3. A causal LTI system has impulse response: \[ h[n]=n\left(\frac{1}{3}\right)^{n} u[n]+\left(-\frac{1}{4}\right)^{n} u[n] . \] For this system determine: - The system function \( H(z) \), including t
Answers
To determine the system function \(H(z)\) for the given impulse response \(h[n] = n\left(\frac{1}{3}\right)^{n} u[n]+\left(-\frac{1}{4}\right)^{n} u[n]\), we need to take the Z-transform of the impulse response.
The Z-transform is defined as:
\[H(z) = \sum_{n=-\infty}^{\infty} h[n]z^{-n}\]
Let's compute the Z-transform step by step:
1. Z-transform of the first term, \(n\left(\frac{1}{3}\right)^{n} u[n]\):
The Z-transform of \(n\left(\frac{1}{3}\right)^{n} u[n]\) can be found using the Z-transform properties, specifically the time-shifting property and the Z-transform of \(n\cdot a^n\) sequence, where \(a\) is a constant.
The Z-transform of \(n\left(\frac{1}{3}\right)^{n} u[n]\) is given by:
\[\mathcal{Z}\{n\left(\frac{1}{3}\right)^{n} u[n]\} = -z \frac{d}{dz}\left(\frac{1}{1-\frac{1}{3}z^{-1}}\right)\]
2. Z-transform of the second term, \(\left(-\frac{1}{4}\right)^{n} u[n]\):
The Z-transform of \(\left(-\frac{1}{4}\right)^{n} u[n]\) can be directly computed using the formula for the Z-transform of \(a^n u[n]\), where \(a\) is a constant.
The Z-transform of \(\left(-\frac{1}{4}\right)^{n} u[n]\) is given by:
\[\mathcal{Z}\{\left(-\frac{1}{4}\right)^{n} u[n]\} = \frac{1}{1+\frac{1}{4}z^{-1}}\]
3. Combining the Z-transforms:
Applying the Z-transforms to the respective terms and combining them, we get:
\[H(z) = -z \frac{d}{dz}\left(\frac{1}{1-\frac{1}{3}z^{-1}}\right) + \frac{1}{1+\frac{1}{4}z^{-1}}\]
Simplifying further, we can obtain the final expression for the system function \(H(z)\).
Visit here to learn more about impulse response brainly.com/question/30426431
#SPJ11
Related Questions
how many square miles of land did the railroad companies get for each mile of track they laid?
Answers
The amount of land the railroad companies received for each mile of track they laid varied depending on the specific circumstances and agreements.
However, a common benchmark used during the construction of railroads in the United States was the granting of land through the Homestead Act of 1862. Under this act, railroad companies were granted approximately 6,400 acres (10 square miles) of land for every mile of track laid. This land was typically located alongside the tracks and served as an incentive for the companies to expand and develop the rail network across the country.
Learn more about square miles here: brainly.com/question/24759478
#SPJ11
The number of milligrams D (ht) of a certain drug that is in a patient's bloodstream h hours after the drug is injected is given by the following function.
D(h) = 25e -0. 4
When the number of milligrams reaches 6, the drug is to be injected again. How much time is needed between injections?
Round your answer to the nearest tenth, and do not round any intermediate computations.
Answers
The time is needed between injections is 3.6 hours, i.e., the drug is to be injected again when the number of milligrams reaches 6 mg.
We have the exponential function of number of milligrams D (ht) of a certain drug that is in a patient's bloodstream h hours after the drug is injected is
\(D(h)=25 {e}^{ - 0.4 h}\)
We have to solve for h (the numbers of hours) that would have passed when the D(h) (the amount of medication in the patient's bloodstream) equals 6 mg in order to know when the patient needs to be injected again.
\(6 = 25 {e}^{ - 0.4h} \)
\( \frac{6}{25} = \frac{25}{25} {e}^{ - 0.4h} \)
\(0.24= {e}^{ - 0.4h} \)
Taking logarithm both sides of above equation , we get,
\( \ln(0.24) = \ln( {e}^{ - 0.4h)} \)
Using the properties of natural logarithm,
\( \ln(0.24) = - 0.4h\)
\( - 1.427116356 = - 0.4h\)
\(h = \frac{1.42711635}{0.4} = 3.56779089\)
=> h = 3. 6
So, after 3.6 hours, the patient needs to be injected again.
