18 Questions
What is a condition for the Fourier transform representation x̂(t) to converge pointwise for all t?
The signal x(t) satisfies the Dirichlet conditions
What happens to the Fourier transform representation x̂(t) at points of discontinuity of the signal x(t)?
It converges to the average of x(t) on the two sides of the discontinuity
What is a consequence of a signal x(t) satisfying the Dirichlet conditions?
The Fourier transform representation x̂(t) converges to x(t) for all t, except at points of discontinuity
What is a sufficient condition for the convergence of the Fourier transform representation of a signal x(t)?
The signal x(t) satisfies the Dirichlet conditions
What happens to the Fourier transform representation x̂(t) at points where the signal x(t) is continuous?
It converges to x(t)
What is a characteristic of a signal x(t) that has a valid Fourier transform representation?
It may violate the Dirichlet conditions and still have a valid Fourier transform representation
What is the primary function of an ideal discrete-to-continuous-time (D/C) converter?
To reconstruct a continuous-time signal from a discrete-time signal
What is the result of sampling a continuous-time signal with a sampling period T?
A discrete-time signal sequence
What is the purpose of interpolation in signal processing?
To assign values to a signal between its sample points
What is the input to an ideal continuous-to-discrete-time (C/D) converter?
A continuous-time signal
What is the relation between the sampling period T and the discrete-time signal sequence y[n]?
y[n] is sampled with a period of T
What is the function of the ideal discrete-to-continuous-time (D/C) converter in Figure 5.23?
To reconstruct a continuous-time signal from a discrete-time signal
What is the primary purpose of interpolation in signal processing?
To construct a continuous-time signal from a discrete-time one
What is the condition required for the convergence of the Fourier Transform?
The signal must have finite energy
What is the consequence of undersampling a signal?
Aliasing occurs
What is the primary difference between the ideal C/D converter and the ideal D/C converter?
One is used for sampling, the other for reconstruction
What is the condition required for a signal to be integrable?
The signal must have finite energy
What is the consequence of a signal having discontinuities?
The Fourier Transform does not converge
Understanding the Fourier transform representation of signals that satisfy the Dirichlet conditions, including convergence and behavior at discontinuity points.
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