Fourier series representations of finite duration non-periodic signals
Fourier series representations of finite duration non-periodic signals.
The DTFS and Fourier Series are the Fourier representations of periodic signals. We represent finite-duration nonperiodic signals, the primary motivation for doing this has to do with the numerical computation of Fourier representations. As we know that the DTFS is the only Fourier representation that can be evaluated numerically. As a result, we often apply the DTFS to signals that are non-periodic. It is important to understand the implications of applying a periodic representation to nonperiodic signals.
Another benefit is to increase our understanding of the relationship between the Fourier Transform and corresponding Fourier series representations.
As we know the sampling of a signal in time generates shifted replicas of the spectrum of the original signal in the frequency domain. In order to prevent overlap or analyzing of these shifted picas in time, we do the frequency sampling interval Ω to be less than or equal 2pi/M. So, the result corresponds to the sampling theorem applied in the frequency domain.
The DTFS provides samples of the DTFT of the length-M sequence. The practice of choosing N > M when evaluating the DTFS is known as zero-padding, since it can be viewed as augmenting or padding the M available values of x[n] with N – M zeros. Though, the zero padding does not overcome any of the limitations associated with knowing only M values of x[n]; it only samples the underlying length M DTFT more densely.
The Discrete-time Fourier series approximation to the Fourier Transform.
The DTFS involves a finite number of discrete-valued coefficients in both the frequency and time domains. All the other Fourier representations are continuous in either the or frequency domain or both. Hence, the DTFS is the only Fourier representation that is evaluated on a computer, and it is widely applied as a computational tool for mandating signals. We will be using the DTFS to approximate the FT of a continuous-time signal.
The Fourier Transform applies to continuous-time nonperiodic signals. The DTFS coefficient is computed by using N values of a discrete-time signal. In order to use the DTFS to approximate the FT, we need to sample the continuous-time signal and retain at most N samples. We assume that the sampling interval is Ts and that M < N samples of the continuous-time signal are retained. Both the sampling and windowing operations are potential sources of error in the approximation.
An efficient algorithm for evaluating the DTFS
The role of the DTFS as a computational tool is greatly enhanced by the availability of efficient algorithms for evaluating the forward and inverse DTFS. These algorithms are collectively termed fast Fourier transform (FFT) algorithms. These algorithms exploit the “divide and conquer” principle by splitting the DTFS into a series of lower-order DTFSs and using the symmetry and periodicity properties of the complex sinusoid e^jk2pin. It requires less computation to evaluate.
Exploring concepts with MATLAB
DECIMATION AND INTERPOLATION
As we know that decimation reduces the effective sampling rate of a discrete-time signal, while interpolation increases the effective sampling rate. Decimation is accomplished by sub-sampling a low-pass filtered version of the signal, while interpolation is performed sorting zeros in between samples and then applying a low-pass filter. MATLAB’s Processing Toolbox contains several routines for performing decimation and interpolation. All of them automatically design and apply the low-pass filter required for both operations. The command y = d e c i m a t e(x, r) decimates the signal represented by x by a positive integer factor r to produce the vector y, which is a factor of r shorter than x. Similarly, y = i n t e r p(x, r) interpolates x by a positive integer factor producing a vector y as long as s. The command y = r e s a m p l (x, p, q) resamples the signal in vector x at p / q times the original sampling rate, where p and q are positive integers. This is conceptually equivalent to first interpolating by a factor p and then decimating by a factor q. The vector y is p / q times the length of x. The values of resampled sequence may be inaccurate near the beginning and end of y if x contains large deviations from zero at its beginning and end.
Periodic and nonperiodic signals often interact in the context of interaction between signals and LTI systems, for example, filtering, and maybe in performing some basic manipulation of signals. The mixture of continuous and discrete-time classes of signals is encountered in sampling continuous-time signals or in reconstructing continuous-time signals from samples. The use of DRFS to numerically approximate the FT also involves the mixing of signal classes, as each class has its own Fourier representation.
Reference
Fourier series representations of finite duration non-periodic signals