adds her to during the podcast that are not in my voice or placed by third party agencies outside of my control and should not imply an endorsement by Weird Darkness or myself. Stories and content in Weird Darkness can be disturbing for some listeners and is intended for mature audiences only. Parental discretion is strongly advised. Welcome Weirdos, I'm Darren Marlar and this is Weird Darkness. Here you'll find stories of the paranormal, supernatural, legends, lore, the strange and bizarre, crime, conspiracy, mysterious, macabre, unsolved and unexplained. Coming up in this episode. If you're looking for some creepy places to visit this Halloween season, you can't get much creepier than cemeteries and graveyards. I'll share some of the most haunted ones that you might want to check out if you have ice in your veins. It's one of the most gruesome ways movie and television villains dispose of their victims, the acid bath. The body being tossed in a tub, barrel or bin, slowly dissolving into soup, leaving no evidence behind. But the truth of acid baths is far more gruesome. Even more so, those who choose to use the method to get rid of a body. The scenery may be beautiful when driving this road, but locals have nicknamed it La Saratera de la Muerta, the highway of death. We'll look at Mexico's La Rumorosa Highway. Many considered Edward Rulloff to be a genius, a man ahead of his time. Others had quite the opposite opinion of him, thinking him a fraud or a con man. But nobody expected they would also be adding the word killer to his description. But first, in 1972, Defiance, Ohio was the location for one of the best true werewolf stories most people never talk about. We'll begin with that story. If you're new here, welcome to the show and if you're already a member of this weirdo family, please take a moment and invite someone else to listen. Recommending Weird Darkness to others helps make it possible for me to keep doing the show. And while you're listening, be sure to check out WeirdDarkness.com where you can find the show on Facebook and Twitter, and you can also join the Weird Darkness Weirdos Facebook group.  Midway through 1972, Defiance, Ohio became what can only be termed werewolf central. Over the course of a hysteria-filled two months, July to September of 72, sightings of a rampaging, hair-covered man-beast with a pronounced muzzle and dressed in rags were made. The local media quickly picked up on a sinister saga, as did the town's police, who even opened an official file on Defiance's very own equivalent of Nessie, the Chupacabra and Bigfoot. To say that Defiance was gripped by terror would not be an understatement. Many of the sightings of the creature were made around a series of old railroad tracks and usually late at night. A couple of guys working on the tracks, Ted Davis and Tom Jones had an encounter of the very close kind. A close call, one might say. Davis told the local newspaper, the blade, I was connecting an air hose between two cars and was looking down. I saw these huge hairy feet. Then I looked up and he was standing there with that big stick over his shoulder. When I started to say something he took off for the woods. For weeks people were on edge. The Defiance dog man was major news and then, like a definitive specter of the night, it was no more. The legend, however, never really died away. The monster may be long gone, but memories of those days and nights of the early seventies still persist among those who lived through that brief, turbulent time of terror. The Klawikend says, the werewolf of Defiance is one of those local legends that sounds like the beginning of a classic horror movie. Two railway workers hooking up train cars late at night, one looks over to see a towering werewolf bangs bared who proceeds to attack the hapless victim. Hollywood is it might sound, that's the story that Ted Davis reported to police in 1972. The werewolf may have attacked him with a club rather than his claws, but otherwise the story sounds like so many other tall tales, except that Davis wasn't the only one seeing dog man in the wild. Another employee of the railway also reported a wolf man stalking the rails. Then, a week later, a grocer driving home sees a lichen throat in his headlights. Astonishing legends had the following important thing to say about the wolfish madness. Chief Donald Brekler said to the news media, we don't know what to think. We didn't release it to the news media, but we got the first report about a week ago, but now we're taking it seriously. We're concerned about the safety of our people. Slight panic began to enter the town and several newspapers including the Toledo Blade and the Defiance Crescent News covering the streamed sightings. In general, it was agreed that the werewolf was large, at least six feet, and humanoid in some way. Some reports even said it was wearing jeans. It was also reported as being bipedal but severely hunked over. Newspapers did report, interesting enough, that none of the events happened during a full moon. Cryptozoologist Colin Schneider had his view on it all, saying, the Crescent News, the local newspaper and the Toledo Blade ran a total of four articles about the incident. The logs to the articles focused on the local police force's investigation into the incident. Headed by Donald Brekler, the police chief at the time, the police searched extensively for the werewolf. Although the general public thought it to be a bored teenager's prank, Chief Brekler, considered the club-wielding animal-faced assailant was a threat to the community. Eventually, after days of looking and no results, the search soon ended. And by September 1972, the whole thing was over. But then you can never be too sure when it comes to canines and cryptids. They both seem to have a knack for finding their way back, even if they've been gone for a while. When Weird Darkness returns, it's one of the most gruesome ways movie and television villains dispose of their victims, the acid bath. But the truth of acid baths is far more gruesome, even more so those who choose to use the method to get rid of a body. But first, if you're looking for some creepy places to visit, you can't get much creepier than cemeteries and graveyards. I'll share some of the most haunted ones that you might want to check out if you have ice in your veins. That story is up next on Weird Darkness. By April 13th, we'll be treated to a Roger Corman crapfest from 1958, Teenage Caveman starring Robert Vaughn. Did he just say dirt that eats men? Yep, I guess so. Mistress Malicious and her Mistress Peace Theater will keep us entertained throughout the film as we watch this caveman teenager with great hair go into the jungle to fight prehistoric monsters like dogs and an armadillo. Whatever, they're prehistoric creatures. Our Weirdo Watch Party is always free to watch online, so grab your popcorn, candy and soda and jump into the fun, and even get involved in the live chat as we watch the movie. Plus, during this Weirdo Watch Party, I'll be giving away a creepy crate to one lucky winner, full of scary surprises like horror collectibles, true crime-themed accessories, books, terrifying trinkets and more, with some Weird Darkness swag added in. You won't know what's in the creepy crate until you open it. And I'll be giving instructions on how to win the creepy crate inside the chat during the movie, so you have to tune in to win. It's Teenage Caveman, Saturday, April 13th, hosted by Mistress Peace Theater. The show begins at 10 p.m. Eastern, 9 p.m. Central, 8 p.m. Mountain and 7 p.m. Pacific. You can watch a trailer for the film and watch horror hosts and schlocky B movies anytime, day or night on the Monster Channel page at WeirdDarkness.com. Hope to see you on Saturday, April 13th! CEMETERIES They always have that creepy feel to them, but these have just too much history behind them to ignore. RESURRECTION CEMETERY Housing the famous Bloody Mary, or Resurrection Mary as the locals call her, this cemetery is in Chicago, Illinois, believed to be haunted by a girl called Mary who lost her life in an argument with her boyfriend and was killed by a hit-and-run driver while trying to hitchhike home from the cemetery. Her body was left by the roadside and she has been in the area ever since, perhaps to avenge her death or settle things with her boyfriend. The name Bloody Mary has passed in and out of folklore, but this blonde-haired ghost has appeared in front of locals to the cemetery more often than most ghosts trying to hitch a ride from passing drivers. Local cab drivers have even refused fares passing through the area. BOOTHILL CEMETERY Located in Tombstone, USA, the BOOTHILL CEMETERY got its name because the souls resting there were buried with their boots on. BOOTHILL is the resting ground of notorious outlaws like Frank McLaury and Billy Clanton, and given their violent life, it's no surprise that the area is full of poltergeist stories, from fast-moving shadows and strange lights to whispers from the unmarked graves and apparitions chasing visitors. BOOTHILL has a reputation and not a peaceful one. A journalist who visited the area to cover the story stepped on an unmarked grave once and had a voice hiss, it was nice of you to do that in his ear. St. Louis Cemetery No. 1 This New Orleans cemetery is home to over 700 tombs and 100,000 souls of the dead. Given its immense size, this cemetery always has an eerily quiet atmosphere as dusk starts to fall. The locals nicknamed it the City of the Dead, and while there are countless reports of vague figures showing up there, there are a few known figures who seem to stay in the area. Marie, the voodoo queen levaux, was born in 1801 and became the most feared voodoo practitioner in New Orleans history. Her ghost is repeatedly seen in the cemetery, and those not showing respect to the dead have been scratched and shoved to the ground. The tomb of the voodoo queen was visited so frequently by visitors to ask for her blessing or forgiveness that the section had to be closed to the public. The Old Burying Point Salem, Massachusetts, it has one of the oldest cemeteries in America, and with the area's turbulent history, it won't surprise anybody that the area has a fair share of hauntings. A lot of activity surrounds the gravestones of John Hathorn, a vicious judge during the Salem witch trials with over 400 reports of spirits seen running with anger towards the site of the grave. Stull Cemetery Some paranormal investigators consider Stull one of the most haunted spots in the world, and it is considered one of the seven gates to hell. Located in Douglas County, the cemetery sits next to the deceased evangelical community that passed on long ago. As the church was deserted, local witches and occult groups made the sanctuary their own, and legend has it, they were trying to open the door to hell. Eventually, both the church and the neighboring tree that witches were hanged from were bulldozed, but the landmark still attracts a lot of paranormal activity, and on the spring equinox, locals continue to see stairs emerging from the