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Sungwoo, S. Shimizu, T. Ida, H. Ishihara, T. Matsuoka, K. Taniguchi, A. Sugimori, and H. ELIN Externally Linear Internally Nonlinear filters offer a good bandwidth-linearity compromise, low power, small area, and electronic tunability, but they suffer from secondary effects and are also very prone to noise because of the compression used. All of the above discussed filters are infinite impulse response IIR filters and cannot be used for applications such as matched filters, which require the implementation of finite impulse response FIR filters.

FIR filters can be implemented using newly developed techniques that use current-mode analog circuits or switched-current circuits with distributed arithmetic. See V Srinivasan, G. Rosen, and P. Sirisuk, A. Worapishet, S. Chanyavilas, and K. In the case of matched filters MFs , floating-gate analog CMOS and charge-domain operations techniques are preferred as they offer the design for low power and area compared to other techniques such as the switched-C implementations.

Yamasaki, T. Taguchi, and T. Passive mixers, such as diode mixers and passive field effect transistor FET mixers, have good linearity, good noise performance, and can operate at frequencies up to 5 GHz. However, active mixers are preferred for low-power integrated circuits as they provide conversion gain, require less power at the local input port, and have a broader design space. Mixers are generally based on the classic Gilbert cell.

Several modifications can be made to this basic circuit to increase its frequency response, decrease its nonlinearity, and limit its noise level. An important issue is limiting the feed-through of a signal on an undesired path isolation since the mixer is essentially a three-port network. These techniques involve the use of matching networks and emitter degeneration.

A CMOS ultra-wideband active mixer fabricated using 0. Tsai and H. Wide bandwidth is obtained through emitter degeneration of the transconductance stage. We will investigate the different modulator topologies and their associated tradeoffs in depth.

Linear feedback shift registers LFSR generate pseudorandom sequences and codes, such as maximal length sequences, etc. This type of register is commonly implemented using high-speed flip-flops. These include double edge-triggered flip-flops, fully differential circuit techniques, and current mode logic. Flip-flops operating at 17 GHz have been reported.

In linear feedback shift registers we consider the output of a single flip-flop, and thus we are not concerned with input clock jitter nor glitches in internal stages transitions. The primary nonideality we will be concerned with is the loading effect on the output flipflop and the associated degradation of rise and fall times.

We will model this effect in conjunction with the mixer nonidealities in order to assess the impact on the system performance. Analog-to-digital converter ADC performance is affected by several parameters. Due to the non-linearity of the active components, difficulties matching between the various components, process variations, circuit noise, and finite conversion time, the actual performance of a fabricated ADC is less than ideal.

In addition, clock jitter appears as an excess noise that further degrades performance. These nonidealities appear as offset, gain, absolute accuracy, integral non-linearity INL , and differential non-linearity DNL errors. Increasing the sampling frequency enlarges the nonidealities.

In this section, we provide details on our AIC inventions. For illustration only, recall Radar scenario 1—Surveillance in entertainment noise wherewe aim to detect and estimate the parameters of a number of linear-FM radar chirps in the presence of a small number of strong sinusoidal TV interferers and noise. First, assume that the chirps and sinusoids are chosen from known dictionaries of possibilities.

Ultimately, the number of recoverable signal elements depends on the incoherence, which in turn depends on the minimum guaranteed separation of parameters. Our first measurement system see FIG. From this reduced set of samples, it is possible to recover information about the signal x t using greedy pursuit.

Design of such a system will be facilitated by considering its analogous discrete-time system for more see Section 4. This is analogous to the discrete-time measurement process, in which each measurement is collected as an inner product. In our setting, each measurement corresponds to an inner product of x against a shifted and time-reversed copy of h. A collection of distinct inner products can be obtained by sampling the output of the filter at different times. The discrete-time setting provides a useful visualization; see Section 4.

Consider a discrete-time signal f, and suppose that we filter f using a discrete-time filter h. For this illustration we ignore the border artifacts caused by finite signal lengths. This inner product can be minimized in variety of ways. Other possibilities for h include wavelet scaling functions, which are orthogonal at scaled integer shifts, or coding sequences such as Barker or other codes see F.

Such a structured measurement system could also offer computational benefits during reconstruction. Greedy algorithms could save on computing inner products by exploiting both the sparsity and the fact that the columns are also shifts of each other. Such a system could progressively recover the signal from left to right in time as more measurements are received.

Of course, the exact design of the system must consider the nuances of the analog front end. In order to design the described hardware for generic incoherent filtering, we utilize a filter that pseudo-randomly changes the magnitudes and phases of the input signal spectral components as shown in FIG. Such incoherent analog filters can be implemented by either a single complex filter or a bank of simple parallel filters selected in a pseudorandom fashion by employing accurate switches with minimized jitter.

