One of the most important equations in wireless communications

 

One of the most important equations in wireless communications, if not the most important, is the Shannon–Hartley theorem.

It states that the channel capacity, C, equals the bandwidth, B, multiplied by the base-two logarithm of one plus the signal-to-noise ratio.

C=B\log_2(1+S/N)

In simple terms, this equation tells us the theoretical maximum rate at which information can be transmitted reliably over a communication channel of a given bandwidth and signal-to-noise ratio.

What makes this equation so important is that it establishes a fundamental limit. No matter how sophisticated our modulation, coding, or communication system becomes, for a given bandwidth and signal-to-noise ratio, there is a theoretical limit to how much information that channel can reliably carry.

Exploring Zadoff–Chu (ZC) Sequences and 3GPP Limitations

• Exploring Zadoff–Chu (ZC) Sequences and 3GPP Constraints

  A Zadoff–Chu (ZC) sequence is a complex-valued sequence with constant amplitude and zero cyclic auto-correlation, meaning that the sequence does not correlate with any of its cyclic shifts. When a ZC sequence is applied to a signal, the resulting waveform maintains constant amplitude. Furthermore, by applying cyclic shifts, it is possible to derive multiple orthogonal signals from a single root sequence, since different cyclically shifted ZC sequences remain mutually uncorrelated.

  These unique correlation properties make ZC sequences extremely valuable in wireless communication systems, particularly in scenarios that require robust synchronization, channel estimation, and multiple-access capabilities.

  Applications in Wireless Communications

  Several 3GPP-defined physical channels and signals use ZC sequences, including:

  • PSS (Primary Synchronization Signal)

  • PRACH (Random Access Channel)

  • PUCCH (Physical Uplink Control Channel)

  • PUSCH (Physical Uplink Shared Channel)

  • SRS (Sounding Reference Signal)

  Beyond ZC sequences, other orthogonal and error-correcting coding techniques commonly appear in modern systems, such as:

  • Walsh–Hadamard codes

  • LDPC (Low-Density Parity-Check) codes

  Zadoff–Chu Sequence Definition

  A ZC sequence of length N is defined using a root index $u$, where $u$ must be coprime with $N$. For a given sample index $k$, the sequence is expressed as follows.

  For odd N:

   x(k)=, 0≤N<k

   For even N:

   x(k)=,  0≤N<k

  Where:

• $N$ = sequence length

• $u$ = ZC root index (coprime with $N$)

• $k$ = sequence sample index

• $j=\sqrt{-1}$

           Continued ...

From horsepower to watts

Evening bazaar street, as shop owners begin to bring the goods back inside the shops as the day approached climax, goods that were on display all day I stopped my motorcycle at a street food stall. The bazaar street ran between a bus station and a crowded taxi stand. In recent years I have witnessed an increase in the number of buses, local and government operated that embarked from this bus station to major cities. At this hour, there were also a noticeable number of cabs and personal cars on the road as well. Far as I can remember and still there seldom seen in the crowd is the hatchback Maruti 800 with 796 cc that produced 35 brake horsepower. I still often see Fiat Premiere and Hindustan Motors’ Ambassador, whenever I get a chance to travel to urban townships.

The changing landscape of automobile sector, this maybe is most likely a story of every small town when it tries coming to grip with the pace of the rest of world. Here we are talking about automobiles. At first, there were only government operated buses but you must have a compelling case to ensure that even with government help, service can sustain a justifiable number of passengers on the route and keep the service profitable. This also provided means of employment for not just only for drivers and bus conductors, but also to mechanics; employs manufacturing and supply chain.

In the decade of 1990, there were a considerable number of villages and small townships that could not afford a bus service. If the government didn’t not find the case reasonable a group of motivated individuals opted to purchase and operate their own bus service. It went something like this, earlier, it was a tough case to sell to people to leave their villages and comfort of the known and join the rest of the world. Today, the operational structure of transport rarely makes you question, how it all began. If your town had a bus service, it meant it’s going to run until it’s done. These buses are lifeline of Indian towns and for the end of time small town boys and girls will hold dearly a dream of becoming a bus driver.

In small villages there are usually buses first and then came the road. The prime presence of bus manufacturers in India is still Tata. At a traffic signal on a busy workday in Kolkata, I once saw Tata, Ashok Leyland, Mercedes, Eicher and Volvo passenger buses within just a few minutes. This amazingly beautiful example of industrial revolution in motion is rarely seen in any other city. The improving living standards have enabled the society to purchase personal automobiles motorcycles and cars. Affordability meant opting for bigger cars and townships waiting for to pass the bill to widen the road, this is an ongoing debate. So far it has been a widely successful business plan, a car for each home, maybe two cars per household or may be three cars per household. I rarely foresee based on the current circumstances people opting for electric vehicles due to lack of infrastructure and initiative, but a change is due not long from now. I would make a case for the increasing gas prices, inflation, and competition between the private and government ecosystems but times are changing for better.

I tend to applaud when I see an electric vehicle pass by me, this is a great but only the owner can weigh in on the pros and cons of the electric vehicle ownership experience. Objectively, it begs to question the sustainability of existing petroleum-based transport system and challenges conventional consumer purchase behavior. Usually automobiles run you a lifetime, so is replacing a fully completely functional automobile with an electric car feasible? Undoubtedly, long from now there will still be petroleum engine enthusiasts. There is a good enough case for introducing change into transport infrastructure it certainly portrays the initiation into the modern era as we embrace a progressive status quo. The problem with pollution and it’s impact will probably become explicitly evident by the next century and the global warming debate would turn into applicable norms and standards. However, like any change, we must find a way to create an ecosystem where petrol and electric, both can co-exist. The prices will soon flip, meaning, driving a petroleum engine vehicle will become more expensive than driving an electric vehicle, that, once we overcome the initial infrastructure costs. What tomorrow holds is an opportunity for us to experience a different world.

