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