B. LTI Systems Flashcards
What is a system?
A system is something that transforms an input signal into an output signal.
What does it mean when a system is memoryless?
A system is memoryless when the output only depends on the current value of the input.
The output is a scaled version of the input
y(t) = a*x(t)
Example of a non-memoryless system: y(t) = x(t) + x(t-2)
(depends on output at a previous time point)
When is a system causal?
A system is causal if the output depends only on the current and previous values of the input, a change in the output can only occur after a change in the input.
x(t) = 0 for t < t0 → y(t) = 0 for t < t0
Example of a non-causal system: y(t) = x(t) + x(t+2)
(depends on future output)
What are inverse systems?
Inverse systems are systems that when put in series, the output is equal to the input.
In the image, S1 and S2 are inverse systems if x(t) = z(t).
When is a system stable?
A system is stable when bounded input yields a bounded output.
A small change in the input yields a small change in the output
|x(t)| < Bx → |y(t)| < By
When is a system linear?
A system is linear if it follows the superposition principle.
T{ax1(t) + bx2(t)} = aT{x1(t)} + bT{x2(t)} = ay1(t) + by2(t)
When is a system time-invariant?
A system is time-invariant when the system response is independent of time.
Shifting before or after the systems yields identical results
y(t) = T{x(t)} → y(t - t0) = T{x(t - t0)}
What do we mean when we talk about LTI systems?
LTI systems are systems that are both linear as well as time-invariant.
In this course, we will mainly focus on LTI systems.
What is meant by decomposition?
Decomposition is a method which eases subsequent processing by a linear system. You decompose the signal in smaller pieces.
For example, a DT signal can be decomposed into a weighed sum of shifted impulses.
What is the definition of convolution?
The convolution is the response of a Linear Time-Invariant (LTI) system.
Notation: x[n] ∗ h[n] or x[n] ⊗ h[n]
Note: convolution is commutative so x[n] ∗ h[n] = h[n] ∗ x[n]
Give the convolution sum of a discrete time signal
∑_(k=-∞)^∞▒〖x[k]h[n-k]〗
= ∑_(k=-∞)^∞▒〖h[k]x[n-k]〗
What is the impulse response?
In the convolution sum, h[n] is the impulse response which is the response of the LTI system to input δ[n]. The impulse response lets us compute the response of the LTI system to any input x[n] by the convolution sum. The impulse response characterizes a LTI system completely.
Note: If it is given that a certain system has an impulse response, the system is automatically a LTI system.
Name two different methods of calculating the convolution and briefly describe them.
Convolution by stamping
Find y[n] by calculating the convolution sum for each individual value of k.
Note: See slide 14 for an illustration.
Convolution by sliding filter
Find y[n] by ‘slindig’ the impulse response h[n-k] over the n-axis, the value of y[n] is then obtained by calculating the overlap between x[k] and h[n-k] (which is basically how convolution works)
Note: See slides 15-18 for an illustration.
What is the impulse response of an LTI system when the output is always identical to the input?
The unit impulse function.
The delta function is the neutral element of convolution.
What is a method for calculating finite geometric series?
∑_(i=0)^N▒〖x^i= (1-x^(N+1))/(1-x)〗
