Fourier representations for four classes of signals
The four classes of Fourier representations are defined by the periodicity properties of a signal and whether the signal is continuous or discrete in time. The Fourier series applies to discrete-time periodic signals. Nonperiodic signals have Fourier transform representations. The Fourier transform applies to a signal that is continuous in time and nonperiodic. The discrete-time Fourier transform or DTFT applies to a signal that is discrete in time and non-periodic.
Periodic signals: Fourier series representations.
Let us consider representing a signal as a weighted superposition of complex sinusoids. As the weighted superposition must have the same period as the signal, each sinusoid in the superposition must have the same period as the signal. It implies that the frequency of each sinusoid must be an integer multiple of the signal’s fundamental frequency.
If x[n] is a discrete-time signal with fundamental period N, then we will be representing x[n] by the DTFS.
x’[n] = Σ A[k]e^iΩ(n-k)
When, Ω= 2pi/ N is the fundamental frequency of x[n]. The frequency of the kth sinusoid in the superposition is kΩ. All the sinusoids have a common period n. Similarly, if x(t) is a continuous-time signal of fundamental period T, we represent x(t) as:
x’[t] = Σ A[k]e^ikwt.
Where, w= 2pi/T is a fundamental frequency of x(t). Here the frequency of the kth sinusoid is kw, and each sinusoid has a common period T. A sinusoid whose frequency is an integer multiple of a fundamental frequency is said to be a harmonic of the sinusoid at the same fundamental frequency.
Therefore, e^ikwt is the kth harmonic of e^ikwt. In the equations, A[k] is the weight applied to the kth harmonic, and the (‘) denotes approximate value, as we do not assume that either x[n] or x(t) can be represented exactly by series of the form shown. The variable k indexes the frequency of the sinusoids, so we can say that A[k] is a function of frequency.
How many terms and weights can we use in each sum? The answer to the question becomes clear for the DTFS as described in the equation. As we know that complex sinusoids with distinct frequencies are not always distinct. In particular, the complex sinusoids e^ikwt are N-periodic in the frequency index k, as shown by the relationship as follows:
e^i(N+k)Ωn = e^iNΩn . e^ikΩn = e^ikΩn.
As there are only N distinct complex sinusoids of the form eîkΩn. A unique set of N distinct complex sinusoids is obtained by letting the frequency index k vary from k = 0 to k = N-1. It can be written as follows:
x’[n] = Σ A[k]e^iΩ(n).
The set of N consecutive values over which k varies is arbitrary and can be chosen to simplify the problem by exploiting symmetries in the signal x[n] For example if x[n] is an even or odd signal, it is simpler to use k = – (N – 1)/2 to (N – 1) / 2 if N is odd.
In contrast to the discrete-time case, continuous-time complex sinusoids e^ikwt with distinct frequencies kw are always distinct. Therefore, there are potentially an infinite number of distinct terms in the series of equations given above and we express x(t) as:
x’(t)= Σ A[k]e^ikwt.
The weights or coefficients A[k] such that x’[n] and x’(t) are good approximations to x[n] and x(t) respectively. This is accomplished by minimizing the mean-square error or MSE between the signal and its series representation.
As the series representations have the same period as the signals, the MSE is the average squared difference over one period or the average power in the error. In the discrete-time case, we have:
MSE= 1/N Σ|x[n] - x’[n]|^2.
Similarly, in the continuous-time case,
MSE= 1/T ∫|x[t] - x’[t]|^2.
The DTFS and FS coefficients minimize the MSE. The determination of these coefficients is simplified by the properties of harmonically related complex sinusoids.
In contrast to the case of the periodic signal, there are no restrictions on the period of the sinusoids used to represent nonperiodic signals. Therefore, the Fourier transform representations employ complex sinusoids having a continuum of frequencies.
Where the variable of integration is the sinusoid’s frequency the signal is represented as a weighted integral of complex sinusoids. Discrete-time sinusoids are used to represent discrete-time signals in the DTFT, while continuous-time sinusoids are used to represent continuous-time signals in the FT. It can be represented as follows:
x’(t) = 1/2pi ∫ X(jw) e^jwt dw.
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