To learn more about exponential function, refer:
https://brainly.com/question/2456547
#SPJ4
if all else remains constant and the values of two variables move in the same direction it indicates a
Answers
If all else remains constant and the values of two variables move in the same direction, it indicates a positive correlation.
What is correlation?Correlation is a statistical measure that expresses the relationship between two variables. The correlation coefficient, which ranges from -1 to +1, is used to quantify the correlation.
A positive correlation is when the values of the variables move in the same direction. A correlation of +1 indicates that the variables are perfectly positively correlated, implying that as one variable increases, the other also increases.
A correlation of -1 indicates that the variables are perfectly negatively correlated, implying that as one variable increases, the other decreases. A correlation of zero indicates that there is no correlation between the variables.
Hence a positive correlation exists when the values of two variables move in the same direction while all other factors remain constant.
To learn more about correlation visit: https://brainly.com/question/29972433
#SPJ11
When would you use a pictograph?
more than one answer
A. if the data is numerical
B. to compare change over time
C. to save space for a large set of data
D. if the data is categorical
Answers
Answer:
The answer to your problem is, D. If the data is categorical
Step-by-step explanation:
What a pictograph is:
Remember that a pictographs are less useful for comparing change over time or for presenting large sets of numerical data, as they can become cluttered and difficult to read.
Here is a little example, if you want to show how the population of a city has changed over the years, a line graph or a bar graph would be a better choice than a pictograph
Thus the answer to your problem is, D. If the data is categorical
evaluate to 212 base 3 - 121 base 3 + 222 base 3
Answers
Answer:
212 bp^3 (base pairs cubed) - 121 bp^3 (base pairs cubed) + 222 bp^3 (base pairs cubed) = 313 bp^3 (base pairs cubed
Step-by-step explanation:
True or False?
-2.08 would be just to the right of -2 on the number line.
Answers
A rectangle has an area of 45x2 - 42x - 48 and a
width of 5x - 8. What is the
length of the
rectangle?
To find the length of the rectangle, use
45x- 42x - 48 as the
v and 5x 8 as
the
Answers
Therefore the length of rectangle is 9x + 6.
The area of a rectangle (A) is the product of its length 'a' and width or breadth 'b'. So, Area of Rectangle = (a × b) square units.
Quadratic factorization using splitting of middle term : In this method splitting of middle term in to two factors. In Quadratic Factorization using Splitting of Middle Term which is x term is the sum of two factors and product equal to last term.
Given that a rectangle has an area of 45x^{2} - 42x - 48 and a
width of 5x - 8.
Therefore
Area of rectangle = A = 45x^{2} - 42x - 48
Width of rectangle = b = 5x - 8
Length of rectangle = a
We know that area of rectangle = a × b
⇒ 45x^{2} - 42x - 48 = a × (5x - 8)
⇒ 3(15x^{2} - 14x - 16) = a × (5x - 8)
⇒ 3(15x^{2} - 14x - 16) = a × (5x - 8)
⇒ 3(15x^{2} -24x + 10x - 16) = a × (5x - 8)
⇒ 3[3x(5x - 8) + 2(5x - 8)] = a × (5x - 8)
⇒ 3(5x - 8)(3x + 2) = a × (5x - 8)
⇒ 3(3x + 2) = a
⇒ 9x + 6 = a
Hence the length of rectangle = 9x + 6
To know more about splitting of middle term visit:
brainly.com/question/13874704
#SPJ1
Use proportions or scale factors to determine the missing side length.
Answers
eeee helppp !!!!!
i got like 5 moreee leftt !!
