# Signal to Noise Ratio of a Coded Slit Hyperspectral Sensor
## 2 Materials and Methods: The Signal to Noise (SNR) Model
### The SNR Model for Assessing the Utility of CA Multiplexed Sensing
Let \(\boldsymbol{\varphi}=H^{-1}(\boldsymbol{y}_{tot}-d-r)\) which represents the estimation of \(\boldsymbol{f}\) from the measurement \(\boldsymbol{y}_{tot}\). According to (1) and (2) it can be seen that \(\langle\boldsymbol{\varphi}\rangle=\langle f\rangle\). Furthermore, it can be demonstrated that \(\text{Var}(\boldsymbol{\varphi})\
eq\text{Var}(\boldsymbol{f})\). Let \(H_{i,j}^{-1}\) denote the element of \(H^{-1}\) of row \(i\) and column \(j\); let the asterisk symbolise the row or column of a matrix, i.e., \(H_{j,*}\) represents the element of \(H\) for the \(j^{\text{th}}\) row in \(H\); and \(\varphi_{n}\) denotes the \(n^{\text{th}}\) element of \(\boldsymbol{\varphi}\). Then,  
\[\text{Var}(\varphi_{n})=\sum_{j=1}^{N}\left(H_{n,j}^{-1}\right)^{2}\text{Var} (\boldsymbol{y}_{tot}-\langle d\rangle-\langle r\rangle)=\sum_{j=1}^{N}\left( H_{n,j}^{-1}\right)^{2}\left(H_{j,*}\langle\boldsymbol{f}\rangle+\langle d \rangle+\sigma^{2}\right) \tag{3}\]  
\(H^{-1}\), which represents the inverse of the binary Hadamard Matrix, has the property [6] of having the same absolute value for all its element of: \(H_{i,j}^{-1}=\pm\frac{2}{N+1}\) which implies that \(\left(H_{i,j}^{-1}\right)^{2}=\frac{4}{\left(N+1\right)^{2}}\) thus Equation (3) can be written:  
\[\text{Var}(\varphi_{n})=\frac{4}{\left(N+1\right)^{2}}\sum_{i=1}^{N}\left(N \left(\langle d\rangle+\sigma^{2}\right)+\langle f_{i}\rangle\sum_{j=1}^{N}H _{j,i}\right)\]  
Note that the sum of any row (also for any column) of the binary Hadamard matrix of size \(N\) can be given by \(\left(N+1\right)/2\) (that is \(\sum_{j=1}^{N}H_{j,i}=\frac{N+1}{2}\)). Thus:  
\[\text{Var}(\varphi_{n})=\frac{2}{N+1}\sum_{i=1}^{N}\langle f_{i}\rangle+ \frac{4N}{\left(N+1\right)^{2}}\left(\langle d\rangle+\sigma^{2}\right) \tag{4}\]  
Figure 5: (**a**) table illustrating a portion of \(F\), showing the indices of each element, with shading to represent an aperture code. The same element of the aperture code is applied to every band from a particular spatial location. (**b**) table illustrating the spatial arrangement of different elements of \(F\) after spectral dispersion. Again, shading indicates the pattern of aperture coding applied. The sum of each column (with the value of \(F\) multiplied by the aperture code) gives the signal at detector element \(j\). (**c**) illustrates which element of the aperture code is applicable to each element in (**b**). Thus the SNR of the sensor (i.e., \(\langle\varphi_{n}\rangle/\sqrt{\text{Var}(\varphi_{n})}\)) can be computed according to Equation (4) and the definition of \(\varphi\). If the sensor views a target that is uniform across its field of view, \(n\) denotes the waveband for which the SNR is estimated. Note that Equation (4) does not depend on \(n\), hence it is seen that the noise is wavelength independent, although the signal source varies with wavelength. This observation is very different from that of a typical dispersive sensor. Additionally, since \(\text{Var}(\varphi_{n})\) exhibits an approximately linear behaviour relative to the signal level (in the form of \(\sum_{i=1}^{N}\langle f_{i}\rangle\)), the SNR seems not to increase with increasing signal strength, unlike that of the conventional dispersive sensor where the SNR is often approximately dependent on the square-root of signal strength (assuming that photon noise is significantly larger than readout noise and other noise sources, which is common for practical applications). For the conventional dispersive pushbroom hyperspectral imager the SNR term is given by \(\text{Var}(\varphi_{n}^{c})=f_{n}+\langle d\rangle+\sigma^{2}\), which is seem to be different from that shown in Equation (4). However, it is necessary to consider how signal levels would differ in the CA multiplexed and the conventional pushbroom sensing, before their SNR characteristics could be compared under practical situations. Due to the finite well depth of the focal plane array of detectors, it is necessary to employ different integration times for each of these two systems to avoid pixel saturation.