A Gabor sampling theorem and some time-bandwidth implications

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

Multi-window Gabor expansion for evolutionary spectral analysis

Current generation radars employ a first order approximation to compensate the frequency shift due to target's velocity. This approach is inadequate when high velocity targets are considered; however, real targets do not possess such high velocities. For that reason, this research analyzes how the increment of the Time-Bandwidth (TBW) product and the target velocity would impact the analysis of radar signals when a first order approximation is implemented to compensate the Doppler shift. The common approach to improved resolution and performance of radar system is to increase the time-bandwidth product of the transmitted signal. The problem of using the first order approximation to compensate the Doppler is that it is limited only to the first two terms of a power series expansion of a full Doppler compensation. As a consequence, an increment on the target's velocity or the time-bandwidth product of the transmitted signal will result in a significant error at the output of the matched filter. On this research a Linear FM (chirp) signal with a large TBW is considered as the transmitted signal. First, to observe the effects of increasing the time-bandwidth product and target velocity the received signal is modeled using a full Doppler compensation and a first order approximation. Second, each signal is applied to the input of one matched filter in which the transmitted signal is used as a reference. Finally, the outputs from both matched filters are analyzed in order to observe the effects of using the first order approximation to model the Doppler induced on the reflected signal. This analysis was performed assuming that the target was moving at a constant velocity. By increasing the time-bandwidth product of the transmitted signal the output of both matched filters are compared and analyzed to observe the differences between modeling the reflected signal using the first order compensation and the full Doppler compensation. The simulation results showed that, by increasing the time-bandwidth product of the transmitted signal the output of the matched filter using the first order approximation deviates significantly with respect to the matched filter that contains the signal modeled using the full Doppler compensation. From these results it is concluded that a dramatic increase in time-bandwidth product of the received signal, results in a significant error at the output of the matched filter if the first order approximation is used to model the reflected signal instead of the Full Doppler compensation.

The use of a self-consistent adjustment of the local acoustic beamwidth in the design of slanted array compressors (SACs) is effective for the case of wide bandwidth, moderate to high time-bandwidth (TB) product devices. Difficulty arises in the design of devices of relatively small fractional bandwidth and low TB product, in particular, when spectral weighting is also required. A solution to this difficulty requires the inclusion of predetermined peripheral circuit elements during design synthesis. This effects a different distribution among the various loss factors, and reduces the burden of beamwidth adjustment, thus alleviating the need for extreme source contour incline angle and curvature. Several peripheral circuit types are considered for the design of an 8% fractional bandwidth, 50 TB product, weighted L-band SAC pulse compressor. These include a single shunt resistive element, and a shunt resistor with a shunt capacitor. A post-fabrication selectable on-chip printed resistor is discussed. Results from devices achieving 26-dB midband insertion loss and greater than 26-dB suppression of the compressed pulse time sidelobes are shown. >

: A method is presented for linearly generating large time-bandwidth signals by tapped delay line filter synthesis techniques. The method is de veloped by an example wherein a filter with a constant amplitude and linear time delay transfer function of arbitrary time-bandwidth product is synthesized. Special emphasis is placed on equipment simplicity and ease of implementation. The essential elements employed by the technique are quartz delay lines, bandpass filters, band pass phase shifters and resistive weighting net works. The techniqu1es not make use of many elements which are found in most conventional pulse compression systems. In particular, dis persive networks, mixers, frequency synthesizers, and precision bandpass filters are not required. Consequently, the total number of components is small, even for time-bandwidth products of a few thousand, thereby making it feasible to obtain good peak signal to 'hash' sidelobe levels at the matched filter output.

This letter investigates the effect of compactness of a pulse shaping filter on the inter-symbol interference (ISI) by presenting a theoretical equation which directly expresses the bit error rate (BER) in terms of quantified compactness of the pulse. We numerically compute time-bandwidth product (TBP) of the considered pulses as the measure of compactness. To compute the TBP, we use four definitions of pulse width (i.e., time duration in time domain and bandwidth in frequency domain) including root mean square (RMS), full width at half maximum (FWHM), 99% energy concentration and - 40 dB bounded energy duration. We show that the standard pulses poses different compactness properties based on the defined TBP and the - 40 dB definition is the most effective definition to study ISI versus TBP. We also propose a new pulse which offers the least TBP based on - 40 dB definition among all standard pulses and show that this pulse outperforms these well-known pulses in terms of BER and bandwidth efficiency.