Answers
Answer
4: 22 degrees
3: (9,2)
Step-by-step explanation:
30,098 in standard form
30,098 to four significant figures
Answers
Significant figures = 30100
how would the line look
Answers
the y is 2, so you point the 2 in the y’s place.
next you go down 4 points, this is because the 4 is negative. if the number is negative you go down, if it’s positive it goes up!
next, you move that point one place to the right because there’s a invisible 1 under the 4. (-4/1)
here’s how ur graph should look like! :)
Which ordered pair is the solution to the system of linear equations y= -7x+2 and. y=9x - 14
A (-5,1)
B (1,-5)
C (5, -1)
D (-1,5P
Answers
The answer is (1,-5), If you put those two on a graph, that point is where they collide.
A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 14°52'. When the boat stops, the angle of depression is 45°10'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.
Answers
The boat's travel from when it was first noticed until it stopped is 554.89 ft.
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of a right-angle triangle.
We have:
A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly.
From the given information we can draw a right-angle triangle.
14°52' = 14 + 0.86 = 14.86 degree
45°10' = 45 + 0.16 = 45.16 degree
In the right-angle triangle ACD
tan14.86 = 200/AC
AC = 753.772 ft
In the right-angle triangle DBC:
tan45.16 = 200/CB
CB = 198.886 ft
AB = AC - CB
AB = 753.772 ft - 198.886 ft
AB = 554.886 ≈ 554.89 ft
Thus, the boat's travel from when it was first noticed until it stopped is 554.89 ft.
Learn more about trigonometry here:
brainly.com/question/26719838
#SPJ1
help pls by pst sunday december 4, 2022 12:00 images below
Answers
X+8 then you need to divide that all by x as it’s how many TIMES.
(X+8)/x
(Let R, denotes the reflection on y-axis and R2 denotes a rotation of 90° about the center O(0.0), then for any point A(3. 4). find RR, (A).)
Answers
The solution of RR(A) is (-3, -4).
What is the reflection and rotation on a point? Reflection and rotation are two types of transformations that can be applied to a point. Reflection is when a point is mirrored across a line of symmetry. The line of symmetry acts as a mirror and the point is flipped across it. This produces a reflection in which the point is now at the exact opposite side of the line of symmetry. Rotation is when a point is spun around a fixed axis. The point can be rotated in either a clockwise or counter-clockwise direction. When the point is rotated, it is still in the same location, only rotated around the axis. Both of these transformations can be used to move a point in a certain direction or to change the shape of an object. Reflections and rotations can be used to create symmetry in a shape or to move a point to a different location. They are both useful transformations that can be used to manipulate the position or shape of an object.To learn more about reflection on y-axis refer to:
https://brainly.com/question/17686579
#SPJ1
If mXWZ = 90, what is x?
Answers
Answer:
x=11
Step-by-step explanation:
To Solve this equation, you need to understand the rules of complementary angles. In this case, XWY and YWZ make up XWZ. In this case, we would add them
5x+5+2x+8=90
7x+13=90
7x=77
x=11
How do you write 2.3 x 103 in standard form?
Answers
Answer:
2300
Step-by-step explanation:
\(2.3 \times {10}^{3} \\ \\ = 2.3 \times 1000 \\ \\ = 2300\)
Zahra plans to draw a triangle with angles measuring 50 degrees, 60 degrees, and 70 degrees. Which is true about Zahra’s plan?
Zahra cannot make a triangle with these angle measures.
Zahra can only draw one unique triangle with these angle measures.
Zahra can make exactly two triangles with these angle measures.
Zahra can make more than one triangle with these angle measures.
Answers
Answer:
Zahra cannot make a triangle with these angle measures.
Step-by-step explanation:
A passenger flies on a heading of N40W with an airspeed of 240 km/h. His actual
ground velocity is 250 km/h at a bearing of N60E. Calculate the speed and direction of
the wind.
Answers
A passenger flies on a heading of N40W with an airspeed of 240 km/h. His actual ground velocity is 250 km/h at a bearing of N60E. the speed and direction of the wind are V=245.15kw/h at N83.46E. This is further explained below.
What is the speed and direction of the wind.?Generally, the equation for the cosine rule is mathematically given as
a2 = b2 + c2 – 2bc cos ∠x.
