The electronic hardware of small animal imaging (SAI) systems requires a certain number of analog-to-digital converters (ADCs) to digitize the pulses generated by the gamma ray detectors. These pulses must be digitized by ADCs that are appropriate to the purpose, the cost of which can become prohibitive in very large SAI systems. This post presents a method to reduce the number of ADCs to design a cost-effective SAI implementation.

The signals that an SAI system has to digitize and analyze are very short pulses generated by the instrument’s array of detectors. These pulses are digitized using an analog-to-digital converter (ADC) that produces a sequence of samples, which are numbers representing the instantaneous amplitude of the pulses at discrete intervals in time. The digitized output of the ADC is typically sent to an FPGA (field programmable gate-array) device. ## Determining The Optimal Sampling Rate

One of the major factors that has to be determined is the optimal rate at which the ADC must generate its digitized samples. It must be fast enough to accurately represent the pulses from the sensor, but not so fast as to require an expensive ADC or to create an unnecessarily large quantity of data to be processed by the FPGA. The optimal sampling rate can be determined from the shape of the detector’s pulses, more specifically from the fastest part of the pulse, which typically corresponds to its rise time. By convention, the rise time, Tr, is measured by the time taken by the pulse to rise from 10% to 90% of its maximum amplitude. From the value of the rise time, it is possible to determine the actual frequency bandwidth of the pulses (the range of frequencies or frequency spectrum occupied by the pulses). The bandwidth can be computed by the following simple formula

: Based on the Nyquist sampling theorem , an analog signal can be faithfully converted to a digital equivalent if it is digitized at a sampling frequency that is at least twice its bandwidth: To be conservative, a certain tolerance margin should be taken into account, so the bandwidth factor could be increased from 0.7 to 1.0: Assuming, for example, that a pulse has a rise time of 100 nsec, the ADC sampling rate to accurately digitize the pulse would have to be higher than 10 MHz: ## Multiplexing The Received Small Animal Imaging Signals To Achieve Cost Saving

SAI systems simultaneously analyze signals originating from a multitude of sensors (detectors). These signals must be digitized prior to being processed by digital signal processing device such as an FPGA or standard computer. A typical multi-channel acquisition system might look like the following, where multiple ADCs are used in parallel: Considering that the number of channels in an SAI system can sometimes be very large, the total cost of the ADCs could eventually become an important factor in the system’s final selling price. In order to minimize the number of ADCs, some of all of the ADCs could be used to digitize multiple analog input channels. Such a configuration is shown below. This process is known as multiplexing. The input of the ADC is driven by an analog selector switch that routes the selected input channel to the ADC. Control lines are used to determine which channel is selected. These control lines can be binary-encoded (as shown in the figure above) to minimize the number of lines, or they can simply be assigned individually for each channel, providing that there is only one active line at a time. The control lines are presented to the FPGA so that it can identify which analog channel the ADC is currently digitizing. The FPGA could also be driving the control lines, depending on where the SAI system’s control is centralized.

Selection of the input channel can be performed in several different manners. In a typical SAI configuration, the control lines are driven by an external circuit that detects a gamma ray pulse on a certain channel and then automatically “points” the ADC, using the selector switch, to the corresponding channel by activating the appropriate control lines.

In other configurations, the ADCs and the multiplexers are used on a continuous basis. It should be noted that in cases where the ADC is multiplexed in a repetitive sequence so that the channels are digitized in a round-robin fashion, the ADC should be capable of operating at a sampling frequency that is proportionally higher in order to respect the bandwidth of each individual input signal, as discussed previously.

The total number of channels in an SAI system sometimes needs to be very large, while at the same time a certain number of events needs to be simultaneously detected and recorded. To address these two requirements, a hybrid configuration, using both parallel ADCs and channel multiplexing, can be used. Such a configuration is shown below. Configurations like the one shown above have been successfully used in PET systems supporting hundreds of detectors and only a handful of ADCs . The proper ratio between ADCs and multiplexing is determined based on the needs of the specific SAI system configuration. This kind of parallel/multiplexed configuration offers an excellent compromise between the number of simultaneous events that can be detected and the number of ADCs required to design an SAI system. It can significantly reduce the size, cost and power usage of a system by reducing the number of ADCs used by the system.

In this post, we looked at the parameters used to determine an appropriate number of ADCs for a given SAI system. We looked at the different possible configurations for the ADCs (parallel, multiplexed and parallel/multiplexed), and determined that the parallel/multiplexed configuration offers many advantages in reducing the size, complexity, power and cost used in the design and production of an SAI system.

References

1. Hu, Wei, et al. 2011. “Free-running ADC-and FPGA-based signal processing method for brain PET using GAPD arrays.” Nuclear Instruments and Methods in Physics Research A. doi:10.1016/j.nima.2011.05.053
2. McNeil, John A. 2011. “Bandwidth – Risetime Relationship”. Course handout. Worcester Polytechnic Institute, Worcester MA. . Accessed February 28, 2013. http://ece.wpi.edu/~mcneill/handouts/BW-tr.pdf
3. Wikipedia. 2013. “Nyquist-Shannon sampling theorem.” Accessed February 28, 2013. http:/https://nutaq.com.wikipedia.org/wiki/Sampling_theorem – Critical_frequency