Time-frequency analysis in waveform engineering can be applied to many detection and imaging systems, such as radar, sonar, and ultrasound to improve their Signal-to-Noise Ratio (SNR). Recently, photoacoustic imaging systems have attracted researchers’ attention. However, the SNR optimization problem for photoacoustic systems has not been fully addressed. In this paper, the one-dimensional SNR optimization of the photoacoustic response to an input waveform with finite duration and energy was considered. This paper applied an eigenfunction optimization approach to find the waveform for optimal SNR for various photoacoustic absorber profiles. SNR gains via the obtained optimal waveform were compared with simple square-pulse and pulsed sinusoidal waveforms in simulations. Results showed that by using the optimal waveform, SNR can be enhanced especially if the input wave duration is comparable with the absorber time profile duration. The optimal waveforms can achieve 5–10% higher SNR than square pulses and over 100% higher SNR compared with pulsed sinusoids. The symmetry between time and frequency domains assures similar behavior when temporal durations of the input waveforms are too short or too long compared with the absorber.

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.

Recent developments in photoacoustics have witnessed the implementation of a radar matched-filtering methodology into the continuous wave photoacoustic modality. The main merit of using matched filtering in continuous photoacoustics is the improvement in signal to noise ratio (SNR), but the correlation process may result in a loss of resolution. It is possible to enhance both SNR and resolution by matched-filtering and pulse compression with a frequency chirp. However, the theory behind the effect of the chirp parameters on both SNR and resolution is still not clear. In this paper, the one-dimensional theory of the photoacoustic radar with a pulse compressed linear frequency modulated sinusoidal laser chirp is developed. The effect of the chirp parameters on the corresponding photoacoustic signal is investigated, and guidelines for choosing the chirp parameters for resolution and SNR optimization are given based on theory and simulations. The results show that by judiciously manipulating the center frequency, bandwidth, and duration, the resolution and SNR can be easily enhanced.

Recent developments in imaging systems have seen the implementation of a radar matched-filtering approach. The goal of the imaging system is to obtain information about an unknown object embedded in the system, by controlling the parameters of the input and measuring the response to the known input. The main merit of using matched filtering in imaging systems is the improvement of Signal to Noise Ratio (SNR). However, the correlation process used in matched filtering may result in a loss of resolution. One way to compensate for lost resolution is via pulse compression. Linear frequency modulated sinusoidal waveforms (chirps) have the property of pulse compression after correlation. Hence, both SNR and resolution can be enhanced by matched-filtering and pulse compression with a chirp. However, the theory behind the effect of chirp parameters on resolution is still not clear. In this paper, a one-dimensional theory of matched-filter imaging with a pulse compressed linear frequency modulated sinusoidal chirp is developed. The effect of the chirp parameters on the corresponding signal is investigated, and guidelines for choosing the chirp parameters for resolution considerations are given based on the developed theory and simulations. The results showed that by manipulating the center frequency, bandwidth, and duration of the chirp, the resolution can be easily enhanced.

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