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EE 67003: Final Exam, Fall 2008
17 December, 2008
Note: You are allowed only a calculator, writing utensils, and two sheets of paper, A4 or 8. x 11 inches, with both sides containing whatever notes you wish to bring into the exam.
Calculators are to be used only for simple arithmetic manipulations, not polynomial factor- ing, etc. Make sure you show all your steps if you want credit for your results.
Good luck and Merry Christmas!
Problem 1 (10)
Problem 2 (30)
Problem 3 (30)
Problem 4 (30)
Problem 5 (10)
Total (110)
Name
- (10 pts.) An LTI DT system has the transfer function
H(z) =
1 + z −^1 − 0. 75 z −^2 1 − 0. 6 z −^1
Find whether this system is minimum-phase. If not, find the transfer function (in simplest form similar to the above expression) of the minimum-phase alternative.
(b) (10 pts.) If this predictor were implemented via a lattice filter, what would be the reflection coefficients K 1 , K 2 , K 3 , K 4?
(c) (10 pts.) What is the correlation coefficient between the forward (f 1 (n)) and backward (g 1 (n)) prediction error at the output of first stage of the lattice filter of (b)?
- A polyphase (two phases) decomposition of a transfer function may be written as
H(z) = H 0 (z 2 ) + z −^1 H 1 (z 2 ).
(a) (10 pts.) Starting from the definition of the z-transform, find the form of the functions H 0 (z) and H 1 (z).
(b) (10 pts.) For the specific case of
H(z) =
1 − 0. 8 z −^1
find the forms of H 0 (z) and H 1 (z).
- Zero-mean white noise (w(n)) with variance 1.0 is the input to a system with transfer function H(z) = 1 + z −^1 + z −^2. (a) (10 pts.) If we call the output of the system x(n), find the power spectral density Γ (^) XX (f ). What is the variance of x(n)?
(b) (10 pts.) Find the values of the minimum mean-squared error prediction coefficients for a second-order AR model of this signal.
(c) (10 pts.) Write the difference equation of a system which will whiten the signal x(n) from (a). Is this system stable? What is the variance of the “re-whitened” signal? Can you reconcile these attributes of the signal and system?
Lattice filter recursion:
K (^) m = am (m)
am− 1 (k) =
am (k) − K (^) m bm (k) 1 − |K (^) m | 2
Characteristic function: φ (^) X (ω) = E[ejωX^ ]
Moment generating function: θX (s) = E[esX^ ]
Joint Gaussian RVs:
p (^) X (x) =
(2π) N/^2 |ΛX | 1 /^2
exp
{ − 1 /2(x − μ (^) X ) T^ ΛX−^1 (x − μ (^) X )
}
φ (^) X (ω) = exp
{ jω T^ μ (^) X − 1 / 2 ω T^ ΛX ω
}
Exponential distribution: p (^) X (x) = θe−θx^ u(x), E[X] = θ −^1 , V ar(X) = θ −^2.
Poisson distributed counts: P (X = k) = e^ −θ (^) θ k k! ,^ E[X] =^ θ,^ V ar(X) =^ θ. Binomially-distributed RV:
P (X = k) = B(k; n, p) =
( n k
) p k^ (1 − p) n−k
Correlation coefficient of X and Y : ρXY = E[(X−μX^ )(Y^ −μY^ )
∗ (^) ] σX σY
Chernoff bound: P(X ≥ a) ≤ e−at^ θX (t), with θX (t) the moment generating function.
Bienayme-Chebyshev: P(X ≥ a) ≤ E[ ϕϕ((aX) )]for ϕ(x) non-negative, symmetric, increasing on x ≥ 0.
Integration by parts:
∫ u dv = uv −
∫ v du
[ a b c d
]− 1
[ d −b −c a
]
(ad − bc)