Because active sonar systems are typically peakpower limited, and because sonar targets can often appear at speeds within the doppler spread of reverberation, pulses having large time-bandwidth (TW) products are called for. The resultant high resolution in range and doppler creates a potential problem because of multipath. Multipath is caused by multiple reflections from the two rough surfaces -- the ocean surface and the ocean bottom -- from which signals are reflected, refraction, eand by multiple reflections from the target itself. Due to motions of the echo ranging ship, the target, and the ocean surface, the multipath exhibits doppler spreading as well as time spreading. A model is postulated for the overall spreading of the signal seen at the receiver in which the magnitudes of time spreading and doppler spreading are represented by L and B. Detection performance of signals with large pulse length (T) and bandwidth (W) is derived as function of WL and BT. Performance is measured in terms of input signal level required to produce specified probabilities of detection and false alarm at the processor output. Two basic signal codes are considered -- random modulation in amplitude and/or phase (RM) and linear frequency modulation (LFM). Detection performance of a single correlator, tuned to one range and doppler, is given. Next, a processor is postulated consisting of a multiplicity of correlators covering the entire expected range and doppler spread of the signal, the outputs of which are combined incoherently.

A phase modulator is described which can directly deviate a crystal-controlled carrier by 1000 rad or more. Frequency multiplication or auxiliwry automatic frequency control (AFC) circuits are therefore not required. The modulator uses an accurate frequency source, a wide-band voltage controlled oscillator, a delay line, and two mixers The achievable phase deviation is proportional to the time-bandwidth (TB) product of the delay line, while the modulation bandwidth is inversely proportional to the time delay. Suitable delay lines are available using either bulk wave or surface acoustic wave transnission. Signal-to-noise ratio (SNR) and distortion are compatible with requirements for broadcast service. Design examples are given for monophonic and stereophonic FM broadcast modulators.

This paper describes the Impulse Radio Ultra-Wideband (IR-UWB) system using linear chirp UWB pulses as symbols. The novel method of coherent or differentially coherent detection of chirp pulses in multipath channels is introduced. The method divides detection in the receiver between its analog and digital part; in the analog part of the receiver, received signal is compressed in frequency by mixing with locally generated chirp pulse and low-pass filtering. In this way, the time-bandwidth (TB) product of the resulting signal is reduced compared to the received signal and made approximately independent of duration of the transmitted signal. This results in relatively low complexity digital back-end of the receiver that is able to capture a large portion of the multipath energy, regardless of duration of the transmitted waveform. The work includes derivation of analytical properties of the system, including derivation of loss parameters that quantify system multipath performance. Numerical results include discussion on the system multipath loss performance with varying chirp duration. As well, a comparison is made with error performance of sampling receivers featuring the same computational complexity. Analysis of the robustness of the system's performance to different chirp pulse implementation errors is also provided.

The applications of microwave optoelectronics are extremely large since they extend from the Radio-over-Fibre to the Homeland security and defence systems. Then, the improved maturity of the optoelectronic components operating up to 40GHz permit to consider new optical processing functions (filtering, beamforming, ...) which can operate over very wideband microwave analogue signals. Specific performances are required which imply optical delay lines able to exhibit large Time-Bandwidth product values. It is proposed to evaluate slow light approach through highly dispersive structures based on either uniform or chirped Bragg Gratings. Therefore, we highlight the impact of the major parameters of such structures: index modulation depth, grating length, grating period, chirp coefficient and demonstrate the high potentiality of Bragg Grating for Large RF signals bandwidth processing under slow-light propagation.

Recent results obtained with helical surface-acoustic-wave (SAW) delay lines of large time delay and bandwidth are described. Both unguided and guided propagation are involved, with time delays up to one millisecond and bandwidths up to 65 MHz being observed. Fiber delay lines of both capillary and cladded types are also discussed. The potential for future application of SAW delay lines of large time-bandwidth product to high-speed signal-and data-processing systems is considered.

