ECE 6250: Advanced Topics in DSP Syllabus for Fall 2009
Instructor: Justin Romberg Updated: August 24, 2009
ECE 6250 is a general purpose, advanced DSP course designed to follow an introductory DSP course.
A senior-level introductory
We will start by reviewing one of the foundational results of digital
signal processing: the Shannon-Nyquist sampling theorem. We will
use this result as a rst example of how continuous-time signals can be
systematically discretized (translated into a dis
A little more on the X(ej ) X(ejT ) step .
What we are doing is taking a discrete sequence x[n] (with DTFT
X(ej ) and turning it into a function xa(t) (with CTFT Xa(j) =
X(ejT ) of a continuous time variable.
x[n] (t nT )
Suppose the spectrum of xc(t) looks like
Compare the outputs of these two systems:
1 | /T
0 | > /T
Which is closer to xc(t)?
That is, which is smaller:
|xc(t) x1(t)| dt or
|xc(t) x2(t)| dt ?
Georgia Tech ECE 6250
1. When can you reconstruct xc(t) perfectly from its samples?
2. How do you do it?
1. When xc(t) is bandlimited, i. e. when
Xc(j) = 0 for all | > /T
where Xc(j) is the continuous time Fourier transform (CTFT)
Just as before, we can generate the basis function b2(t) from the lower
b2(t) = (b1 b0)(t) = (b0 b0 b0)(t).
In general, any th order polynomial spline x(t) is uniquely represented by a list of numbers cfw_k , k Z, which correspond to the
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
There are essentially two things going on here:
1. Xc(j) T Xc j T
dilates the spectrum
makes this dilation periodic (w/ period 2).
Graphically, this is what happens for B < /T :
Georgia Tech ECE 6250 Notes by J. Romb
Mathematically, we can write the output as:
xr (t) =
sin(t nT )/T )
(t nT )/T
x[n] hT (t nT )
shifts of the sinc
hT (t) =
HT (j) =
Georgia Tech ECE 6250 Notes by J. Romberg
T, | T ,
0, | > T
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from cvxopt import solvers, matrix
" A Support Vector Machine classifier object. "
def _init_(self, kernel=linear_k):
from _future_ import division
import numpy as np
self.features = 
with open(filename) as f:
data = f.read().splitlines()
lines = data[3:]
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