DSP is Everywhere

    Sound applications

    Compression, enhancement, special effects, synthesis, recognition, echocancellation,

    Cell Phones, MP3 Players, Movies, Dictation, Text-to-speech,

    Communication Modulation, coding, detection, equalization, echo cancellation,

    Cell Phones, dial-up modem, DSL modem, Satellite Receiver,

    Automotive ABS, GPS, Active Noise Cancellation, Cruise Control, Parking,

    Medical Magnetic Resonance, Tomography, Electrocardiogram,

    Military Radar, Sonar, Space photographs, remote sensing,

    Image and Video Applications DVD, JPEG, Movie special effects, video conferencing,

    Mechanical Motor control, process control, oil and mineral prospecting,

    l b dd d

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    Digital

    Radiographic

    Imaging

    Ultrasound

    Medical

    Imaging

    SpySatellite

    Imaging

    Military

    Appls

    Real-Time

    Video-Camera

    Cell-Phones

    Video

    Communications

    Space

    Imaging

    Appls

    Optical

    Wearable

    Computers

    Web wireless

    technology

    Data Storage

    & Transmission

    Car Awakewarning system

    Real- Time DSP

    Speech

    Recognition

    DSP in Real Time Embedded Systems

    http://www.euspaceimaging.com/content/Downloads/samples/madrid/madrid_(1280x960).jpghttp://www.ieee.com/
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    Digital Signal Processing Represent signals by a sequence of numbers

    Sampling or analog-to-digital conversions

    Perform processing on these numbers with a program or HW

    Digital signal processing

    Reconstruct analog signal from processed numbers

    Reconstruction or digital-to-analog conversion

    A/D DSP D/Aanalogsignal

    analogsignal

    digitalsignal

    digitalsignal

    Analog input analog output

    Speech in Mobile Phone

    Analog input digital output

    Speech to text

    Digital input analog output

    Text to speech

    Digital input digital output

    Compression of a file on computer

    Data minning

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    EEG electroencephalogram signal

    Records the electrical activity of the brain obtained from

    electrodes placed on the scalp.

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    A computer aided system capable of processing biological signals of learners in real-time to

    monitor their level of attention, cognition and engagement.

    d l

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    Intra-cardiac Signals

    f

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    fMRI

    fMRI measures brain activity by detecting associated changes in

    blood flow

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    3-D Accelerometers

    Powergloves: body pose estimation using a network of 3D

    accelerometers

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    Thallium scans

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    Fundus Image

    The fundus of the eyeis the interior surface of the eye, opposite the lens, and

    includes the retina, optic disc, maculaand fovea,

    http://en.wikipedia.org/wiki/Human_eyehttp://en.wikipedia.org/wiki/Lens_(anatomy)http://en.wikipedia.org/wiki/Retinahttp://en.wikipedia.org/wiki/Optic_dischttp://en.wikipedia.org/wiki/Maculahttp://en.wikipedia.org/wiki/Foveahttp://en.wikipedia.org/wiki/Foveahttp://en.wikipedia.org/wiki/Maculahttp://en.wikipedia.org/wiki/Optic_dischttp://en.wikipedia.org/wiki/Retinahttp://en.wikipedia.org/wiki/Lens_(anatomy)http://en.wikipedia.org/wiki/Human_eye
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    Projects related to signal processing

    DSP in CARE / EME

    I t lli t M di l E i t b d U ifi d N t k d

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    i-Hospital

    Cardiac ECG

    Intra Cardiac

    Holter Monitor

    Nuclear

    Cardiac & CT

    Angiography

    Diabetic

    Retinopathy,

    Maculopathy,OCT

    Hess

    Charting

    Gastroenterology

    National

    Repository &

    Analytics

    Use ICT as a Catalyst

    Intelligent Medical Equipment based Unified Networked

    Hospital

    Digitization and net-enabling of medical equipment

    Incorporation of intelligent diagnostics

    Image and signal processing & artificial intelligence

    The virtual hospital and workflows in SW

    Need to add payment system in the SW

    National Storage for Decision Aiding

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    Eye Care Diabetic Retinopathy (DR) Diabetic Maculopathy (DM)

