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Best teacher: Sabrina Alcala. She saved my life without even realizing it. I don’t remember a single thing that she taught me in classes. But I do remember how she made me feel about learning, and that kept me in school long enough to prevent me from getting sucked into the drug trade.Sabrina taught me math, chemistry, and physics at the private high school I attended in the 10th grade with a very low student:teacher ratio (the now-defunct North Toronto Academy). The school itself was very informal, with teachers often teaching multiple courses to groups of students at a time (for instance, she would be at a table with three students… I would be doing 10th grade math, the guy on my left might be doing 9th grade biology, and the guy on my right might be doing 12th grade physics, and she would alternate between us throughout each period).Sabrina was brilliant in that not only could she juggle this group of students/courses perfectly and flawlessly, but she taught me the importance of practical analogies by using them frequently in her teaching. I was very close to not wanting to ever attend school again, and was closer than ever to dropping out that year - but Sabrina’s teaching methods were so relatable, and actually made learning enjoyable. For instance, how a certain math formula, or chemical compound might be used in the real world, in a way that I could understand and relate to. It was refreshing to actually learn things for a practical reason, rather than to just try and understand the theoretical aspect with no reasons given other than “just learn it because we say so”, which is what I was used to in school, and this was a major reason why I rejected it all so much.My family couldn’t afford to keep me in that school for the following year, which ultimately resulted in me dropping out of high school about nine weeks into the 2008–2009 school year at an alternative public school, because I couldn’t handle it anymore, and didn’t see any value in staying in school.To be fair, my dropping out of school was essentially decided even around the time I started the 10th grade. The decision was essentially made, but at 15, I had no idea how to go about it (and it never occurred to me that simply not showing up was an option). How to get out of school was at the top of my mind for a while, but as I began to appreciate Sabrina’s teaching methods, I felt like it kind of held me back in place, since I was actually enjoying learning when she was the one teaching me.If it weren’t for Sabrina, I’d have easily dropped out in that school year (late 2007, or early-mid 2008), instead of in the following school year (I stopped attending on Monday November 17th, 2008). No matter when I dropped out, I knew I'd have a hard time finding something to do with my life. Knowing several people in the drug trade, and how they did quite well at it (and many had no education), it was always a very tempting option, especially since I had connections who could get me started.What prevented me from going that route was landing my first call centre job in May of 2009, six months after I dropped out. In the weeks that led up to that, I had hit rock bottom so hard that I believed I had nothing to lose anymore, and the opportunities that I’d have as someone who moved illicit substances were much greater than anything I could do legally. If I dropped out that year, I wouldn’t have been able to land a job like I did at the call centre, and would almost certainly have gone into the drug trade with no legitimate occupation to turn me in the right direction.Looking back, considering how close I was to entering the drug trade shortly after I left school in late 2008, I think that Sabrina may have saved my life. If I had left school several months earlier, that would have been several more months for frustration to set in, and bring me to the conclusion of “screw it, I’ll deal drugs”. As a naive 15 year old with an attitude, I'd probably have been dead in a matter of weeks. Me dropping out was inevitable, but Sabrina is the main reason I stuck around as long as I did… which is something I could very well owe my life to.Sabrina was the best teacher I ever had because she was able to keep an inevitable dropout interested in school just long enough so that he wouldn’t make a bad (and possibly fatal) decision too soon. There was nothing that could possibly have gotten me interested in learning about math or science. But Sabrina did what dozens of other teachers failed at doing - and that is teaching the student, not teaching the subject. I doubt she’ll ever know how impactful her teaching was on me. But I hope that an anecdote like this can persuade at least one teacher somewhere to go that extra mile with your students. It isn't just about getting the grades. In cases like mine, it's possibly a matter of life and death. Consider the gravity of that.Worst teacher: Too many to mention. But here’s a few of them that I will never forget…Greg Boyd - Glenview Senior Public School (7th grade math, 2004–2005). Greg Boyd taught math in his own way. He wilfully, and even openly disregarded the standard ways of teaching math, in favour of his own methods. Which has its own place in academia - but not in a classroom of 11 and 12 year olds who all learned math the same way in previous years. He was especially mean to me because I believe he perceived my asking him questions as arrogance. Admittedly, I did later get diagnosed with some learning disabilities, but these problems with Greg Boyd (I refuse to refer to him with the “Mr.” prefix) were ones that virtually all students experienced… and, like most permanent full-time teachers, he basically couldn’t be fired for anything, no