In this framework we present a new method for measurement of the UWB impulse train based on the parallel sampling of the cascaded identical RC filters. We show that the amplitudes and time locations of p sequential impulses can be reconstructed from simultaneous measurement of the outputs from 2p cascaded identical RC filters. The parallel sampling scheme has a wide range of applications including the detection of the ultra wideband (UWB) impulses. Due to identical analog RC filters and buffer amplifiers, the parallel sampling scheme is flexible to implement in VLSI applications.
The conventional sampling methods are based on the sequential measurement of signals at equidistant intervals. If the signal contains abrupt changes, such as edges, impulses and other discontinuities, the signal bandwidth is not limited below the Nyquist frequency and the Shannon’s sampling theorem is not warranted. Recently the problem to measure and reproduce non-band limited signals has been widely studied in signal processing society. Excellent articles [1-8] concern on the prefiltering and reconstruction of signals under nonideal sampling constraints. The sampling scheme based on the finite rate of innovation (FRI) recovers a variety of transient signal families, which are not band limited, such as impulses, piecewise linear and amplitude modulated edges.
The measurement and recovery of short term transient signals (Diracs) plays a main role in wireless ultra wideband (UWB) technology, where the information is coded to impulse sequences. The Diracs are fed to an analog circuit, which has a specific impulse response. The purpose of the sampling filter is to lengthen the UWB impulses for equidistant sampling by an analog-to-digital converter. Various methods have been developed for reconstruction of the original signal from discrete samples.
Our research group has introduced a sampling scheme, where the signal is fed to the parallel RC filters, whose outputs are sampled simultaneously [9,10]. A variant of the parallel sampling scheme is tailored for detection of edges [11,12]. The parallel sampling scheme can be applied to reproduce transient signals without ad hoc knowledge of the signal waveform [
In this framework we introduce a new method for measurement and reconstruction of the impulse trains based on the parallel sampling of the identical cascaded RC filters. We show that the amplitudes and locations of p Diracs can be reproduced from simultaneous measurements of 2p outputs of cascaded identical RC filters. We compare the reconstruction performance of the present method with the MCS arrangement using an analog electronic circuit simulator.
We consider the impulse train consisting of p sequential Dirac distributions
where are amplitudes and the time locations. The impulse train is fed to the N identical cascaded RC filters (
of the cascaded RC filter network are obtained as where the unit step function for and for and. The output signals
In Laplace transform domain we have
The inverse Laplace transform then gives
where. Now we can write the convolution integral
The prominent idea in this work is to measure the RC filter outputs simultaneously at a time. By denoting the time difference and rearranging we obtain
where the notations and are used.
In the following we develop a reconstruction algorithm for the amplitudes and time locations of the Diracs based
on formulation (7). The z transform of the sequence with respect to k yields
Let us define the pole cancellation filter as
and the polynomial as
Clearly the roots of the pole cancellation filter correspond to the roots of the polynomial. The impulse response of the can be computed from convolution
The roots of the are obtained by setting for. This yields the matrix/vector equation
We may observe that the solution of the coefficients of the pole zero cancellation filter requires the knowledge of the 2p values of the sequence. The polynomial
has roots.
The time locations of the Diracs are then obtained as. For solution of the amplitudes we may write (7) in matrix/vector form
where # denotes the pseudoinverse matrix. Now the amplitudes of the impulses are obtained as. To summarize, the reconstruction of p impulses requires the knowledge of the sequence. This needs the simultaneous measurement of at least samples from the outputs of the cascaded RC filters. An alternative singular value decomposition (SVD) based null space method is presented in Appendix for the solution of the roots of the pole zero cancellation filter (9).
The above formulation (8)-(13) is valid only in noise free situation. The noise was rejected by the singular value decomposition (SVD) based subspace method. Let us construct a Hankel matrix
where the antidiagonal elements are identical. The singular value decomposition of the matrix is, where and are unitary matrices. is a diagonal matrix consisting of the singular values in descending order. We may decompose as
contains the smallest singular values and hence the matrix can be considered to belong to the noise subspace. The matrix is then related to the noise free signal subspace. The dimension of the signal subspace can be evaluated by several methods [
In the case the impulse rate is limited so that only one impulse arrives at the cascaded RC network, the reconstruction algorithm simplifies notably and the sampling of the outputs of the first two RC filters is enough to recover the amplitude and time location of the impulse. Based on (6) we obtain
In practice the outputs of at least three RC filters is advisable to measure, since we may then construct the Hankel matrix for SVD based noise cancellation.