    Detection of AMD using OCT

    Glaucoma Detection

    Diagnosis of Paralytic Strabismus using HessScreening

    CASE/College of EME

    Dr Shafaat A Bazaz

    Dr Usman Akram

    Dr Waheed

    Dr Shahzad

    AFIO

    Dr Mazhar Ishaq

    Dr Ubaidullah Yasin

    Dr Yasir

    Funded by

    Human Retina and Fundus Image

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    Human Retina and Fundus Image

    The retina is the layer of tissue at the back of the inner eye

    Optic Disc - brightest circular spot

    Macula - main central part of retina responsible for fine

    details and sharp vision

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    Retinal Image

    Blood Vessels

    Optic Disc

    Candidate Lesions

    FeatureExtraction

    Microaneurysms

    Soft Exudates

    Haemorrhage

    Hard Exudates

    C

    L

    A

    SS

    I

    F

    I

    E

    R

    G

    R

    A

    D

    I

    N

    G

    Normal

    Severe

    Mild

    Moderate

    Feature Selection

    Feature Set

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    AFIC / NIHD

    National ICT R&D Fund

    Corporate Social Responsibility

    A Tele-Cardiac &

    i-Diagnostic System

    Deployed System in AFIC

    Doctor on the move Tele-cardiac node at

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    p y y

    49

    ONT

    ONT

    Doctors

    TerminalOperatorPatient

    Ambulance

    WirelessInternet

    WiFi Router

    PDA

    District hospital

    WebSerices

    and NMSServer

    Local Area Network

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    ECG Wave

    50

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    ECG View

    52

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    Advanced ECG View

    53

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    Advanced ECG View

    54

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    ECG Streaming View

    55

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    Doctors Comment View

    56

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    Decision Aiding Tools

    57

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    ECG Wave

    58

    ECG i l

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    Total

    Power

    VLF

    ULF

    LF10m

    HF

    LF

    HF10m

    Frequency Domain

    Parameters

    LF/ HF

    Alpha

    ECG signal

    Morphology Comprehension and location of fiducial points

    SDNN

    SDANNRMSSD SDNN

    index

    NN50

    countpNN50

    HRV

    Triangular

    index

    TINNDifferent

    ial index

    Logarithm

    ic indexDispersion

    Time Domain Parameters

    ApEn MSE

    SEn

    TWA

    DFA MFA

    IBSI

    Nonlinear Analysis

    Parameters

    MKLT

    HRT

    Input Space Classifier 3rdStage

    Output Space

    RR ST QT PP PR P QRS T HR

    1 2 3 4 5 6 NMembership functions

    StartPerform Analysis

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    Acquire Digital ECG dataEctopic Beat

    Calculate RR

    intervals (Heart

    Rate Variability)

    Calculate QT

    interval

    Calculate ST

    segment duration

    Calculate P, QRS and

    T wave Statistical

    parameters (onset, Offset,

    duration, amplitude etc)

    Calculate T wave

    alternans

    Calculate Heart

    Rate Turbulence

    Perform Analysis Perform Analysis Perform Analysis Perform Analysis Perform Analysis Perform Analysis

    Morphology

    Analysis

    A Hybrid Classifier

    Compute Abnormality Factor

    Generate Report

    Abnormality

    Found? Local

    Database

    End

    Classify Ectopic Beat

    Get total no. of

    Ectopic beats

    Eliminate Ectopic Beats

    Repopulate signal

    of normal beats

    yesSet sampling and bit resolution parameters

    ECG preprocessing and noise removal

    Locate fiducial points

    Locate P wave, QRS complex and T wave

    Beat Classification

    no

    yes

    no

    Transmit

    Data

    y

    Perform Time Domain Analysis

    Perform Frequency Domain Analysis

    Perform Nonlinear Analysis

    Template

    Matching

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    Acquire ECG Data

    Bandpass Filter

    0.5/1 Hz < fs < 40Hz

    Bandpass Filter

    0.05 Hz < fs < 150Hz

    Signal Smoothing through

    polynomial fitting

    Signal Smoothing through

    higher order polynomial fitting

    Gaussian Modeling for

    high quality filtering

    Monitoring Mode ECG Raw ECG Diagnostic Quality ECG

    Remove Baseline Wandering

    Remove Powerline Interference

    Remove Wideband Noise (AWGN)

