# Back-End Hardware Design

Several receiver designs were considered, and were simulated mathematically in Mathematica. A random set of numbers were generate and interpolate to formulate an amplitude and phase modulating function. This function was then used to vary a carrier signal at a specific known frequency. Mixer functionality was obtained via the multiplication operation, and filtering (specifically low-pass) was obtained by taking a Fourier transform of the signal, multiplying by a filter function (to weight the high frequency components), and then taking the inverse Fourier transform. All of the models were conducted numerically using discrete mathematics and DFTs. Additionally, LOs with sine waves and square waves were tested, and showed no significant difference, leading to the conclusion that for our intended system, a frequency synthesizer would not be needed. The following receiver designs were tested:

• IQ Modulation:
1. Break into LO and LO90 Components (phase offset of 90 degrees)
2. Filter the two paths with a low pass filter
3. Apply another LO and LO90 to the I and Q modulations, respectively
4. Apply another low pass filter to both channels
5. Scale and compare the resultant channels
• 1-bit Multi-sample (Old VLBI):
1. Down convert with a single LO
2. Apply a low-pass filter
3. Take a series of small window samples on the transient signal
4. Assign either a 1 or -1 to the signal based on if the signal at the sample time is >0 or <0
5. Take the small windows, and perform a Fourier transform, taking the Re and Im part corresponding to the frequency of interest
6. Recombine the Re and Im samples in time and compare the resultant channels
• 2-bit Multi-sample (New VLBI):
1. Down convert with a single LO
2. Apply a low-pass filter
3. Take a series of small window samples on the transient signal
4. Assign either a 2, 1, -1 or -2 to the signal based on if the signal at the sample time is >0 or v0 or < v0, where v0 is a threshold voltage
5. Take the small windows, and perform a Fourier transform, taking the Re and Im part corresponding to the frequency of interest
6. Recombine the Re and Im samples in time and compare the resultant channels
• FX Channelizer:
1. Down convert with a single LO
2. Apply a low-pass filter
3. Take a series of small window samples on the transient signal
4. Take the small windows, and perform a Fourier transform, taking the Re and Im part corresponding to the frequency of interest
5. Recombine the Re and Im samples in time and compare the resultant channels Plots comparing the four aforementioned receiver types. The phases are negatively flipped due to their normalization, where their structure is the most important part to compare.

The figure above shows the four receiver types, comparing the input signal and the received signal for the different paradigms. It can be seen that the 1-bit and 2-bit multi-sample methods provide similar results, while the IQ modulation and FX Channelizer provide similar results, with the IQ modulation being the closest. The biggest issue here comes into processing power for the required signal, where in either case, an FPGA (or a more advanced poly-phase filter bank) would be needed. Further investigation is required, with the FX Channelizer being the most promising result (using a low sample FFT method). Note that the correlation would be done post-facto, with the computer recording the Re and Im bits after either FFT is performed. Furthermore, it should be noted that the IQ modulation would be far superior in terms of signal detection, as it could read the Re and Im components from a logarithmic amplifier.