ELEN E4810: Digital Signal Processing Week 1: Introduction - PowerPoint PPT Presentation

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ELEN E4810: Digital Signal Processing Week 1: Introduction

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ELEN E4810: Digital Signal Processing Week 1: Introduction Course overview Digital Signal Processing Basic operations & block diagrams Classes of sequences – PowerPoint PPT presentation

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Title: ELEN E4810: Digital Signal Processing Week 1: Introduction


1
ELEN E4810 Digital Signal ProcessingWeek 1
Introduction
  1. Course overview
  2. Digital Signal Processing
  3. Basic operations block diagrams
  4. Classes of sequences

2
1. Course overview
  • Digital signal processingModifying signals with
    computers
  • Web site http//www.ee.columbia.edu/dpwe/e4810/
  • BookMitra Digital Signal Processing (2nd
    ed.)2001 printing
  • Instructor dpwe_at_ee.columbia.edu

3
Grading structure
  • Homeworks 20
  • Mainly from Mitra
  • Collaboration
  • Midterm 20
  • One session
  • Final exam 30
  • Project 30

4
Course project
  • Goal hands-on experience with DSP
  • Practical implementation
  • Work in pairs or alone
  • Brief report, optional presentation
  • Recommend MATLAB
  • Ideas on website

5
Example past projects
  • Speech enhancement
  • Audio watermarking
  • Voice alteration with the Phase Vocoder
  • DTMF decoder
  • Reverb algorithms
  • Compression algorithms

on web site
W
6
MATLAB
  • Interactive system for numerical computation
  • Extensive signal processing library
  • Focus on algorithm, not implementation
  • Access
  • Engineering Terrace 251 computer lab
  • Student Version (need Sig. Proc. toolbox)
  • ILAB (form)

7
Course at a glance
8
2. Digital Signal Processing
  • SignalsInformation-bearing function
  • E.g. sound air pressure variation at a point as
    a function of time p(t)
  • DimensionalitySound 1-DimensionGreyscale
    image i(x,y) 2-DVideo 3 x 3-D r(x,y,t)
    g(x,y,t) b(x,y,t)

9
Example signals
  • Noise - all domains
  • Spread-spectrum phone - radio
  • ECG - biological
  • Music
  • Image/video - compression
  • .

10
Signal processing
  • Modify a signal to extract/enhance/ rearrange the
    information
  • Origin in analog electronics e.g. radar
  • Examples
  • Noise reduction
  • Data compression
  • Representation for recognition/classification

11
Digital Signal Processing
  • DSP signal processing on a computer
  • Two effects discrete-time, discrete level

x(t)
xn
12
DSP vs. analog SP
  • Conventional signal processing
  • Digital SP system

Processor
p(t)
q(t)
pn
qn
Processor
A/D
D/A
p(t)
q(t)
13
Digital vs. analog
  • Pros
  • Noise performance - quantized signal
  • Use a general computer - flexibility, upgrde
  • Stability/duplicability
  • Novelty
  • Cons
  • Limitations of A/D D/A
  • Baseline complexity / power consumption

14
DSP example
  • Speech time-scale modificationextend duration
    without altering pitch

  • M

15
3. Operations on signals
  • Discrete time signal often obtained by sampling a
    continuous-time signal
  • Sequence xn xa(nT), n-1,0,1,2
  • T samp. period 1/T samp. frequency

16
Sequences
  • Can write a sequence by listing values
  • Arrow indicates where n0
  • Thus,

17
Left- and right-sided
  • xn may be defined only for certain n
  • N1 n N2 Finite length (length )
  • N1 n Right-sided (Causal if N1 0)
  • n N2 Left-sided (Anticausal)
  • Can always extend with zero-padding

Right-sided
Left-sided
18
Operations on sequences
  • Addition operation
  • Adder
  • Multiplication operation
  • Multiplier

19
More operations
  • Product (modulation) operation
  • Modulator
  • E.g. Windowing multiplying an infinite-length
    sequence by a finite-length window sequence to
    extract a region

20
Time shifting
  • Time-shifting operation
  • where N is an integer
  • If N gt 0, it is delaying operation
  • Unit delay
  • If N lt 0, it is an advance operation
  • Unit advance

21
Combination of basic operations
  • Example

22
Up- and down-sampling
  • Certain operations change the effective sampling
    rate of sequences by adding or removing samples
  • Up-sampling adding more samples
    interpolation
  • Down-sampling discarding samples decimation

23
Down-sampling
  • In down-sampling by an integer factor
  • M gt 1, every M-th samples of the input sequence
    are kept and M - 1 in-between samples are
    removed

24
Down-sampling
  • An example of down-sampling

25
Up-sampling
  • Up-sampling is the converse of down-sampling
    L-1 zero values are inserted between each pair of
    original values.

