Useful signal: s(t), is trapezoidal signal.
signal duration: T
signal delay: T0
observation segment: TL
In this observation segment the useful is corrupted by the additive white noise, we want to estimate the duration of the useful signal in the background of white noise.
The algorithm for the duration estimation is based on auto-convolution.
The known information about the useful signal:
the rising time of the trapezoidal signal is:
the falling time of the trapezoidal signal is the same as the rising time:
the constant time of trapezoidal signal is:
the amplitude A, the arrival time T0, the signal duration T are unknown.
the noise signal n(t): additive white (gaussian) noise
the input signal of the algorithm: r(t)= s(t)+n(t) in the certain segment
the auto-convolution algorithm:
step 1: compute the convolution of r(2TL-t) and r(-t), find the peak pint of the convolution t*, so the middle point of the trapezoidal signal M is ;
step 2: from location of point M, divide the pulse into left and right halves. Compute the auto-convolution of the left half pulse and find the peak point of tl*, the ;
step 3: obtain the signal duration which is .
The simulation of MATLAB:
- assume a certain trapezoidal signal and the appropriate noise power (not too large) to verify the algorithm are feasible to estimate the duration of signal. The duration of the trapezoidal signal is in the range of (10 microsecond to 100 millisecond).
- assume a certain trapezoidal signal and for the different SNR (2dB:2dB:20dB) do the simulation experiments. Each experiment performs N independent trials on a certain SNR to obtain the variance of the estimated signal duration. Each trial involves processing a data segment TL. Choose the number of N which makes the variance of estimated signal duration converge to a certain level (for instance, 200). In the end plot the logarithm figure of the estimated duration variances in dependence of different SNR. (increase the efficiency of the algorithm, the sampling frequency should exceed MHz).