earth below. Highgate Cemetery Heading over to the UK now, Highgate is one of the most haunted places in London. In the 1800s, the city was overpopulated and with such a high death rate, graves were crammed between buildings and dug into the street itself. Illegal funerals and shallow graves were so common, is it any surprise that souls buried at the time are still restless? Built in 1839, Highgate Cemetery was the place for the rich and powerful to be put to rest to separate them from the shallow graves of the poor. But as the years went on, and the cemetery's fortunes waned, the pristine landscapes turned to an overgrown jungle and the luxurious buildings began to tumble down. That's when the stories started. Locals and visitors say the spirits of well-dressed noblemen glare at them through gaps in the fence. Tombstones have reportedly exploded. Watches have stopped working and visitors have been chased as the spirits seem to be angry that their resting place has not been maintained. Rookwood Cemetery Close to Sydney, Australia, Rookwood Cemetery has over a million souls buried there, and is the largest operating Victorian Cemetery to this day. Aside from the victims of war and atrocities being buried in Rookwood, two well-known spiritualists, the Davenport Brothers, were buried there. In life they were said to work as a link between the living and dead, and that doesn't seem to have changed in death. And then there is Bachelors Grove Cemetery. Bachelors Grove, Chicago. It was made for a settlement established by just four men. Established in 1840 with only 82 graves. As gang member bodies started being unceremoniously dumped there in the 1920s, the grounds were quickly abandoned as the ghost stories started to emerge. Aside from the usual ghost orbs seen in most cemeteries, the locals reported phantom vehicles. What would have been exotic gangland cars at the time have been silently driving around the graveyard ever since. In the 1980s and 1990s, the reports became so frequent that paranormal researchers began to arrive and continually communicated with spirits like the white lady who carries an infant between the graves looking for something. In 2012, Bachelors Grove Cemetery was featured in an episode of Ghost Adventures. Bachelors Grove Cemetery is considered the most haunted cemetery in the world. Tales of human remains dissolving in acid fill us with a special dread. Given that homicide investigations need bodies, acid baths will always be a hit among murderous fiends. In some cases, unfortunate people accidentally dissolve themselves in acid. Acid steals our identity and reduces us to undifferentiated organic matter, reminding us that eventually we return to nothingness. It was the most horrific thing I've ever encountered in police work in my life, noted Detective Jeremy McCord. About the night in November 2016, he entered the residence of Knoxville's Golden View Lane. Inside, McCord discovered blood-smeared walls and the stench of death mingled with chemicals. Body parts were scattered around in plastic containers containing acid. In a pot on the stove, a woman's head boiled. The smells never going to leave me. Tennessee charged Joel Guy Jr. 32 with the double homicide of Joel Guy Sr. 61 and Lisa Guy 55. Prosecutors alleged he stabbed and dismembered his parents over Thanksgiving break because they were tired of bankrolling his shiftless lifestyle. After hacking apart the bodies, Guy Jr. allegedly placed the remains in an acid-based solution in an attempt to destroy evidence. Multiple space heaters, which raised the temperature to 93 degrees, were set up to speed decomposition. In December 2013, a Karen's man was found guilty of murdering his wife and dissolving her body. Klaus Andrus, 70, admitted to disposing the body of his wife, Lee Ping Chow, in a wheelbarrow full of acid. He then poured her remains down a storm drain outside his home in Brimstead. The only thing police recovered, Chow's prosthetic teeth. Andrus met the single mother online after a visit to her home in China, the Australian and Chow married and moved to Karen's. When confronted about her disappearance, Klaus initially claimed Chow accused him of having an affair and went back to China. CCTV footage showed Andrus using her Commonwealth Bank card to purchase 60 liters of acid. Even with its enormous amount of acid, Detective Sergeant Brad McLeish determined it took several days to dissolve Chow's body. Andrus forged his wife's signature, so her Center League payments transferred to his account. In June 2016, a 23-year-old man dissolved in a boiling pool of acidic water in Yellowstone. Caleb Scott and his sister, Sable, ventured 200 meters off the trail to Hot Pot, or illegally take a soak in the National Park's thermal baths. Sable caught cell phone footage of Caleb accidentally tumbling into the 4x10-foot pool while he attempted to test the water temperature. She went for help, but never saw her brother again. In very short order, there was a significant amount of dissolving, noted Deputy Chief Ranger Lauren DeVaresse. Search and rescue were called in, but lightning prevented them from recovering Caleb's remains. The following day, Caleb, his flip-flops and wallet had completely dissolved into what are not only Yellowstone's hottest but most acidic waters. Since 1890, at least 22 people are known to have died from hot spring-related injuries in Yellowstone, which is set above a geologically active supervolcano. In April 2018, a Mexican rapper admitted to dissolving the bodies of three film students in acid. Christian Palma Gutierrez, 24, raps under the name QBA, and has over half a million views on his YouTube channel. The students, ranging from 20 to 25, were kidnapped on March 19 from a Tanola rank. Investigators believe the Jalisco New Generation Cartel mistook Javier Salaman-Acivez Castellum, Jesus Daniel Diaz, and Marcos Avalos for members of a rival organization. When Castellum died under torture, the cartel opted to dispose of the other two. This is not the first time QBA dissolved victims of Mexico's fastest growing cartel. He received 3,000 pesos, or $160 per week, as a cook, disposing of bodies and acid baths in water tanks at a cartel safe house. After soaking the bodies for 48 hours, QBA opened the valve and dumped remaining oddments in the fields. Authorities charged QBA with aggravated kidnapping. You know what they say about good intentions? In August 2017, a pair of vigilantes committed one of the most gruesome crimes ever recorded in Montana. Operating on a rumor that an acquaintance had sexually assaulted a young girl, Tiffany Rae Pierce, 24, and Augustus Standing Rock, 27, broke into a Missoula home and proceeded to slaughter its inhabitants. Standing Rock stabbed Jackson Wiles, 24, to death. Pierce slashed the throat of 15-year-old Marilyn Pickett. The vigilantes hacked apart the corpses and dissolved them in acid. A neighbor who heard the victims' screams alerted police. In the basement, authorities stumbled upon a vat of liquidating remains. In April 2019, Judge James Wiles sentenced Pierce and Standing Rock each to two life terms. Beyond the count of deliberate homicide, Pierce faced an additional charge of attempted homicide for an earlier incident, during which she barged into a house and stabbed a woman eight times. In July 2015, three advanced mathematics students dissolved Eva Borsot's body in a plastic-trunk acid bath. The 23-year-old art history student's body was discovered by her mother in a plastic bin at her apartment in Toulouse. Investigators believed the three-man, ages between 19 and 23, ventured to Borsot's residence to collect a drug debt. There was a fierce outburst of violence, one marked by the use of brass knuckles and a crowbar, alleged prosecutor Pierre Yves Couillet. Inspired by events from the TV series Breaking Bad, the trio left her to dissolve in acid. They returned to the scene of the crime several times, covering the stench of disintegrating flesh with air freshener. When Borsot's mother failed to hear from her for two weeks, she went to the apartment. Police indicated the victim's body had been breaking down for approximately 10 days and was in advanced stages of decomposition. In August 2018, a North Carolina woman received a sentence of 20 years when a Texas jury discovered she tried to cover up a murder by dissolving the victim in acid before feeding it to alligators. Four years prior, Amanda Hayes and her husband Grant Hayes were convicted of the murder of Grant's ex-wife, Laura Ackerson, 27. Grant received a life sentence. Amanda initially got 13 to 16 years. After killing Ackerson at the couple's Raleigh apartment, Amanda and Grant attempted to dissolve Ackerson's remains in muriatic acid. When that proved too slow, the couple dismembered Ackerson with a power saw, packed her remains in a cooler, rented a U-Haul and drove to Amanda's sister's house in Texas. On January 4, 2011, investigators found Ackerson's torso and lower leg floating in Oyster Creek. A dive team later found some, but not all, of Ackerson's remains in the alligator-infested waters. When a suspicious neighbor requested a welfare check on a Laredo apartment block in February 2019, police came upon an overwhelming scene.  Rebecca's parents, Monica Dominguez and Gerardo Zavala Laredo admitted to trying to dispose of the body. They claimed the toddler drowned while unsupervised. Both pled guilty to tampering with evidence of a human corpse. Zavala Laredo received a 14-year sentence. Texas authorities charged Dominguez with violating parole from a previous conviction for an incident in which her nine-month-old incurred six bone fractures. With additional charges of endangering a child and abuse of a corpse, she received 20 years. The couple's remaining children were taken into custody. Neither parent was charged with murder. The coroner could not determine a cause of death, given the damage the acid caused to Rebecca's remains. In September 2021, East to West News reported that a Bolshoi theater battle arena may have been chopped up and dissolved in acid. In 2014, Olga Domina 25 went missing. She had long been presumed dead. Her manager, Malka Zavoev 40, has been the prime suspect in her disappearance. He allegedly threatened to release sexually compromising photos unless Domina paid up. Domina's mother claims that she was completely at his mercy and sold her Prusio and took out loans to pay off Zavoev. Despite the payments, the photos emerged online. In 2017, on an Interpol warrant, officials extradited Zavoev from Germany, where he posed as a Kurdish refugee. He is currently being held in Moscow. According to law enforcement sources, while behind bars, Zavoev admitted that he dismembered the dancer and dissolved her in acid. Sulfuric acid does not have to dissolve everything to zero. A search for any evidence of Domina's demise continues. Also, in September 2021, a Belgian court announced that a single tooth of Conga's first prime minister will be returned to Africa. It is all the remains of Patrice Lamumba. On January 17, 1961, Lamumba and two associates, Joseph Aquito and Maurice Impolo, were taken from prison and executed by firing squad in a forest outside Lubumbashi. The next evening, brothers Gerard and Michael Sowett exhumed the corpses, hacked them to pieces, and dissolved them in sulfuric acid. Gerard notes that they needed to get drunk, stone drunk over the two-day corpse disposal, during which they did things an animal wouldn't do. While the Belgian court ruling only mentions one tooth, Gerard claims to have taken two home as a trophy. While admitting to undeniable responsibility in the events that led to Lamumba's death, the Belgian government refuses to accept full responsibility and offered a pardon to citizens involved in the assassination. Coming up, the scenery may be beautiful when driving this road, but locals have nicknamed it La Seretera de La Muerta, the Highway of Death. We'll look at Mexico's La Rumorosa Highway. Plus, many considered Edward Rolof to be a genius, a man ahead of his time.  