Therefore
V^2 = (240^2 + 250^2) – (2(240*250)* cos 100)
V^2=140937.7813
V=375.42kw/h
In conclusion, the sine rule is mathematically given as
Sin x/240=sin60/375.42kw/h
Therefore
sin x=sin100*250/375.42kw/h
x=40.981
Hence
\(\theta= 180-2*48.27\)
Direction N83.46E
Read more about Speed
https://brainly.com/question/4931057
#SPJ1
Nick has a $45 Starbucks gift card. He purchased the same latte six times and has $18 left on the card. What was the price of each latte? What does the variable represent in this problem?
Answers
Answer:
The price of each latte was $4.5.
Step-by-step explanation:
6x + 18 = 45
6x = 45 - 18
6x = 27
x = 27/6
x = 4.5
The variable represents the cost of each latte.
the mean of the underlying distribution for chi-square is a. always 1.00 if the null is true b. always 0.00 if the null is true c. equal to the df for that particular test d. cannot be determined without a critical values table
Answers
The mean of the underlying distribution for the chi-square is equal to the degrees of freedom (df) for that particular test.
The chi-square distribution is characterized by its degrees of freedom, which determine the shape of the distribution. The mean of the chi-square distribution is dependent on the degrees of freedom. The degrees of freedom represent the number of independent pieces of information used to estimate a parameter. In the case of the chi-square test, it is the number of categories or cells in a contingency table minus 1.
Therefore, the mean of the chi-square distribution is equal to the degrees of freedom for that specific test. Option (c) is the correct choice. The mean of the underlying distribution for the chi-square is not always 1.00 if the null is true (option a) or always 0.00 if the null is true (option b). Additionally, the mean cannot be determined without a critical values table (option d) since it is directly related to the degrees of freedom.
To learn more about chi-square click here: brainly.com/question/31871685
#SPJ11
14) Which example best shows how a community issue may affect a small pottery manufacturing business?
Question 14 options:
A small fire in the kiln caused several people to be slightly sick from the fumes.
An employee was injured when a shelf of pottery broke and crashed down on top of her.
The pottery business was cited for improperly training employees on kiln use after an injury.
Inflation in the town has caused unemployment, so few people buy items like decorative pottery.
Answers
The "best-example" which shows how "community-issue" affects a small pottery manufacturing business is (d) Inflation in the town has caused unemployment, so few people buy items like decorative pottery.
The Inflation and unemployment causes negative impact the spending power of the local community, which causes a decrease in demand for non-essential items like decorative pottery.
So, as a result, the small pottery business experiences a decrease in sales, revenue, and potentially even have to lay off employees. This is an example of how macroeconomic factors can have a negative effect on small businesses in the community.
The correct option is (d).
Learn more about Inflation here
https://brainly.com/question/30724318
#SPJ1
The given question is incomplete, the complete question is
Which example best shows how a community issue may affect a small pottery manufacturing business?
(a) A small fire in the kiln caused several people to be slightly sick from the fumes.
(b) An employee was injured when a shelf of pottery broke and crashed down on top of her.
(c) The pottery business was cited for improperly training employees on kiln use after an injury.
(d) Inflation in the town has caused unemployment, so few people buy items like decorative pottery.
Mark walk miles in 5 hours how many miles did Mark walk in 1 hour?
Answers
3x+1/x^(2)-1=2/x-2+x/x-1
Answers
Answer:
x = 0 and 3 other real roots. See below.
Step-by-step explanation:
This is what you wrote:
\( 3x + \dfrac{1}{x^2} - 1 = \dfrac{2}{x} - 2 + \dfrac{x}{x} - 1 \)
I don't think that is what you meant to write.
I think you meant to write this:
\( 3x + \dfrac{1}{x^2 - 1} = \dfrac{2}{x - 2} + \dfrac{x}{x - 1} \)
I'm going to answer the second equation because that is what I think you meant.