The bustle of the evening bazaar street probably won’t be the same.

An account and review of "The Brief History Of Time"

“Aristotle believed that all the matter in the universe was made up of four basic elements: earth, air, fire, and water,” a quote from the popular science book The Brief History of Time, an exceptionally well-written book by Stephen Hawking.

What sparked my interest in this book was a picture I took at an elevation of about 5300 meters above sea level. The picture came out bleak and barely discernible; to the naked eye, the sky was illuminated in all its phosphorescence, full of stars and constellations. However, after carefully reviewing the image, I dejectedly rendered it useless and stowed it away with indifference.

This book is divided into eleven chapters, along with a "Conclusion" chapter. The author explores various theories ranging from "Space and Time" to "The Origin and Fate of the Universe." The publication discusses the theory of relativity, touches upon philosophical ideas from time to time, and delves into "The Big Bang Theory." The author reviews, inquires into, and reminisces about methods of identifying black holes, their characteristics, and the challenges involved in identifying them, all in detail.

The desire of humankind to find answers to the phenomena of the universe—the unknown—has been explicitly evident throughout the chapters and remains a key theme and motivation. The author elaborates in an engaging and clear manner that never becomes dull. I found his dynamic writing style descriptive and apt. The Brief History of Time is a well-researched publication, and the writing, I believe, is still relevant. Stephen Hawking, who received a first-class BA in Physics from University College, Oxford, and later earned a PhD in Applied Mathematics and Theoretical Physics from Trinity Hall, Cambridge, in 1966, summarizes that there is no single unified theory to explain the events of the universe but instead a collection of theories that help us understand it. If we were to find a complete theory, “for then we would know the mind of God.”

Research has come a long way since then, but the underlying principles remain very much the same. If this book had been written in this century, it would surely have included observations on the "Higgs Boson." For a casual reader interested in learning about science, it is a great resource; for a science student, it serves as a mirror into the probable history of the universe and a memento of what we have discovered so far.

After reading this book, I dug out the picture I had put away earlier and decided to restore it as much as I could. With some basic image processing and programming tools, I recovered the image. I still question the couple of green dots on what is now a mostly black-and-white image.

Rain pouring behind my window, as imagined.

Rain pouring behind my window prompted me to ponder. This occasion sounds as good as any that would rile oneself to pitter patter. I mean, who hasn’t felt the rain, I mean, really! It’s a rhetorical question asking to draw upon an experiential memory. If on a quiet evening, it’s a little inconvenience and you start thinking about umbrellas and rain covers, can’t go outside; it’s a concession as well, a discount on the constant demand of conscience to be a part of the people segment. It rains and evenings progressively turn colder, especially at places where there are trees, mountains and wide deserts. Usually, tropical places feel breezier, hot terrain turn  bearable, at least for some time and the day after it would become humid if sun chooses to shine.

Miles upon miles of dirt tracks become even muddier, first etching and eventually all tracks fading into an even wash. Puddles start building up, holes and ditches alike start filling up with water. Something does take place; air becomes pleasant to breathe, the overarching rush of getting somewhere has to take a back seat. I can’t assume the same for the tourists caught off-guard on the market streets as panic sets in. Apparently, no one likes getting wet. Everyone in the sane mind absolutely denying the experience for drier consequence begs a hysterical tangent of thought. Although, cultured fans still romanticize a soccer ball kicked around on a muddy field. Water evaporates, steam goes up and it rains.

 Superstitious and conventionalists call it a sign of change, farmers too convey their fortune with rain. Rain-shelters are a real thing, bus-stop rooftops keep the wagers dry as they wait for buses and taxis on their way to work or on their way home. Cafeterias and restaurants take in passersby. Garbs take quite some effort to dry and if you are stuck at a snow mountain the dry change of clothes quickly turn into a "humanitarian" crisis. “Cheapskates” on a weekend night know cold feet like that very first “nostalgic” frozen black ice. If that does not stop you, you must have seen stars on a “fall mountain”. Rain makes corn, you know that? Rain also makes sleet what has sleet ever made?

Fashion quickly takes refuge in comfort. What makes you stand out in rain on a non-fashion week is the color of your umbrella. And such is the scene of rainy marketplace! If you are interested in making a purchase there are only prime sects, ponchos, rain covers and umbrellas, I don’t think either objects have anything against each other but surely they surely don’t like rain.

 According to a folklore enjoying a bowl of soup is encoded in our DNA, DNA is what constitutes our genetic makeup, besides other things. Rain pours and some of us are off to the stores to purchase our favorite bowl of soup. If you live in a forest, usually, uninvited guests tend to treat your dwelling respectfully theirs, some chose to stay overnight and disappear without notice, the likes of earth worms, grasshoppers, praying mantis, beetles, bugs that smells nice and insects of many other species that have celebrity like countenances.

But either because a snow mountain is melting somewhere up high or because of rain, rivers enjoy a stream of water, sometimes more, other times not so much but rain makes rivers and who knew there is a lot more of science to weather than just water steaming up to form clouds.

Is euphemisim dead?

 Is euphemism dead? As an adult I always felt a responsibility to stow away personal bias before declaring attitude towards a matter. As the holidays sling joy, a naive age-old question edges up with childlike innocence. “Will there be snow for Christmas this year?”

The dilemma sets in, paralyzing the foresight and rational thought for few moments, I answered “No, it’s not going to snow”, as I slid the phone back into the right pocket and slouched. The telly was still on, it spun the reel, and I kept weaving the recaps of what I had just said, and weighing it against the tone because, I had just killed euphemism.