matter how many problems he was causing. So all of us students were stuck with him. I nearly failed seventh grade math because of him (despite earning top math grades in all prior years of school), and he just didn’t care.I also later found out that I’m indirectly related to him (by marriage, not by blood). On top of that, I also found out that a relative of mine was abusive to him as a kid. I can’t help but think that this might have had something to do with the way he treated me specifically. This is speaking only speculatively, of course - I have to make it very clear that this is only my opinion, since I could be toeing the line of slander here. But from what I learned of Greg Boyd’s character, I would not put it past him as something he would do. He's a first-class jerk, and I have less than zero respect for him. You'll see in my two other mentions below that I'm able to say at least one positive thing, even about teachers who I would classify as the “worst”. Greg Boyd? Can't say a single positive thing. You could wave $500 in my face to say one positive thing about Greg Boyd, and I’m certain I could not come up with even one thing.Howard Gerhard - George Brown College (second year, 2013–2014). He taught an elective class called Muskets, Maps, and Model Ts, which went over the history of various technologies (to include, as you may be able to guess, muskets, maps, and Model T Ford vehicles). Admittedly, this class, when viewed the right way, was pretty interesting. For instance, it never occurred to me that so many generations of people didn’t even know what they looked like, before mass production of mirrors came about. It was a very cool course, in theory.What wasn’t very cool, sadly, is the way that Howard taught this course. With templated PowerPoint presentations, a monotone speaking voice, and virtually no class engagement, it was difficult to sink my teeth into the details that surrounded each point. It also didn’t help that he tried to give an abbreviated version of very dense information, but then expect us to perform on tests with a non-abbreviated understanding of the subject matter. Smart guy, terrible teacher.Now, this sounds like a typical case of an under-trained prof without a personality, right? Not exactly. Aside from being really bad at teaching, he’s also really bad at communicating with people. Which happens to include unclearly stating timeframes for tests to be taken in the computer lab - on top of the fact that the timeframes actually deviated from the standard timeframes in which lab-based tests were taken. Inevitably, as a result of this, I missed a test worth 25% of my grade. He refused to re-administer the test to me, claiming that missing the test was my fault, and it had nothing to do with him. I asked him if other students missed the test because of the lack of clarity in times - which, of course, he ignored, knowing that the answer would only strengthen my case.So, I ran this up the administration flagpole. I got his boss to order him to attend a meeting of the three of us, and we talked it out. Eventually, we arrived to agreeable terms where he would not permit me to write the test right away, but he would if it would be the factor that made me pass the course. His exact words were “if it makes the difference between you passing and failing the course, I will let you write the test”. So I had that insurance policy. Great. It was much needed, since I wasn’t doing very well in that class.As it turned out, I needed to cash in that policy, as my grade after the last scheduled test was a 45/100. Therefore, being able to write that test would make the difference between me passing and failing. I emailed him to make this request, expecting him to go along with it. After all, he didn’t just make this promise to me… he made this promise with a witness. A witness who happened to be his boss. I thought this would be a slam dunk.Sneakily, this asshole changed two grades I had at 4/5, from their existing grade to 5/5 - which I could see on my student portal. I was then at 47/100, after suddenly gaining two marks. He sent me a terse email saying that I’m close enough to passing that he can “bump” me to 50%, and therefore there would be no need to re-write the test. I told him I wanted to take the test, because it was 25% of my mark, and I could get a final mark of a lot more than 50% if I did well on the test. Then he suddenly pretended not to know what I was talking about, because supposedly he “offered me the privilege of a possible re-write based on good behaviour”. But my behaviour “did not warrant this privilege”. Which doesn’t add up with the promise he made.Then, he was bold enough to change my artificially inflated marks back to 4/5 (again, could be seen in my student portal), and email me to say that he has now FAILED me in the course, with no opportunity to re-write the test. That, right there, was his death warrant. I was now out for blood.To make a long story short, as soon as I threatened to take legal action against the school, and to press criminal charges against Howard personally for fraudulently tampering with my grades to fail me, virtually nothing went down in writing. There were a lot of meetings, and a lot of informal conversations, undoubtedly to ensure that no written record of this existed. The net result was that I got to take the test due to an “administrative misunderstanding” (bull fucking shit). I scored something like 30% on it, and ended up getting a slightly higher overall course grade of 50-odd percent.Aside from being a worse than horrible teacher, Howard will always go down as one of the worst teachers I ever had, specifically because he told multiple lies, just to fail me in his course. That’s not bad teaching. That’s