Discrete and continuous-time non-periodic signals
Discrete-time non-periodic signals: the discrete-time Fourier transform.
The DTFT or discrete-time Fourier transform is used to represent a discrete-time nonperiodic signal as a superposition of complex sinusoids. Its representation of a time-domain signal involves an integral over frequency,
x[n] = 1/2pi ∫|X[e^iΩ]e^iΩ dΩ.
Where, X(e^iΩ) = Σx[n] e^iΩn, is the DTFT of the signal x[n].
x[n]<—> X(e^iΩ).
The transform X(e) describes the signal x[n] as a function of a sinusoidal frequent and is termed the frequency-domain representation of x[n]. The equation: x[n] = 1/2pi ∫|X[e^iΩ]e^iΩ dΩ is termed the inverse DTFT since it maps the frequency-domain representation back to the time domain.
The DTFT is used primarily to analyze the action of discrete-time systems on discrete-time signals.
The infinite sum in the eq. X(e^iΩ) = Σx[n] e^iΩn converges if x[n] has a finite duration and is finite value If x[n] is of infinite duration, then the sum converges only for certain classes of signal
Σ[x[n] < infinity
(i.e., if x[n] is absolutely summable), then the sum in the eq. X(e^iΩ) = Σx[n] e^iΩn converges uniformly continuous function off. If x[n] is not absolutely summable but does satisfy (i.e., if x[n] has finite energy), then it can be shown that the sum in Eq. (3.32) converges in a mean-square error sense, but does not converge pointwise.
We now consider several examples illustrating the determination of the DTFT for common signals.
Many physical signals encountered in engineering practice satisfy these conditions. However, several common nonperiodic signals, such as the unit step u[n], do not. In some of these cases, we can define a transform pair that behaves like the DTFT by including impulses in the transform. This enables us to use the DTFT as a problem-solving tool, even though, it does not converge.
Continuous-time non-periodic signals: the Fourier transform
The Fourier transform or FT is used to represent a continuous-time nonperiodic signal as a superposition of complex sinusoids. The continuous nonperiodic nature of a time signal implies that the superposition of complex sinusoids used in the Fourier representation of the signal involves a continuum of frequencies ranging from –∞ to ∞. Therefore, the Fourier Transform representation of a continuous-time signal involves an integral over the entire frequency interval; that is:
x(t) = 1/2pi ∫ X(jw) e^jwt dw.
Where, X(jw) = ∫ x(t) e^iwt dt, is the Fourier Transform of the signal x(t). In Eq. x[n] = 1/2pi ∫|X[e^iΩ]e^iΩ dΩ, we have expressed x(t) as a weight on each superposition of sinusoids having frequencies ranging from – ∞ to ∞. The superposition is an integral, and the weight on each sinusoid is (1/(2pi))X(jw). We say that x(t) X(jw) are a Fourier Transform pair and represent:
x(t) ←> X(jw).
The transform X(jw) describes the signal x(t) as a function of frequency w and is termed the frequency-domain representation of x(t). Equation (3.35) is termed the inverse F since it maps the frequency-domain representation X(jo) back into the time domain.
The integrals in the above-mentioned equations, that is: x(t) = 1/2pi ∫ X(jw) e^jwt dw and X(jw) = ∫ x(t) e^iwt dt, may not converge for all functions x(t) and X(jw). An analysis of convergence is beyond the scope of this book, so we simply state several convergence conditions on the time-domain signal x(t). If we define
The Fourier Transform is used to analyze the characteristics of continuous-time systems and the interaction between continuous-time signals and systems. The Fourier Transform is also used to analyze interactions between discrete- and continuous-time signals, such as occur in sampling.
Zero squared error does not imply pointwise convergence [ie.. x(1)-(1) at all values of it does, however, imply that there is zero energy in the difference of the terms
Pointwise convergence is guaranteed at all values of r except those correspond discontinuities if x(t) satisfies the Dirichlet conditions for nonperiodic signals
- x(t) is absolutely integrable:
- ∫ |x(t)| dt <∞.
x(t) has a finite number of maxima, minima, and discontinuities in any finite inter
- The size of each discontinuity is finite.
Almost all physical signals encountered in engineering practice satisfy the second and conditions, but many idealized signals, such as the unit step, are neither absolutely square-integrable. In some of these cases, we define a transform pair that satisfies FT properties through the use of impulses. In this way, we may still use the FT as a problem-s tool, even though, in a strict sense, the FT does not converge for such signals.
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