This embodiment of the invention provides an efficient implementation of a measurement matrix with the qualities required for successful CS reconstruction, such as incoherence and randomness. The incoherent analog filter can be built such that there is a large randomness in its phase response.

This is required in order to reach a sufficient incoherence with the dictionary. This would require a very high order transfer function. Such very high order transfer function is very difficult to be implemented in VLSI without degrading the frequency response, linearity, power consumption, and the area of our system.

The transfer function of such filter will be constant and hence the randomness will be fixed and that is may degrade the performance of the DSP part of our system. This family of filters is well known in wireless communications. SAW filters rely on the conversion of an electrical signal into acoustic waves through the use of piezoelectric substrate and Inter-Digital Transducers IDT. In this type of filters the frequency response of the filter can be specified by building the IDTs in the shape of the time response of the filter.

Therefore, by building the IDTs in discontinuous shapes, we can have random phase without affecting the magnitude response of the filter. The randomness of the phase depending on the amount of discontinuity in the IDT's shape and hence cause the randomness to be limited by the fabrication shape. A third approach uses a bank of parallel low order different transfer function analog filters, which can be connected to the circuit via switches.

This architecture randomly connects different combinations of filters to generate a pseudo-time variant transfer function with sufficient randomness in phase. Such bank of analog filters can be built using Gm-C, switched-C, and ELIN circuits depending on the required frequency response, linearity and complexity area. Other possibilities filters include wavelet scaling functions, which are orthogonal at scaled integer shifts, or coding sequences such as Barker or other codes, which have peaked autocorrelations.

The sampler after the incoherent analog filter operates at the measurement rate which can be much lower than the Nyquist sampling rate. The measurement rate is basically determined by the sparsity of the signal, rather than the Nyquist frequency. Our system design incorporates a sigma-delta ADC for this purpose. In this section, we discuss the use of discrete-time random filters as a method for incoherent filtering in the purely discrete setting.

Although this section focuses on the setting of discrete signal reconstruction, a mathematical framework for continuous signals has been presented in Section 4. The extension of the AIC 1 implementation described in this section to such case will be obvious from the description, or may be learned by practice of the invention.

We now describe random filters as a new paradigm for discrete-time compressive signal acquisition. Our approach captures a discrete-time signal s by convolving it with a random-tap FIR filter h and then downsampling the filtered signal to obtain a compressed representation y. Reconstruction of s involves a nonlinear algorithm. An alternative approach shown in FIG. At first glance, one might think this method would convert a signal into garbage.

In fact, the random filter is generic enough to summarize many types of compressible signals. At the same time, the random filter has enough structure to accelerate measurement and reconstruction algorithms. Our method has several benefits:. Measurements are time-invariant and nonadaptive. Measurement operator is stored and applied efficiently. Can trade longer filters for fewer measurements. Generalizes to streaming or continuous-time signals. This section Section 4.

After providing some background information, we discuss two different methods for signal acquisition that take full advantage of the structure of the random filter. We then present a reconstruction algorithm, based on Orthogonal Matching Pursuit OMP , that uses the structure of the random filter to accelerate computations. We then report on extensive numerical experiments, which confirm that random filters are effective tools for signal acquisition and recovery. This model covers many signal classes, including i Poisson processes and spike trains, ii piecewise polynomials and splines, and iii signals in weak l p balls.

Results from the CS literature provide a benchmark for studying the performance of random filters. The costs for CS encoding and decoding depend significantly on the type of measurement matrix. Gaussian and Rademacher matrices require storage and computation O dN for encoding.

Fourier measurement matrices improve storage to O d and encoding times to O d log d. Two different algorithms, OMP and l 1 minimization, are commonly used for signal reconstruction. Reconstruction costs via l 1 minimization have not been reported, but one expects them to be O d 3.

This section defines more precisely what we denote as random filter, and it compares two different methods for determining the compressed signal. In particular, we are interested in the cases where the taps are drawn i from the. Draw a random filter h of length B. Note that the filter requires just O B storage. To take N measurements of a signal s of length d, we must calculate. Method 1: The first method for calculating the measurements performs linear convolution and downsampling simultaneously.

Computing N measurements requires O BN arithmetic operations. This method can be applied in systems where the input s is streaming, since the measurements are localized in time and also time-invariant. Method 2: The second method uses FFTs to calculate the convolution. In this case, we compute. The cost of computing the measurements is O d log d , independent of the filter length or the number of measurements. Compared to Method 1, this calculation may be faster if the filter has many taps.

Note, however, that the entire signal must be presented at once. In practice, these two encoding methods are at least as efficient as anything described in the CS literature. We also note that filtering can be performed with other standard methods, such as overlap-add, but we omit this discussion; see A.

V Oppenheim, R. Schafer, and J. The number of measurements required to capture a class of signals depends on several different factors:. Explaining these tradeoffs theoretically is a major project. One expects that signals sparse in the time domain, i. Yet we present empirical evidence below that random filters are effective for recovering time-sparse signals: a random filter of length d performs as well as a fully Gaussian matrix.