malicious use of power. I really wish I did carry out those legal actions, because I truly wish I could see Howard out of a job. He doesn’t deserve to work anywhere, in my opinion. I hope he’s reading this, and I hope he knows what almost happened to him. Lucky guy. I am not the kind of person who you cross.Ojelanki Ngwenyama. Ryerson University (grad school, 2016). Let’s start out with the positives… he introduced me to one of the most perspective-changing books I’ve ever read (it’s An Introduction to Reasoning by Stephen Toulmin, Richard Rieke, and Allan Janik). He’s done some really impressive research. He’s also got a really cool name that’s kind of fun to say. Sadly, I cannot say anything else positive about him. He’s singlehandedly responsible for making me leave grad school in 2016, and that’s because he’s beyond horrible as a teacher.The short version of the issues I had with him is as follows…He ignored my emails asking for clarification about basic things (that really should have been made clear in the first place), and diverted my questions when I asked why he didn’t replyHe forced his meditation habits on the class, making virtually everyone in the room uncomfortable, often coming up to students and staring them in the eye until they hesitantly broke contactWhen I told him I couldn’t do a certain assignment because of reading comprehension issues that I had (and that I’d provide the diagnosis in whatever form the school may require), he said “that’s irrelevant and in the past, just do the assignment”He would copy emails that students sent him that he perceived as being silly, and paste them into broadcasted emails to the entire class calling the students out on things that the said that he disagreed with, just to embarrass them.None of my encounters with him make for nearly as rich of a story at the one with Howard, so I can’t go into a lot more detail. But the point is that his habitually unprofessional and insensitive behaviours made for nothing less than a horrific experience for me as a student - and if it strengthens my case any, I know I’m not alone from his RateMyProfessors ratings… http://www.ratemyprofessors.com/ShowRatings.jsp?tid=487454Sadly, he’s greatly accomplished, so it will take a lot to get him forced out of teaching. He also earns over $200,000 CAD per year according to the Sunshine List, so I’m sure he’s got more than enough money to retire on if he ever is thrown out. It’s difficult to say that he’s not the #1 person who I’d like to see fired (since Howard Gerhard has distinctly earned that position with honours). But he is definitely right up there. I hope you’re reading this too, Ojelanki. There’s a whole population of people who want you out of teaching because of how lousy you are at it, and I have no issue sharing my sentiments publicly about it. I sleep better at night knowing I never have to work with or even see you again - and even if I ran into you somewhere, I’d have nothing to say to you.

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal).A sample is a value or set of values at a point in time and/or space.A sampler is a subsystem or operation that extracts samples from a continuous signal.A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.Contents[hide]1Theory2Practical considerations3Applications3.1Audio sampling3.1.1Sampling rate3.1.2Bit depth3.1.3Speech sampling3.2Video sampling3.33D sampling4Undersampling5Oversampling6Complex sampling7See also8Notes9Citations10Further reading11External linksTheory[edit]See also: Nyquist–Shannon sampling theoremSampling can be done for functions varying in space, time, or any other dimension, and similar results are obtained in two or more dimensions.For functions that vary with time, let s(t) be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every T seconds, which is called the sampling interval or the sampling period.[1]Then the sampled function is given by the sequence:s(nT), for integer values of n.The sampling frequency or sampling rate, fs, is the average number of samples obtained in one second (samples per second), thus fs= 1/T.Reconstructing a continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant (T), the sequence of delta functions is called a Dirac comb. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with s(t). That purely mathematical abstraction is sometimes referred to as impulse sampling.[2]Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when s(t) contains frequency components whose periodicity is smaller than two samples; or equivalently the ratio of cycles to samples exceeds ½ (see Aliasing). The quantity ½ cycles/sample × fssamples/sec = fs/2 cycles/sec (hertz) is known as the Nyquist frequency of the sampler. Therefore, s(t) is usually the output of a lowpass filter, functionally known as an anti-aliasing filter. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process.[3]Practical considerations[edit]In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion.Various types of distortion can occur, including:Aliasing. Some amount of aliasing is inevitable because only theoretical, infinitely long, functions can have no frequency content above the Nyquist frequency. Aliasing can be made arbitrarily small by using a sufficiently large order of the anti-aliasing filter.Aperture error results from the fact that the sample is obtained as a time average within a sampling region, rather than just being equal to the signal value at the sampling instant. In a capacitor-based sample and hold circuit, aperture error is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have non-zero width.Jitter or deviation from the precise sample timing intervals.Noise, including thermal sensor noise, analog circuit noise, etc.Slew rate limit error, caused by the inability of the ADC input value to change sufficiently rapidly.Quantization as a consequence of the finite precision of words that represent the converted values.Error due to other non-linear effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the pass band, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations.Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of low-pass filtering. The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with a proposed nonlinear function.Applications[edit]Audio sampling[edit]Digital audio uses pulse-code modulation and digital signals for sound reproduction. This includes analog-to-digital conversion (ADC), digital-to-analog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discrete-time, discrete-level analog of a previous electrical analog. While modern systems can be quite subtle in their methods, the primary usefulness of a digital system is the ability to store, retrieve and transmit signals without any loss of quality.Sampling rate[edit]A commonly seen measure of sampling is S/s, which stands for "samples per second". As an example, 1 MS/s is one million samples per second.When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing,[4]such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (CD), 48 kHz, 88.2 kHz, or 96 kHz.[5]The approximately double-rate requirement is a consequence of the Nyquist theorem. Sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early professional audio equipment manufacturers chose sampling rates in the region of 50 kHz for this reason.There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz[6]This is in contrast with laboratory experiments, which have failed to show that ultrasonic frequencies are audible to human observers; however in some cases ultrasonic sounds do interact with and modulate the audible part of the frequency spectrum (intermodulation distortion).[7]It is noteworthy that intermodulation distortion is not present in the live audio and so it represents an artificial coloration to the live sound.[8]One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for Compact Disc (CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering.[9]A more complete list of common audio sample rates is:Sampling rateUse8,000 HzTelephone and encrypted walkie-talkie, wireless intercom and wireless microphone transmission; adequate for human speech but without sibilance (ess sounds like eff (/s/, /f/)).11,025 HzOne quarter the sampling rate of audio CDs; used for lower-quality PCM, MPEG audio and for audio analysis of subwoofer bandpasses.[citation needed]16,000 HzWideband frequency extension over standard telephone narrowband 8,000 Hz. Used in most modern VoIP and VVoIP communication products.[10]22,050 HzOne half the sampling rate of audio CDs; used for lower-quality PCM and MPEG audio and for audio analysis of low frequency energy. Suitable for digitizing early 20th century audio formats such as 78s.[11]32,000 HzminiDV digital video camcorder, video tapes with extra channels of audio (e.g. DVCAM with four channels of audio), DAT (LP mode), Germany's Digitales Satellitenradio, NICAM digital audio, used alongside analogue television sound in some countries. High-quality digital wireless microphones.[12]Suitable for digitizing FM radio.[citation needed]37,800 HzCD-XA audio44,056 HzUsed by digital audio locked to NTSC color video signals (3 samples per line, 245 lines per field, 59.94 fields per second = 29.97 frames per second).44,100 HzAudio CD, also most commonly used with MPEG-1 audio (VCD, SVCD, MP3). Originally chosen by Sony because it could be recorded on modified video equipment running at either 25 frames per second (PAL) or 30 frame/s (using an NTSC monochrome video recorder) and cover the 20 kHz bandwidth thought necessary to match professional analog recording equipment of the time. A PCM adaptor would fit digital audio samples into the analog video channel of, for example, PAL video tapes using 3 samples per line, 588 lines per frame, 25 frames per second.47,250 Hzworld's first commercial PCM sound recorder by Nippon Columbia (Denon)48,000 HzThe standard audio sampling rate used by professional digital video equipment such as tape recorders, video servers, vision mixers and so on. This rate was chosen because it could reconstruct frequencies up to 22 kHz and work with 29.97 frames per second NTSC video - as well as 25 frame/s, 30 frame/s and 24 frame/s systems. With 29.97 frame/s systems it is necessary to handle 1601.6 audio samples per frame delivering an integer number of audio samples only every fifth video frame.[9]Also used for sound with consumer video formats like DV, digital TV, DVD, and films. The professional Serial Digital Interface (SDI) and High-definition Serial Digital Interface (HD-SDI) used to connect broadcast television equipment together uses this audio sampling frequency. Most professional audio gear uses 48 kHz sampling, including mixing consoles, and digital recording devices.50,000 HzFirst commercial digital audio recorders from the late 70s from 3M and Soundstream.50,400 HzSampling rate used by the Mitsubishi X-80 digital audio recorder.88,200 HzSampling rate used by some professional recording equipment when the destination is CD (multiples of 44,100 Hz). Some pro audio gear uses (or is able to select) 88.2 kHz sampling, including mixers, EQs, compressors, reverb, crossovers and recording devices.96,000 HzDVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, HD DVD (High-Definition DVD) audio tracks. Some professional recording and production equipment is able to select 96 kHz sampling. This sampling frequency is twice the 48 kHz standard commonly used with audio on professional equipment.176,400 HzSampling rate used by HDCD recorders and other