These stories are up next on Weird Darkness. The Chilling True Terror of the Black-Eyed Kids, a monster compilation by G. Michael Vasey. The Black-Eyed Kids are an urban legend of vast proportions. The stories of small children turning up on people's doorsteps all across the world, spreading fear and terror, have only increased over time. This compilation of G. Michael Vasey's books on this scary phenomena include new material and new true stories as well as the complete texts of the Black-Eyed Demons are coming and the Black-Eyed Kids. Supernatural expert G. Michael Vasey carefully investigates this truly terrifying phenomenon using real-life encounters with these scary supernatural beings. The result is an unsettling and sometimes terrifying book that will have you fearfully anticipating that knock at your door late at night. Who and what are these mysterious visitors to the doorstep? Are they demons? Aliens? What do they want? Why do they need to enter your home? And what happens if they do? Small kids that ask to use your phone or for a ride and yet those who encounter them are scared to death even before they notice their Black-Eyes. The chilling true terror of the Black-Eyed Kids, a monster compilation by G. Michael Vasey, narrated by Weird Darkness host Darren Marlar. Here a free sample on the audiobooks page at WeirdDarkness.com. Winding through the Sierra de Juarez mountains at an elevation of 4042 feet between Decate and Mexicali in Baja, California, Mexico is a treacherous pass called La Rumorosa. It is undoubtedly a beautiful yet haunting stretch of road, meandering through spectacular vistas of surreal rock formations like the surface of some alien world. And here, there are also various archaeological sites and cave paintings by the ancient Kumi'i people, but it is also a forbidding and dangerous place to be. Featuring countless hairpin turns, twists, steep grades, high winds and perilous sheer drops along the entire 24-mile route, the once two-lane road is considered to be one of the most dangerous in the world. The fact being abundantly clear by the husks of rusting car wrecks littering the rocks below. Indeed, drivers have often nicknamed it La Seretera de la Muerta, the highway of death, making this a sought-after road trip for those who want a challenge, but not for novices or the faint of heart. This wind-lashed road is also well known as a haunted and even cursed place, with numerous tales of the paranormal and other strangeness orbiting it. Many of the ghostly tales revolve around the many people who have lost their lives on this expanse of deadly road, and perhaps the most famous of these is what is called the Ghost Truck Driver. The story starts with a harrowing experience of a truck driver named Francisco Vasquez, who was said to have headed towards Mexicali to see his wife who was expecting a child when he lost control of his rig on one of the road's deep turns and went careening into some rocks. Although shaken, the driver crawled from his overturned vehicle to find that he was miraculously free of injury. He then decided to walk off looking for help and would never be seen again. Three days later, the truck was found, but no sign of Francisco ever was. However, this is not the end of the story. Some years later, a young man was supposedly driving through the area when he was flagged down by a man by the side of the road. Since this was pretty much the middle of nowhere, he stopped to provide assistance, and the man introduced himself as Francisco Vasquez. He then told the young man that he had crashed and did not want to leave his truck behind until help arrived, so he asked the young man to see his wife to let her know that he was okay. The young man agreed, and Francisco handed him a piece of paper with his wife's address on it, and he was shocked to find that the stranger's hand was as cold as ice. He got back on the road, and when he reached Mexicali, he looked for the wife as promised, but she did not live at the address he had received. He asked around town and was finally shown a house where the woman was said to now live. When the young man knocked on the door, a woman answered who confirmed that she was Francisco's wife, but she went pale when the man told her that he was all right because her husband had died five years before. The shocked young man went back out to the same spot along the road, but Francisco was gone. It is said to this day that the truck driver wanders along that same stretch, waving down drivers and asking them to go see his wife, and the common advice is to just keep on driving because one sinister aspect of the legend is that anyone who does not fulfill their promise to visit the wife will drop dead. There are numerous other specters thought to be the victims of accidents along the road, including the ghost of a nurse who appears in people's passenger seats, the apparition of an old woman who will ask people for a ride only to vanish, another old lady who will guide trucks through the pass and that of a young man on a bicycle who is said to have been hit by a car here. Numerous others will appear as shadow figures, glowing wraiths or merely disembodied voices whispered into the ear, and there are countless stories of seeing ghosts along the road. In fact, there are so many wraiths and ghosts ambling about along the road that they are often blamed for causing many of the accidents along the lonely stretch. The area is also known as a hotspot of UFO sightings, to the point that there is now an annual UFO camp held here which began in 2018. One rather infamous strange story surrounding La Rumorosa is that it was the source of a sort of curse on the production of the James Bond film Licensed to Kill starring Timothy Dalton, which filmed a car chase portion along the road. According to witnesses, there were all manner of strange occurrences on set while they were there, including shadow figures lurking about at night, freak accidents or malfunctions of equipment and others. Actor Timothy Dalton was allegedly almost killed when a truck he was driving narrowly missed going over the edge of a cliff when another rogue truck that shouldn't have been there crashed into him. A man fixing telegraph poles two miles from where they were filming was hit and injured by a missile that went astray from the set and a truck moved by itself and parked itself somewhere else. According to second unit director Albert Wooster, a stuntman was photographing some of the explosion scenes on his own camera and when the pictures were developed one showed a mysterious flaming clawed hand reaching from the fire, something that none of the other cameras had picked up. Adding to this were far more accidents and mishaps that would have been expected and it seems that whatever forces might reside there did not appreciate the film crew's presence. In recent years the road has been widened and is considered to be safer than it once was, but the tales of supernatural phenomena along LaRumarosa certainly persist. Is there anything to such tales or is it all superstition and myth? If there is something to it then why should these forces be drawn to this place and what drives them? For now, LaRumarosa remains a curiosity and focal point of a local legend that will probably never be truly understood, that road stretching off into the distance and past the reality we think we know. Edward H. Rulloff was considered by many to be a genius, a man of great intellect, ahead of his time, ready to revolutionize the study of philology and just as many thought him a fraud and a con man. He was well versed in medicine, law and language and an educator respected by his students. He was also a thief and a swindler who had trouble leaving a city without a run-in with the law. When an 1870 burglary in Binghamton, New York went bad, leaving three men dead, the public would face the paradox of the man of two lives. The early morning of August 17, 1870, Gilbert Burroughs, a clerk at Halbert's Dry Good Store in Binghamton, New York rushed from the store into the street to sound the alarm. Fred Merrick, another clerk, had been shot dead during a burglary. The store had been rowed before and Burroughs and Merrick were serving as night watchmen to prevent it from happening again. About 2.30 am, the clerks were awakened by three men who had broken into the store. A fight ensued and one of the burglars started shooting. Merrick had a pistol and tried to shoot back but the gun jammed. He was shot in the head at close range. Burroughs ran out of the store and the burglars escaped. Later that morning, two corpses were found floating in the Shadango River not far from Halbert's store. Three men had been seen fleeing the store and heading into the river. Apparently two of the burglars had drowned trying to get away. Police began to round up suspicious individuals and before noon had three suspects in custody. One was a mysterious man, respectively dressed, carrying a satchel and an umbrella. He refused to stop when approached by the police and escaped them by running across a railroad track just before a train passed, blocking the road. They eventually found him crouching in the outhouse of a nearby farm. The man first told the police his name was Charles Augustus. Then, later said he was George Williams. He was taken to see the two drowned men but did not recognize either of them. The bodies had been on public display all morning and hundreds of curiosity seekers had passed by. Among them was Judge Ransom Balcom who initially recognized the suspect and said to him, You are Edward H. Rulloff.  