Homework #3: Filter Design and Revisit of Comb Reverberator

Homework #3: Your First Experience of Filter Design and Revisit of Comb
Reverberator
1. Pseudo parking sonar: estimate the echo time of the sonar sound propagating back
and forth through the distance between a car and an obstacle – experiencing the
magic of the linear FM or chirp signal.
Generally, the parking sonar will prefer to employ a windowed/weighted linear FM
signal as a transmit signal, e.g., x provided in the provided sample codes, and then
will receive an echo signal contaminated by noise, as y given in the MATLAB file –
SonarSignalWithWhiteGausNoise.mat. y is contaminated by Gaussian white noise
and is recorded at a sampling rate of Fs in Hz, also provided in the same file. With the
unprocessed y, you will find out it is impossible to figure out the echo time. Note that
there is only one perfect reflector for y. To perform the echo time estimation, the first
step will be noise removal to improve signal-to-noise ratio (SNR). Note that the SNR
is defined as the ratio of the peak signal to the root-mean-square value of noise (will
be explained in the class) and we get used to converting it into dB by 20*log10(SNR).
Ideally, the SNR improvement can be simply done by filtering y with a filter which
passband exactly matches its signal frequency band.
(a) Based on the moving average filter, design a proper filter with a proper filter
length (i.e., the length of the impulse response) to improve the SNR of y so that
the echo time can be estimated. The passband of the filter has to match the signal
frequency band of the echo, which should be the same as that of x. You can
perform Fourier analysis of x with the sample codes provided in your HW1.
Please elaborate how your design your filter (Hint: see slide 90 and slide 107 in
Topic2_ReviewOfFreqDomainAnalysis_Part2_HandWriting0330_2017.pdf and
Associative property in Table 2.3, slide 56, in
Topic2_ReviewOfFreqDomainAnalysis_Part1_HandWriting0315_2017.pdf) and
how you determine the filter length. Process y with the designed FIR filter h, plot
the filtered signal, compare the filtered signal with y to see if the SNR is improved,
and then estimate the echo time in sec. The filtering can be done with the
MATLAB function conv().
(b) Is the filter you designed in (a) a linear phase system? Is the echo waveform of the
filtered signal similar to the original transmit signal x? Please provide a measure2/2
to quantify the similarity and justify your answers. If the provided signal y and the
filtered waveform are too noisy for you to answer these questions, you may try to
simulate a noise-free echo signal and check if the waveform distortion occurs after
filtering. Please define what kind of distortion happened (e.g., amplitude distortion
and phase distortion, see slide 95 and slide 96 in
Topic2_ReviewOfFreqDomainAnalysis_Part2_HandWriting0330_2017.pdf).
(c) Based on x, design a matched filter. Note that the passband of the matched filter
will exactly matches the signal frequency band of the echo. Perform matched
filtering (i.e., cross-correlation or normalized cross-correlation) over y, plot the
filtered signal, compare the filtered signal with y to see if the SNR is improved,
and then estimate the echo time in sec. Implement the matched filtering using the
MATLAB function conv().
(d) Is the matched filter a linear phase system? Is the echo waveform of the matched
filtered signal similar to the transmit signal x? Please use the measure you provide
in (b) to quantify the similarity, and justify your answers. Again, if the provided
signal y and the filtered waveform are too noisy for you to answer these questions,
you may try to simulate a noise-free echo signal and check if the waveform
distortion occurs after filtering. Please define what kind of distortion happened
(e.g., amplitude distortion and phase distortion). Please comment which filter (the
one in (a) or the one in (c)) you prefer for echo time estimation.
2. Revisit of the comb reverberator: given an output of the comb reverberator you
implemented in your HW2, please provide your strategy (or algorithm, i.e., signal
processing flow with reasoning) to estimate the system parameter D of the comb
reverberator in your HW2. Better you can justify your strategy using simulation
(will be explained in the class) or your output in HW2.