Convolution

Question 1

if there is a linear system p(t) = q(t), what is the differentiation of p(t) equal to

Answer 1

d/dt[p(t)] = d/dt[q(t)]

Question 2

if there is another linear system f(t) = y(t), what is the sum of f(t) and p(t) equal to

Answer 2

f(t) +p(t) = q(t) + y(t)

Question 3

what does 5f(t) + 3p(t) equal

Answer 3

5f(t) + 3p(t) = 5y(t) + 3q(t)

Question 4

what does a basic unit step function H(t) look like on a graph

Answer 4

• the line stays at H(t) = 0 for t < 0
• then when t = 0 it instantly shoots up to H(t) = 1
• and the line remains at H(t) = 1 forever

Question 5

what does a delta / impulse function δ(t) look like on a graph

Answer 5

a vertical spike at a given x-axis value

Question 6

what is the area of an impulse function spike

Answer 6

1

Question 7

what is the derivative of a step function or step response

Answer 7

an impulse function or impulse response

Question 8

what is the response of a function in this case

Answer 8

• the less ideal, more realistic version of the function’s behaviour
• because step and impulse functions act in very short time periods they are ‘modelled’ to have instantaneous reactions
• but step / impulse responses show the more ‘capacitor-like’ behaviour of these movements

Question 9

if you have an impulse function δ(t-b), where is the spike

Answer 9

at t = b

Question 10

if you have an arbitrary function f(t), what is the product of δ(t-b)*f(t)

Answer 10

δ(t-b)*f(t) = f(b)

Question 11

what is the important trick to remember when evaluating whether the integration of the product of an arbitrary function and an impulse function is automatically 0

Answer 11

• work out the value of t in the impulse function bracket to get δ(0)
• if this value of t does not lie within the integration boundaries, the integration is automatically 0

Question 12

if the value of t lies within the integration boundaries, what is another trick for making the integration easy

Answer 12

• simply input that value of t into the arbitrary function and thats your answer
• no actual integration is needed

Question 13

for finding the step response to a differential equation that equals f(t), what is the first step

Answer 13

set f(t) = 1 and find the general solution

Question 14

what is the second step

Answer 14

• set the boundary conditions y(0) = 0 and solve for the constant
• if its a second order system set y’(0) = 0 too

Question 15

what is the third step

Answer 15

• write in the curly brackets that y = 0 for t < 0

- and y = the solution found for t => 0

Question 16

if you have a system input composed of impulses f(t) = 3σ(t - 1) + 4σ(t - 2), how many impulse responses are there and at what times

Answer 16

• 3 impulse responses at t = 1

- 4 impulse responses at t = 2

Question 17

what is the convolution integral formula

Answer 17

y(t) = int[g(t - T)*f(T) dt] from limits t to -∞

Question 18

what is the difference between t and T (actually tau but the symbol doesn’t exist apparently)

Answer 18

• t is the time as it relates to the output of the system y(t)
• T is the time as it relates to the input of the system f(T)
• also t is not the integration variable, T is

Question 19

where do you get the value for f(T)

Answer 19

• it is the value(s) in the curly brackets for the input

- if those values are a function themselves like sin(t), replace the t with a T when adding it to the integral

Question 20

what do you do if f(t) is a discontinuous function

Answer 20

• work out the convolution integral for each section of the function
• piece them together in the end

Question 21

what is the overall step by step process for convolution when starting with a differential equation

Answer 21

• get the step response by equating it to 1 and using y(0) = y’(0) = 0
• differentiate the step response to get the impulse response g(t)
• IF the step response is the input, use the convolution integral using the step and impulse response
• if the input is simply given, use that in the convolution integral instead
• the answer for both is the output y(t)

Question 22

if you get a question like “find the impulse response of … = f(t) and hence find the output when input f(t) = H(t)*e^t, what is the important thing to remember about what this means

Answer 22

• it means the required input f(T) in the convolution integral is the e^t part (actually e^T in the integration)
• so if it just says f(t) = H(t), your input is just 1

Question 23

what is the name of the other method we can use for evaluating convolution integrals when g(t) is too complicated

Answer 23

the alternative convolution integral

Question 24

how do you set up the alternative convolution integral (this makes no sense as to why it works but anyway)

Answer 24

• set T(2) = t - T(1)
• T1 is the tau we’ve been using this whole time and T2 is a “‘new’ tau but has no subscripts, they look exactly the same
• sub into the convolution integral so f(T) = t - T and g(t - T) = g(T)
• giving y(t) = int[f(t - T)*g(T) dT] from limits t to 0

Question 25

what is a spatial convolution

Answer 25

a system with a time varying input and output

Question 26

for the string of length L that has a point load on it a distance a from the datum, what does the function g(x,a) represent

Answer 26

• g(x,a) represents the displacement at position x

- due to the unit load at position a

Question 27

for the point load F on a string example, what would the diagram you draw need to look like for you even get started

Answer 27

• the two angles made on either side are labelled to be different
• T1 runs through one side of the string while T2 runs through the other
• the vertical distance the string reaches is labelled d
• the distance where the ‘peak’ occurs is at distance ‘a’ from the datum
• the point force F is assumed to = 1 unless specified

Question 28

when resolving forces horizontally, what do you usually figure out

Answer 28

• T1 = T2 = T

- because of small angle approximations

Question 29

when resolving vertically what you do usually figure out

Answer 29

• if Tsin(x1) + Tsin(x2) = 1
• dividing by cos(x1) or cos(x2) gives T(tan(x1) + tan(x2)) = 1
• because of small angle approximations

Question 30

how do you convert that equation into a formula for d

Answer 30

• write the tans in terms of the dimensions of the diagram

- rearrange for d = something

Question 31

how does this formula relate to g(x,a)

Answer 31

• you can write equations for 2 straight segments of the g(x,a) function
• one when x < a and another then x > a
• x = distance from datum btw, different from the angles

Question 32

what tends to be the input function in these kinds of questions

Answer 32

• a continuous load on the rope (like its weight or something)
• so f(t) tends to just be a constant