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This is our joint work with colleagues from TU Braunschweig. Prof. H. G. Matthies had an excellent idea to develop a Bayesian surrogate formula for updating not probability densities (like in classical Bayesian formula), but PCE coefficients of the given random variable. Bojana Rosic implemented the linear case. I (with help of Elmar Zander) implemented non-linear case. Later on Elmar significantly simplified the algorithm.
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To give the complete list of Uq(sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.
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Linear Bayesian update surrogate for updating PCE coefficientsAlexander Litvinenko
This is our joint work with colleagues from TU Braunschweig. Prof. H. G. Matthies had an excellent idea to develop a Bayesian surrogate formula for updating not probability densities (like in classical Bayesian formula), but PCE coefficients of the given random variable. Bojana Rosic implemented the linear case. I (with help of Elmar Zander) implemented non-linear case. Later on Elmar significantly simplified the algorithm.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
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To give the complete list of Uq(sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.
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1. Signals and Systems Assignment Help
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2. Discrete-Time Sampling
Recommended Problems
P19.1
Consider Figures P19.1-1 and P19.1-2, and determine X(Q), P(Q), x,[n), and
X,(Q).
P19.2
x[n] has a transform X(Q). Determine in terms of X(Q) the transforms
of the signals in parts (a) and (b).
(a) x[n/2]1, n even, (a)x[n]=0,) n odd
(b) (b) xd[n] = x[2n], i.e., xd[n] is x[n] decimated.
Problem
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3. P19-2
(c) If X(Q) is as given in Figure P19.2, sketch X,(Q) and Xd(Q) for parts (a)
and (b).
P19.3
Consider the system in Figure P19.3-1
(a) If p[n] is given by Figure P19.3-2 sketch P(Q) for N =1, 2, and
L, an arbitrary integer.
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4. (b) For each of the discrete-time spectra in Figures P19.3-3 and P19.3-4, determine
the maximum sampling period N such that x[n] is reconstructible from its samples
x,[n] using an ideal lowpass filter
In each case, specify the associated cutoff frequencies for the lowpass
filter.
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5. P19.4
Suppose the signal x(t) is processed as shown in Figure P19.4-1.
(a) The system in Figure P19.4-1 can be replaced by the one in Figure P19.4-2. Find
T1 .
(b) Let X(w) be given as in Figure P19.4-3. Find X(Q) and Y(Q).
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6. P19.5
As discussed in Section 8.7 and illustrated in Figure 8.40 of the text as well as in
Figure P19.5-1 below, the procedure for interpolation or upsampling by an integer
factor N can be thought of as the cascade of two operations. The first system,
system A, corresponds to inserting (N - 1) zero sequence values between each
sequence value of x[n], so that
For exact bandlimited interpolation, H(Q) is an ideal lowpass filter.
(a) Determine whether system A is linear.
(b) Determine whether system A is time-invariant.
(c) For Xd(2) as sketched in Figure P19.5-2, with N = 3, sketch X(Q).
(d) For N = 3, Xd(Q) as in Figure P19.5-2, and H(Q) appropriately chosen for exact
bandlimited interpolation, sketch X(Q).
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7. P19.6
Consider the sampling systems in Figure P19.6-
1.
Let x(t) andX(w) be given as in Figure P19.6-2.
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8. Let y(t) and Y(w) be given as in Figure P19.6-3.
(a) Draw x,(t) and Y,(w).
(b) Find X,(w) and y,(t).
(c) Is y,(t) periodic? Does Y,(w) reflect this property? Optional Problems
P19.7 Consider a discrete-time sequence x[n] from which we form two new
sequences, x,[n] and xd[n], where x,[n] corresponds to sampling x[n] with
sampling period 2 and xd[n] corresponds to decimatingx[n] by a factor of 2, so
that
(a) If x[n] is as illustrated in Figure P19.7-1, sketch the sequences x,[n] and xd[n].
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9. (b) If X(Q) is as shown in Figure P19.7-2, sketch X,(Q) and Xd(Q).
Consider the system in Figure P19.8-1, where X(Q) is as shown in Figure P19.8-
2.
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10. There is a range of values for N such that, with an appropriate choice for H(Q), y[n] will
equal x[n]. For each allowable positive integer value of N,
(a) Draw X,(Q).