26
Up-sampling
  • An example of up-sampling

not inverse of downsampling!
27
Complex numbers
  • .. a mathematical convenience that lead to simple
    expressions
  • A second imaginary dimension (j?v-1) is added
    to all values.
  • Rectangular form x xre jxim where
    magnitude x v(xre2 xim2) and phase q
    tan-1(xim/xre)
  • Polar form x x ejq xcosq j xsinq
  • ( )

28
Complex math
  • When adding, real and imaginary parts add (ajb)
    (cjd) (ac) j(bd)
  • When multiplying, magnitudes multiply and phases
    add rejqsejf rsej(qf)
  • Phases modulo 2?

29
Complex conjugate
  • Flips imaginary part / negates phaseconjugate
    x xre jxim x ej(q)
  • Useful in resolving to real quantities x x
    xre jxim xre jxim 2xre xx x
    ej(q) x ej(q) x2

30
Classes of sequences
  • Useful to define broad categories
  • Finite/infinite (extent in n)
  • Real/complex xn xren jximn

31
Classification by symmetry
  • Conjugate symmetric sequencexcsn xcs-n
    xre-n jxim-n
  • Conjugate antisymmetricxcan xca-n
    xre-n jxim-n

32
Conjugate symmetric decomposition
  • Any sequence can be expressed as conjugate
    symmetric (CS) / antisymmetric (CA) parts xn
    xcsn xcanwhere xcsn 1/2(xn
    x-n) xcs -n xcan 1/2(xn
    x-n) -xca -n
  • When signals are real,CS ? Even (xren
    xre-n), CA ? Odd

33
Basic sequences
  • Unit sample sequence
  • Shift in time dn - k
  • Can express any sequence with da0,a1,a2..
    a0dn a1dn-1 a2dn-2..

34
More basic sequences
  • Unit step sequence
  • Relate to unit sample

35
Exponential sequences
  • Exponential sequences eigenfunctions
  • General form xn Aan
  • If A and a are real

a gt 1
a lt 1
36
Complex exponentials
  • xn Aan
  • Constants A, a can be complex A Aejf a
    e(s jw)? xn A esn ej(wn f)

37
Complex exponentials
  • Complex exponential sequence can project down
    onto real imaginary axes to give sinusoidal
    sequences
  • xren en/12cos(pn/6) ximn
    en/12sin(pn/6)M

xren
ximn
38
Periodic sequences
  • A sequence satisfying
  • is called a periodic sequence with a period N
    where N is a positive integer and k is any
    integer.
  • Smallest value of N satisfyingis called the
    fundamental period

39
Periodic exponentials
  • Sinusoidal sequence and
    complex exponential sequence
  • are periodic sequences of period N only if
  • with N r positive integers
  • Smallest value of N satisfying
  • is the fundamental period of the sequence
  • r 1 ? one sinusoid cycle per N samplesr gt 1 ?
    r cycles per N samples M

40
Symmetry of periodic sequences
  • An N-point finite-length sequence xfn defines a
    periodic sequence xn xfltngtN
  • Symmetry of xf n is not defined because xf n
    is undefined for n lt 0
  • Define Periodic Conjugate Symmetric xpcs n
    1/2(xn xlt-n gtN) 1/2(xn
    xN n) 0 n lt N

n modulo N
41
Sampling sinusoids
  • Sampling a sinusoid is ambiguous
  • x1 n sin(w0n) x2 n sin((w02p)n)
    sin(w0n) x1 n

42
Aliasing
  • E.g. for cos(wn), w 2pr w0 all r appear the
    same after sampling
  • We say that a larger w appears aliased to a
    lower frequency
  • Principal value for discrete-time frequency 0
    w0 p i.e. less than one-half cycle per sample
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