# Enhancing the Quality of Sampled Phase-Only Hologram (SPOH) Based on Time-Division Comb Filtering
## 3 Proposed Method to Generate an Enhanced SPOH with Time-Division Comb Filtering
### Comb Filtering  
The transfer function of a comb filter takes the form of periodic band-pass and band-stop sections. Due to the simplicity of comb filter (which is easy to implement with hardware circuits or software) and effectiveness of attenuating signals at regular intervals on the frequency spectrum, it has been adopted in a lot of signal processing applications. Referring to Figure 3, it can be inferred that comb filter can be applied to reduce the replica frequency bands resulted from down-sampling. The principles of comb filtering, and its application in reducing the replica frequency bands, are explained as follows. For a one-dimensional signal \(J(m)\), comb filtering is realized by summing the signal and its delayed version as  
\[J_{C}(m)=0.5\times[J(m)+J(m-T)] \tag{9}\]  
where \(T\) is the delay. In the frequency space, we have  
\[\overline{J}_{C}(\omega)=0.5\times\overline{J}(\omega)\times[1+exp(-i\omega T)] \tag{10}\]  
The transfer function is given by  
\[\overline{C}(\omega)=\frac{\overline{J}_{C}(\omega)}{\overline{J}(\omega)}=0.5 \times[1+exp(-i\omega T)] \tag{11}\]  
The magnitude of the transfer function in Equation (11) takes the form of a periodic band-stop filter, with notches (i.e., frequencies at which \(\overline{C}(\omega)=0\)) at frequencies \(\omega=\frac{(2j+1)\pi}{T}\Big{|}_{0\leq j<T}\). By setting the delay \(T\) to be half of the down-sampling factor (i.e., \(T=\frac{\pi}{2}\)), the comb filter can be used to suppress the odd replicas in the spectrum of a down-sampled signal. To illustrate this, the spectrum of a four times down-sampled sequence (i.e., \(\tau=4\)), and the transfer function of a comb filter with \(T=\frac{\pi}{2}=2\), is shown with the continuous line and the dotted line in Figure 5a, respectively. It can be seen that the notches of the filter transfer function are aligned with the first and the third replica spectrum. The result of filtering the down-sampling signal with the comb filter is shown in Figure 5b. After filtering, the number of replica spectrum is reduced to 2, and the spectrum is similar to a signal that has been down-sampled by two times, i.e., half of the original down-sampling factor.  
Figure 4: Proposed method for enhancing the reconstructed image of a SPOH with time-division comb filtering.  
The above principle can be extended to a 2-D signal. For a 2-D down-sampled signal \(J(m,n)\), comb filtering is first applied to each row of pixels to reduce the down-sampling factor along the horizontal direction, i.e.,  
\[J_{\text{Cr}}(m,n)=0.5\times\left[J(m,n)+J(m-T,n)\right] \tag{12}\]  
Subsequently, comb filtering is applied on \(J_{\text{Cr}}(m,n)\) along the vertical direction as  
\[J_{\text{Cxy}}(m,n)=0.5\times\left[J_{\text{Cr}}(m,n)+J_{\text{Cr}}(m,n-T)\right] \tag{13}\]  
Equation (13) can be rewritten as  
\[J_{\text{Cxy}}(m,n)=0.25\times\left[J(m,n)+J(m-T,n)+J(m,n-T)+J(m-T,n-T)\right] \tag{14}\]  
Equation (14) shows that the output of the comb filter is the sum of four components. The first component is the original signal \(J(m,n)\). The remaining three components are translation of the original signal along the horizontal, vertical, and diagonal directions.