Correlation-processing techniques for recovering information from a signal contaminated by noise are being used very fruitfully today in fields ranging from radar to medicine. Processing of this type allows us to make use of a priori information which we may have, or of the results of repeated observations. Our ability to use a priori information or repeated experiments is in general limited only by the number of degrees of freedom or the time-bandwidth product of the correlator. This paper discusses: (1) the general principle involved in correlation processing, (2) a variety of correlators which are based upon delay lines and (3) two interesting applications, one in medicine and one in radar. In conclusion, a chart is shown which summarizes in a time-band width plane plot, the characteristics of most of the delay-line correlators known to the writer. This diagram shows regions where improvements in the technology will be of immediate profit in information processing applications.

A symmetric acousto-optic time-integrating architecture to compute the cross ambiguityfunction is described. An application of this new architecture is the generation of range-Dopplerimages from laser radar returns. This symmetric architecture has several advantages over theasymmetric time-integrating architecture, particularly in lowered optical power requirements andthe elimination of background ridges. Specific advantages of the reduced optical powerrequirement are that 1) laser diodes are more likely to be able to satisfy the coherent optical powerrequirement, and 2) the risk of optical damage to the Bragg cells is diminished.1. INTRODUCTIONWideband laser radar has the potential to provide high resolution range-Doppler imaging. It ishighly desirable to be able to generate a steady stream of range-Doppler images from such a radarhowever, the computational burden is enormous, especially for digital solutions. Acousto-optics is ideallysuited to provide this computational capability, particularly if the application places tight constraints on thesize, weight, and power consumption of the processor.For a wideband radar signal withcomplex modulation (ideally pseudo-random noise modulation),the cross ambiguity function (CAF) of the the radar return signal and reference signal is the desiredrange-Doppler image, where the reference signal is an exact replica of the transmitted radar signal.To obtain range-Doppler images which have high resolution in both range and Doppler the signaland coherent processing duration must equal the inverse of the desired Doppler resolution, and the signaland processing bandwidth must equal the inverse of the desired range resolution. As a result, the requiredproduct of coherent processing time and processor bandwidth requirement may far exceed 1000, thetypical upper limit for acousto-optic (AO) space-integrating CAF architectures. In such cases, an AO time-integrating CAF architecture is appropriate because time-bandwidth products of up to a million areobtainable.The key factor in the design trade-off of an AO processor is dynamicrange. To maximize dynamicrange, enough light must reach the photodetector array of the AO processor to nearly saturate thosephotosites corresponding to a target. For limited duration radar pulses this requirement can lead to larger

White light diffraction phase microscopy with slightly off-axis blind two-step phase-shifting

The design and performance of a fully-synchronous multi-GHz analog transient waveform recorder I.C. ("SST") with fast and flexible trigger capabilities is presented. The SST's objective is to provide multi-GHz sample rates with intrinsically-stable timing, Nyquist-rate sampling and high trigger bandwidth, wide dynamic range and simple operation. Containing 4 channels of 256 samples per channel, the SST is fabricated in an inexpensive 0.25 micrometer CMOS process and uses a high-performance package that is 8 mm on a side. It has a 1.9V input range on a 2.5V supply, exceeds 12 bits of dynamic range, and uses ~128 mW while operating at 2 G-samples/s and full trigger rates. With a standard 50 Ohm input source, the SST exceeds ~1.5 GHz -3 dB bandwidth. The SST's internal sample clocks are generated synchronously via a shift register driven by an external LVDS oscillator running at half the sample rate (e.g., a 1 GHz oscillator yields 2 G-samples/s). Because of its purely-digital synchronous nature, the SST has ps-level timing uniformity that is independent of sample frequencies spanning over 6 orders of magnitude: from under 2 kHz to over 2 GHz. Only three active control lines are necessary for operation: Reset, Start/Stop and Read-Clock. When operating as common-stop device, the time of the stop, modulo 256 relative to the start, is read out along with the sampled signal values. Each of the four channels integrates dual-threshold trigger circuitry with windowed coincidence features. Channels can discriminate signals with ~1mV RMS resolution at >600 MHz bandwidth.