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enter image description here

I have a signal, s, given at the end of this message. I need to remove the square wave shape from it. The desired signal is actually very small and it has been contaminated by a strong square-shape noise.

The desired signal is just couple peaks similar to an ECG or EEG but we have a very large square-shape noise that randomly affects some of the scan results.

Central Freq = 10MHz, Sampling Freq = 125MHz

(Please note that these are the frequencies related to the red signal and robot's transducer that transmits the ultrasonic pulse. These are not the frequencies of the blue signal nor that of the square-wise wave.)

s = [0.00463923811912537,0.00524966418743134,-0.000122085213661194,-0.00683677196502686,-0.00671468675136566,0.00183127820491791,0.00610426068305969,0.00256378948688507,-0.00500549376010895,-0.00647051632404327,-0.000610426068305969,0.00549383461475372,0.00476132333278656,-0.00329630076885223,-0.00805762410163879,-0.00256378948688507,0.00622634589672089,0.00915639102458954,0.00354047119617462,-0.00366255640983582,-0.00305213034152985,0.00488340854644775,0.00866805016994476,0.00476132333278656,-0.00439506769180298,-0.00720302760601044,-0.000732511281967163,0.00598217546939850,0.00402881205081940,-0.00390672683715820,-0.00915639102458954,-0.00317421555519104,0.00402881205081940,0.00598217546939850,0.000976681709289551,-0.00598217546939850,-0.00366255640983582,0.00561591982841492,0.0101330727338791,0.00720302760601044,-0.000732511281967163,-0.00183127820491791,0.00378464162349701,0.0102551579475403,0.00598217546939850,-0.00366255640983582,-0.00891222059726715,-0.00341838598251343,0.00134293735027313,0.00207544863224030,-0.00476132333278656,-0.0123306065797806,-0.00830179452896118,0.00427298247814179,0.0134293735027313,0.0155048221349716,0.0133072882890701,0.0153827369213104,0.0225857645273209,0.0220974236726761,0.00866805016994476,-0.0102551579475403,-0.0233182758092880,-0.0201440602540970,-0.0126968622207642,-0.00512757897377014,-0.00537174940109253,-0.00866805016994476,-0.00415089726448059,0.00366255640983582,0.00341838598251343,-0.00427298247814179,-0.0119643509387970,-0.0106214135885239,0.00109876692295074,0.0111097544431686,0.0129410326480865,0.00683677196502686,0.00280795991420746,0.00964473187923431,0.0168477594852448,0.0153827369213104,0.00586009025573731,-0.00427298247814179,-0.00586009025573731,-0.00341838598251343,-0.00573800504207611,-0.0152606517076492,-0.0238066166639328,-0.0242949575185776,-0.0181906968355179,-0.0152606517076492,-0.0229520201683044,-0.0310096442699432,-0.0290562808513641,-0.0115980952978134,0.00586009025573731,0.0212428271770477,0.0324746668338776,0.0376022458076477,0.0427298247814179,0.0583567321300507,0.0894884616136551,0.113783419132233,0.121230617165565,0.114271759986877,0.126236110925674,0.157612010836601,0.171529725193977,0.160542055964470,0.126968622207642,0.0648272484540939,0.00512757897377014,-0.0288121104240417,-0.0553046017885208,-0.0919301658868790,-0.142473444342613,-0.180197775363922,-0.187767058610916,-0.192040041089058,-0.219509214162827,-0.240019530057907,-0.246612131595612,-0.231717735528946,-0.172506406903267,-0.0767915993928909,0.0205103158950806,0.108289584517479,0.209742397069931,0.248199239373207,0.242339149117470,0.246490046381950,0.257843971252441,0.264314502477646,0.266267865896225,0.256012707948685,0.210719078779221,-0.00427298247814179,-0.169332191348076,-0.236234888434410,-0.236845314502716,-0.247100472450256,-0.255280196666718,-0.257111459970474,-0.253326833248138,-0.252228051424027,-0.256012707948685,-0.258820652961731,-0.259186923503876,-0.256989389657974,-0.231107309460640,0.0240507870912552,0.172506406903267,0.246245875954628,0.253326833248138,0.258088141679764,0.253937244415283,0.243071660399437,0.241240382194519,0.260285675525665,0.278110116720200,0.272372126579285,0.240141615271568,-0.0282016843557358,-0.180441945791245,-0.240996211767197,-0.235746547579765,-0.243315830826759,-0.257721900939941,-0.269197911024094,-0.268831640481949,-0.267732888460159,-0.269442081451416,-0.269808322191238,-0.262849479913712,-0.109632521867752,0.136491268873215,0.231595650315285,0.253326833248138,0.252472221851349,0.259919434785843,0.264802843332291,0.264192402362824,0.261018186807632,0.257966071367264,0.260041505098343,0.264802843332291,0.262727379798889,0.160908311605454,-0.106458306312561,-0.217189595103264,-0.247466728091240,-0.237821996212006,-0.250396788120270,-0.271151274442673,-0.278598457574844,-0.276034682989121,-0.266756206750870,-0.261750698089600,-0.255280196666718,-0.244414597749710,0.0291783660650253,0.194481745362282,0.263459891080856,0.253448903560638,0.246490046381950,0.247100472450256,0.258820652961731,0.270785003900528,0.278354287147522,0.271883785724640,0.259675264358521,0.220241725444794,-0.0590892434120178,-0.198022216558456,-0.266756206750870,-0.273470878601074,-0.274081319570541,-0.268343299627304,-0.269075810909271,-0.275058001279831,-0.275912582874298,-0.265413254499435,-0.250640958547592,-0.241484552621841,-0.000122085213661194,0.173849344253540,0.258332312107086,0.271151274442673,0.273470878601074,0.270907104015350,0.262116968631744,0.254181414842606,0.253082662820816,0.255036026239395,0.255036026239395,0.250640958547592,0.234281525015831,-0.0394335240125656,-0.193260893225670,-0.267854958772659,-0.273715049028397,-0.276767194271088,-0.271639615297318,-0.259431093931198,-0.246123790740967,-0.240874126553535,-0.243682086467743,-0.244292512536049,-0.234525695443153,-0.0302771329879761,0.171529725193977,0.256012707948685,0.267488718032837,0.267854958772659,0.271761685609818,0.272738367319107,0.264314502477646,0.251251369714737,0.2441704273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enter image description here