    Remove Muscle Artifacts such as EMG signals

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    Start

    ECG Signal

    Baseline wandering free signal

    Overall Median Calculation

    Sample by Sample shifting in

    accordance with median

    Fourth Degree Polynomial Fit

    Calculate deviation of the signal

    with respect to zero line

    Measure discrepancy

    and eliminate

    R waves and RR intervals

    are detected

    Median corrected in

    each RR interval

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    Start

    Acquire ECG Signal

    Powerline Interfaceremoved signal

    Compute power spectral density

    Slide a 1Hz window on power

    spectrum ranging from 40Hz to 70Hz

    Compute frequency component

    fnotch with highest energy

    in the sliding window

    Filter ECG

    Create a digital notch filter with

    sampling frequency fs, notch

    frequency fnotch and attenuationconstant of at least 10dBs

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    Start

    Acquire ECG data

    Diagnosed ECG signal

    Select a mother wavelet e.g. db6

    Decompose into subbands

    Modify the coefficients by

    applying a threshold function

    Reconstruct the signal

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    Start

    Acquire Digital ECG data

    Diagnosed ECG

    Pilot Signal EstimationQRS Detection

    Inverse Transform

    Beats Splitting

    and Assignment

    Inter-beat

    decorrelating transform

    Intra-beat

    decorrelating transform

    Noise Estimation

    Transform Domain

    Wiener Filter

    Inverse TransformBeats Merging

    Start

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    Digital ECG data

    Multiscale Zero-crossing

    point and edge detection

    QRS onsetand Q point

    QRS offsetand S point

    Detection of

    Q point

    Detection of

    S point

    Detection of

    QRS onset

    Detection of

    QRS offset

    Calculation of Time

    Window before R peak

    Calculation of Time

    Window after R peak

    Multiscale Zero-crossing

    point and edge detection

    Multiscale Zero-crossing

    point and edge detection

    Detection of P

    wave onset

    Detection of P

    wave offset

    Detection of T

    wave onset

    Detection of T

    wave offset

    Detection of P pointDetection of

    T point

    Detection of

    Modulus Maxima Pair

    Detection of

    Modulus Maxima Pair

    P point T pointP wave Onset P wave Offset T wave Onset T wave OffsetR point

    Multiscale Wavelet Transform

    using a Quadratic Spline wavelet

    Determination of Modulus

    Maximum Lines of R waves

    Elimination of Redundant andIsolation Modulus Maximum Lines

    Detection of R peak

    Calculation of

    Singular Degree

    Selection ofCharacteristic Scales

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    ECG signal with fiducial points marked with cross. P

    wave in pink, QRS in blue and S wave in red color 67

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    AR coefficients

    Calculate Y1 APC/

    PVC/ NSR/ SVT or VT/ VF

    APC/ PVC/ NSR/ SVT VT/ VF

    Calculate Y3 APC/ PVC/ NSR/ SVT Calculate Y2 VT/ VF

    APC/ PVC/ NSRSVT VT VF

    Calculate Y4 APC or PVC

    APC/ NSR PVC/ NSR

    Calculate Y5 APC or NSR Calculate Y6 PVC or NSR

    APC NSR PVC NSR

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    RR intervals

    Clean RR intervals

    HRV Preprocessing

    DWT

    Soft/ hard Thresholding

    iDWT

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    P

    Q

    S

    R

    T

    P

    Q

    S

    R

    TP

    Q

    S

    R

    T

    P

    Q

    S

    R

    TT1T2 T2T4

    T1T3T3T4

    If (|T1 T3| < |T1 T2|) and (|T2 T4| < |T3 T4|) and (|T2 T4| < |T1 T2|) and (|T1 T3| < |T3 T4|)

