Reference no: EM132397910

Complete the following problems

Chapter 15.

Question 5 A glottal pulse can be modeled by
g ( n ) = { 1 + cos ( Π n T ) n = 0 , ... , T - 10 otherwise
• (a) Sketch g(n).
• (b) Calculate the Discrete-time Fourier Transform (DtFT) of g(n).

Question.7 If we define the short-time spectrum of a signal in terms of its short-time Fourier transform as

S_m (e^jω) = |X_m (e^jω)|2

and we define the short-time autocorrelation of the signal as

R_m ( k ) = ∑∞n = - ∞ x m ( n ) x m ( n + k )
where x(n) = x(n)ω(n-m), then show that for
) = ∑ ∞n = - ∞ x ( n ) ω ( n - m ) e - j ω n

R_m(k) and S_m (ejω) are related as a normal (long-time) Fourier transform pair. In other words, show that S_m (ejω) is the (long-time) Fourier transform of R_m (k), and vice versa.

Chapter 16.

Q1. To mitigate the effects of multipath propagation, we can use an equalizer at the RX. A simple example of an equalizer is the linear zero-forcing equalizer. Noise enhancement is, however, one of the drawbacks to this type of equalizer. Explain the mechanism behind noise enhancement and name an equalizer type where this is less pronounced.

Q3. The "Wiener-Hopf" equation was given by Re_opt = p. For real-valued white noise nm with zero mean and variance σ_n². calculate the correlation matrix R = E(u*u^T) for the following received signals:
(a) u_m = a sin(wm)+ n_m
(b) u_m = bu_m-1 + n_m; a. b ∈ R. b ≠ ± 1.

Q5. An infinite-length ZF equalizer can completely eliminate ISI, as long as the channel transfer function is finite in the transform domain - i.e., Eq. (16.40). Here, we investigate the effect of using a finite-length equalizer to mitigate ISI.

(a) Design a five-tap ZF equalizer for the channel transfer function described in Table 30.1 -i.e., an equalizer that forces the impulse response to be 0 for i = -2,-1,1,2 and 1 for i = 0.
Hint: A 5x5 matrix inversion is involved.

(b) Find the output of the above equalizer and comment on the results.

Table 30.1 Channel transfer function

n	fn
-4	0
-3	0.1
-2	-0.02
-1	0.2
0	1
1	-0.1
2	0.05
3	0.01
4	0

Q7. In general, the MSE equation is a quadratic function of equalizer weights. It is always positive, convex, and forms a hyper parabolic surface. For a two-tap equalizer, the MSE equation takes the form:
A e 21 + B e 1 e 2 + C e 21 + D e 1 + E e 2 + F

with A, B, C, D, E ε R.

For the following data, make a contour plot of the hyperbolic surface formed by the MSE equation:

R = [ 1 0.651 0.651 1 ] p = [ 0.288 0.113 ] T , and σ 2S = 0.3