Then he turned to the coroner's jury assembled near the corpses and said, This man understands his rights better than you do and will defend them to the last. An 1871 biography of Edward Rulloff was entitled The Man of Two Lives. This was an understatement. Rulloff, also known as James Nelson, E. C. Howard, James Dalton, Edward Liero, etc., had been a doctor, a lawyer, a schoolmaster, a photographer, a carpet designer, an inventor, and a phrenologist. Most notably, Rulloff was a phelologist who could speak Latin, Greek, and six modern languages and was working on a manuscript, method in the formation of language, which he believed would revolutionize the field. The dichotomy of Edward Rulloff's life was the fact that he financed his research by theft and did much of his philological work in prison. Rulloff started both sides of his life early, working in a law firm and spending two years on the penitentiary for theft, both before age 20. In 1842, he moved to the town of Lansing, in New York's Finger Lakes region, where he taught school and began studying botanical medicine with Dr. Henry W. Bull. A year later, he married Harriet Shute, but the marriage was in trouble from the start. Rulloff was extremely jealous and accused her of having an affair with Dr. Bull. The birth of their daughter, Priscilla, kept them together for a time, but Rulloff wanted to move west. Harriet did not. Then on June 23, 1844, Harriet and Priscilla disappeared. Rulloff said that she had left him, but the people of Lansing were suspicious. By the time an investigation was launched, Rulloff had left town. Police searched his house and found Harriet's clothing and no evidence that she had moved out. When Rulloff returned to Lansing, he said that he had moved the family to Ohio. This story was not believed, and when Rulloff slipped away again, he was pursued, captured and charged with murder. The problem with the charge of murder, though, was the lack of corpses. The police believed that Rulloff had sunk the bodies in Lake Cayuga, but repeated dredging and turned up nothing. In 1846, he was brought to trial anyway. Rulloff directed his own defense and tried to focus on the lack of evidence, but the jury was predisposed to convict him of something. He was found guilty of abduction and sentenced to 10 years in Auburn prison. On the day of his release in 1856, a warrant was issued against Rulloff for the murder of his wife. Rulloff argued that it was double jeopardy, and rather than argue this point, the district attorney charged him instead with the murder of his daughter. Rulloff was tried again and convicted again. This time he appealed, but the verdict was upheld. While awaiting sentencing in the Ithaca Jail, Rulloff began tutoring Albert Jarvis, the son of the undersheriff in Latin and Greek. He also became close with Albert's mother, Jane Jarvis, who did not believe that Rulloff was a murderer. In 1857, Rulloff escaped from jail. It was fairly obvious that he had inside help since, in addition to a chain around his ankle, there were eight logs between the cell and the outside. Al Jarvis remained a friend and an associate of Rulloff for the rest of his life, and Jane Jarvis would later secretly visit when Rulloff was living in Brooklyn. Rulloff was recaptured and sentenced to hang. He appealed again, and this time succeeded, and a new trial was ordered. Tired of waiting for justice, the people of Ithaca formed a lynch mob ready to storm the jail, but Rulloff was transferred to Auburn before any damage could be done. Prosecutors decided there was little hope of success in trying him again for murder, without corpses, and there was no interest in prosecuting him for escaping jail, so Rulloff was extradited to Erie, Pennsylvania, where he was wanted for jewel theft. Rulloff was not prosecuted there either, though nearly everyone in Western New York believed Rulloff was guilty of abduction, theft, assault, fraud, jailbreaking, and murder, he was now a free man. Rulloff moved to New York City and was soon joined there by Al Jarvis. He spent his time working on his book and robbing stores with Al and other accomplices. He specialized in stealing, sowing silk, an expensive item that was easy to conceal and hard to identify. In 1861, Rulloff was arrested in Poughkeepsie and sentenced to two years in Sing Sing. In prison, Rulloff met Billy Dexter and on his release, Billy joined the gang. Though Al and Billy preferred to work without Rulloff, he had an unfortunate knack for getting caught. He was with them in Binghamton that night in 1870 robbing Halbert's dry good store. Through the contents of their pockets, the drowned men were identified as Al Jarvis and Billy Dexter and traced to the Brooklyn apartment where Rulloff had been living under the name Edward C. Howard. Rulloff was charged with the murder of Fred Merrick. Edward Rulloff's murder trial received extensive press coverage and over 2,000 people a day came to watch in a courtroom built for half that many. The prosecution introduced witnesses from Brooklyn as well as the eyewitness Gilbert Burroughs. Rulloff once again directed his own defense, claiming that he was not in Binghamton the night of the murder and that whoever committed the murder acted in self-defense and not premeditation. The jury deliberated for four and a half hours. The issue of debate was not guilt, but the degree of the crime. Most wanted first degree murder, but some were pressing for second degree or even manslaughter. In the end, Edward Rulloff was convicted of first degree murder and he was sentenced to hang on March 3, 1871. Appeals moved to the hanging date back two and a half months. While awaiting execution, the case became a subject of national debate. Some said it was wrong to take the life of such a learned man who may be on the verge of a great intellectual breakthrough. Boris Greeley, owner of the New York Tribune, wrote, In the prison in Binghamton, there was a man awaiting death who was too curious an intellectual problem to be wasted on the gallows. Others, however, including some academic philologists, believed that Rulloff wasn't intellectual fraud. In a letter to the New York Tribune, Mark Twain satirized Greeley's position, saying, If a life be offered up to the gallows to atone for the murder Rulloff did, well, that suffice. If so, I'll bring forward a man who, in the interest of learning and science, will take Rulloff's crime upon himself and submit to be hanged in Rulloff's place. Outside of court, the people of New York ascribed multiple murders to Rulloff. Rulloff claimed that Al Jarvis and Billy Dexter drowned because they thought the river was shallow enough to walk across and neither man could swim. But the public saw their deaths as murder by Rulloff to prevent identification. He was also still accused of the murder of his wife and daughter and now accused of killing his sister-in-law and her child who died while under Rulloff's medical care shortly before Harriet disappeared.  He did not confess to killing his daughter Priscilla or anyone else. It was reported that Priscilla did not die but was raised by Rulloff's brother. Edward Rulloff was hanged on May 18, 1871. Before his death, he requested that his body be put in a vault so it would not be desecrated. But his request was not honored. Before his lawyer could claim the body, it was placed on public display and the owner of a local art gallery made a plaster death mask. His lawyer gave the body to Dr. George Burr of the Geneva Medical College who promised to bury the body in a private cemetery if he could keep the head for study. After the body was buried, it was dug up and stolen by medical students. Edward Rulloff's brain, however, still exists as part of the Wilder Brain Collection at Cornell University. on the Contact social page at WeirdDarkness.com. Also on the website, if you have a true paranormal or creepy tale to tell, click on Tell Your Story. All stories in Weird Darkness are purported to be true unless stated otherwise and you can find source links or links to the authors in the show notes. Welcome to Werewolf Central was written by Nick Redfern. The world's most haunted graveyards is from RealParanormalExperiences.com. Acid Bath Horrors is by Abram Ringqvist for Listverse. LaRuma Rosa Hauntings is by Brett Swancer. And Genius Conman Killer is by Robert Wilhelm for Murder by Gaslight. Again, you can find links to all of these stories in the show notes. Weird Darkness is a production and trademark of Marlar House Productions. And now that we're coming out of the dark, I'll leave you with a little light. Proverbs 11 verse 12. A man who lacks judgment derides his neighbor. But a man of understanding holds his tongue. And a final thought from Martin Walsh. When you look for the good in others, you discover the best in yourself. I'm Darren Marlar. Thanks for joining me in the Weird Darkness. Shenango, Shenango. Oh, Shenango, I long to hear you away, you rolling river. Well, that's Shenango. That's Shen. No, that's Shenandoah. And did much of his philology philology philology philology philological philological philological philological philological. And now that we're coming out of the dark, I forgot to get that. Fartknuckers.


What's stopping you from investing in the stock market? Student loan debt? A paycheck that doesn't leave much wiggle room? Did the 2008 crisis leave a bad taste in your mouth? Maybe you just don't know where to start. If that's the case, many young investors are jumping into the market by buying up stocks from companies that have products they like to use. But is that a good strategy? Is that a smart strategy? Is this a strategy at all? Let's take a closer look. On the day Snap went public, activity on Robinhood, which is an app that lets users buy stocks commission free, surged. 43% of their users bought shares that day, according to the Wall Street Journal. Guess what? Snap surged right along with it. The company's shares ended the first day of trading up 44%. If you bought Snap because you like using Snapchat, it looked like a good bet. But the good times didn't last much longer than a photo last on Snapchat. The stock quickly tumbled. If you stuck with the stock through August, you and CEO Evan Spiegel were both down 50% while he lost more than you did at the end of the day, he's still a billionaire. But there is something to be said about this strategy. The top holdings of TD Ameritrade's millennial clients for the month of June included Apple, Facebook, and Amazon. Each of these companies makes a compelling argument for buying the companies that you use. Year-to-date, Apple is up 40%. Facebook is higher by 49% and Amazon surged 31%. So is it fair to compare these more established companies to a newcomer like Snap? That's not what we're doing here. We're looking at the buy what you like strategy. If you like Apple, Facebook, or Amazon, you would have done very well. So this is my take. If you love a product enough that you want to invest in the company that makes it, you might strike gold, but you could also end up losing your shirt. Let's face it, people have purchased stocks for worse reasons than liking their products. Just remember, you might love a brand, but Wall Street doesn't necessarily feel the same way. Just look at Twitter. So always do your research. Always look at things like price to earnings ratios. Check the competition and know the industry trends. But if you're not confident or you don't have the time to do the work, the safer bet is often looking at index funds, which allow you to spread your risk across companies that you already love and some that you've never heard of.


