Digital Signal Processing
Week 1: Introduction
1.
2.
3.
4.
Course overview
Digital Signal Processing
Basic operations & block diagrams
Classes of sequences
Based on: http://www.ee.columbia.edu/~dpwe/e4810/
By: Dan Ellis
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1. Course overview
Grading structure
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Homeworks (and small project): 10%
Midterm: 20%
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Presentation (Max. 20 minutes): 10%
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One paper related to course topics
Final exam: 30%
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One session
One session
Project: 30%
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1. Course overview
Course project
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Goal: hands-on experience with DSP
Practical implementation
Work in pairs or alone
Brief report, optional presentation
Recommend MATLAB
Ideas on DSP_Pack CD or past class or
internet
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MATLAB
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Interactive system for numerical
computation
Extensive signal processing library
Focus on algorithm, not implementation
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2. Digital Signal Processing
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Signals:
Information-bearing function
E.g. sound: air pressure variation at a
point as a function of time p(t)
Dimensionality:
Sound: 1-Dimension
Grayscale image i(x,y) : 2-D
Video: 3 x 3-D: {r(x,y,t) g(x,y,t) b(x,y,t)}
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Example signals
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Noise - all domains
Spread-spectrum phone - radio
ECG - biological
Music
Image/video - compression
….
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Signal processing
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Modify a signal to extract/enhance/
rearrange the information
Origin in analog electronics e.g. radar
Examples…
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Noise reduction
Data compression
Representation for
recognition/classification…
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Digital Signal Processing
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DSP = signal
processing on a
computer
Two effects:
discrete-time,
discrete level
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DSP vs. analog SP
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Conventional signal processing:
p(t)
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p(t)
Processor
q(t)
Digital SP system:
A/D
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p[n]
Processor
q[n]
D/A
q(t)
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Digital vs. analog
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Pros
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Noise performance - quantized signal
Use a general computer - flexibility,
upgrade
Stability/duplicability
Novelty
Cons
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Limitations of A/D & D/A
Baseline complexity / power consumption
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DSP example
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Speech time-scale modification:
extend duration without altering pitch
M
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3. Operations on signals
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Discrete time signal often obtained by
sampling a continuous-time signal
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Sequence {x[n]} = xa(nT), n=…-1,0,1,2…
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T= sampling period; 1/T= sampling frequency
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Sequences
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Can write a sequence by listing values:
{x[n]}  {,  0.2, 2.2,1.1, 0.2,  3.7, 2.9,}

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Arrow indicates where n=0
Thus, x[ 1]  0.2, x[0]  2.2, x[1]  1.1,
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Left- and right-sided
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x[n] may be defined only for certain n:
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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
Left-sided
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Right-sided
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3. Basic Operations on sequences
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Addition operation:

x[n]
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Adder
y[n]
y[n]  x[n]  w[n]
w[n]
Multiplication operation
A
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Multiplier x[n]
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y[n]
y[n]  A  x[n]
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More operations
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Product (modulation) operation:
y[n]

x[n]
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Modulator
w[n]
y[n]  x[n]  w[n]
E.g. Windowing: multiplying an infinitelength sequence by a finite-length
window sequence to extract a region
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Time shifting
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Time-shifting operation: y[n]  x[n  N ]
where N is an integer
If N > 0, it is delaying operation
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Unit delay x[n]
z
1
y[n]
y[n]  x[n  1]
If N < 0, it is an advance operation
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Unit advance
x[n]
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z
y[n]
y[n]  x[n  1]
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Combination of basic operations
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Example
 1 x[n]
  2 x[n 1]
y[n] 
  3 x[n  2]   4 x[n  3]
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Up- and down-sampling
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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
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Down-sampling
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In down-sampling by an integer factor
M > 1, every M-th samples of the input
sequence are kept and M - 1 in-between
samples are removed:
xd [n]  x[nM ]
x[n]
M
xd [n]