professional applications for CD production. Four times the frequency of 44.1 kHz.192,000 HzDVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, and HD DVD (High-Definition DVD) audio tracks, High-Definition audio recording devices and audio editing software. This sampling frequency is four times the 48 kHz standard commonly used with audio on professional video equipment.352,800 HzDigital eXtreme Definition, used for recording and editing Super Audio CDs, as 1-bit DSD is not suited for editing. Eight times the frequency of 44.1 kHz.2,822,400 HzSACD, 1-bit delta-sigma modulation process known as Direct Stream Digital, co-developed by Sony and Philips.5,644,800 HzDouble-Rate DSD, 1-bit Direct Stream Digital at 2× the rate of the SACD. Used in some professional DSD recorders.11,289,600 HzQuad-Rate DSD, 1-bit Direct Stream Digital at 4× the rate of the SACD. Used in some uncommon professional DSD recorders.22,579,200 HzOctuple-Rate DSD, 1-bit Direct Stream Digital at 8× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD512.Bit depth[edit]See also: Audio bit depthAudio is typically recorded at 8-, 16-, and 24-bit depth, which yield a theoretical maximum signal-to-quantization-noise ratio (SQNR) for a pure sine wave of, approximately, 49.93 dB, 98.09 dB and 122.17 dB.[13]CD quality audio uses 16-bit samples. Thermal noise limits the true number of bits that can be used in quantization. Few analog systems have signal to noise ratios (SNR) exceeding 120 dB. However, digital signal processing operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution.Speech sampling[edit]Speech signals, i.e., signals intended to carry only human speech, can usually be sampled at a much lower rate. For most phonemes, almost all of the energy is contained in the 100 Hz–4 kHz range, allowing a sampling rate of 8 kHz. This is the sampling rate used by nearly all telephony systems, which use the G.711 sampling and quantization specifications.Video sampling[edit]This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.(June 2007)(Learn how and when to remove this template message)Standard-definition television (SDTV) uses either 720 by 480 pixels (US NTSC 525-line) or 704 by 576 pixels (UK PAL 625-line) for the visible picture area.High-definition television (HDTV) uses 720p (progressive), 1080i (interlaced), and 1080p (progressive, also known as Full-HD).In digital video, the temporal sampling rate is defined the frame rate – or rather the field rate – rather than the notional pixel clock. The image sampling frequency is the repetition rate of the sensor integration period. Since the integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time:50 Hz – PAL video60 / 1.001 Hz ~= 59.94 Hz – NTSC videoVideo digital-to-analog converters operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early games consoles, to 250 MHz or more for the highest-resolution VGA output).When analog video is converted to digital video, a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along scan lines. A common pixel sampling rate is:13.5 MHz – CCIR 601, D1 videoSpatial sampling in the other direction is determined by the spacing of scan lines in the raster. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.Spatial aliasing of high-frequency luma or chroma video components shows up as a moiré pattern.3D sampling[edit]The process of volume rendering samples a 3D grid of voxels to produce 3D renderings of sliced (tomographic) data. The 3D grid is assumed to represent a continuous region of 3D space. Volume rendering is common in medial imaging, X-ray computed tomography (CT/CAT), magnetic resonance imaging (MRI), positron emission tomography (PET) are some examples. It is also used for seismic tomography and other applications.The top two graphs depict Fourier transforms of two different functions that produce the same results when sampled at a particular rate. The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband. The lower graphs indicate how identical spectral results are created by the aliases of the sampling process.Undersampling[edit]Main article: UndersamplingWhen a bandpass signal is sampled slower than its Nyquist rate, the samples are indistinguishable from samples of a low-frequency alias of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable. Such undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.[14]Oversampling[edit]Main article: OversamplingOversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters, such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula.[15]Complex sampling[edit]Complex sampling (I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers.[note 1]When one waveform[math]{\displaystyle ,{\hat {s}}(t),}[/math] is the Hilbert transform of the other waveform[math]{\displaystyle ,s(t),\,}[/math] the complex-valued function, [math]{\displaystyle s_{a}(t)\ {\stackrel {\text{def}}{=}}\ s(t)+i\cdot {\hat {s}}(t),}[/math] is called an analytic signal, whose Fourier transform is zero for all negative values of frequency. In that case, the Nyquist rate for a waveform with no frequencies ≥ B can be reduced to just B (complex samples/sec), instead of 2B (real samples/sec).[note 2]More apparently, the equivalent baseband waveform, [math]{\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t},}[/math] also has a Nyquist rate of B, because all of its non-zero frequency content is shifted into the interval [-B/2, B/2).Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing [math]{\displaystyle {\hat {s}}(t),}[/math] by processing the product sequence[math]{\displaystyle ,\left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right],}[/math][note 3]through a digital lowpass filter whose cutoff frequency is B/2.