How do we know we have an EVEN function?
- 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)
How do we know we have an ODD function?
f(x) = -f(-x)
-all sine functions are ODD
When do we know a signal is memoryless and give an example?
y(t) = (2 + sin t)x(t) is memoryless because y(t) depends only on x(t) and not prior values of x(t)
How to tell if a signal is time invariant?
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.
An examples for when a function is not casual?
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.
An example of a signal invertible?
y(t) = x(2t) is invertible; x(t) = y(t/2).
An example when a signal is not invertible?
y[n] = E _.x[k] is not invertible. Summation is not generally an invertible
operation.
2 examples of a stable system?
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.
An example of not stable?
-If |x[k]| 5 M, ly[n]j 5 M - E_,, which is unbounded. Therefore, y[n] = E”Lx[k] is not stable.
Define the unit step sequence?
u[n] =
is 0 for negative values of n and 1 for positive values of n or values of n = 0
Define the unit impulse sequence?
delta[n]
delta[n]
is 0 for all cases for its argument except when n =0 where the amplitude is 1
Define the first difference (the relationship between the unit step and unit impulse):
delta[n] = u[n]-u[n-1]
How can we rewrite the unit step?
u[n] = n(sun)m=-inf delta[m]
-unit step can be thought of as a summation of unit impulses
Whats time invariance?
-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)]
Whats linearity?
-A set of inputs linear combination should equal a set of outputs of linear combinations
What are 2 LTI systems:
enables us to
- Delayed impulses leads to CONVOLUTION
- Complex exponentials leads to FOURIER ANALYSIS
Convolution approach:
- 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]
Explain how convolution is commutative:
- 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)
Whats the convolution associative property?
- Doesnt matter how we group the terms together
- [h1h2]x(t) = h1[h2x(t)]
Whats the distributive property of convolution?
- x1[h1h2] = [x1h1] + [x1h2]
Define causality?
- 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
What do we know about an LTI system?
- 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)