What's up, guys? My name is Patty Graham. I'm a professional freeskier from the UK, and my favorite thing to do on the mountain is to shred powder. Now, shredding powder isn't all that easy because you have to find it. It's all about being in the right place at the right time. So today, I'm going to show you my best tips on how to get to that place. Needless to say, before you go out chasing powder, make sure you have all the right equipment and experience to do so. Always look at multiple snow forecasts. Now, I like to look at a few websites and kind of gather the data and then see exactly where I want to go. Let's have a look at some of them. We've got Bergfex, a pretty standard one. You get a nine-day forecast. You've got snowforecast.com. There, you can check out any country in the world. I also like to look at WePowder. This guy puts together all his information and tells you exactly where to go. So, from the forecast, you need to figure out the basics of these storms that are coming. Where are they coming from and where are they going to stop? So, figure out where the motherloads come in and you want to get in there. So, once you've got your weather forecast, don't just go to where it's going to get the most snow. You need to look at temperatures and wind because you need low temperatures and less winds to get the best snow. But snow forecasts are great, but what you really want is to get intel on the ground. So, pick up the phone and start dialing. Hey, Sven, you in St Anton? Yeah, what's it like there? So, the forecast has given you a region to hit. Now, be flexible. Find a resort and find the best one for you. Be mindful when you're going to go skiing. You want to have a look at local holidays, school holidays and all things like that because they're always different in different countries. And remember, on the weekends, Saturday is general transfer day, so there's not that many people skiing. And if there are weekend warriors out there, then they usually go home on Sundays. When you're choosing a ski resort, don't just look at the snowfall in the future. You need to look at the snowfall that came in the past. You want to know how much base these resorts have. The bigger the base, usually the better the skiing. Once you've found your resort, you need to explore that and really analyze it to see where you want to go. Lower down is going to be trees. Higher up is going to be more alpine. So you can use aids such as Google Maps or Fatmap and then find some terrain that's really going to suit your skiing and the conditions you have. If the weather's bad, why not go tree skiing? They provide great visibility and there's super fun. So once the storm's cleared, don't just head up to the top of the mountain and ski the gnarliest stuff you can. You're going to want to get a feel of the snow. You're going to want to know what the snow is doing. You're going to want to read the Avalanche reports and know exactly what you're getting yourself into before you do it. So if it hasn't snowed in a while, maybe you want to go ski touring. Ski touring is a great way to access more terrain and there are less people. If you're still looking for soft snow after a snowfall, you can look at those northeastern faces because they're going to be less affected by sun and you might find some fresh snow in there. Oh, that's a good picture, Benny's put on Instagram. Annie's left a location. Looks sick. Let's see if that was today. Hey, Benny, did you have a good day? Was that picture you posted from today? Oh, it was two weeks ago. And a different location tag. What? Dude. Never trust Instagram.


So, this is lecture 2 and the last class what we were seeing was a very high level picture of what a digital communication system is supposed to do, some kind of a definition of the major parameters and all that. So, I am going to remind you of that. So, the problem we have at hand is basically I will do a block diagram kind of description and you will see this block diagram description is very very useful and helpful to imagine and visualize these kind of systems. So, you have a transmitter and the receiver, transmitter gets say a vector of bits b and it is supposed to convert that into a signal x of t which goes through a channel and that the output of the channel you get y of t which is received by the receiver and the receiver has to produce an estimate of what the bits transmitted could have been. So, this is a nice block diagram of what we saw before and this kind of completely defines the problem of digital communication. So, what do we have to do? We have to design two functions if you will. You can think of this Tx and Rx as two function. So, x of t is this transmitter function of this vector b and what is b hat? b hat is whatever the receiver does to y of t to produce the estimate. So, given the certain b and given information about a certain channel, channel is kind of known to you. You have to design Tx and Rx such that you can achieve communication at very low error rate and all that. So, you want b hat to be equal to b without any serious problem. So, that is what I want to capture kind of in this picture hopefully it is clear. So, a few things we saw the main parameters here which we will have to play with and be careful about. The first thing is power of x of t. So, there are ways to write it down mathematically in terms of what it is. So, typically when you have a voltage or current, the power is square of that quantity. You think of x of t as a voltage or current in some circuit, the power would be square of that quantity. So, that goes for the relationship b square by R and i square r. So, power square of that quantity is instantaneous power. So, x square of t would be instantaneous power. So, typically we will be worrying about the average power of x of t, so not the instantaneous power and some systems instantaneous power is also important, but usually the average power. So, when you do average power what should you do? You integrate out x square of t and divide by the total time. So, that will give you a way of calculating the power that x of t has, the transmit signal x of t has. So, this is going to be again, once again remember this is average, not instantaneous, maybe this is some px. So, usually there will be an upper bound on this. So, there will be an upper bound on the transmit power, you will be constrained to transmit at powers less than, average power less than a certain maximum power possible. That is the first thing. And the next thing is bandwidth or spectrum, I will say bandwidth of x of t. This will also be constrained. So, this will also be constrained because of the channel and all kinds of other things that I talked about. There will be a constraint on this also. So, these two guys will be constrained. So, this is what I mean by knowing the channel. So, you know what your constraints are given. These two constraints you have to design these functions t of tx and rx, so that what is your objective? Your objective is to get a reasonable bit rate. So, your bit rate should be some kind of reasonable number or if you want me to be specific on it, maybe you want to have some kind of a trade off between the bit rate and the next quantity of interest to you which is probability of error or error rate of your system. So, if you think of this vector B as B1, B2, so on and B hat as B hat 1, B hat 2, so on. So, this probability of error is probability that B i hat is not equal to B i. So, one assumes this is kind of an identically distributed for each B i. So, all bits are equally are going to have similar behavior and you get a certain probability of error. So, what would you desire for bit rate? You would want as fast a bit rate as possible. What would you desire for probability of error? As low a probability of error as possible. So, those two things are going to be related and there is going to be some kind of a trade off between these two. There is going to be a trade off between these two.  So, all these four quantities are going to be closely interrelated. There is also another quantity which I have not written down here which is the receiver noise. So, that is also very important noise power, it is maybe I will call P n. So, bandwidth of X of t I did not put a number for it, it is a w. So, noise power is also important. We saw noise power is another quantity which affects probability of error and in fact, it is also related to bandwidth and all that. So, you will see it is all these quantities are interrelated. So, our goal in this course roughly is to study given a set of constraints on these quantities. How do you go about designing Tx and Rx, what are the interrelationships? So, one simplification we did to get rid of all the physical properties of the channel is to come up with a mathematical model for the channel itself. So, one model that I wrote down last class which I said which will follow pretty much throughout the course, the most general model is what, it is a linear Gaussian model that I had. So, maybe I will write it right here, I will say Y of t is what, X of t convolved with impulse response H of t, okay, so maybe you cannot see it plus a noise, okay. So, this is my linear Gaussian model and like I said it is a very, very powerful model. You design for such a model and it is directly translatable into several actual physical systems including wireless, wire line, optical, whatever you name it, okay, so it is all possible, so it is a very powerful model. And then we made a few simplifying assumptions to get this linear Gaussian model into a slightly simpler form to start off with, which was the band limited ideal model, okay. So, what is ideal about my channel, I am going to say H of f which is the frequency response of H of t, it is going to be 1 bit in the bandwidth of interest minus W to W for instance, okay. So, there it is going to be 1, so it is an ideal thing but it is band limited, okay, so that is the, that is with that understanding we could move from here to Y of t equals X of t plus N of t, okay. And this was the ideal band limited additive Gaussian noise model for instance, so N of t is going to be Gaussian and all that and that is fine. So, for instance it is quite surprising that with such an abstract model a lot of things are known, for instance the most ideal trade-off between all these quantities is known, okay. And this was found long, long back in 1949 by Claude Shannon in the first papers to ever show up in this area, okay. So he found this very surprising and nice result, suppose you want the probability of error to be arbitrarily low, when you say arbitrarily low it means as low as I want, okay, to be arbitrarily low, then he showed your rate will have to be less than or equal to W log base 2 1 plus Px by Pn bits per second, well this is bits per second, okay. So, quite a surprising and powerful result to have come up very early on in the theory, okay.  