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What is the general form of the transformed signal from a continuous-time signal?
y(t) = x(at + b) where a is a constant for scaling, and b is a constant for shifting.
What transformation is represented by y(t) = x(-t)?
Time Reversal
If the parameter a in time scaling (y(t) = x(at)) is less than 1, what happens to the signal?
The signal is expanded.
What happens to the signal when a in time scaling is greater than 1?
The signal is compressed.
What is the formula for time shifting transformation?
y(t) = x(t - t0), where t0 is a constant.
What is an even signal?
A signal x(t) is even if x_e(t) = x_e(-t), exhibiting symmetry around the y-axis.
Define an odd signal.
A signal x(t) is odd if x_o(t) = -x_o(-t), exhibiting symmetry with respect to the origin.
In terms of periodic signals, what does it mean for a function to be periodic?
A function x(t) is periodic if x(t) = x(t + T) for some period T.
What is the fundamental frequency?
The fundamental frequency f0 is given by f0 = 1/T0 where T0 is the fundamental period.
How can you express any signal in terms of its even and odd parts?
x(t) = x_e(t) + x_o(t), where x_e is the even part and x_o is the odd part.
What property of even functions refers to their sum being even?
The sum of two even functions is even.
How do you find the convolution of two signals x(t) and h(t)?
y(t) = ∫ x(τ) * h(t - τ) dτ.
What characterizes a causal system?
A causal system’s output depends only on present and past inputs.
What is the definition of a stable system in terms of BIBO stability?
A system is BIBO stable if every bounded input results in a bounded output.
What is the convolution of any function with delta function δ(t)?
The convolution yields the function itself.
What indicates a memoryless system?
A memoryless system's output depends only on the present input.
What represents time invariance in a system?
Time invariance means that a time shift in input results in the same time shift in output.
Define linearity in terms of system properties.
A system is linear if it satisfies the principles of superposition, including additivity and homogeneity.
What is the impulse response of a system?
The impulse response is the output of the system when the input is an impulse function (δ(t)).
What is the result of the convolution of two unit step functions?
The convolution of two unit step functions results in a ramp function.
What is the Laplace Transform of a unit step function?
L{u(t)} = 1/s.
How do you express a transformed signal's response in terms of impulses?
y(t) can be expressed based on the system's impulse response convolved with an input signal.
What is the significance of the roots of the characteristic equation in a system?
The roots determine the stability of the system: stability is achieved if all have negative real parts.
In terms of stability, what characterizes a marginally stable system?
A system that has all roots with negative real parts and one root at zero is marginally stable.
What process defines the behavior of an integrator system with an impulse input?
The output is the integral of the input function.
How does the stability of a system relate to the poles of its transfer function?
If all poles are in the left half-plane, the system is stable.
What properties characterize the impulse response of an LTI system?
The impulse response characterizes the system's behavior, defining its output in relation to any input signal.
What is the output of a system with impulse response h(t) under zero initial conditions?
The output can be calculated using convolution with the input signal.
Describe the relationship between input and output in an LTI system.
The output is the convolution of the input signal with the system's impulse response.
What is the graphical method of convolution?
The graphical method involves plotting the input and impulse response and calculating the area under the overlap.
What is the significance of impulse functions in continuous-time system analysis?
Impulse functions allow for examining system response and behavior in terms of convolution.
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