For an LTI system, we can find the output for any input using convolution if we know the system’s ___ response.
impulse
Convolution is defined by ____ in continuous time and ____ in discrete time.
integration; summation
δ(ω-v) ✳ (1/ω) = ?
1/(ω-v)
Given X(w) and O(w) as shown, what is X(w)✳O(w)?
If x[n] the input, h[n] the LTI system impulse response, what is y[n] (full mathematical definition)?
If x(t) is the input and h(t) the LTI system impulse response, what is y(t) (full mathematical definition)?
What property of convolution is indicated by associativity?
Associativity allows for the grouping of functions in convolution without affecting the result.
What does the commutativity property of convolution state?
Commutativity indicates that the order of functions in convolution does not change the outcome.
What is the step response of a system?
The output when input x(t)=u(t) (the step signal)
The step response is the output of a system when the input is a step function.
What is the equivalent impulse response for two systems connected in parallel?
reduced system: h(t), sys 1: h1(t), sys 2: h2(t)
h(t) = h1(t)+h2(t)
This property states that convolution distributes over addition.
What is the equivalent impulse response for two systems connected in series?
reduced system: h(t), system 1: h1(t), system 2: h2(t)
This property states that convolution distributes over addition.
What does the notation x(t) * y(t) represent?
Convolution of x(t) and y(t)
What is the output of an LTI system when given an input x(t)?
h(t) is the impulse response, y(t) the output
What does LTI stand for in the context of systems?
Linear Time-Invariant
LTI systems are characterized by their linearity and time-invariance properties.
Fill in the blank: The output of an LTI system can be expressed as (y(t) = h(t) * _______.
x(t)
This indicates that the output is derived from the convolution of the input signal with the impulse response.
What is the significance of the impulse response (h(t)) in an LTI system?
It characterizes the system’s response to an impulse input
The impulse response is crucial for determining the output for any arbitrary input through convolution.
True or false: An LTI system can have multiple impulse responses.
FALSE
An LTI system has a unique impulse response for a given system, which defines its behavior.
In the equation (y(t) = h(t) ⁕ x(t)), what does the symbol ⁕ represent?
Convolution
Convolution is a mathematical operation used to express the output of an LTI system.
What is the relationship between parallel connections of LTI systems?
The outputs of each system are summed
In parallel connections, each system processes the same input independently, and their outputs are combined.
In the context of LTI systems, what does the term series connection imply?
The output of one system becomes the input to the next
In series connections, the systems are arranged such that the output of one feeds into the next.
What is the time shift property in convolution?
This property indicates how shifting a signal in time affects its convolution with another signal.
What is the duration property of convolution?
If x(t) has duration T1 and h(t) has duration T2, then y(t) has duration T1 + T2
The output signal’s duration is the sum of the durations of the convolved signals.
What is the result of convolving a signal with an unshifted impulse?
y(t) = x(t)
Convolution with an impulse function reproduces the original signal.