So, it's a surprising and nice result and this such results have actually given shape to most of communication theory as it happens today, okay. So, that's just wanted to write it down so that you know what the kind of powerful results you can derive with simple models like this, okay. So, that's the formula and you'll notice a couple of things. First thing is the rate seems to grow, I'm sorry. So, rate seems to grow linearly with the bandwidth W, okay, so that seems to be good enough and the other thing it depends on is only the ratio Px by Pn, okay, which is the power of X of t, average power of X of t divided by the average noise power, that's what it depends on. It doesn't depend on for instance Px separately or Pn separately, okay, so it depends, so you'll see Pn actually depends on W also, okay, so that's something I've not shown here, Pn will depend on W, so it's kind of a pretty good thing, but in terms of the rate it depends only on the ratio, okay, so that ratio is important, we'll define it later on and proceed, okay, so this is roughly the system that we'll be dealing with pretty much throughout this course, our goal will be to change some of these assumptions and design different types of transmitters and then define suitable receivers for them and define suitable quantities to analyze probability of error, all these things. So, we'll keep coming back to this picture over and over again and these kind of block diagrams are very, very powerful tools kind of, almost like a tool, okay, so we'll start with a big block diagram and spread it up into several smaller blocks and then design each block separately, it ends up being a very, very powerful tool, okay, any questions on, comments on any of these things, it's a very provable, accurate theorem, okay, maybe we'll see a proof of it towards the end if we have time, okay, so couple of other things just to give you a big picture in case you're seriously thinking about becoming a communications engineer or getting into some research in this area, so a big picture of how the different fields or courses that you have taken so far interact, okay, so you have, I'm going to put digital communications at the center because you're doing the course right now, so I'll put this at the center, maybe it's not at the center depending on where you are, okay, so there are two related areas which feed into it, which guide the design of digital communication systems, okay, so the one thing is information theory, okay, information theory which is what enables you to get results like that, R less than or equal to W log base 21 plus SNR for instance, those kind of results come from information theory and in fact this is closely related to coding, error control coding and coding theory, so information theory and coding theory feed into the very fundamental basic designs of communication systems, how you go about doing it, what's the right metric to look at, how do you do it and all that, okay, so once you know the high level design, that high level design has to be translated into a lot of algorithms and those algorithms are typically done in signal processing, okay, so you have signal processing kind of feeding into this, okay, in particular today it's just completely digital signal processing, there is some analog part which is very front end but everything else is completely totally digital, okay, so it's all digital signal processing and of course there are parts which you still cannot build without any RF and analog things, particularly in some and high frequency and all these things, so you have the RF analog part feeding into the very front end of these things and then of course for very fast functioning and compactness and all that you have digital VLSI building into, so all these things play together in, so the things on the right hand side are more like tools for translating the design you have into algorithms, implementations and all that, okay, and things on the left are fundamental tools to design the system itself, okay, so there's also an area called communication theory which kind of puts all these things together, okay, so you can imagine communication theory at the intersection of all these three, okay, so how do you build communication systems? Okay, so this is, so depending on where you want to be in the scheme of things, you might want to specialize in one of these things and maybe know the other things as enough to do, okay, but of course they're not strict divisions, people who do analogs claim to know a lot of digital communication also these days, okay, so they do a lot of stuff back and forth, so there's also all kinds of give and take, okay, all right, so that's pretty much as far as introduction and other things are concerned, so the rest of this lecture we're going to spend beginning with preliminaries that you need to be able to understand most of this course, okay, so that's what we'll do now, okay, so these are all basics which you should already know and be very very familiar with at this point, okay, so I'm going to assume that and we'll go through it at a very very fast pace, as fast as possible, I don't want to spend too much time, like I said, it's only to fix notation, okay, nothing, nothing else, most of it I will state and I'll go through very quickly and it's, you may not even want to write these things down because I'm not trying to really make you understand or anything like that, I'm going to quickly state all these things, hopefully it's all very very familiar to you, in case something is not familiar, you've never seen ever before, stop me at that point maybe we'll go into some detail, okay, so the first thing is we'll be dealing with complex vector spaces, okay, the notation I'll use for a complex vector space of dimension n is this crazy c, c with a stroke right in the middle, cn, okay, and a vector in cn I'll denote as z, which will be a column vector but just for compactness I'll write it as z1, z2, zn transpose, okay, so when I say transpose it is, it becomes a column vector, this is an arbitrary vector in z, okay, in cn, okay, you can think of your discrete time signals for instance, okay, so of course you have the infinite sequence length but usually when you deal with it in practice it's always finite length or you deal with it in finite length, so all your discrete time signals will be, will be, will belong to some complex vector space, okay, so what is each of the zi, each of the zi is a complex number, hopefully you're very very familiar with it, a complex number is, is a real ordered pair of real numbers, the root of minus 1 I'll denote j, okay, so x of i plus y of i, you can also write it as in polar form as r theta, okay, so all these things are very very standard, you know how to go from one to the other, okay, it's quite good enough, okay, so a dot product defined in this, in this vector space z1, z2, okay, is actually summation z1i, z2 star i, where star is complex conjugate i equals 1 to n, it also has a more compact notation which is z2 Hermitian z1, okay, what is z2 Hermitian, you do a transpose and then you take conjugate of each element, okay, and you have the norm of vector z which is what, z inner product with itself which will be summation of square, okay, mod square of each element, okay, so that's the complex vector space, hopefully that's very very familiar, the second thing that we'll be dealing with, so that's for discrete time signals, for continuous time signals we need vector space with whose elements are signals, okay, or functions of one variable, usually that variable is time for you, you can also keep it as frequency if you like and you can know, you know how to go from one to the other as well, okay, so that the signals will live in, typically most of the signals we see live in this function space which is called L2, okay, so this L2 is a special function space, it actually collects all finite energy signals together, okay, finite energy is pretty much all the signals that you have, right, so finite energy signals belong to L2, that's a very simple definition for it, X of t, okay, this L2 is set of all X of t such that integral from minus infinity to infinity, absolute value of X of t squared dt is, when I say less than infinity I mean it's finite, okay, so it doesn't go to infinity, what is, so X of t is complex value, okay, it can have a real part as well as an imaginary part, okay, so when absolute value is the absolute value for the complex number, so like I said L2 collects together all finite energy signals, for instance finite signals that are spread over only a finite time and are bounded will belong to L2, likewise signals that are spread over a finite frequency and are bounded in frequency will also belong to L2, okay, so L2 is a very, very nice space, it's very well behaved, lot of things happen very, very comfortably there, you don't have to worry about weird things in L2, okay, so in L2 also you have an inner product which is X of t, Y of t, this is integral minus infinity to infinity, X of t, Y star of t dt, okay, so this is a valid inner product, it satisfies all the properties that you need and it also, you also get a norm from this, okay, so the norm is what's called the 2 norm which is, which will work out to, if you do X of t, comma X of t, it will work out to mod X of t squared dt, so you see, so norm is kind of, so all finite norm, finite 2 norm signals will belong to L2, okay, so that's L2 for you, it's got a nice inner product as well, all right, so often, so this is like I said, what is L2? L2 contains our continuous time signals that we'll be dealing with in this course, okay, sometimes you might also, it's also useful to have signals which are not really finite energy, infinite energy but finite power, okay, several times you deal with it, what's a very good example, finite power, infinite energy signal that you use all the time, sinusoids, okay, sinusoids you use all the time and it's a good abstraction to have and have good theory for it and that we'll say belongs to this function space L infinity, okay, well it contains many more things than sinusoids but I'll just simply define L infinity because that's the only thing that's, it's very nicely defined in theory, okay, so we'll say it's bounded, bounded functions, okay, so the function itself is bounded, it doesn't explode like a sinusoid, right, so sinusoids will be heavier and this is not such a nice space, for instance, there's no inner product here, there's no norm, this can define a norm but the inner, it doesn't come from an inner product, okay, so it's not a very nice space and a lot of strange things can happen in this space but we won't worry too much about it but for us, the main use here will be it contains sinusoids, okay, contains for instance, sorry, contains for instance e to the power j omega t, okay, which is a, which is an indispensable function in any signal's course, sin omega t, cos omega t and all that, okay, so these are the two, our signals will live in these two spaces, we won't take anything else that's not here, okay, then what else, what else, what else, what else, okay, so once you have an inner product, the inner product obeys, okay, I'll just keep listing it out as soon as I go, the inner product obeys what's called the Cauchy-Schwarz inequality, okay, so it's quite useful, several times we'll use it in our proofs, okay, Cauchy-Schwarz, how does Cauchy-Schwarz work? If you have two vectors, say s and r in an inner product space, what do we know is true? What's Cauchy-Schwarz? The inner product, absolute value of the inner product between s and r, okay, so the inner product takes two vectors to a complex number, okay, that's how I'm defining my inner product, okay, so s and r will be less than or equal to, the magnitude will be less than or equal to norm of s times norm of r, when will you have equality here? Equal is too strong, both have to be differ by a constant, right, complex constant, so if s can be, s is equal to a times r for a complex a, then you have equality in this Cauchy-Schwarz, it's definitely, there's no other equality otherwise, okay, so this is something we'll use quite often sometimes to come up with some results in the proofs, okay, so the next thing is, I don't know, I'm going to drop the numbering, I've been numbering but I think I'll lose the numbering pretty soon, the next thing is convolution of functions, okay, so it's quite important, if you have two functions s of t r of t, okay, the convolution is another function of time which is properly denoted as s star r t, the evaluation at t is going to be equal to minus infinity to infinity s of u r of t minus u du, okay, so this is a very standard definition, convolution, so