[note 4]Computing only every other sample of the output sequence reduces the sample-rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.See also[edit]DownsamplingUpsamplingMultidimensional samplingSample rate conversionDigitizingSample and holdBeta encoderKell factorBit rateNotes[edit]Jump up^ Sample-pairs are also sometimes viewed as points on a constellation diagram.Jump up^ When the complex sample-rate is B, a frequency component at 0.6 B, for instance, will have an alias at −0.4 B, which is unambiguous because of the constraint that the pre-sampled signal was analytic. Also see Aliasing#Complex sinusoids.Jump up^ When s(t) is sampled at the Nyquist frequency (1/T = 2B), the product sequence simplifies to [math]{\displaystyle \left[s(nT)\cdot (-i)^{n}\right].}[/math]Jump up^ The sequence of complex numbers is convolved with the impulse response of a filter with real-valued coefficients. That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs.Citations[edit]Jump up^ Martin H. Weik (1996). Communications Standard Dictionary. Springer. ISBN 0412083914.Jump up^ Rao, R. Signals and Systems. Prentice-Hall Of India Pvt. Limited. ISBN 9788120338593.Jump up^ C. E. Shannon, "Communication in the presence of noise", Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10–21, Jan. 1949. Reprint as classic paper in: Proc. IEEE, Vol. 86, No. 2, (Feb 1998)Jump up^ "Frequency Range of Human Hearing". The Physics Factbook.Jump up^ Self, Douglas (2012). Audio Engineering Explained. Taylor & Francis US. pp. 200, 446. ISBN 0240812735.Jump up^ "Digital Pro Sound". Retrieved 8 January 2014.Jump up^ Colletti, Justin (February 4, 2013). "The Science of Sample Rates (When Higher Is Better—And When It Isn't)". Trust Me I'm A Scientist. Retrieved February 6, 2013.Jump up^ David Griesinger. "Perception of mid frequency and high frequency intermodulation distortion in loudspeakers, and its relationship to high-definition audio". Archived from the original (Powerpoint presentation) on 2008-05-01.^ Jump up to:a b AES5-2008: AES recommended practice for professional digital audio - Preferred sampling frequencies for applications employing pulse-code modulation, Audio Engineering Society, 2008, retrieved 2010-01-18Jump up^ http://www.voipsupply.com/cisco-hd-voice[unreliable source?]Jump up^ "The restoration procedure - part 1". http://Restoring78s.co.uk. Archived from the originalon 2009-09-14. Retrieved 2011-01-18. For most records a sample rate of 22050 in stereo is adequate. An exception is likely to be recordings made in the second half of the century, which may need a sample rate of 44100.Jump up^ "Zaxcom digital wireless transmitters". Zaxcom. Retrieved 2011-01-18.Jump up^ "MT-001: Taking the Mystery out of the Infamous Formula, "SNR=6.02N + 1.76dB," and Why You Should Care" (PDF).Jump up^ Walt Kester (2003). Mixed-signal and DSP design techniques. Newnes. p. 20. ISBN 978-0-7506-7611-3. Retrieved 8 January 2014.Jump up^ William Morris Hartmann (1997). Signals, Sound, and Sensation. Springer. ISBN 1563962837.Further reading[edit]Matt Pharr, Wenzel Jakob and Greg Humphreys, Physically Based Rendering: From Theory to Implementation, 3rd ed., Morgan Kaufmann, November 2016. ISBN 978-0128006450. The chapter on sampling (available online) is nicely written with diagrams, core theory and code sample.External links[edit]Journal devoted to Sampling TheoryI/Q Data for Dummies – a page trying to answer the question Why I/Q Data?Sampling of analog signals – an interactive presentation in a web-demo at the Institute of Telecommunications, University of Stuttgart

Post articles that engage your members and create value. Use a chat bot soyou can interact with someone while they’re on your fan page.Most marketers know by now that Facebook is an important business tool for companies of every size and industry. With a daily active user base of 1.13 billion (1.03 billion on mobile alone), you know it can help you reach new audiences you may not have been able to reach otherwise. It can also help you get found more easily in search, create a community around your business, promote the content you create, and develop a strong brand identity. But what about using Facebook for lead generation? Attracting new leads using Facebook -- leads that might eventually turn into paying customers -- is one of the most intriguing reasons to use Facebook marketing. And yet, we find that only about half of marketers use Facebook to source leads. This needs to change. And even if you are generating leads on Facebook, we all could probably use a little boost in our lead generation efforts.Download our complete Facebook guide here for more tips on generating leads and customers from Facebook.To make sure we're all on the same page, let's start with what a lead is (and isn't), and the two types of leads you can generate on Facebook.The 2 Types of Leads You Can Capture on FacebookAlthough definitions can vary, in general, a lead is a person who has indicated interest in your company's product or service by giving you their information in some way. People can show interest in a variety of ways: filling out a form to download an ebook, requesting a demo, or completing an online survey.Unfortunately, simply Liking a status update, photo, or video on your Page doesn't make someone a lead. That type of action doesn't indicate interest in your company or product -- it's possible they just Liked your post because it had a cute puppy in it, ya know?On Facebook, there are two ways you can generate leads: direct leads and indirect leads.Direct LeadsDirect leads are generated by sharing content that links directly back to a form on your website where visitors can share information in exchange for an offer -- whether that be an ebook, coupon, infographic, or any other piece of content. This form is