When the filter length decreases, the number of measurements increases somewhat. For signals sparse in the frequency domain, the number of measurements depends weakly on the filter length; a four-tap filter already yields good reconstruction probability. Reconstructing a signal from the compressed data requires a nonlinear algorithm. Both of these approaches can be accelerated using the properties of random filters, and we believe that random filters will also lead to novel reconstruction algorithms that take full advantage of the localization and time-invariance of the measurements.

In this section, we adapt OMP to reconstruct signals. Using the structure of the measurement process, we can implement Algorithm RFR very efficiently. First, upsample r t so it has length d and then convolve it with the time-reversed filter Rh. Using direct convolution, this step costs O dB arithmetic operations. Using the FFT to implement the convolution, the cost is O d log d. We can also apply the orthogonal projector P t efficiently. Then the marginal cost of calculating P t y drops from O tN to O N , which is significant unless the sparsity level m is very small.

We believe that similar gains can be achieved in algorithms for l 1 minimization by exploiting the structure of the random filter. For illustration purposes, we present results from extensive numerical work, which offer compelling evidence that random filters are a powerful approach to compressive signal acquisition and reconstruction. Three related experiments establish that random filters can capture sparse signals in three different sparsity bases. Let us describe the experimental setup. For each data point, we fix a random filter with.

We begin with signals that are sparse in the time domain, i. Recall that this case is challenging due to high coherence. We now consider signals that are sparse in the Fourier domain, i. We performed the same experiment for signals sparse in the Haar wavelet domain.

The results were slightly superior to the first experiment and somewhat worse than the second experiment because the Haar basis is localized in time—but not as much as the Dirac basis. We omit the figure. Two additional experiments examine the performance of random filters for signals with few degrees of freedom per unit time; see M.

First, we attempt to acquire and reconstruct piecewise constant signals. In each trial, the signal has two uniformly random breakpoints and. Finally, we attempt to acquire and reconstruct discrete Poisson processes using random filters. These signals contain spike trains with geometrically distributed interarrival times.

The results appear in FIG. The following trend matches the data well:. The slope can be viewed as the number of measurements required to increase the number of spikes by one; there is a minimal cost of 2. Sections 4. We show that convolution with a random filter followed by downsampling yields an efficient summary of a compressible signal. This approach has some features in common with proposed methods for Compressed Sensing. In particular, random filters are universal, because they can capture a wide variety of sparse signals using a small number of random linear measurements.

In contrast with CS, random filters require much less storage and computation for measurement and for reconstruction. Moreover, random filters are time-invariant and time-localized, so they can be used to acquire streaming signals, and they have potential for real-time applications. AIC implementation through Random Filtering offers a cornucopia of directions for customization; we now describe a subset.

First, there are tradeoffs between measurement and computation costs. In particular, a minor increase in the number of measurements makes the computation much more feasible, which may be useful in some applications. Second, this work can be extended to settings such as compressive sampling of images and video and other high-dimensional data.

In these settings, possible measurement matrix included banded matrices, non-banded matrices, and matrices with a structure such that measurement is only influenced by a small number of pixels or voxels. Finally, analog filters can be designed to approximate a random filter response. In this case, sampling the signal as described in this section will enable to reconstruct the analog input signal well. Along these lines, we consider a second prototype design.

It operates as follows see FIG. Note that the sequence p t must be known at the receiver to perform recovery; this is easily done by sharing the seed of the pseudorandom generator. Compared to AIC 1, this design features a simpler filter h, because it uses the sequence p to introduce incoherence into the system. The utility of such as system for CS measurement can be visualized as follows.

Multiplying x by p is equivalent to convolving the Fourier transforms X w and P w. Lowpass filtering is equivalent to keeping a subset of the remaining coefficients. In this sense, then, this system is analogous to AIC 1, except that it performs CS measurement calculation in the frequency domain.

The two critical design considerations in this system are the modulation sequence p and the filter h. First, the Fourier transform P w must be sufficiently noise-like. Our preliminary empirical results indicate that this criterion is satisfied for a pseudorandom square wave. This is accomplished by ensuring that the chip rate is on the same order as the Nyquist frequency for x.

However, we stress that the ultimate sampling at the end of the system operates at a rate determined by the sparsity of x. In practice, we believe that an ideal LPF will not be necessary. This system also has an interpretation in the discrete-time domain. The filter h corresponds to a Toeplitz matrix, as explained above.

We note that some basis functions such as delta functions are already incoherent with respect to the LPF. Instead, the modulator is necessary for quasi-Fourier dictionaries such as sinusoids and chirps. We can also follow the modulator with a small bank of different filters, to obtain more diversity.