this is the value taken by the convolution at t, okay, so that's the definition but usually there's an abuse of notation which helps us a lot in terms of ease of working with convolution, you typically write q of t as s of t star r of t but you should know that there is, this is a tricky notation, sometimes it can lead to all kinds of problems but this is an abuse of notation which we will use in this course, okay, so the next thing I want to mention is delta functions, okay, I think enough has been said about delta functions for you as far as we're concerned, one of the properties is very useful as the sifting property for delta functions, what is this integral defined to be s of t naught, okay, so the use of delta function is typically to define Fourier transforms for L infinity functions, when you have sinusoids you can diffuse this delta function notion and define a Fourier transform and so that the whole thing is consistent, for L2 functions in L2 Fourier transform is properly defined, there's no problem, once you go out this L infinity it's not defined properly, so you use delta functions to define Fourier transform and keep it consistent with all the other definitions in L2 as well, so that's useful as far as delta functions are concerned. The other thing I'll talk about often is the linear filter, okay, we'll use these blocks quite often, linear filter is nothing but an LTI system, okay, so an LTI system which takes say in this case a continuous input X of t, it's fully characterized by an impulse response, what does it produce at the other side, X of t convolved with H of t, okay, so that's what happens in a linear filter, we use this quite often, okay, so any term that I talked about which is strange, it's okay, all right, so maybe if you've not seen this L2 and L infinity very formally it's nothing to be really scared about, one thing has an inner product, other doesn't have an inner product, it's the only thing you should know, okay, beyond that it's nothing more to worry about, okay, all right, so the next one small aside here which will be important for us is this computing inner product by filtering, filtering and sampling, okay, so maybe you have not seen this very formally, it's quite important, okay, so how did I define inner product for two functions, okay, so if I have two functions X of t and S of t, okay, so I'm sorry, I'm doing convolution already, okay, the inner product between X of t and S of t is defined as what, right, integral minus infinity to infinity, X of u, S of u, du, well S star of u, du, okay, so that's how I define my inner product, if you stare at it very closely it seems to be quite similar to the convolution integral, so it makes sense that you should be able to compute the inner product by convolution, for instance if you said t equals zero it seems to be almost close and then you would only do some adjustment to fit R of minus u2, S star of u, okay, so that's the only adjustment you have to do and you see here a linear filter is going to convolve whatever input with its impulse response, so if you select a suitable impulse response based on S of t you should be able to achieve inner product calculation through filtering and sampling, sampling at t equals zero, okay, so that's the notion of computing inner product and that's called the matched filter, okay, filter matched to S of t, so how do you do it? You take X of t, filter it with, okay, so something happened here, X of t, you filter it with S star of minus t, okay, so that's the impulse response which is matched to S of t, okay, it turns out here if you sample at t equals zero you end up getting the inner product between X of t and S of t, which will be some complex number, right, some a plus j b, okay, that's what you'll get, okay, so it's easy to prove this, it's not too difficult but the, I'm not going to write detailed proof, okay, try to write down the proof if you're interested, it's easy to do it, but this impulse response, a filter which has impulse response equal to S star of minus t is called a matched filter, matched to, matched to what, S of t, okay, so this S star of minus t, once in a while I'll use this notation just to ease the confusion because S star of minus t has so many other things going on around it, right, there's a conjugation, there's a minus usually when you use it in some other systems people get confused because of the notation, okay, so I'll typically use this notation also, S m f of t, I'll say is S star of minus t, S m f, okay, subscript is m f which is matched filter, star of minus t, okay, so in several receivers you'll have to do correlations or inner product computation and there it's done typically using matched filtering, okay, so this is a very common way of doing this, all right, so a couple of things I want to point out about this because I think if the first time you're seeing it, it's, it requires some effort to translate this into real life, I said all my signals I'm going to imagine are voltage and current in a circuit, what will I do with the complex valued X of t, can you have complex current, complex voltage, okay, so what you do is you have to take two signals, that's all, so when I say a complex X of t what do I mean, I have two real signals X, X r of t which is the real part and X i of t which is the imaginary part and I'm looking at both of them together as an ordered pair, okay, so that makes me a complex signal, okay, so that's the, that's the first thing to keep in mind, so this X of t when it's complex is actually two wires, okay, carrying two different signals X r of t plus j X i of t, so if I have to implement this, I have to actually do a fairly complicated implementation, I'll show you how it works, okay, so suppose and similarly this S m f of t will also be what, real part and a imaginary part, okay, so I'll call this simply S 1 of t plus j S 2 of t, okay, suppose you have such a picture, this entire filtering is quite complex, okay, so it's not, so it's actually X of t convolved with S m f of t, you'll have to do four different convolutions to execute that, okay, so you'll have two different symbols, signals coming in, wires coming in, one will carry X r of t, another will carry X i of t, okay, both of them will get filtered by what, okay, so you'll have, okay, I'll draw four different filters, okay, so maybe you can do it smartly with just two, but S 1 of t, S 2 of t, okay, and then you have S 1 of t and S 2 of t, okay, and then how do you finish up the convolution, you'll have to do, okay, so these two, this has to be added to, added or subtracted, subtracted from this guy, right, am I right, okay, to get the real part of what's going out, this will be Y, okay, I didn't put Y of t here, so Y of t, Y r of t, okay, how will you get Y i of t? You take these two guys and add them, okay, so it's not really impossible to imagine an actual LTI system or something which does all this, okay, so all filters, then addition, subtraction, these things can be done in analog and definitely in digital very, very easily through any problem, okay, so this whole box is basically representative of what, that just that one single small box I write with S star of minus t, okay, so that's also we think of complex signals, okay, two wires carrying two real signals, that's one complex signal, okay, it's just mathematically easy to write down a simple convolution with complex value signals as opposed to four different convolutions adding and keeping everything real, okay, so that's something and when you sample, of course you have to sample both, okay, so sample both to get the sample, all right, any questions on this, okay, okay, so we're slowly going to move towards spectrum and Fourier transform and all that, so before we go there we'll see a few popular signals and give them names, the first one is the sync, what is the sync? The sync in this class is going to be defined like this, sync of t is sine pi t by pi t for t not equal to zero and for t equals zero you make it a continuous extension to get one, okay, so that's my definition for sync and then I'll also define rect, okay, rect in from a to b of t will be one for t between a and b and zero outside of it, okay and next we'll have unit step, I think this is pretty much all we'll need, if we need more I'll just introduce it wherever, okay, so I think you know what unit step is, okay, u of t is defined as one for t greater than or equal to zero and zero else, okay, so that's the, that's the definition for functions, standard functions, sync might be slightly different from maybe what you're used to, maybe some of you have seen sine t by t, it's not a big difference, same thing, but some of the transform properties will change, if you're used to the sine t by t you might want to read up a little bit and make sure you know the sine by t by by t change as well, okay, all right, so next thing we'll define is Fourier transform, well this is continuous time Fourier transform, there's also a discrete time version which we will see a little bit later, okay, so continuous time Fourier transform, you start with x of t which is a possibly complex value which we'll call time domain signal because I'm going to use t as my free variable there, it's it's it's defined as typically I'll denote by capital letter the Fourier transform of the small letter of the signal x of f, I'll use f, I won't use omega, okay, so I'll use f, it might also cause some change in the way you write down the formula, x of t e power minus j 2 pi f t d, okay, so that's the definition for Fourier transform and I'm sure you've read enough about it, you know the value of this and it's quite useful, okay, and also has an inverse relation which is valid for well-behaved good functions, it's minus infinity to infinity x of f e power j 2 pi f t d f, okay, so if you're used to the omega you'll get a 1 by 2 pi in this inverse relation, okay, so you do an f, you don't get it and if x of t is an L2, okay Fourier transform is very well defined, it converges, everything works beautifully, capital X of f will also be an L2 function, okay, in the variable f, okay, so it will also be L2, it's all very well defined, in fact it preserves inner products, inner product between x1 of 2 and x2 of t, x1 of t, x2 of t will be the same as inner product between x1 of f and x2 of f in the variable f, okay, so it preserves inner products, so it's a very, very nice transformation from one's L2 space to another, okay, but very often we'll also use L infinity functions as x of t and expect a Fourier transform and that's defined using delta functions, okay, so there's all kinds of definitions for that, I'm not going to go into detail there, I'm assuming you're familiar with all that, okay, so that's the first quick property, so if x of t is an L2, like I said it's very, very well behaved, you can show the inner product between x1 of t, well you can first show that x of f is also an L2, so I'm going to write L2 here but you should know that this L2 is different from this L2, right, what is, what is, what is different, t has become f, okay, both of them are complex valued except that t has become f, I think more, you have the same functions here also with f, okay, it's no, no, no big difference, okay, so the inner product you can show is the same as x1 of f, x2 of f, in particular if you let x1 equals x2 what happens, you get the familiar Parseval's relation which says the two norm of x, x of t is the same as the two norm of x of f, okay, so all that is valid in, valid for L2, well infinity these relationships are more complex, we won't worry about it except by saying you'll need delta and it's all consistent, all the properties and everything is consistent, okay, so quick roundup of the properties of Fourier transform that we will use, okay, so the first one is a very