Problem 1 : Space of bandpass signals
For a given frequency 0 , consider the space V of signals x(t) L2 (R) that are bandpass in [ 0 , 0 ).
This means that the Fourier transform X () of x(t) is zero at any such that | [ 0 , 0 ) (or,
In this problem, all numbers must be exact, no approximate decimal number will be accepted.
Consider the Euclidean space E = L2 ([0, 1]) with the inner product f (t), g (t) = 0 f (t)g (t) dt. We
call V the subspace of the functio
In a Euclidean space E with an inner product , , we consider an innite family of vectors cfw_k k0
which is not necessarily orthogonal, and for each n 0 we dene the subspace Sn = Spancfw_0 , 1 , , n .
For any vector f S , we dene fn
Consider two real positive numbers T and , and the function (t) = sinc( T ).
a - Prove that the shifted function (t ) satises the Nyquist condition with respect to s =
b - Derive the sequence x[n] such that (t ) =
x[n] (t nT ).
ELEN6860, Fall 2010, Midterm Exam, November 5, 11:00am-12:50pm
Total number of points: 30
Remark: Do not rederive results already obtained in class. Just state them as results
In all the following problems, we consider sequences in 2 (Z) with
For given sequences g[n] and h[n], let gi and hi be the sequences of ztransforms
: G(Z)G(22) - . . G(z2ifz)G(22i-1),
Hi(z) = G(Z)G(z2) . . .G(Z2"2)H(z21*1)
with the convention 91 = g[n] and h1 = Show that
gi+1inl = Z M] shi
ELEN 6860 HW3 solutions
ELEN 6860 HW9 solutions
HW31, Daubechies filters
The filter of length 6 is a better lowpass filter than the one of length 4
a - Derive the Daubechies lowpass lter g6 [n] of length 6 and give numerically its sample values with
6 decimals (choose the minimum-phase solution).
Hint: Find roots of polynomials using numerical software (if you can nd them by s
Consider a scaling function (t) with respect to T , meaning that cfw_(tnT )nZ is orthonormal
and there exists a sequence a[n] such that ( 2 ) = nZ a[n] (tnT ). Express the orthogonal
projection of (2t) onto V = spancfw_(tnT )nZ i
Problem 1: Spline scaling functions
In L2 (R), consider the functions
1, t [0, T )
(k) (t) = (t) (t) (k times).
Note that (1) (t) = (t).
a - Give the explicit expression of (3) (t). Plot (1) (t), (2) (t) and (3) (
If F () is the Fourier transform of f (t), the Fourier transform of f (t) is F ( ).
For a given function (t) L2 (R) and T > 0, we dene V = Spancfw_(tkT )kZ and s =
each of the following cases, show that ( 2 ) V