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Down-sampling
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An example of down-sampling
Input Sequence
1
0.5
Amplitude
Amplitude
0.5
0
-0.5
-1
Output sequence down-sampled by 3
1
0
-0.5
0
10
20
30
Time index n
x[n]
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-1
50
3
0
10
20
30
Time index n
40
y[n]  x[3n]
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Up-sampling
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Up-sampling is the converse of downsampling: L-1 zero values are inserted
between each pair of original values.
 x[n / L], n  0,  L,  2 L,
xu [n]  
otherwise
 0,
x[n]
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L
xu [n]
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Up-sampling
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An example of up-sampling
Input Sequence
1
0.5
Amplitude
Amplitude
0.5
0
0
-0.5
-0.5
-1
Output sequence up-sampled by 3
1
0
10
20
30
Time index n
x[n]
40
-1
50
0
10
20
30
Time index n
40
50
xu [n]
3
not inverse of downsampling!
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Complex numbers
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.. a mathematical convenience that lead
to simple expressions
A second “imaginary” dimension (j√-1)
is added to all values.
Rectangular form: x = xre + j·xim
where magnitude |x| = √(xre2 + xim2)
and phase q = tan-1(xim/xre)
Polar form: x = |x| ejq = |x|cosq + j· |x|sinq
jq
( e  cos q  j sin q )
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Complex math
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When adding, real
and imaginary parts
add: (a+jb) + (c+jd)
= (a+c) + j(b+d)
When multiplying,
magnitudes multiply
and phases add:
rejq·sejf = rsej(q+f)
Phases modulo 2
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Complex conjugate
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Flips imaginary part / negates phase:
conjugate x* = xre – j·xim = |x| ej(–q)
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Useful in resolving to real quantities:
x + x* = xre + j·xim + xre – j·xim = 2xre
x·x* = |x| ej(q) |x| ej(–q) = |x|2
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4. Classes of sequences
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Useful to define broad categories…
Finite/infinite (extent in n)
Real/complex:
x[n] = xre[n] + j·xim[n]
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Classification by symmetry
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Conjugate symmetric sequence:
xcs[n] = xcs*[-n] = xre[-n] – j·xim[-n]
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Conjugate antisymmetric:
xca[n] = –xca*[-n] = –xre[-n] + j·xim[-n]
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Conjugate symmetric decomposition
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Any sequence can be expressed as
conjugate symmetric (CS) /
antisymmetric (CA) parts:
x[n] = xcs[n] + xca[n]
where:
xcs[n] = 1/2(x[n] + x*[-n]) = xcs *[-n]
xca[n] = 1/2(x[n] – x*[-n]) = -xca *[-n]
When signals are real,
CS  Even (xre[n] = xre[-n]), CA  Odd
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Basic sequences
1, n  0
d [ n]  
0, n  0
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Unit sample sequence:
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Shift in time:
d[n - k]
Can express any sequence with d:
{0,1,2..}0d[n] + 1d[n-1] + 2d[n-2]..
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More basic sequences
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Unit step sequence:
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Relate to unit sample:
d[n]  [n]  [n 1]
1, n  0
[ n]  
0, n  0
[n]  k d[k]
n
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Exponential sequences
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Exponential sequences= eigenfunctions
General form: x[n] = A·n
If A and
 are real:
 = 0.9
= 1.2
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Amplitude
Amplitude
40
30
20
5
10
0
10
0
5
10
15
20
Time index n
|| > 1
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0
0
5
10
15
20
Time index n
25
|| < 1
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Complex exponentials
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x[n] = A·n
Constants A,  can
be complex :
A = |A|ejf ;  = e(s + jw)
 x[n] = |A| esn ej(wn +
f)
scale varying
magnitud
e
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varying
phase
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Complex exponentials
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Complex exponential sequence can
‘project down’ onto real & imaginary
axes to give sinusoidal sequences
q
1
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e
x[n]  exp( 12  j 6 )n Imaginary part cos q  j sin q
j
Real part
1
xre[n]
0.5
Amplitude
0.5
Amplitude
1
0
-0.5
-1
xim[n]

0
-0.5
0
10
20
Time index n
30
40
-1
0
10
20
Time
index n
n/12
xre[n] = en/12cos(n/6) xim[n] = e
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sin(n/6)M
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Periodic sequences
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x [n] satisfying ~
A sequence ~
x [ n]  ~
x [n  kN ],
is called a periodic sequence with a
period N where N is a positive integer and
k is any integer.
x [ n]  ~
x [n  kN ]
Smallest value of N satisfying ~
is called the fundamental period
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Periodic exponentials
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Sinusoidal sequence A cos(wo n  f ) and
complex exponential sequence B exp( jwo n)
are periodic sequences of period N only if
w o N  2 r with N & r positive integers
Smallest value of N satisfying wo N  2 r
is the fundamental period of the
sequence
r = 1  one sinusoid cycle per N samples
r > 1  r cycles per N samples
M
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Symmetry of periodic sequences
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An N-point finite-length sequence xf[n]
defines a periodic sequence:
“n modulo N”
x[n] = xf[<n>N]
Symmetry of xf [n] is not defined
because xf [n] is undefined for n < 0
Define Periodic Conjugate Symmetric:
xpcs [n] = 1/2(x[n] + x*[<-n >N])
= 1/2(x[n] + x*[N – n]) 0 ≤ n < N
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Sampling sinusoids
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Sampling a sinusoid is ambiguous:
x1 [n] = sin(w0n)
x2 [n] = sin((w0+2)n) = sin(w0n) = x1 [n]
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Aliasing
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E.g. for cos(wn), w = 2r ± 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 ≤ 
i.e. less than one-half cycle per sample
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