Although the design description focuses on the setting of continuous signal reconstruction, the design can be extended to discrete time signals using the standard CS mathematical framework. The extension of the AIC 2 implementation described in this section to such case will be obvious from the description, or may be learned by practice of the invention. Consider a smooth signal consisting of the sum of 10 sine waves; this corresponds to 10 spikes in the Fourier domain.

We perform several tests; for clarity, the following figures show the results in the Fourier domain. The recovery is correct to within machine precision mean squared error is 2. We next apply noise to the sparse vector see FIG. This demonstrates that the system still performs reasonably well in substantial amounts of additive noise, but more measurements may be required to produce a higher quality result. We consider the case of wideband signals that are time-frequency sparse in the sense that at each point in time they are well-approximated by a few local sinusoids of constant frequency.

As a practical example, consider sampling a frequency-hopping communications signal that consists of a sequence of windowed sinusoids with frequencies distributed between f 1 and f 2 Hz. It is well known that signals that are localized in the time-frequency domain have compact transformations under the Gabor transform, which is defined as. Academic Press, We will leverage this compact nature during the reconstruction of the signal to obtain a representation directly in the time-frequency domain, without performing reconstruction of the original time signal.

The conventional tool for this class of signals is a spectrogram. A spectrogram is assembled using the magnitude of short-time Fourier transforms STFT that performs Fourier analysis of shifted windowed versions of the input signals to establish frequency content at local time neighborhoods. The STFT is written as. This tool provides a visual representation of the Fourier spectrum of a signal over time.

The spectrogram can be thought of as a uniform sampling of the coefficients of the signal under the Gabor transform. In this fashion, our sparse reconstruction of the signal will be obtained directly in the time frequency domain—we observe the spectrogram directly without requiring reconstruction of the original signal. An example is shown in FIG. We see that for small ranges of time, the signal is well identified by its carrier frequency, but when we consider the whole signal length there are many carriers to isolate.

The spectrogram pictured in FIG. The carriers in the reconstruction are easily identified. The noise appears due to the non-sparse structure of the input signal; however, its compressibility allows us to recover the largest components.

As a bonus, when the we reconstruct the sparse representation a from our measurements y, the values in a directly correspond to the coefficients in the spectrogram. Then the output from the multiplier is fed to the integrator for a second manipulation stage, with its output sampled by a low-rate ADC to obtain the low-rate representation of the information. This feature allows the decoder to re-generate the pseudo-random sequence in the reconstruction algorithm.

The MLFSR is reset to its initial state every time frame, which is the period of time that is captured from the simulations and fed to the frame-based reconstruction algorithm. The time-frame based operation imposes synchronization between the encoder and the decoder for proper signal reconstruction. To identify the beginning of each frame, header bits can be added in the beginning of each data frame in order to synchronize the decoder; the overhead in the number of data bits is much smaller than the data rate compression of the decoder.

However, the system is time-varying because the random number generator has different values at different time steps. The clock frequency of the random number generator is MHz. The output of the demodulator is low-pass filtered as shown in FIG. In FIG. The SFDR is the difference between the original signal amplitude and the highest spurs.

Higher SFDR values can be obtained by increasing the sampling frequency. Transistors M 1 through M 6 , , utilize a double balanced Gilbert cell, which is the core for the modulator multiplier. Transistors M 1 through M 4 , are connected to the high-frequency modulating signal, which is the random number pattern at 2 GHz. Transistors M 5 and M 6 are connected to the low-frequency analog input signal that has to be modulated.

We chose the double balanced Gilbert cell topology because of its ability to operate as a four quadrant multiplier as well as its high common mode rejection ratio. In order to realize the low-pass filtering, we utilized a simple integrator circuit, as shown in FIG. Although the integrator is the simplest low-pass filter, it is very effective in the reconstruction process. We used differential input differential output RC-active integrator in order to minimize the noise effect on the signal.

The values of the resistors and the capacitors determine the closed loop gain value and the cutoff frequency of the integrator. The main source of non-ideality in the integrator circuit is the finite time constant that arises from the limited amplifier gain. The example shown in FIG. The modulator output in FIG. The integrator is used to smooth the fast variations in the randomized signal and provide a signal that carries the sparse signal information in a much lower bandwidth FIG.

In order to demonstrate the implementation, we built the transistor level implementation for each block and developed the designs to be suitable for best linearity and highest frequency operations. While the CS literature has focused almost exclusively on problems in signal reconstruction, approximation, and estimation in noise see E. E Duarte, D. Such tasks, such as detection, do not require a reconstruction of the signal, but only require estimates of the relevant sufficient statistic for the problem at hand.

Our key finding is that it is possible to directly extract these statistics from a small number of random projections without ever reconstructing the signal. Both of these bode well for many applications. As in reconstruction, random measurements are universal, in the sense that with high probability the sufficient statistics can be extracted from them regardless of the signal structure.

Although this section focuses on the setting of discrete signals, a mathematical framework for continuous signals has been presented in Section 4. The extension of the framework described in this section to such case will be obvious from the description, or may be learned by practice of the invention.