The practical UWB transmission protocol consists of wireless RF pulses each containing three impulses (
contains the output due to the impulses occurring at time interval reduced by the pile-up signal due to the previous UWB pulses at time interval. The approximation in (17) is exact for. However, in cases for the exponential is mainly responsible for the descending tail and the approximation error is negligible.
The reconstruction algorithm was tested by an extensive simulation study. The cascaded network comprised eight identical RC filters which were constructed using an ana-
log electronic circuit simulator (Spice). The parameter varied in the range 0.1 - 0.8. In the absence of noise the reconstruction algorithm (12, 13) recovered the amplitudes and time locations of the impulse train consisting of 1 - 4 Diracs with machine precision. When the random noise was added to the input of the network, the algorithm could reconstruct 1 - 3 Diracs with good accuracy, when the SVD based subspace method was applied to noise cancellation. The error in amplitudes varied between 0.8 - 1.2 percent and in time locations 0.3 - 0.7 percent. In the case of four sequential Diracs the reconstruction algorithm yielded inaccurate results or totally failed.
We made a preliminary comparison with the multichannel sampling (MCS) arrangement [
The cascaded RC network was implemented using metallic foil resistors and low-noise capacitors. Eight RC filters were separated by unity gain buffer amplifiers (
The reconstruction of Diracs using the SVD based null space method (Appendix) gave almost identical results, which are not repeated here. The slight difference is probably due to that the matrices in (14) and in (17) are different.
In our previous work [
The formulation of the reconstruction algorithm was warranted by extensive simulation study using Spice toolbox. In the presence of noise the SVD based noise cancellation method had to be applied. The computational complexity of the SVD algorithm is, which restricts the real-time applications of the present method to a relatively low transmission rate. However, since the information is coded to both the amplitude and the time locations of the Diracs, the number of transmitted pulses can be significantly lower compared with the conventional UWB methods. As an advantage lower number of transmitted pulses reduces the RF radiation load.
We showed in Spice simulation study that in noise free conditions the p Diracs can be reconstructed from sampling the outputs of the 2p cascaded identical RC filters. In the presence of noise this theoretical “2p rule” is not valid and it seems evident that preferable “2p + 1 rule” should be applied.
The high-speed method (16) suits best for reconstruction of single Diracs, whose amplitudes are quantized to only a few levels. For example the impulses with 15 discrete levels can be produced by a 4-bit digital-to-analog converter, which the method reproduces perfectly.
In MCS arrangement the signal is modulated by a set of sinusoidal waveforms, followed by a bank of integrators [
In most of the UWB devices information is transmitted by monocycle Gaussian pulses. The FCC restricted the UWB frequency band between 3.1 - 10.6 GHz in year 2002 [
In this work we have concentrated on the UWB communication device, which transmits pulses consisting of most 1 - 3 impulses. The information is coded to the amplitudes and time locations of the Diracs. Such pulse generators are relatively easy to implement in VLSI [
This work was supported by the National Technology Agency of Finland (TEKES).
Let us write the convolution (11) in the matrix/vector form
In the following we describe the singular value decomposition (SVD) based null space solution of the unknown vector in (18). The SDV of the matrix in (18) yields
where matrices and contain the left and right singular vectors (column vectors) and matrix the singular values in descending order. Matrices and are unitary, i.e. and, where is the identity matrix. Applied to (19) we obtain, which yields
Finally we may write
for. Equation (21) forms the basis for the SVD based null space method. By searching very small singular value, the right singular vector equals vector in (18) yielding the solution for the roots of the pole cancellation filter (9). In the presence of noise the dimensions of the and matrices should be selected so that there appears only one tiny singular value. This can be also achieved by zeroing the rest of the tiny singular values. It should be pointed out that the SVD based null space method does not yield the coefficients of the pole zero cancellation filtering (9) in normalized form, i.e.. However, we may apply post normalization as for.