ECE 6250: Advanced Topics in DSP Syllabus for Fall 2009
Instructor: Justin Romberg Updated: August 24, 2009
Summary
ECE 6250 is a general purpose, advanced DSP course designed to follow an introductory DSP course.
Prerequisites
A senior-level introductory
A rst look at basis expansions
In the last lecture, we looked at the sampling theorem from the point
of view of frequency-domain transformations. This is a very good way
to understand it in the context of classical signal processing. We can,
however, thin
Example: Fourier series
Recall that any periodic signal can be written as a (possibly innite)
superposition of harmonic sinusoids. If x(t) has period T , we
can write
k e j2kt/T ,
x(t) =
(1)
k=
1
where k =
T
T /2
x(t) ej2kt/T dt.
(2)
T /2
(The integral ab
To see this, suppose that x(t) is zero outside of [T /2, T /2], so (1)
is building it up only inside this interval. Then its Fourier transform
is
T /2
x(t) ejt dt.
X(j) =
T /2
Notice that the Fourier series coecients k in (2) are samples of the
Fourier tr
3.5
3
2.5
2
1.5
1
0.5
0
0.5
1
1.5
6
4
2
0
2
4
6
8
10
Any th order polynomial spline can be written as a superposition
of B-spline functions (the B is for basis!).
The piecewise constant function above can be written as
4
k b0(t k),
x(t) =
b0(t) =
k=1
1 1/
We know how to compute these complementary (t), but they are
b
not easy to write down as nice expressions.
Bases and discretization
All of the examples above have a common theme: we take a signal in
a certain class (bandlimited, zero outside of [0, T ], p
[1/2, 1/2], we can write
t
k tk ,
e =
k=0
k
k t
log(1 + t) =
k=0
where k =
(1)k+1
where k =
,
k
k tk where k =
sin(2t) =
1
,
k!
k=0
(1)(k+3)/2 (2)k+1
(k+1)!
k odd
k even
0
Here are the three examples above with the series truncated to the
rst six terms:
e
Example: Polynomials
It is almost too obvious that any mth order polynomial can be written
as a super position of the m + 1 functions 1, t, t2, . . . , tm. (Indeed,
this is pretty much the denition of polynomial.)
For example:
1
t3 17 t2 + 5 t 12
12
8
17
Linear combinations and spans
Let M = cfw_v 1, . . . , v N be a set of vectors in a linear space S.
Denition: A linear combination of vectors in M is a sum of
the form
a1v 1 + a2v 2 + + aN v N
for some a1, . . . , aN F.
Denition: The span of M is the set
4. S = cfw_x(t) : x(t) is periodic with period 2
A basis for S is B = cfw_ejkt
k= (Fourier Series)
S is innite dimensional.
5. S = GF (2)3 (length 3 bit vectors with mod 2 arithmetic).
A basis for S is
1
v 1 = 1 ,
0
0
v 2 = 1 ,
0
0
v 3 = 0 .
1
How w
Bases
Denition: A basis of a linear vector space S is a (countable) set
of vectors B such that
1. span(B) = S
2. B is linearly independent
The second condition ensures that all bases of S will have the same
(possibly innite) number of elements.
The dimens
Linear dependence
A set of vectors cfw_v j N is said to be linearly dependent if there
j=1
exists scalars a1, . . . , aN , not all = 0, such that
N
an v n = 0
n=1
Likewise, if n anv n = 0 only when all the aj = 0, then cfw_v nN is
n=1
said to be linearly
Question: What is the span of cfw_v 1, v 2, v 3 for
1
v 1 = 1
0
0
v 2 = 1
0
1
v 3 = 1
0
?
What about if
1
v 1 = 1
0
0
v 2 = 1
0
0
v 3 = 1
1
?
Example:
S = cfw_x(t) : x(t) is periodic with period 2
M = cfw_ejktB
k=B
Then span(M) = periodic, bandlim
Example 2:
S
v1
v2
v3
= C([0, 1])
= cos(2t)
= sin(2t)
= 2 cos(2t + /3)
Find a1, a2, a3 such that
a1v 1 + a2v 2 + a3v 3 = 0
Repeat for
v 3 = A cos(2t + )
for some A > 0, [0, 2).
Suppose that cfw_v 1, v 2, . . . , v N are linearly dependent. Then
an v n =
4. For each vector x S, there is a unique vector (called
x) such that
x + (x) = 0
Scalar multiplication must obey the following four rules for all
a, b F and x, y S:
1. a(x + y) = ax + ay
(a + b)x = ax + bx
(distributive)
2. (ab)x = a(bx)
(associative)
3
Here is an example of something which is not a vector space:
5. Bounded, continuous functions f (t) on [a, b] such that
|f (t)| 2.
Why is this not a linear vector space?
Linear subspaces
A (non-empty) subset T of S is call a linear subspace of S if
a, b
Linear Algebra has become as basic and as applicable
as calculus, and fortunately it is easier.
Gilbert Strang
Linear signal spaces (vector spaces)
A vector space is simply a collection of things that obeys certain
abstract (but mostly familiar) algebrai
b1 (t k), k = 6, . . . , 10
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.02
6
4
2
0
2
4
6
8
10
Notice that we can generate b1 by convolving b0 with itself:
b1(t) = (b0 b0)(t).
The expansion for the piecewise quadratic spline above is a little
more complicat
It is important to realize that Taylor series is not the only way to
build up a function as a sum of polynomials, and despite its convenience, it has a few unsatisfying properties (e.g. there are innitely
dierentiable functions whose Taylor series converg
in the next section with precise (but abstract) denitions of linear
vector space, norm, and inner product.
24
Georgia Tech ECE 6250 Notes by J. Romberg
BETTER DESIGN
COPY designs
Good artists copy,
great artists steal.
Pablo Picasso
INFOGRAPHICS
WEBSITE
DESIGN
Visual Complexity
The New York Times
Smashing Magazine
Pinterest
D3 Gallery
just copy
and paste
ENSURE ALIGNMENT
GRAMENER
Every item should have
Frequency domain interpretation
Like many things, it is illuminating to look at sampling and reconstruction in the frequency domain.
First, we will relate the discrete time Fourier transform (DTFT)
of x[n] to the CTFT of xc(t):
jn
j
X(e ) =
x[n] e
n=
xc(n
The Fundamental Theorem of DSP
If xc(t) is bandlimited to B (Xc(j) = 0 for | > B), then it can
be perfectly reconstructed from samples spaced T /B apart.
x[n]hT (t nT ),
xc(t) =
x[n] = x(nT ).
n=
1. This is known as the Shannon-Nyquist sampling theorem
2.
Mathematically, we can write the output as:
xr (t) =
x[n]
n=
sin(t nT )/T )
(t nT )/T
x[n] hT (t nT )
=
n=
shifts of the sinc
Recall that:
sin(t/T )
hT (t) =
t/T
CTFT
HT (j) =
3
Georgia Tech ECE 6250 Notes by J. Romberg
T, | T ,
0, | > T
There are essentially two things going on here:
1
1. Xc(j) T Xc j T
dilates the spectrum
2.
1
Xc
T
j T
1
T
k
Xc j
T
+ 2k
T
makes this dilation periodic (w/ period 2).
Graphically, this is what happens for B < /T :
7
Georgia Tech ECE 6250 Notes by J. Romb
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I. Signal Discretization using
Basis Decompositions
Georgia Tech ECE 6250 Fall 2015; Notes by J. Romberg. Last updated 16:29, August 25, 2015
We will start by reviewing one of the foundational results of digital
signal processing: the Shannon-Nyquist samp
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