(b) Find an appropriate H(Q) such that y[n] = x[n].
P19.9 Consider the system with input x[n] and output y[n] related by
(a) For the sequence x[n] in Figure P19.9, sketch y[n].
(b) Express the system as a combination of filtering and decimation. 6
P19.10
Consider the system in Figure P19.10, where
Find a constraint on h[n] such that y[kN] = x[k], for all k.
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12. x[n] is given by
x[n] = (-1)" = e'"
Hence, the Fourier transform of x[n] is
Now p[n] can be written as
Hence, its Fourier transform is given by
It is clear that x,[n] = p[n]. Hence
X,(Q) = P(Q)
S19.2
(a) x,[n] is x[n] "stretched" by interspersing with zeros, as indicated
in Figure S19.2-1.
Solutions
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14. 'X(Q/2) is indicated in Figure S19.2-3. Therefore, is as shown in Figure S19.2-4.
(a) For N = 1, p[n] = 1. Hence
as shown in Figure S19.3-1.
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15. For N = 2,
Hence
shown in Figure S19.3-2.
ForN =L
Hence
shown in Figure S19.3-3.
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16. (b) X,(Q), the spectrum of x,[n], is proportional to the periodic convolution of P(Q) and
X(Q). Consequently, with P(Q) as indicated in Figure S19.3-3 and X(Q) as indicated in
Figure S19.3-4, X,(Q) is shown in Figure S19.3-5. In order that x[n] be reconstructible
from x,[n] using an ideal lowpass filter, aliasing must be avoided, which requires that
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17. (i) Qm = 37r/10. Therefore, to avoid aliasing,
Since N must be an integer, we require that N s 3. For N = 3, the cutoff frequency of the
lowpass filter must be greater than 3r/10 and less than
(ii) Q, = 37r/ 5 . To avoid aliasing,
Since N must be a positive integer, this requires that N = 1, i.e., x[n] cannot be
sampled.
S19.4
(a) The sampling period Ti is 3 ms for the system in Figure P19.4-2 to be
equivalent to the one in Figure P19.4-1.
(b) (b) X(Q) is sketched in Figure S19.4-1.
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18. From the result of part (a), Y(Q) is as shown in Figure S19.4-2.
S19.5
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19. So system A is linear. (b) Take xd,[n] as shown in Figure S19.5-1, with
N = 4.
Then x,,[n] is as shown in Figure S19.5-2.
Take Xd2[n] = Xd,[n + 1]. Then x,,[n] is as shown in Figure S19.5-3.
Hence, system A is not time-invariant.
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20. (d) X(Q) is as shown in Figure S19.5-5 for exact bandlimited interpolation.
(a) x,(t) is sketched in Figure S19.6-1, and Y,(w) is sketched in Figure
S19.6-2.
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21. (b) X,(w) is sketched in Figure S19.6-3, and y,(t) is sketched in Figure S19.6-4.
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22. (c) Yes, y,(t) is periodic and this is reflected in Y,(w), which contains
impulses.
Solutions to Optional Problems
S19.7
This is sketched in Figure S19.7-1.
Similarly, xd[n] = x[2n], as shown in Figure S19.7-2.
(b) X,(Q) is obtained as follows:
and
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23. which are shown in Figures S19.7-3 and S19.7-4. See Problem
P19.2(b).
S19.8 (a) We know that the Fourier transform of p[n] is given by
Aliasing will be just avoided when the sampled spectra will look as shown in
Figure S19.8-1.
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24. Hence, we require that
Consequently, aliasing is avoided if 1 S N 3. X,(Q) for N = 1, 2, and 3 are shown in
Figure S19.8-2.
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25. (b) An appropriate H(Q) is shown in Figure S19.8-3.
Hence, y[n] can be sketched as in Figure S19.9.
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26. (b) If
z[n] = [x[nj + x[n + 11 + x[n + 2]], for all n
and
y[n] = z[3n],
we have expressed the processing as a combination of filtering and decimati
If h[O] = 1 and h[n] = 0 for n = kN, k # 0, it is easy to see that the samples
xo[n] that came from x[n] will be unaffected. Hence,
y[kN] = x[k], for all k
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