# Towards a Multi-Perspective Time of Flight Laser Ranging Device Based on Mirrors and Prisms
## 2 Materials and Methods
### Walk Error  
When transmitted light is focused in a parallel beam, the amount of the returned light follows an inverse square law with increasing distance to the target. Total received power for diffuse targets may be estimated using a modified radar Equation (1) [23], where \(P_{Rx}\) is received power, \(G_{Rx}\) is receiver gain, \(G_{Tx}\) transmitter gain, \(P_{Tx}\) power of the transmitted signal, \(r\) is the radius of receiving lens (\(\pi r^{2}\) is its area), and \(d\) the distance between measuring device and the target. \(2\pi\) in the equation represents the solid angle, at which the light is reflected. In this case it is assumed that the target behaves as a perfect point source that reflects light evenly in the hemisphere towards the light source. \[P_{Rx}=\frac{G_{Tx}G_{Rx}P_{Tx}\pi r^{2}}{2\pi d^{2}} \tag{1}\]  
The equation does not take into account the target's albedo, which provides information on what portion of light is reflected off of the target. This factor could be added to the equation as a multiplicative factor in the fraction's numerator. The possibility for the target to be non-Lambertian is omitted as well. In such a case, another orientation dependent factor that addresses the material's physical reflective properties should be added to the equation. Since real materials rarely have a uniform reflection pattern, Equation (1) is only good to obtain a rough estimate for diffuse targets, which remains valid so long as the target's orientation in regard to the illuminator and the observer is fixed. To summarize the main takeaways, the further the target or lower its albedo, the less signal is captured by the receiving diode. For real signals with finite rise and fall times, the reduction in amplitude effectively shrinks the pulse width after the comparator. Figure 4a shows the return signal, captured after TIA with the target at different distances. It should be noted that TIA is saturated for signal responses from targets at 80 cm or closer. To make the effect more pronounced, a dark target with low albedo was selected when gathering data for Figure 4a. Traces of different colors represent return signals for measurements with targets at different distances. Which color corresponds to which distance can be read from the legend. For strong return signals, the pulse width can exceed 150 ns, which is notably wider than the transmitted pulse, set at 100 ns. The returned signal's amplitude plays a huge role in when and if the signal will cross the detection threshold, which introduces the walk error. Ideally, neighboring traces in Figure 4a would cross the threshold, set at 1.68 V, 667 ps apart, the same time it takes the light to traverse the roundabout distance between subsequent measurements. In our case, spread between traces was greater than this and varied by signal strength, as observed in Figure 4a. Walk error is a systemic error, inherent to ToF based ranging devices [24]. Some laser ToF ranging devices are constructed so that they do not need additional walk error compensation. More often than not, that is due to having an analog compensation built into the receiver or having a design that is inherently less sensitive to walk error. When that is not the case, walk error compensation has to be implemented to obtain accurate measurements. Common methods for measurement correction for systems with PIN photodiode or APD receivers are based on the signal's time over threshold (TOT) or rise time. The TOT-based compensation function relies on detected pulse width [25], and the later one is based on signal's rise time, measured by the time it takes for the received signal to rise from first threshold to the next [26]. The ranging device used in this article used TOT-based walk error compensation. The measured pulse width is a systemic parameter that only depends on the target reflectivity and distance to the target, assuming that the ranging device's operating parameters, such as supply voltage and temperature, are fixed.