    Then T wave Alternans in this window is present

    Here Tn is the nth T wave in a Four Beat Window (n=1,2,3,4)

    Type PPInterval

    Variation

    (s)

    PRInterval

    Variation

    (s)

    PPInterval

    Duration

    (s)

    RRInterval

    Duration

    (s)

    AtrialRate

    (1/s)

    VentricularRate (1/s)

    P-wave PRInterval

    Duration

    (s)

    QRSInterval

    Duration

    (s)

    T wave QTInterval

    Duration

    (s)

    STsegment

    shift

    (mV)

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    (s) (s) (s) (s) (s) (s) (s) (mV)

    Normal 0.16 0.33

    0.6

    0.33

    0.6

    Normal Normal Normal >0.2 Normal Normal Normal -

    TDB - - Normal >1 Normal 1

    Ischemia - - - - - - - - - - - 1 >1 0.16 >0.16 >0.33 >0.33 Normal Normal Normal Normal Normal Normal Normal -

    SB Normal Normal >1 >1 0.1 - - -

    VA - - - > 1.5 - - - - - - - -

    PAC > 0.16 > 0.16 0.33

    0.6

    0.33

    0.6

    - - - - Normal - Normal -

    PVC - - - - - - Absent - >0.1 Opposite in

    direction to

    QRS

    - -

    AT Normal Normal 0.24

    0.4

    - 150 -

    250

    - - - Normal - Normal -

    AFr Normal Normal - Normal 250 -

    400

    Normal Abnorma

    l

    - Normal - - -

    AFn - - - 0.33 >400 100 - 180 Abnorma - Normal - - -

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    STM: Coherent Burst Satellite Receiver

    72

    32 kbps 512 Kbps BPSK/QPSK

    DSP Algorithms

    Burst Receiver

    FEC

    Carrier recovery and PLL

    DSP Software

    Embedded Software

    Glue Logic in Xilinx

    System Board

    Integration and testing

    HSP52014

    B-bit DAC & LFP

    68332

    A/D

    DDS

    Flash

    SRAM

    Amplifier &

    squarer

    SBSRAM

    TMS320C6201

    Output

    Bitsream

    Square

    wave outpt

    49.162 MHz Sine

    wave clock

    Xlllnx 4062

    From RF

    Board

    To RF

    Board

    73

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    A Pakistani Engineering Team Making History

    Voice processing @ 32 ms LEC

    VZM1004L, 256 x G.711 or 168 x G.726

    VAD, CNG & LPC (Lost Packet Compensation)

    Voice band signaling: CAS, DTMF, MF

    ~6 mw/channel in .18m Standard Cell

    World Highest DensityMedia Processing Chip

    Developed in Pakistan in 2000

    RTOS Abstraction Layer

    Media

    Processing

    Apps &

    Devices

    Cell &

    Packet

    Processing

    Apps &

    Devices

    Call

    Control &

    Signaling

    Protocols

    & Apps

    AVAZ

    SNMP

    Agent

    & MIB

    Utilities

    User Applications

    AVAZ System Components & Resources Manager

    74

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    VoIPPlatform for PTCL

    VoIP platform with multiple applications in

    telecom like call centers, IVR, CTI, ACD,

    PBX etc

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    Passive Navigation System

    75

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    Universal Radio Receiver COMINT System

