Experimental
1. Computer Simulation
The computer simulation was done entirely within MATLAB program. The noise was generated by filtering a (stationary) white Gaussian noise with Infinite Impulse Response (IIR) filters, which have feedback mechanism and thus are able to introduce self-dependency in our noise. The IIR filters we have tested in our project were Butterworth, Chebyshev type I and type II filters. The test signal was an exponentially decaying sine wave. We then ran our AR model optimization program on the combined noise-corrupted signal to obtain an optimal model. Following the process described in theory section, we were able to determine the onset time of the signal and extract the shape of the signal for a few periods in various simulations.
2. Audio Experiment
Due to time constraint, we only have one test case worked where a 14th order Butterworth bandpass filtered white Gaussian noise was used. The test signal involved was a sinusoidal wave with onset time 3.502s, duration of 0.1s and signal noise ratio (SNR) about 1. The reason for using sine wave as the test signal instead of the exponentially decaying sine as in the computer simulation is that the frequency component of the sine wave is much simpler than its exponentially decaying counterpart. With the rather low quality of our audio system, the broad frequency spectrum of the exponential decaying sine wave would be severely affected by the band passing effect of the speaker-microphone system, resulting in rather low fidelity. To obtain the signal to do the comparison, we sent the signal out of the speaker by itself multiple times and recorded by the microphone. Negligible deviation in the overall shape of the signal was observed. But we also recognized a slight deviation in the onset time of the signal due to our program initializing the microphone and the speaker at slightly inconsistent times. This deviation in the onset time was set as 2ms.
A picture of our apparatus is included in below. The noise-contaminated signal was generated from a MATLAB program as in the computer simulation. The data was then transferred to a LabVIEW interface, which sent the noise-contaminated signal to a REALISTIC amplified speaker system connected to an output portal on the BNS-2110 DAQ board. And a BEHRINGER ECM 8000 microphone was connected a MINI MIC modeling preamp which then sent data back to LabVIEW through an input portal of the DAQ board.
Figure 2. The picture of our experimental setup