2 convolution

Question 1

How do we know we have an EVEN function?

Answer 1

• All cosine functions are even
• Evaluated at at a positive f value, the result is mirrored in the y axis and is identical.
• f(x) = f(-x)
• delta function is an even function sigma(x) = sigma(-x)

Question 2

How do we know we have an ODD function?

Answer 2

f(x) = -f(-x)

-all sine functions are ODD

Question 3

When do we know a signal is memoryless and give an example?

Answer 3

y(t) = (2 + sin t)x(t) is memoryless because y(t) depends only on x(t) and not prior values of x(t)

Question 4

How to tell if a signal is time invariant?

Answer 4

y(t) = (2 + sin t)x(t) = T[x(t)], T[x(t - T0)] = (2 + sin t)x(t - T0 )
9y(t-T0)=(2+sin(t- T0))x(t-T0) Therefore, y(t) = (2 + sin t)x(t) is not time-invariant.

Question 5

An examples for when a function is not casual?

Answer 5

y(t) = x(2t),
y(1) = x(2)
The value of y(-) at time = 1 depends on x(-) at a future time = 2. Therefore,
y(t) = x(2t) is not causal.

Question 6

An example of a signal invertible?

Answer 6

y(t) = x(2t) is invertible; 
x(t) = y(t/2).

Question 7

An example when a signal is not invertible?

Answer 7

y[n] = E _.x[k] is not invertible. Summation is not generally an invertible
operation.

Question 8

2 examples of a stable system?

Answer 8

1) If Ix(t) I < M, Iy(t) I < (2 + sin t)M. Therefore, y(t) = (2 + sin t)x(t) is stable.
2) If |x(t)| < M, |x(2t)I < M and ly(t)| < M. Therefore, y(t) = x(2t) is stable.

Question 9

An example of not stable?

Answer 9

-If |x[k]| 5 M, ly[n]j 5 M - E_,, which is unbounded. Therefore, y[n] = E”Lx[k] is not stable.

Question 10

Define the unit step sequence?

Answer 10

u[n] =

is 0 for negative values of n and 1 for positive values of n or values of n = 0

Question 11

Define the unit impulse sequence?

delta[n]

Answer 11

delta[n]

is 0 for all cases for its argument except when n =0 where the amplitude is 1

Question 12

Define the first difference (the relationship between the unit step and unit impulse):

Answer 12

delta[n] = u[n]-u[n-1]

Question 13

How can we rewrite the unit step?

Answer 13

u[n] = n(sun)m=-inf delta[m]

-unit step can be thought of as a summation of unit impulses

Question 14

Whats time invariance?

Answer 14

-can be applied to both continuous and discrete time
- For any input and output, the output time shifts by the same amount.
continuous time:
input = x(t) output = y(t)
input = x(t-t(0)) output = y(t-t(0))

Discrete time:
input = x[n] output = y[n]
input = x[n-n(0)] output = y[n-n(0)]

Question 15

Whats linearity?

Answer 15

-A set of inputs linear combination should equal a set of outputs of linear combinations

Question 16

What are 2 LTI systems:

enables us to

Answer 16

• Delayed impulses leads to CONVOLUTION

- Complex exponentials leads to FOURIER ANALYSIS

Question 17

Convolution approach:

Answer 17

• Decompose an arbitrary sequence into a linear sequence of delays
• By reflecting x[n] about the origin, shifting, multiplying, and adding, we see that y[n] = x[n] * h[n]

Question 18

Explain how convolution is commutative:

Answer 18

• We can convolve x with h or h with x.
• The order in which this is done doest affect the output result
• h(t)x(t) = x(t)h(t)

Question 19

Whats the convolution associative property?

Answer 19

• Doesnt matter how we group the terms together

- [h1h2]x(t) = h1[h2x(t)]

Question 20

Whats the distributive property of convolution?

Answer 20

• x1[h1h2] = [x1h1] + [x1h2]

Question 21

Define causality?

Answer 21

• The system can’t anticipate the output (can’t anticipate some time later if the inputs are gonna change from each other)
• If 2 inputs are identical up until some time, then so will be the outputs
• A LTI system is causal if the system remains at rest until there’s an input

Question 22

What do we know about an LTI system?

Answer 22

• Its linear and time invariant
• Any delay in the input is reflected in the output
• Impulse response is used to define the LTI system

-We use Laplace transform or Fourier transform as its hard to go from output to input in time domains, so we should convert to frequency domain. To get back to the time domain we can use the ILP or IFT to give impulse response h(t)