Fourier Transform of Discontinuous Signals

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Questions and Answers

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 (correct)

The signal x(t) is a polynomial

The signal x(t) has no discontinuities

The signal x(t) has infinite energy

What happens to the Fourier transform representation x̂(t) at points of discontinuity of the signal x(t)?

It converges to zero

It diverges

It converges to x(t)

It converges to the average of x(t) on the two sides of the discontinuity (correct)

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 (correct)

The signal x(t) has finite energy

The signal x(t) is periodic

The signal x(t) is integrable

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 Signup and view all the answers

What happens to the Fourier transform representation x̂(t) at points where the signal x(t) is continuous?

It converges to x(t) Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

What is the result of sampling a continuous-time signal with a sampling period T?

A discrete-time signal sequence Signup and view all the answers

What is the purpose of interpolation in signal processing?

To assign values to a signal between its sample points Signup and view all the answers

What is the input to an ideal continuous-to-discrete-time (C/D) converter?

A continuous-time signal Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

What is the primary purpose of interpolation in signal processing?

To construct a continuous-time signal from a discrete-time one Signup and view all the answers

What is the condition required for the convergence of the Fourier Transform?

The signal must have finite energy Signup and view all the answers

What is the consequence of undersampling a signal?

Aliasing occurs Signup and view all the answers

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 Signup and view all the answers

What is the condition required for a signal to be integrable?

The signal must have finite energy Signup and view all the answers

What is the consequence of a signal having discontinuities?

The Fourier Transform does not converge Signup and view all the answers

Study Notes

Fourier Transform and Dirichlet Conditions

A signal ( x(t) ) meeting Dirichlet conditions ensures pointwise convergence of its Fourier transform ( \hat{x}(t) ) everywhere except at discontinuities.
At discontinuity points ( t = t_a ), the Fourier transform converges to the average value: ( \hat{x}(t_a) = \frac{1}{2} [x(t_a^+) + x(t_a^-)] ).
The average considers the left-side value ( x(t_a^-) ) and the right-side value ( x(t_a^+) ) of the signal.
Valid Fourier transform representation is attainable even if finite-energy and Dirichlet conditions are violated, indicating that these conditions are sufficient, not necessary.

Signal Reconstruction and Sampling

An ideal continuous-to-discrete-time (C/D) converter system is used to sample the continuous-time signal ( x(t) ) at periodic intervals.
Interpolated continuous-time signal ( \hat{x}(t) ) is generated from discrete samples ( y[n] ), using a specific interpolation function ( f(y[n]) ).
The form of function ( f ) varies based on interpolation methods, with emphasis on bandlimited interpolation in subsequent discussions.

Interpolation and Impulse Train

The interpolation system is modeled as an ideal discrete-to-continuous-time (D/C) converter.
Interpolation determines values of a signal between sample points, facilitating the reconstruction of a continuous-time signal from its discrete-time samples.
Generally, exact reproduction of the continuous-time signal from samples is difficult, barring special cases.

Frequency Spectrum and Aliasing

Impulse-train sampling impacts the frequency spectrum of the original signal ( x(t) ), highlighting potential aliasing phenomena.
Two scenarios exist: one without aliasing (preserving original frequency components) and one with aliasing (overlapping frequency components).

Conclusion

Understanding Dirichlet conditions is essential for Fourier analysis.
Signal sampling and interpolation play critical roles in digital signal processing, impacting data integrity and reconstruction fidelity.

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Description

Understanding the Fourier transform representation of signals that satisfy the Dirichlet conditions, including convergence and behavior at discontinuity points.