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" This is Sampling in frequency = aliasing in time" example ļi, oenz M-1 Let n...
Frequency-domain sampling using Fourier series
Frequency-domain sampling. Consider the following discrete-time signal$$ x(n)= \begin{cases}a^{|n|}, & |n| \leq L \\ 0, & |n|>L\end{cases} $$where \(a=0.95\) and \(L=10\).(a) Compute and plot the signal \(x(n)\).(b) Show that$$ X(\omega)=\sum_{n=-\infty}^{\infty} x(n) e^{-j \omega n}=x(0)+2 \sum_{n-1}^{L} x(n) \cos \omega n $$Plot \(X(\omega)\) by computing it at \(\omega=\pi k / 100, k=0,1, \ldots, 100\).(c) Compute$$ c_{k}=\frac{1}{N} X\left(\frac{2 \pi}{N} K\right), \quad k=0,1, \ldots, N-1 $$for \(N=30\).(d) Determine and plot the signal$$ \tilde{x}(n)=\sum_{k=0}^{N-1} c k e^{j(2 \pi / N) k n} $$What is the...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m,...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3} a. List all numbers of S. b. List all ordered pairs in R. c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)? d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges. e. Give each edge in...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of hea...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved?
Suppose...
1. In this problem, you are going to numerically verify that the Central Limit Theorem is valid e...
1. In this problem, you are going to numerically verify that the Central Limit Theorem is valid even when sampling from non-normal distributions. Suppose that a component has a probability of failure described by a Weibull distri- bution. Let X be the random variable that denotes time until failure; its probability density is: for a 2 0, and zero elsewhere. In this problem, assume k 1.5, 100 a) Simulate drawing a set of N-20 sample values, repeated over M 200...
An m×n array A of real numbers is a Monge array if for all i,j,k, and l such that 1≤i<k≤m and ...
An m×n
array A
of real numbers is a Monge array if for all i,j,k,
and l
such that 1≤i<k≤m
and 1≤j<l≤n
, we have
>A[i,j]+a[k,l]≤A[i,l]+A[k,j]>
In other words, whenever we pick two rows and two columns of a
Monge array and consider the four elements at the intersections of
the rows and columns, the sum of the upper-left and lower-right
elements is less than or equal to the sum of the lower-left and
upper-right elements. For example, the following...
2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation:...
2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation: w = 1. There are exactly N distinct N-th roots of unity. Let w be a primitive root of unity, for example w = exp(2 i/N). Show the following: N, if N divides m k=0 10, otherwise N -1 N wmk 2. Fix and integer N > 2. Let f = (f(0), ..., f(N − 1)) a vector (func- tion) f : [N] →...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diago...
Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diagonal entry is one of the form mii for some i. Swapping rows i and j of the matrix M denotes the following action: we swap the values mik and mjk for k = 1,2, ... , n. Swapping two columns is defined analogously. We say that M is rearrangeable if...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved?
Suppose...
1. In this problem, you are going to numerically verify that the Central Limit Theorem is valid even when sampling from non-normal distributions. Suppose that a component has a probability of failure described by a Weibull distri- bution. Let X be the random variable that denotes time until failure; its probability density is: for a 2 0, and zero elsewhere. In this problem, assume k 1.5, 100 a) Simulate drawing a set of N-20 sample values, repeated over M 200...
An m×n
array A
of real numbers is a Monge array if for all i,j,k,
and l
such that 1≤i<k≤m
and 1≤j<l≤n
, we have
>A[i,j]+a[k,l]≤A[i,l]+A[k,j]>
In other words, whenever we pick two rows and two columns of a
Monge array and consider the four elements at the intersections of
the rows and columns, the sum of the upper-left and lower-right
elements is less than or equal to the sum of the lower-left and
upper-right elements. For example, the following...
2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation: w = 1. There are exactly N distinct N-th roots of unity. Let w be a primitive root of unity, for example w = exp(2 i/N). Show the following: N, if N divides m k=0 10, otherwise N -1 N wmk 2. Fix and integer N > 2. Let f = (f(0), ..., f(N − 1)) a vector (func- tion) f : [N] →...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
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