housed on a landing page dedicated to that specific offer.Indirect LeadsIndirect leads are generated by using Facebook on the path to conversion. For example, if you shared a blog post that had a call-to-action to a landing page at the bottom of the post, your initial Facebook share is helping direct visitors to that landing page.While directly promoting landing pages is an instant gratifier of leads generated, providing content without a form makes your Facebook presence a friendlier home for content that your fans will want to come back for again and again.Now, let’s dive into 12 ways you can capture leads, whether they are direct or indirect.12 Types of Facebook Posts to Help You Generate Leads From Your Business Page1) Post landing pages for offers directly to Facebook.One of the best ways to generate leads on Facebook is simply to send people directly to landing pages for lead-generating offers. (If you don't have many lead-generating offers yet, read this blog posts for ideas.)When you do this, make sure the offer has a compelling featured image that's getting pulled into the Facebook post. To ensure Facebook pulls the right image from your blog post into your Facebook posts, you'll need to first optimize the image size for Facebook and then add the proper open graph tags to your website, which you can learn how to do here.You'll also want to make sure it's clear to the reader where you're sending them. If they think they're clicking into a blog post and find themselves needing to fill out a form, they could get confused or frustrated. Use verbal phrases like "Download your ebook" or "Get your cheat sheet" to indicate where you're sending them.Here's an example from IBM's Facebook Page, which reads, "Explore our 2015 Corporate Responsibility Report":In addition to using clear language, you may want to nix the stock photo from that image in favor of your own, custom image. Even the least design-savvy of marketers can easily create a custom image in PowerPoint that includes the name of the offer, just like we did in the example below. (Click here to browse through and download our collection of 100 free social media image templates.)2) Post the blog posts that generate the most leads.Another way of generating leads from the content your team is producing is to simply pick the blog posts that generate the most leads, and post those ones to Facebook. (Learn how to do a blog lead generation analysis here.) The topic and title of the blog post will intrigue your audience to click and read, and then they'll find a CTA within that post -- preferably high up, near the intro -- to either a solution to a problem they're having or to something they want to learn more about.Pro Tip: Our social media managers have found they're able to generate more leads from Facebook by posting blog posts containing anchor text CTAs in the introduction. If you aren't using anchor text CTAs yet, you may want to read up on the study we did on anchor text CTAs on the blog and consider adopting them yourself. In every single post we tracked for that study, the anchor text CTA was responsible for between 47% and 93% of a post's leads.Here's an example of a Facebook post linking to a blog post that includes an anchor text CTA in the introduction:And here's that anchor text CTA, indicated by the red arrow:3) Include links to landing pages in your image captions.Most marketers understand the importance of using visuals like images and videos in your Facebook strategy. For example, Facebook posts with images see 2.3X more engagement than those without images. To turn these higher engagement rates into lead generation opportunities, consider including links to your website in the descriptions for your images -- especially your profile picture and cover photo descriptions.Whether it's to a blog post, a piece of lead gen content, or just an "About Us" page, links are opportunities for interested folks to get to know your company better, and the descriptions of your profile picture and cover photo are prime real estate to do it. That way, any time people view your cover photo directly, they can access the download link.Make sure you shorten your links and add UTM codes so you can track clicks on them. Shortening and tracking features are available in the HubSpot Marketing Platform and in tools like bitly..4) Use videos to promote lead gen offers.Facebook's organic reach has dropped to 52% so far in 2016, thanks to the tweaks in Facebook's algorithm to help mitigate the increasing amount of content on its platform. But videos are the big exception here. In fact, posting videos has actually helped neutralize some of that pain for marketers.Why? To start, Facebook's algorithm favors video content. As a result, video posts have 135% greater organic reach than photo posts. So if you're trying to increase your lead gen efforts on Facebook, you'll want to start using videos to help introduce and promote those lead-generating content, whether they're offers, events, courses, or something else.In addition to the text CTA you can add in the video's description, remember to add a verbal CTA to the video to "register" or "download," both earlier in the video and at the very end.Check out how L.L. Bean used a video to encourage sign-ups for their course:Here's another example from us here at HubSpot, in which we used a how-to video to introduce a gated offer:We've also created videos specifically to promote lead-generating content, like we did here for our career assessment called The Next Five:5) Use Facebook Live videos to remind people to register.Videos can be pretty time-intensive to create. (Not to mention intimidating.) But you don't necessarily have to pull together the time and resources to create a perfectly scripted