This section of the document is organized as follows. We first provide background on CS. Next, we state our detection problem and formulate a greedy algorithm for CS detection. We then present a case study involving wideband signal detection in narrowband interference. We also give ideas for extensions to classification, and we end with suggestions for some straightforward extensions of these ideas.

The amount of oversampling c required depends on the nonlinear reconstruction algorithm. Most of the existing literature on CS see E. Greedy reconstruction algorithms build up a signal approximation iteratively by making locally optimal decisions, see J. Duarte, M. Mallat and Z. Signal Processing , Vol. Update the residual and the estimate of the coefficient for the selected vector. Increment t.

The implications of CS are very promising. CS with random measurements is advantageous for low-power and low-complexity sensors such as in sensor networks because it integrates sampling, compression and encryption of many different kinds of signals, see J.

However, several significant challenges to CS-based signal reconstruction remain. Greedy algorithms use fewer computations, but even larger c. While the CS literature has focused almost exclusively around signal reconstruction, we now show that incoherent measurements can also be used to solve signal detection problems without ever reconstructing the signal. In the process, we will be able to save significantly on both the number of measurements and computational complexity see FIG.

That is, we must decide between two composite hypotheses:. To begin, suppose that the signal x is provided directly. This is analogous to multiuser detection, a classical problem in communications which is known to be NP-hard, see S. The practical near-optimal iterative decoding algorithm known as successive cancelation or onion peeling is very similar in spirit to MP.

This suggests that for our detection problem we should employ a greedy algorithm such as MP from Section 4. This strongly motivates an MP approach to solving the sparse detection problem under incoherent measurements.

Furthermore, it may be possible to extract the sufficient statistics of interest from a smaller number of incoherent projections; as in reconstruction, random measurements can give us a universal representation of the sufficient statistics, i. Rather, we generally only need to know whether there is a significant contribution from these elements. This allows us to considerably reduce the number of measurements and computation required when detecting compared to reconstructing.

Additionally we anticipate that a minimal number of projections will be necessary to accurately reconstruct the necessary sufficient statistics. Second, we replace Step 5 with the following:. However, it may still contain sufficient information for detection.

Indeed, the detection process can succeed even when M is far too small to recover x using any convergence parameter. Thus, the number of measurements can be scaled back significantly if detection, rather than reconstruction, is the ultimate goal. IDEA is very well suited to detecting signals in the presence of interference and noise when the signals and interference can be sparsely represented in distinct, incoherent dictionaries. We formalize the problem as follows. We aim to distinguish between two hypotheses.

IDEA offers several advantages in this detection scenario. First, the sparsest approximation of y will tend to correctly separate the contributions from the signal and interference components thanks to the incoherency of the two dictionaries. Second, the additive noise is attenuated during sparse approximation since its energy is distributed over all of the expansion coefficients, see S. Wideband Signals in Strong Narrowband Interference. As a concrete example, we study the problem of detecting from random measurements the presence of weak wideband signals corrupted by strong interfering narrowband sources and additive noise.

This is a potentially difficult problem: The weakness of the wideband signal precludes an energy detection approach, and if the wideband and narrowband signals overlap in the frequency domain, then bandpass interference suppression may damage the signal beyond detectability. We seek to detect wideband signals that are frequency modulated chirps.

Chirps are sparsely represented in the chirplet dictionary see R. Hence, we can apply IDEA directly. To illustrate the ability of such measurement systems to preserve the information in x, we give a simple example in the traditional discrete CS setting. Donoho and Y. This measurement rate is 6 times lower than the highest digital frequency, indicating the promise for significant savings over Nyquist for large problem sizes.

Detection vs. Given the same measurements, we also attempt to detect the presence of a wideband component s; detection P e as a function of M averaged over 10, trials is also given in FIG. Effect of interference: We now focus exclusively on detection performance. However, with few measurements, some of the interference energy is incorrectly assigned to the signal components, which corrupts performance.

We see a graceful performance degradation as the SNR decreases; indeed, when the power of the noise becomes comparable to that of the signal to be detected, most detection methods suffer. Effect of quantization: FIG. Note in particular that the detection performance is remarkably robust with 4-bit 16 level quantization; we expect the acceptable level of quantization to be dependent on the SIR and SNR of the received signal. The properties of incoherent measurements allow us to formulate a simple algorithm for sparse signal classification.

Our goal is to determine the class to which the signal best fits. If the signal were available, then we could perform sparse approximation using each one of the bases and choose the class that gives the sparsest representation. This requires a set of N signal samples to make the decision. We can also use random projections; due to their universal quality, cK measurements will suffice to find the sparsest representation of a signal that belongs to any of the C classes. Algorithms such as Orthogonal Matching Pursuit see J.