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  • $\begingroup$ If the square noise is LF, what about a simple high pass filter? Otherwise I would consider rectifying and band pass filtering. How important is waveform shape vs. magnitude response? $\endgroup$ – Emanuel Landeholm Mar 2 '15 at 20:53
  • $\begingroup$ Hi Emanuel, I have applied couple of filtering approaches, however, the problem is that the desired signal is just like a very short damping pulse. It also has a very small amplitude compared to the square wave. Therefore, traditional filtering approaches usually destroy the desired signal. It could help if one could first extract the square wave correctly and then try to subtract it from the original to reach the desired signal. $\endgroup$ – Alberto Castillo Graza Mar 2 '15 at 21:38
  • $\begingroup$ There were no obvious peaks in your samples that I could distinguish from line noise. The period of your square wave is $25$ samples, giving it $12$ to $13$ samples between each peak. You could try a noise filter and create a square wave at your sample rate $/25$. You could programmatically, manually identify the square wave and subtract it off. That could leave your original signal. I would normalize the wave first. This would be a comment but I'm a newb here. $\endgroup$ – ChocoBilly Mar 3 '15 at 3:48
  • $\begingroup$ Update: I received a comment from a colleague stating that ICA (independent component analysis) might help for separation. Any detailed suggestion in that regard? $\endgroup$ – Alberto Castillo Graza Mar 3 '15 at 16:42
  • $\begingroup$ @AlbertoCastilloGraza Can you mark the desired signal on plot of your data? Do you have any additional information about your signal? Shape? amplitude? Is signal periodic? etc. Do you have information about square-shape noise without desired signal? If your answers are NO for those questions you need to change your measurements scheme and eliminate noise. There is not silver bullet for such signals. $\endgroup$ – SergV Mar 4 '15 at 4:15
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Your question is very interesting now after adding full information. (I upvoted question now)

I'm pretty skeptical about attempts to use digital signal processing techniques for such signals. Focus on reducing the "square wave noise" in the recorded signal before any digital signal processing. Firstly examined the nature of this kind of noise - it may be sound (ie the sound of the robots, which is registered by the receiver of the scanner) or it may be electrical (eg noise from power supply of robots propagate to the scanner power supply or directly to the receiver circuits, etc.). Depending on the outcome of this study try to reduce this noise on scanner signals. For example, if it's the sound from robots - try adding sound absorbers in front of the receiver. You can try to record the sound from outside the scanner and subtract it from the signal of the scanner. If this pickup from power circuits of robots - improve the protection of electrical circuits of scanner. In experimental physics there are many methods and techniques to solve these problems, but often it's more like Art.

Good luck!

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  • $\begingroup$ Thank you SergV. I appreciate your comments and valuable suggestions. We will consider them all. $\endgroup$ – Alberto Castillo Graza Mar 9 '15 at 15:18
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If you know the source of the square wave signal, or can separately lock onto it, sum the inverted, amplitude-tweaked, version of that square wave with your noisy input prior to any other filtering. You might also use the principle of synchronous detection on the remaining signal to detect the square wave contribution, then subtract it DC-wise after the desired signal detection. I have a GoogleDocs collection on synchronous detection here:

Let me know if you have trouble accessing it.

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  • $\begingroup$ Thank you for your suggestions and the link you provided. I will be reading the document for possible clue. $\endgroup$ – Alberto Castillo Graza Mar 9 '15 at 15:29
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I'm reminded of an image-processing technique called "unsharp-masking" Your goal is to increase the sharpness (high frequencies) in an image. It is accomplished by blurring the image (increasing low frequencies). Then you subtract the blurred version from the original since original freq - low freq = high freq. Maybe that could be applied here since your noise is of much lower frequency than your signal and since your signal appears to be 0 mean.

You make a small sliding window average of the waveform(this is equivalent to blurring). Then you subtract the sliding window average reconstruction from the original. Leaving only the information you want.

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  • $\begingroup$ Thank you Andrew. We did a similar thing: We tried to approximately re-estimate the signal itself with some curve fitting techniques and then find the difference with the original. Window-averaging, as you suggest might help and we will take that into consideration. Thank you. $\endgroup$ – Alberto Castillo Graza Mar 9 '15 at 15:23

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