\( 3x + \dfrac{1}{(x - 1)(x + 1)} = \dfrac{2}{x - 2} + \dfrac{x}{x - 1} \)
\( 3x(x - 1)(x + 1)(x - 2) + (x - 2) = 2(x - 1)(x + 1) + x(x + 1)(x - 2) \)
\( (3x^2 - 6x)(x^2 - 1) + x - 2 = 2x^2 - 2 + x(x^2 - x - 2) \)
\( 3x^4 - 3x^2 - 6x^3 + 6x + x - 2 = 2x^2 - 2 + x^3 - x^2 - 2x \)
\( 3x^4 - 7x^3 - 4x^2 + 9x = 0 \)
\( x(3x^3 - 7x^2 - 4x + 9) = 0 \)
\( f(x) = 3x^3 - 7x^2 - 4x + 9 \)
\( f(-3) = -123 \)
\( f(-2) = -35 \)
\( f(-1) = 3 \)
\( f(0) = 9 \)
\( f(1) = 1 \)
\( f(2) = -3 \)
\( f(3) = 15 \)
One root is x = 0.
There is a root between x = -2 and x = -1.
There is a root between x = 1 and x = 2.
There is a root between x = 2 and x = 3.
Plot the graph of f(x) = 3x^3 - 7x^2 - 4x + 9 and try to read the other three roots.
The graph of y = f(x) is transformed to the graph of y = 2f(x - 1).
Describe fully the two single transformations which have been combined to give the resulting transformation.
Answers
Answer:
Stretches vertically by 2, shifts right by 1
Step-by-step explanation:
changes to y are direct, but you have to take the inverse for changes to x. This is because for a given value of y, with x-1, you would have to add one to x to make it the same value(to get the same point before the transformation)
A boy's weight increased by 15% between his fifteenth and sixteenth birthdays. If he weighed 55kg on his fifteenth birthday, what did he weigh on his sixteenth birthday?
Answers
Answer:
63.25 KG
Step-by-step explanation:
Convert 15% to a decimal, then add 1 and multiply 15.
\(55 \times (1 + 0.15) = 63.25\)
What is the slope of the following line?
Answers
The Sugar Sweet Company is going to transport its sugar to market. It will cost $6500 to rent trucks, and it will cost an additional $175 for each ton of sugar transported.
let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported. Write an equation relating C to S. Then use this equation to find the total cost to transport 16 tons of sugar.
Answers
The equation is C = 6500 + 175S
The total cost to transport 16 tons of sugar is $9300
From the question, we have
C = total cost
S represent the amount of sugar (in tons) transported.
C = 6500 + 175S
total cost to transport 16 tons of sugar = 6500 + 175(16)
= 6500 + 2800
= $9300
The total cost to transport 16 tons of sugar is $9300.
Addition:
The phrase "the addition" refers to combining two or more numbers. Adding two numbers is indicated by the plus sign (+), therefore adding three is written as three plus three. Additionally, the number of times the plus symbol (+) is used is up to you. For example, 3 + 3 + 3 + 3. Mathematical addition is a crucial component of the learning curriculum for kids. In primary school, students learn simple addition sums such as one-digit facts, Math addition sums, and double-digit Math addition sums. One of the most fundamental and interesting Math concepts for elementary school children is addition sums. Math is a fascinating subject, and children first learn basic mathematical concepts like adding numbers in their early years.
To learn more about addition visit: https://brainly.com/question/680614
#SPJ1
consider a symmetric matrix a. if the vector v is in the image of a and w is in the kernel of a, is v necessarily orthogonal to w?
Answers
Yes, the vector v is necessarily orthogonal to w if v is in the image of the symmetric matrix A and w is in the kernel of A.
Yes, the vector v is necessarily orthogonal to w if v is in the image of the symmetric matrix A and w is in the kernel of A. This is due to the fact that for any symmetric matrix A, the image (column space) and kernel (null space) are orthogonal subspaces. Since v is in the image of A and w is in the kernel of A, their dot product must be zero, indicating orthogonality.
To know more about Symmetric matrix visit:
https://brainly.com/question/14405062
#SPJ11
Which angle is formed by EA−→
and EG−→−?
Select all that apply.
Answers
Answer:
Step-by-step explanation:
\(\angle 2,\angle AEG, \angle GEB\)