The incoherence between the bases for different classes implies that only one class will have a sparse representation for the given signal. IDEA provides reliable detection performance from just a few incoherent measurements when the signals of interest are sparse or compressible in some basis or dictionary. In addition to its efficiency gains over CS reconstruction in terms of the number of measurements and computations required, IDEA shares the many known benefits of CS reconstruction, including the universality of random measurements, progressivity, and resilience to noise and quantization; see E.

The extension of IDEA to other signal and interference scenarios is straightforward. When the sparse signal decomposition can be parameterized, i. In particular, detection experiments with the efficient random filtering approach of Section 4. While our examples here have used 1-D signals, IDEA applies to signals of any dimension, including images, video, and other even higher-dimensional data.

One potential application would be using IDEA to detect edges in incoherent measurements of images via sparsity in the 2-D wavelet or curvelet domains. The foregoing description of the preferred embodiments of the invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the invention.

The embodiments were chosen and described in order to explain the principles of the invention and its practical application to enable one skilled in the art to utilize the invention in various embodiments as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto, and their equivalents. The entirety of each of the aforementioned documents is incorporated by reference herein.

A method for approximating a continuous time signal s using compressed sensing, comprising the steps of: in a scheme where said signal s has bandwidth n, taking a set of measurements y for the signal s, where y comprises a vector with only m entries, where m is less than 2n.

A method for approximating a continuous time signal s using compressed sensing according to claim 1 , further comprising the step of: from said set of measurements y, producing an exact reconstruction of said signal s. A method for approximating a continuous time signal s using compressed sensing according to claim 2 , where the reconstruction of said signal s comprises one or more of the following methods: sparse approximation algorithms; l 0 minimization; l 1 minimization; greedy algorithms; optimization algorithms; complexity-regularization algorithms; homotopy-based algorithms; group testing algorithms, Viterbi algorithms and belief propagation algorithms.

A method for approximating a continuous time signal s using compressed sensing according to claim 1 , further comprising the step of: from the said set of measurements y, producing an approximate reconstruction of said signal s. A method for approximating a continuous time signal s using compressed sensing according to claim 1 , wherein said signal s is well approximated by a k-parameter representation. A method for approximating a continuous time signal s using compressed sensing according to claim 5 , further comprising the step of: from said set of measurements y, producing an exact reconstruction of said k signal parameters.

A method for approximating a continuous time signal s using compressed sensing according to claim 5 , further comprising the step of: from said set of measurements y, producing an approximate reconstruction of said k signal parameters. A method for approximating a continuous time signal s using compressed sensing according to claim 5 , further comprising the step of: from said set of measurements y, producing an approximation to said signal s with quality similar to that given by said k parameter signal representation.

A method for approximating a continuous time signal s using compressed sensing according to claim 5 , wherein said set of k parameters correspond to the coefficients for a linear combination of k functions that returns said approximation to said signal s. A method for approximating a continuous time signal s using compressed sensing according to claim 9 , wherein said k functions comprise one or more of the following types of functions: Dirac deltas, sinusoidal functions, wavelets, linear-FM chirps, chirplets, binary chirp sequence signals, polynomial functions, phase coded waveforms, Barker code pulses, PN sequences, communication signals.

A method for estimating the value a functional f on a continuous time signal s using compressed sensing, comprising the steps of: taking a set of measurements y for the signal s, and processing said set of measurements y to obtain an estimate of the value of the functional f on a signal s. A method for estimating the value of a functional f on a continuous time signal s using compressed sensing according to claim 11 , wherein said processing of said set of measurements y performs reconstruction or approximation of said signal s, followed by evaluation of said functional f on said reconstruction or approximation.

A method for estimating the value of a functional f on a continuous time signal s using compressed sensing according to claim 11 , wherein said functional f applied to said signal s is designed according to the signal processing task desired on the signal s.

A method for estimating the value of a functional f on a continuous time signal s using compressed sensing according to claim 13 , wherein said signal processing task comprises one or more of the following: detection, classification, approximation, dimensionality reduction, parameter estimation, manifold learning, reconstruction.

A method for estimating the value a functional f on a continuous time signal s using compressed sensing according to claim 11 , wherein said number of measurements m taken for said signal is adjusted accordingly to the signal processing task desired. A method for estimating the value of a functional f on a continuous time signal s using compressed sensing according to claim 11 , wherein said processing of said set of measurements y comprises the use of one or more of the following:.

An apparatus for taking measurements y of a discrete time signal s comprising: means for manipulating the signal s of length n;. An apparatus for taking measurements y of a discrete time signal s according to claim 17 , wherein said means for manipulation comprises one or more of the following: modulation, convolution, mixing, multiplication, integration, filtering.