    76

    SDPROLIANT

    1850R

    DRSE DemodulatorLoad BatteryLine

    O n O nBattery

    SmartBoost

    ReplaceBattery

    Test

    SD

    DATAX

    iZ 9200

    SD R D

    PORT ASD R D

    PORTBA

    ONLINEB

    BWD - ENTER

    POR T SEL D ISC D AT A

    +

    DRSE Demodulator

    Universal Demodulator

    Universal Demultiplexer

    Ethernet

    Control & Operator

    Console Computer

    IF up

    to 70

    MHz

    Multi Frame Generation Channel Activity Status

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    Software Defined Radio

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    SDR

    Th T h l

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    The Technology

    Analog

    input

    1 0 1 0 0 1 0

    Analog to

    DigitalConverter Bits Encoded

    Bits

    SourceEncode

    1 0 1 1 0

    Encrypt

    Encrypted

    Data

    0 1 1 0 1

    Bit to Sym.& Pulse

    Modulate

    Pulse

    modulated

    waveform

    Digital Bandpass

    waveformBandpassmodulate

    Multiplex

    0 1 1 0 1

    0 1 0 1 0

    1 0 1 0 1

    From Other

    Channels

    Multiplexed Data

    ChannelEncode

    Channel

    Encoded

    Data

    1 0 0 1 1 0 1

    Scrambler

    Scrambled

    data

    1 0 0 0 1

    Equalizer,

    Timing andSym. to Bits

    Bits

    Decrypted

    Bits

    1 0 1 1 0DecryptAnalog

    output

    D/A

    De-modulate

    Digital

    Baseband

    waveform

    Digital

    Bandpass

    waveformChannel

    Decode

    Channel

    Decoded

    Data

    0 1 1 0 1

    Source

    Decoded

    Bits

    1 0 1 0 0 1 0

    SourceDecode

    De-Multiplex

    To otherChannels

    De-

    multiplexed

    Bits

    Descrambled Bits

    1 0 0 0 1

    De-scramble

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    Software Defined Radio (Frequency

    Hopping)

    80

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    Software Defined Radio (Equilizer)

    81

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    SDR (Equalizer)

    82

    Speaker Identification System: Speech Playback

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    Selected Speech File

    for Playback and

    Spectrum Analysis

    Speech playback with

    Spectrum Analysis

    Speaker Identification System: Speech Playback

    KLT (PCA)

    Ei i l

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    Eigenimagesexamples:

    3 eigenimages and the individual variations on those components

    Facial

    image

    set

    Corresponding

    eigenfaces

    Face

    aproximation,

    from rough to

    detailed, as more

    coefficients areadded

    COURSE OUTLINE

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    COURSE OUTLINE

    &LEARNING OUTCOMES

    85

    Adv DSP:

    Focus on DSP Software Desi n

    Course Learning Outcomes

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    1. To review basic concepts of DSPa. Use basic concepts of convolution, system analysis,

    transformations and design in solution of DSP Problems2. To build advanced concepts in DSP

    a. Learn advanced topics in DSP relating to Multi-rate systems,Bandpass sampling, Statistical Signal Processing, Wavelettransform

    b. Use these advanced concepts in designing signal processingsystems

    3. To learn key areas in DSP Software Designa. To design DSP SW Systems relating to 1 and 2

    b. To develop software relating to 1 and 2 for mapping it on DSP

    Processors

    Course Outline

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    Basic Concepts Fundamentals of discrete-time, linear, shift-invariant signals

    and systems in Representation and Analysis:sampling, quantization, Fourier and

    z-transform;

    Implementation:filtering and transform techniques;

    Design:filter & processing algorithm design. Efficient computational algorithms for FFT and their implementation.

    Course Outline

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    Advanced Topics

    Sampling rate conversion in multirate systems,multirate signal processing, bandpass samplingand advanced transforms

    Signal Modeling, Least Square Method, PadeApproximation, Pronys Method, Finite DataRecord and Stochastic Models

    Levinson Recursion

    Course Outline

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    DSP SW Design Hybrid System: DSPs, GPPs and FPGAs

    Fixed Point & Floating Point Arithmetic

    DSP Architectures and HW interfaces

    DSP Processor Programming

    DSP BIOS, Programming DMA & Interrupts DSP Software Engineering Processes

    DSP Software Architecture Design

    Course Outline

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    DSP Fundamentals

    Band-pass

    sampling

    Multi-rate

    SignalProcessing

    Signal

    Modeling

    DSP SWDesign

    Prerequisite

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    A fundamental course in signalprocessing

