Fourier transforms the representation of periodic signals
Introduction
The applications of Fourier representations to situations in which the classes of signals are mixed are discussed intensively below.
Mostly the signals mixed will be:
- Periodic and non-periodic signals
- Continuous-time and discrete-time signals.
The mixing of periodic and non-periodic signals, and continuous-time and discrete-time signals occur most commonly when one uses Fourier methods to either analyze the interaction between signals and systems or to numerically evaluate properties of signals or the behavior of a system. It can be illustrated with an example, such as, if we apply a periodic signal to a stable LTI system, the convolution representation of the system output involves the mixing of nonperiodic and periodic signals. Another example is that a system that samples continuous-time signals involve both continuous-time and discrete-time signals.
To use Fourier methods to analyze interactions involving the mixing of signals, we must establish a relation between the Fourier representations of different classes of signals. The Fourier Transform and DTFT are most commonly used for applications involving analysis. Therefore, we develop Fourier Transform and the Discrete-Time Fourier Transform representations of continuous-time and discrete-time periodic signals, respectively. The analysis of continuous-time applications that involves a mixture of periodic and nonperiodic signals is done using the Fourier Transform. The Discrete-Time Fourier Transform is used to analyze mixtures of discrete-time periodic and nonperiodic signals. The Discrete-Time Fourier Series is the primary representation used for computational applications, therefore, it is important to know the relations in which the DTFS represents the Fourier Transform, Fourier Series, and Discrete-Time Fourier Transform.
Fourier Transform representation of periodic signals.
The Fourier Series and Discrete-Time Fourier Series have been derived as the Fourier representations of period signals, neither the FT nor the DTFT converges for periodic signals. Although, by incorporating impulses into both of them in the appropriate manner it is possible to develop FT and DTFT representations of such signals. These representations can satisfy the properties expected of the FT and DTFT; therefore, they can be used along with the properties of the FT or DTFT to analyze problems involving mixtures of periodic and non-periodic signals. This establishes the relationship between Fourier series representations and Fourier transform representations, and is formulated below:
Relating FT to FS
The FS representation of a periodic signal x(t) is:
x(t) = Σ X[k]e^ikwt
In the above-established equation, w is the fundamental frequency of the signal. As we know, 1 ←> 2pi d(w-kw).
Using this result and the frequency shift property of signals, we can obtain the inverse Fourier Transform of the frequency-shifted impulse as a complex sinusoid with the frequency of kw.
The Fourier Transform is obtained from the Fourier Series by placing impulses at integer multiples of Ω and then weighting them by 2pi times the corresponding Fourier Series coefficient. Given a Fourier Transform consisting of impulses that are uniformly spaced in Ω, then to obtain the corresponding Fourier Series coefficients the impulse strengths are divided by 2pi.
The series of impulses for the case of the Fourier Transform of a periodic signal are spaced according to the fundamental frequency, where the kth impulse has strength 2pi X [k], given X[k] is the kth Fourier Series coefficient. It also shows how to convert between Fourier Transform and Fourier Series representations of periodic signals.
It is to be noted that the fundamental frequency corresponds to the spacing between impulses.
Relating the DTFT and the DTFS
The method for deriving the DTFT of a discrete-time periodic signal parallels that given in the previous situation. The DTFS expression for an N-periodic signal x[n] is:
x[n] = Σ X[k]e^ikΩn
The inverse Discrete-Time Fourier Transform of a frequency-shifted impulse for the case of Fourier Transform is a discrete-time complex sinusoid. The inverse DTFT can be expressed as a frequency-shifted impulse either by expressing one period or by using an infinite series of shifted impulses separated by an interval of 2pi, as the DTFT is a 2pi- periodic function of frequency.
The series of impulses for the Discrete-time Fourier Transform representation of a periodic signal is spaced by the fundamental frequency Ω.
The shapes for the DTFS X[k] and the corresponding DTFT X(e^iΩ) are similar. The impulses are placed at the integer multiples of Ω and then weighed by 2pi times the corresponding DTFS coefficients, to obtain the DTFT representation.
In the same way, we can just reverse the process to obtain the DTFS coefficient from the DTFT representation, if the DTFT consists of impulses that are uniformly spaced in Ω then we obtain DTFS coefficients by dividing the impulse strengths by 2pi.
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