On VSB Modulation
Philliph Schniter
February 24, 1998
VSB modulation still has wide applications in analog communication, but not much
any more for data communication.
Gitlin, Hayes, and Weinstein, 1992.
1
Introduction
The ATSCs recent adoption of the VSB
EE 511 Solutions to Problem Set 8
1. E [Yt ] = E [Xt ] cos 2fc t. E [Yt ] is periodic with period 1/fc . RY (t, t + ) = E [Xt+ cos 2fc (t + )Xt cos 2fc t] = RY (t, t + ) is periodic with period 1/(2fc ). Therefore, Yt is wide-sense cyclostationary with pe
EE 511 Solutions to Problem Set 7
1. (i) SX (f ) = |H (f )|2 SX (f ) where H (f ) = j sgn(f ). Therefore, SX (f ) = SX (f ). (ii) Suppose h( ) is the impulse response of the Hilbert transformer, h( ) = h( ) is the impulse response of the inverse Hilbert t
EE 511 Solutions to Problem Set 6
1. The power spectral density of the output noise process is equal to N0 /2 for |f | B and equal to 0 otherwise. Therefore, the variance of the output noise process (zero-mean) is the area under the PSD = (2B )(N0 /2) = N
EE 511 Solutions to Problem Set 4
1. (i) X (s) = es
2 /2
. (ii) We have P [X a] = eas X (s) for all s > 0.
This upper bound should be minimized with respect to s to obtain the Cherno bound. eas X (s) = eas es Setting the derivative with respect to s to 0,
EE 511 Solutions to Problem Set 3
1. (a) FY (y |X = 1/4) = P [Y y |X = 1/4] = P [X + N y |X = 1/4] = P [1/4 + N y |X = 1/4] = P [N y 1/4|X = 1/4]. Since X and N are independent, we have P [N y 1/4|X = 1/4] = P [N y 1/4] and FY (y |X = 1/4) = FN (y 1/4). T
EE 511 Solutions to Problem Set 2
1. P (AB |A) = P (AB ) P (A) and P (AB |(A + B ) = P (AB (A + B ) P (AB ) = . P (A + B ) P (A + B )
Since P (A + B ) P (A), P (AB |(A + B ) P (AB |A). 2. Since A and B are mutually exclusive, AB = and P (AB ) = 0. For A a
EE 511 Solutions to Problem Set 1
1. (i) A + A = S and AA = . Therefore, P (A) + P (A) = P (S ) = 1 and P (A) = 1 P (A). (ii) P (A) 0. Therefore, P (A) 1. (iii) + S = S and S = . Therefore, P () + P (S ) = P (S ) and P () = 0. (iv) B = BS = B (A1 + . + An
EE 511 Problem Set 8
1. Let Xt be a W.S.S. random process and let Yt = Xt cos 2fc t and Zt = Xt cos (2fc t + ) where is uniformly distributed in [0, 2 ] and independent of Xt . Show that Yt is wide-sense cyclostationary and Zt is W.S.S. 2. Let Xt (transmi
EE 511 Problem Set 7
Due on 16 Nov 2007
1. Let Xt be the Hilbert transform of the W.S.S. random process Xt . Show that (i) SX (f ) =
t ] = 0, and (iv) for Zt = Xt + j Xt , determine
SX (f ), (ii) SX X (f ) = SXX (f ), (iii) E [Xt X
SZ (f ) in terms of SX
EE 511 Problem Set 6
Due on 5 November 2007
1. White noise of power spectral density N0 /2 is ltered using an ideal low pass lter of
bandwidth B . What is the variance of the output noise process?
2. Suppose Xt is a Wiener process dened for t 0, i.e., a G
EE 511 Problem Set 5
Due on 17 October 2007 1. An experiment has four equally likely outcomes 0, 1, 2, and 3, i. e., S = cfw_0, 1, 2, 3. If a random process Xt is dened as Xt = cos (2st) for all s S , then (a) Sketch all the possible sample functions. (b)
EE 511 Problem Set 4
1. Consider a Gaussian random variable X with E [X ] = 0 and var(X ) = 1. (i) Determine
the moment generating function X (s), and (ii) Using the result of part (i), determine
the Cherno bound for P [X a]. (iii) Use the Chebyshev inequ
EE 511 Problem Set 3
1. The receiver of a communication system receives random variable Y which is dened as
Y = X + N in terms of the input random variable X and the channel noise N . X takes on
the values 1/4 and 1/4 with P [X = 1/4] = 0.6. Let fN (n) de
EE 511 Problem Set 1
Due on 27 Aug 2007
1. From the axioms of probability, derive for any event A, (i) P (A) = 1 P (A), (ii) P (A) 1,
(iii) P () = 0. A collection cfw_Ai n is a partition of the sample space S . (iv) Prove for any
i=1
event B , P (B ) = n