t tao are real numbers) is a basis set for the analog complex signal space 1:...
An analog signal is given as below x(t) = 10sin 4rtt The signal is sampled by two different frequencies f, = 1Hz, f, = 10Hz respectively, and the output are yı, Yz. (i) Sketch signal x(t) in the time domain. (3 marks) (ii) Sketch frequency spectrum of x(t). (3 marks) (iii) After sampling, the continuous signal is converted to a discrete signal. Draw the two discrete signals Yı, Y2: (4 marks) (iv) Discuss whether f1, f, can successfully sample the...
A set of orthonormal basis functions for a particular signal
space are shown below.
(a) Given that x(t) = ,
calculate the signal space vector (a1, a2, a3) such that x(t) =
.
(b) Given that y(t) can be represented by the vector (1,1,-2) in
signal space using the orthonormal basis set above. Sketch y(t)
below in time domain.
4,0< t<2 0, otherwise alwl(t) +a2w2(t) + a33(t) i(t) 0.5 wg(,) 10.5 2 2 -0.5 0.5 0.5
4,0
advanced linear algebra, need full proof thanks
Let V be an inner product space (real or complex, possibly
infinite-dimensional). Let
{v1, . . . , vn} be an orthonormal set of vectors.
4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
1. A 4-ary signal set is given as So(t)-0,0StsT SI(t)-A cos(oot), OStsT Si(t)A sin(oot) + A cos(oot), 0StsT Assume (oT = 2m for some integer n. The signals are transmitted over an AWGN with noise power spectral density = N-2. The receiver uses the ML decision rule (ie, he assumes all signals are equiprobable) Choose a suitable set of Orthonormal basis functions and find (and sketch) the geometric representation of this signal set (called signal constellation). Clearly identify the decisions...
QUESTION 1 Is the set of complex numbers {α i complex numbers? lal =r) where r is a real positive number a subfield of the field C of
QUESTION 1 Is the set of complex numbers {α i complex numbers? lal =r) where r is a real positive number a subfield of the field C of
151 4.8 Problems 4.17 The autocorrelation function Txx(t) of a real analog signal x (t) is defined by xx(t) x(t)x (r-t) dt, (4.73) obtained from Eq. (4.70) by replacing y(t) with x (t). Thus the function is a cross-correlation of x(t) with itself. One application of the autocorrelation function is to detect the period of a periodic signal that has been corrupted by noise. Show that rxx(t) is an even function of r.
151 4.8 Problems 4.17 The autocorrelation function...
3. Find at least two different sets of orthonormal basis signals for a 4-dimensional signal space
3. Find at least two different sets of orthonormal basis signals for a 4-dimensional signal space
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that j-1 for some nonnegative numbers a,, j-1,.,k, that sum up to 1
Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that...
2) The set S of all real-valued functions f(x) of a single real variable z is a vector space. (a) Show that the set L of all real-valued linear functions f(x) = mx + b of a single variable x is a subspace of S. (b) Show tha (f(x), g(x))= | f(z)g(x)dx is an inner product on L. (c) Find an orthonormal basis for C with respect to the inner product defined in (b)