On the Time Frequency Compactness of the Slepian Basis of Order Zero for Engineering Applications

Abstract

Time and frequency concentrations of waveforms are often of interest in engineering applications. The Slepian basis of order zero is an index-limited (finite) vector that is known to be optimally concentrated in the frequency domain. This paper proposes a method of mapping the index-limited Slepian basis to a discrete-time vector, hence obtaining a time-limited, discrete-time Slepian basis that is optimally concentrated in frequency. The main result of this note is to demonstrate that the (discrete-time) Slepian basis achieves minimum time-bandwidth compactness under certain conditions. We distinguish between the characteristic (effective) time/bandwidth of the Slepians and their defining time/bandwidth (the time and bandwidth parameters used to generate the Slepian basis). Using two different definitions of effective time and bandwidth of a signal, we show that when the defining time-bandwidth product of the Slepian basis increases, its effective time-bandwidth product tends to a minimum value. This implies that not only are the zeroth order Slepian bases known to be optimally time-limited and band-concentrated basis vectors, but also as their defining time-bandwidth products increase, their effective time-bandwidth properties approach the known minimum compactness allowed by the uncertainty principle. Conclusions are also drawn about the smallest defining time-bandwidth parameters to reach the minimum possible compactness. These conclusions give guidance for applications where the time-bandwidth product is free to be selected and hence may be selected to achieve minimum compactness.

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

“…Here, N is the number of points of Slepian basis and NW is the time-bandwidth product. With the mapping method in [50], the Slepian basis can be mapped to a discrete time vector and consequently, NW = 2TB.…”

“…Here, N is the number of points of Slepian basis and NW is the time-bandwidth product. With the mapping method in [50], the Slepian basis can be mapped to a discrete time vector and consequently, 2 NW TB = . Figure 2 shows some examples of Slepian basis n  with different number of points plotted against time vectors.…”

“…Simulations were used to determine the number of points required for the Slepian basis to reach less than 2% error between PSWFs for different time-bandwidth products (see Figure 13). Figure 13a shows the error when the Slepian basis reaches less than 2% error with the PSWF; Figure 13b shows the number of points when the Slepian basis reaches less than 2% error for different timebandwidth products (note here that the horizontal axis is taking "time-bandwidth products as TB for comparison with [50]. "); Figure 13c,d show detailed plots of the vectors to provide a better view of how different they look from each other.…”

“…The number of points in the Slepian basis needed to reach less than 2% error tends to be a constant value after a certain time-bandwidth threshold. In [50], it was shown that when TB of the Slepian basis of order zero is greater than a certain lower threshold (approximately 0.5~1 depending on different measures used to define effective duration and bandwidth), the effective energy-based or variance-based time-bandwidth In Figure 13, the duration of the vectors are kept the same as T = 1 s, while B ranges from 0.3 Hz to 3.2 Hz. The number of points in the Slepian basis needed to reach less than 2% error tends to be a constant value after a certain time-bandwidth threshold.…”

“…The number of points in the Slepian basis needed to reach less than 2% error tends to be a constant value after a certain time-bandwidth threshold. In [50], it was shown that when TB of the Slepian basis of order zero is greater than a certain lower threshold (approximately 0.5~1 depending on different measures used to define effective duration and bandwidth), the effective energy-based or variance-based time-bandwidth products tend to achieve a minimum value. Here, we demonstrate that a similar trend can also be observed with higher order eigenvectors, as shown in Figures 14 and 15 Evaluating PSWFs with large time-bandwidth products ( 10 c  ) is notoriously difficult [36].…”