    Liner System analysis and transformanalysis

    convolution and filtering Fourier transforms

    Laplace and z transforms

    Textbook

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    Oppenheim, Schafer and

    Buck, Discrete-TimeSignal Processing, 2ndedition (Prentice-Hall,

    1999) Refrences: Hayes, Digital Signal

    Processing (Schaums

    Outlines Series), 1999 McClellan, Schafer, &

    Yoder, DSP First

    Text

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    Monson H. Hayes

    Statistical Digital Signal Processing andModeling

    John Wiley & Sons, Inc

    J. G. Proakis, C. M. Rader, F. Ling, & L. NikiasAdvanced Digital Signal Processing

    eferences

    Ifeachor JervisDigital Signal Processing- A Practical Approach

    Prentice Hall

    Marks Distribution

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    Grading Sessional #1: 20%

    Sessional #2: 20% Quizzes 5%

    Assignments 5%

    Term Paper 2%

    Term Project: 5%-10%

    Final: 40-45%

    Homework: Due at the beginning of next week class fromthe date of issuing of assignment Problems from the book / previous exams

    MATLAB simulations Code of DSP Processors

    DSP FUNDAMENTALS

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    DSP FUNDAMENTALS

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    Focus on DSP Software Desi n

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    Signals

    Basic Types

    Discrete-Time Signals: Sequences

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    Discrete-time signals are represented by sequence of numbers The nthnumber in the sequence is represented with x[n]

    Sampling of continuous time signal x[n]is value of the analog signal at xc(nT)

    Where T is the sampling period

    0 20 40 60 80 100-10

    0

    10

    t (ms)

    0 10 20 30 40 50-10

    0

    10

    n (samples)

    Basic Signals

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    Basic Signals

    Unit sample (impulse) sequence

    Unit step sequence

    Exponential sequences

    0n1

    0n0]n[

    0n1

    0n0]n[u

    nanx ][

    -10 -5 0 5 100

    0.5

    1

    1.5

    -10 -5 0 5 100

    0.5

    1

    1.5

    -10 -5 0 5 100

    0.5

    1

    Sinusoidal Sequences

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    Sinusoidal Sequences

    Sinusoid

    A complex exponential

    nAnx ocos

    ][

    nj oAenx

    njAnAnx oo sincos

    sindemo

    Sine and Exp Using Matlab

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    % sine generation: A*sin(omega*n+theta)

    % exponential generation: A^n

    n = 0: 1: 50;

    % amplitude

    A = 0.87;

    % phase

    theta = 0.4;

    % frequency

    omega = 2*pi / 20;

    % sin generation

    xn1 = A*sin(omega*n+theta);

    % exp generation

    xn2 = A.^n;

    Basic Operations

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    operations

    Operations in Matlab

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    xn1 = [1 0 3 2 -1 0 0 0 0 0];

    xn2 = [1 3 -1 1 0 0 1 2 0 0];

    yn = xn1 + xn2;

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    x[n] via impulse functions

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    Input: sum of weighted shifted impulses

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    Time Domain Analysis

    Linear Time-Invariant Systems

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    Linear

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    linearity

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    Linear Time-Invariant SystemsLinear Time-Invariant System

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    Linear Time-Invariant System

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    Input: sum of weighted shifted impulses

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    Using Linearity and Time-Invariance for the impulses

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    Sum of wt. Shifted impulses sum of wt. Shifted impulse responses

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    LTI System

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    The Spatial Filtering Process

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    j k lm n o

    p q r

    Origin x

    y Image f (x, y)

    eprocessed= n* e +

    j* a + k* b + l* c +

    m* d + o* f +

    p* g + q* h + r* i

    Filter (w)Simple 3*3

    Neighbourhoode 3* 3 Fil ter

    a b cd e f

    g h i

    Original Image

    Pixels

    *

    The above is repeated for every pixel in the

    original image to generate the filtered image

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