An apparatus for taking measurements y of a discrete time signal s according to claim 17 , wherein said means for manipulation comprises the use of one or more of the following: deterministic signals, random signals, pseudorandom signals, binary sequences, binary communication codes, error correcting codes. An apparatus for taking measurements y of a discrete time signal s according to claim 17 , wherein said means for measuring comprises one or more of the following: uniform downsampling, non-uniform downsampling, random downsampling, uniform quantization, non-uniform quantization.

An apparatus for estimating the value of a functional f of a discrete time signal s comprising: means for manipulating the signal s of length n;. An apparatus for taking measurements y of a continuous time signal s comprising: means for manipulating the signal s in a continuous fashion;. An apparatus for taking measurements y of a continuous time signal s according to claim 22 , wherein said means for manipulation comprises one or more of the following: modulation, convolution, mixing, multiplication, filtering.

An apparatus for taking measurements y of a continuous time signal s according to claim 22 , wherein said means for manipulation comprises the use of one or more of the following: deterministic signals, random signals, pseudorandom signals, binary sequences, binary communication codes, error correcting codes.

An apparatus for taking measurements y of a continuous time signal s according to claim 22 , wherein said means for measuring comprises one or more of the following: uniform sampling, non-uniform sampling, random sampling, sample-and-hold sampling, integrate-and-sample sampling, uniform quantization, non-uniform quantization.

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Self-check sparseness self-adaption matching pursuit arithmetic based on compressive sensing. Data system for interfacing with a remote data storage facility using compressive sensing and associated methods. System with sub-nyquist signal acquisition and transmission and associated methods. Low-pass filtering of compressive imaging measurements to infer light level variation. Apparatus and method for channel estimation using compressive sensing based on Taylor series expansion.

System and methods of compressed sensing as applied to computer graphics and computer imaging. Compensation of compressive imaging measurements based on measurements from power meter. Method for compressive sensing , reconstruction, and estimation of ultra-wideband channels. Modulation and equalization in an orthonormal time-frequency shifting communications system. Method and apparatus for super-resolution video coding using compressive sampling measurements. Methods and apparatuses for detecting anomalies using transform based compressed sensing matrices.

Methods and apparatuses for detecting anomalies in the compressed sensing domain. Method and apparatus for arbitrary resolution video coding using compressive sampling measurements. Method and system for modifying compressive sensing block sizes for video monitoring using distance information. System and method for implementing orthogonal time frequency space communications using OFDM.

Signal separation in an orthogonal time frequency space communication system using MIMO antenna arrays. Systems and methods for symplectic orthogonal time frequency shifting modulation and transmission of data.

Iterative two dimensional equalization of orthogonal time frequency space modulated signals. System and method for providing wireless communication services using configurable broadband infrastructure shared among multiple network operators. Orthogonal time frequency space modulation over a plurality of narrow band subcarriers.

Channel acquistion using orthogonal time frequency space modulated pilot signal. Inconspicuous multi-directional antenna system configured for multiple polarization modes. Iterative channel estimation and equalization with superimposed reference signals. Precoding in wireless systems using orthogonal time frequency space multiplexing.

LFMCW vehicle-mounted radar distance and speed measurement method based on rapid slope mode. Implementation of orthogonal time frequency space modulation for wireless communications. Achieving synchronization in an orthogonal time frequency space signal receiver. Apparatus, method and computer program for processing a piecewise-smooth signal. Method for digitizing analog signals and reconstructing the respective signal therefrom and signal processing path for carrying out the method.

CNB en. Sensor-based wireless communication systems using compressed sensing with sparse data. Compressed sensing reconstruction algorithm based on pseudo-inverse multiplication. Device for processing received signal, and method for processing received signal. Over-complete dictionary constructing method applicable to voice compression sensing. Multichannel random harmonic modulation sampling radar receiver and method thereof. MXB en. Ultrasonic signal time-frequency decomposition for borehole evaluation or pipeline inspection.

System and method for acquisition and processing of seismic data using compressive sensing. A kind of distributed video compressed sensing fast reconstructing method based on degree of rarefication estimation. Adaptive processing system having an array of individually configurable processing components. Estimating a signal based on samples derived from dot products and random projections. Radio frequency RF sampling apparatus with arrays of time interleaved samplers and scenario based dynamic resource allocation.

Method for a radio frequency RF sampling apparatus with arrays of time interleaved samplers and scenario based dynamic resource allocation. Apparatus and method for compressive sensing tap identification for channel estimation. Dedicated power meter to measure background light level in compressive imaging system. System and method for two-dimensional equalization in an orthogonal time frequency space communication system.

Variable latency data communication using orthogonal time frequency space modulation. Integrated circuit implementation of methods and apparatuses for monitoring occupancy of wideband GHz spectrum, and sensing respective frequency components of time-varying signals using sub-nyquist criterion signal sampling. Multiple access in wireless telecommunications system for high-mobility applications.

Channel acquisition using orthogonal time frequency space modulated pilot signals. WOA3 en. Marques et al. Ragheb et al. A prototype hardware for random demodulation based compressive analog-to-digital conversion. Pfetsch et al.