standard Fourier transform pair which is used at, used for deriving pretty much every other Fourier transform pair that you want, okay, so this, okay, so this is one more thing, so if we write two arrows and put ft means it's a pair, okay, so t sink ft, okay, so this is a very common thing and delta of t, okay, very often I'll omit the ft, so whenever it's clear I guess it's not really necessary, okay, so ft, so clearly one for all f is not L2 function, right, so you can't integrate, so that's the problem, okay, so that's the first thing and this pair, this pair along with the properties that I'm going to, well some of the properties that I'm stating and additional properties that I will not be stating will give you pretty much every other Fourier transform pair that you want, okay, so this is just, this is good enough, so you should remember this by heart, the thing else you can do it, okay, so the second property is if you have x of t being a Fourier transform pair with x of f, then whole bunch of other pairs can be obtained immediately, the first one is remember it's all complex valued, okay, so I'm all writing complex valued things, so this conjugation and all these things have a meaning, okay, first thing is what's called duality, you can show x of minus t will be a Fourier transform pair with x of f, okay, so if you take the capital X function with t as the variable instead of f, you get the small x of f, okay, that's duality and then if you conjugate t, you get what, you get the negative, the mirror image conjugate for capital X, okay, if you do a match function transformation on t on x, okay, you do x star of minus t, which basically you make this the match filter response for x of t, then you get x star of f, okay, remember this, this is very important, okay, the match filter match to x will have a response which is the conjugate of the original response, okay, so when I say conjugate what happens, the real part remains the same, the imaginary part becomes minus, okay, alright, so if we know that x of t is real valued, then it turns out there is a nice symmetry, okay, so what is x star of t if x of t is real value, equal to x of t, so what should happen, x of f should be the same as x star of minus f, so what this means basically is the positive frequencies are enough to completely describe the spectrum, the negative one can be obtained easily from the positive one by conjugation and time, then frequency reversal, okay, so you do that, so this actually translates into two conditions for real and imaginary part of x of f, real part of x of f equals real part of x of minus f, so this is even symmetry for the real part and you will have odd symmetry for the imaginary part, imaginary part of x of f is minus imaginary part of, okay, so I think this is called conjugate symmetry, okay, so this is conjugate symmetry and quite a few other properties, I'll just quickly list them as we go along, x1 of t convolved with x2 of t is going to be a Fourier transform pair with x1 of f, x2 of f, use it along with duality, you can also get the Fourier transform pair for the product, it will be x1 of f convolved with x2 of f, okay and then the next properties which will be useful are delay and multiplication by exponential, these are very useful for us, if you delete a time domain signal, so to speak by t0, you would get a multiplication by 2 pi ft0 in the frequency domain, okay, so what's more important to us is this guy x of t e power j 2 pi ft0 will be what, no f0 t, I'm sorry, okay, f0 t, okay, will be what? x of f minus f0, okay, so this is extremely useful in building systems, particularly communication systems because like I said, you will, the x of t several times the constraint will be is that it should be in a particular frequency band, okay and that frequency band may not be amenable to a small circuit board that you might want to build at the transmitter side, so what do you do? You build your signal first in base band in whatever system you have and then you use this property to multiply by a suitable sinusoid and shift it to whatever frequency you want just before transmission, okay, so this is a very useful trick you can use that simplifies your design a lot, okay, so this is a property that's used a lot, another one is scaling property, I don't want to write it down, okay, so I think that's enough, that should be more than enough, x of a t becomes 1 by mod a, x of f by a and all that, okay, all right, so the next thing is autocorrelation and I don't know what number I'm at, okay, maybe I'll say I'll say 3, okay, so autocorrelation and energy spectrum, okay, maybe this is not all that familiar, okay, this is particularly useful when you go to random signals but for now we are just dealing with deterministic signals but even there, let me define it formally so that we'll know what to do when these things show up even for deterministic signals, okay, so you have an x of t, okay, how do you define the autocorrelation, you have an x of t being a deterministic function, the autocorrelation evaluated at tau, okay, so the autocorrelation you don't use t usually, you say tau just for just to differentiate it from t, okay, this is minus infinity to infinity x of u x star of u minus tau du, okay, right, so once again this is closely related to inner product and convolution obviously, right, so you see for instance a tau equals 0 you pretty much get the norm of x, okay and for every other tau what do you get, you get the inner product of x of t with a delayed version of x of t, okay, so the autocorrelation basically is an evaluation of inner products for all time delays, okay and it gives you a lot of information about the function, you can show like I said it's very closely related to convolution, you can show this is x of tau convolved with x star of minus tau, okay, so you send x of t through a filter matched to itself, you get the autocorrelation at the outside, okay, right, makes a lot of sense, so this is x mf and based on this property one can quickly show the Fourier transform of rx of tau will be what, mod x of f square, okay because x of t, x of tau will go to x of f and x star of minus tau will go to x star of f, we multiply the two you get mod x of f square, so in particular if you have an autocorrelation function its spectrum will be what, yeah real value but positive also, okay, so it's a very good property to have for spectrum, okay, so that's the thing, okay, so what else, okay, so the next thing is the energy spectrum, okay, so it's a very simple definition but I want to just write it down, okay, suppose you take x of t and pass it through a very, very narrow band filter at f naught, okay, so I am going to pass it through a very narrow band filter centered at f naught and one edge being f naught plus delta f by 2, the other edge being f naught minus delta f by 2, okay, so this is my filter maybe this height is 1, okay, you would get y of t, right, so the energy spectrum at f naught is defined as, okay, once again this is much more useful for random signals, we will come to that at a later time, at f naught is the two norm of y of t, basically the energy in y of t divided by delta f and you have to tend delta f to 0, okay, so delta f is seem to be very, very small, okay, so what will happen if you have a reasonably well behaved x of t, this will tend to mod x of f square, f naught square, okay, so hopefully you can see it all the way at the bottom, so the energy spectrum is very, very closely related to the Fourier transform of the autocorrelation function, okay, so you can see how you define similar things for the random signals, okay, so when you have a random signal x of t, each realization is not fixed, right, it's random, so you can't, there's no point in computing energy for each realization separately, so what you do, you average it out and compute the average autocorrelation, okay, so in the autocorrelation instead of this you will do an expectation around that and you get a average autocorrelation and then once you get an autocorrelation function it's a deterministic function of time, for that you compute the Fourier transform, you get the spectrum, okay, so in the random case this makes a lot of sense in extension, but in real, real deterministic case also one can make similar definitions and nothing stopping it, okay, so I think I've done reasonably okay as far as time is concerned, pretty much finished up all that I wanted to say, the next thing we'll see is baseband and passband signals, basically I think we have about five minutes left, so maybe I should just quickly define those quantities, we'll proceed further later, okay, so okay, so this is again definitions which you might already know, but in this context it becomes very, very useful, okay, passband signals, okay, so when I introduce the ideal band limited AWGN channel I said I'm going to assume my H of t is going to have a response which is flat between minus w and w, okay, so typically that may not be the case, okay, so based on the requirement you have your H of t might actually be flat in some other frequency band, okay, so you might be required to operate only in a certain frequency band, okay, so then you will need a certain passband characteristic for X of t, okay, X of t should be non-zero in spectrum only in some passband, okay, so for that these passband signals are very, very useful, okay, so let me formally define it, X of t is said to be a baseband signal, okay, if the spectrum of X of f equals in absolute value goes to zero for mod f greater than w for some w, okay, so this is a baseband signal, so by definition all signals will be baseband, right, so you also need the other condition, so X of f should not be zero within the band, okay, only then it becomes a real proper baseband signal and you also want to think of w as a very reasonably small number, okay, compared to the actual frequency at which you might be transmitting, so it should be a small number, okay, so that all that is understood here, so it is band limited to a small band in the baseband, okay, and the next and similar definition you have for an LTI system, okay, an LTI filter with impulse response H of t, when is it supposed to be baseband?  If H of t is passband, okay, so throughout this course, okay, we will consider real baseband signals and real passband signals, okay, I have been talking about complex signals also, you will see why soon enough, okay, turns out real passband signals can also be thought of as complex baseband signals, okay, so that is why it is useful to have complex signals in our midst, but still in our entire course we will be concerned with real baseband and passband signals, this is good enough, okay, so for communication purposes there is really no reason why you should consider anything other than this, okay, so it turns out this is good enough and these passband signals we will represent as complex baseband, okay, I will do this derivation maybe in next class, okay, so you can show a complex baseband signal is representative of a real passband signal, so in essence what will we be dealing with? Baseband signals, real and complex baseband signals, okay, so those are the only things that we will be dealing with in this class, okay, so there is a lot of advantage in doing this passband to baseband jump, okay, so the reason is you will see this passband signal while it seems to be dependent on FC, once you come to baseband there is no FC in the picture anymore, all you have to do is some multiplication at the end to get any FC you want, okay, so you can do design independent of the center frequency if you concern yourself only with the baseband signal, so that is an advantage in doing this, okay, so we will stop here if there are any questions, it is a good time to ask, fine, okay, so we will begin next class