Discrete and continuous-time periodic signals
The discrete-time Fourier series or the DTFS representation of a periodic signal x[n] with fundamental period N and fundamental frequency Ω = 2pi/N is as follows:
x[n] = Σ X[k]e^iΩn
They are the DTFS coefficients of a signal x[n]. It can be said that x[n] and X[k] are a DTFS pair and the relationship can be denoted as follows:
X[n] <——> X[k].
From N values of X[k], we can determine x[n] and from N values of x[n], we can determine X[k]. X[k] or x[n] will provide a complete description of the signal. It is seen that in some problems it is advantageous to represent the signal using its time-domain values x[n], while in others the DTFS coefficients X[k] offer a more convenient description of the signal. The DTFS coefficient X[k] is termed a frequency-domain representation for x[n], because each coefficient is associated with a complex sinusoid of a different frequency. The variable k determines the frequency of the sinusoid associated with X[k], so we say that X[k] is a function of frequency. The DTFS representation is exact.
The DTFS is the only Fourier representation that can be numerically evaluated and manipulated in a computer. This is because both the time-domain, x[n], and frequency domain, X[k], representations of the signal are exactly characterized by a finite set of N numbers. The computational tractability of the DTFS is of great practical significance. The series finds extensive use in numerical signal analysis and system implementation and is often used to numerically approximate the other three Fourier representations.
Continuous-time periodic signals: the Fourier series
Continuous-time periodic signals are represented by the Fourier series (FS). The FS of a signal x(t) with fundamental period T and fundamental frequency w, 2pi/T can be written as:
x(t) = Σ X[k]e^ikwt
Where,
X[k] = 1/T∫|x[t]e^-jkwt dt.
are the FS coefficients of the signal x(t). x(t) and X(k) are an FS pair and the relationship can be denoted as:
x(t) ←> X[k].
From the FS coefficients X[k], we may determine x(t). and from x(t), we may determine X[k]. In many problems, it is advantageous to represent the signal in the time domain as x(t), while in others the FS coefficients X[k] offer a more convenient description. The FS coefficients are known as a frequency-domain representation of x(t) as each Fourier series coefficient is associated with a complex sinusoid of a different frequency. As in the DTFS, the variable k determines the frequency of the complex sinusoid associated with X[k]. It is most often used in electrical engineering to analyze the effect of systems on periodic signals.
The infinite series in the equation: x(t) = Σ X[k]e^ikwt is not guaranteed to converge for all possible signals, therefore, in this regard we define:
x’(t) = Σ X[k]e^ikwt
Where, k= – infinity,
and choose the coefficients X[k] according to the equation: X[k] = 1/T∫|x[t]e^-jkwt dt.
To find out where x’(t) actually converges to x(t), we do the following. Firstly, if x(t) is square-integrable, then the MSE between x(t) and x’(t) is zero.
This is a useful result that applies to a broad class of signals encountered in engineering practice. Opposite to the discrete-time case, an MSE of zero does not imply x(t) and x’(t) are equal pointwise, or x(t)= x’(t) at all values of t; it only implies that there is zero power in their difference.
Pointwise convergence of x(t) to x(t) is guaranteed at all values of t except the corresponding to discontinuities if the Dirichlet conditions are satisfied:
- It has a finite number of discontinuities in one period.
- It has a finite number of maxima and minima in one period.
- It is bounded.
If a signal x(t) satisfies the Dirichlet conditions and is not continuous, then x’(t) converges to the midpoint of the left and right limits of x(t) at each discontinuity. As with the DTFS, the magnitude of X[k] is known as the magnitude spectrum of x(t), while the phase of X[k] is known as the phase spectrum of x(t). As x(t) is periodic, the interval of integration in Eq. X[k] = 1/T∫|x[t]e^-jkwt dt may be chosen as an interval one period in length. Choosing the appropriate interval of integration simplifies the problem.
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