On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion. JPB2 en. Sundman et al. Landau et al. Time Scope performs analysis, measurement, and statistics including root-mean-square RMS , peak-to-peak, mean, and median. Using data cursors to measure time and amplitude differences between two points of a waveform in Time Scope. Spectrum Analyzer computes the frequency spectrum of a variety of input signals and displays its frequency spectrum on either a linear scale or a log scale.

The spectrogram mode view of Spectrum Analyzer shows how to view time-varying spectra and allows automatic peak detection. DSP System Toolbox provides an additional family of visualization tools you can use to display and measure a variety of signals or data, including real-valued or complex-valued data, vectors, arrays, and frames of any data type including fixed-point, double-precision, or user-defined data input sequence.

Some of the visualization tools can show a 3D display of your streaming data or signals so that you can analyze your data over time until your simulation stops. Measuring the frequency and power of spectral peaks generated by applying a nonlinear amplifier model to a chirp signal. You can use DSP System Toolbox with Fixed-Point Designer to model fixed-point signal processing algorithms, as well as to analyze the effects of quantization on system behavior and performance.

You can configure MATLAB System objects and Simulink blocks in the system toolbox for fixed-point modes of operation , enabling you to perform design tradeoff analyses and optimization by running simulations with different word lengths, scaling, overflow handling, and rounding method choices before you commit to hardware.

DSP System Toolbox automates the configuration of System objects and blocks for fixed-point operation. The FFT Simulink block dialog box provides options for fixed-point data type specification of accumulator, product, and output signals, which requires Fixed-Point Designer right. In DSP System Toolbox, filter design functions and the Filterbuilder app enable you to design floating-point filters that can be converted to fixed-point data types with Fixed-Point Designer.

This design flow simplifies the design and optimization of fixed-point filters and lets you analyze quantization effects. Fixed-point filter design analysis of quantization noise where the filter design constraints are not met, and the stop band attenuation is insufficient because of the 8-bit word length left. Experimenting with different coefficient word lengths and using bit word length is sufficient, and the filter design constraints are met right. The generated code can be used for acceleration, rapid prototyping, implementation and deployment, or for the integration of your system during the product development process.

You can generate efficient and compact executable code, a MEX function, tuned for performance to speed up computation-intensive algorithms in your simulation. To accelerate frame-based streaming simulations, dspunfold uses DSP unfolding to distribute the computational load in the generated MEX function across multiple threads. Because this standalone executable runs on a different thread than the MATLAB code or Simulink model, it improves the real-time performance of your algorithm.

The generated C code of your signal processing algorithms can be integrated as a compiled library component into other software, such as a custom simulator, or standard modeling software such as SystemC. A key benefit is an immediate increase in performance when compared to standard C code. You can also perform code verification and profiling using processor-in-the-loop PIL testing.

You can also automatically create VHDL and Verilog test benches for simulating, testing, and verifying generated code. Select a Web Site. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location.

Toggle Main Navigation. DSP System Toolbox. Search MathWorks. Close Mobile Search. Get a free trial. View Pricing. Get Started:. How to Process Signals as Frames in Simulink. Signal Processing Blocks for DSP System Design, Implementation, and Validation Simulink blocks for signal processing support double-precision and single-precision floating-point data types and integer data types.

Simulink model of a multistage decimation filter for a sigma-delta analog-to-digital converter. Frequency response of the individual stages of a multistage digital down converter. How to Filter Signals in Simulink. Sorry, your browser doesn't support embedded videos. Signal Scopes, Analyzers, and Measurements. Fixed-Point Modeling and Simulation.

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Sigma delta converter matlab torrent | We believe that similar gains can be achieved in algorithms for l 1 minimization by exploiting the structure of the random filter. We aim to distinguish between two hypotheses. Some of these approaches have been adopted in the computer science literature for solving related problems. Detection vs. CS with random measurements is advantageous for low-power and low-complexity sensors link as in sensor networks because it integrates sampling, compression and encryption of many different kinds of signals, see J. |

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Qbittorrent webui bannedstory | Even worse, the current pace of ADC development is incremental and slow. Our preliminary empirical results indicate that this criterion is satisfied for a pseudorandom square wave. This type of register is commonly implemented using high-speed flip-flops. Graduated Doctoral Students. However, we stress that the ultimate sampling at the end of the system operates at a rate determined by the sparsity of x. Gerardo Gomez-Martinez. |

Stephen dorff somewhere soundtrack torrent | Other possibilities for h include wavelet scaling functions, which are orthogonal at scaled integer shifts, or coding sequences such as Barker or other codes see F. See J. These blocks process streaming input signals as individual samples or as collections of samples called frames. Rather, we generally only need to know whether there is a significant contribution from these elements. The primary limitation of such an approach is that it pertains only to a limited class of signals. Kurishima, T. |

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