and edited marketing video to leverage the power of video on Facebook.Facebook Live is Facebook's live video platform that lets anyone broadcast live videos from their mobile device straight to their Facebook News Feed. The best part about these live videos is that they're meant to be a little scrappier and more spontaneous than normal marketing videos -- that's what makes live videos special.What's more, Facebook Live has proven itself pretty incredible for engagement rates. Facebook's initial data revealed that people comment 10X more on Facebook Live videos than on regular videos.So get the conversation going about your lead gen offers by creating a live video to promote them. You might promote an event by showing the setup live, for example. Or, you might promote an offer by hosting an open Q&A on live video where you actually interact with Facebook commenters live and on camera.Just like you'd do with your normal videos, add a verbal CTA to the video in addition to the text CTA. In a live video, though, you'll want to repeat that CTA even more than you would with a pre-recorded CTA. Why? Because when you first start live streaming, you may have zero people watching. Even a few seconds in, you could only have a handful of viewers.As people find your video on their News Feeds, they'll join in -- but that means you'll want to repeat the CTA a few times to catch people up. You can also add a text CTA in the video's description.6) Pin posts that link to lead gen offers to the top of your feed.Pinning a post to the top of your Page's Timeline allows you to highlight what would otherwise be a typical post. It'll stay at the top of your Timeline for up to seven days, after which it'll return to the date it was published on your Page's Timeline. A pinned post is signified by a small blue-and-white pushpin icon on the top right of the post.Here's an example from Apttus' Facebook Page:.You can pin any type of post, from text to images to videos; even live videos. If you pin a Facebook Live video, that video will simply show up at the top of your profile with the whole recording, indicating that the Page "was live" at a certain point. Here's an example of what that looks like from Refinery29's Facebook Page:.7) Add a call-to-action button to your Facebook Page.Alright, this one isn't technically a type of Facebook post, but it's a pretty crucial lead generation tactic that no marketers will want to miss out on. Back in late 2014, Facebook added a feature to its business Pages allowing users to place a simple call-to-action button at the top of their Facebook Pages. This button is simple but powerful, and it can help drive more traffic from your Facebook Page to your website -- including landing pages, contact sheets, and other lead generation forms. You can learn how to install and use the Facebook CTA button here.You'll find you have seven pre-made button options to choose from: "Sign Up," "Shop Now," "Contact Us," "Book Now," "Use App," Watch Video," and "Play Game." Once you choose a button and link it to a page your website, the button you chose will appear up at a fixed location right below your cover photo and to the right.While some marketers choose a CTA and keep it the same for weeks and months at a time, consider taking your marketing game a step further and switching up that button -- and the web page it links to -- to match your team's and business' goals and the campaigns you're running at the time. For example, you might align the CTA both with your cover photo design and a pinned post around a single campaign.8) Ask for input on your products.One way to feed two birds with one scone (as my colleague Carly Stec would say) is to post a status update to your business' Timeline asking for feedback on one of your products or tools and linking to a landing page where people can sign up for a trial -- or, if it's free, to simply download the tool. You'll encourage sign-ups by linking directly to the landing page, and your followers will love the opportunity to give their two cents.The obvious risk here is that you'll be opening up the floodgates for negative commenters, so be selective on the tools and products you post for feedback. Make sure you're posting something you're proud of and ready to receive feedback for. You'll also want to have at least one or two people ready to respond to Facebook comments as they roll in -- both the positive and the negative.If you do receive negative feedback, respond as quickly as you can to show you care, and prevent them from turning into something more serious. If you get complaints about the product, use the "customer is always right" approach and say you're sorry. You'll get respect from other customers for being upfront. Share you appreciation for folks' feedback. Finally, ask how you can help -- and then actually help. Take notes on the feedback you get and send it to the people who can make things happen. (Read this post for more tips on dealing with negative comments on social media.)9) Run a contest or giveaway.People love contests and giveaways. Not only are they fun for your followers, but they can also teach you a lot about your audience -- all the while engaging them, growing your reach, driving traffic to your website, and (drum roll, please) generating leads.If the goal of your contest is to generate leads, publish posts on Facebook (in addition to your other social media accounts) that include an attractive featured image or video, language that's compelling and simple, and a link to your contest page where they can fill out a form. Read this post to learn more about running successful social media contests.(Before you start your Facebook contest, though, make sure you can actually run it legally by reading through their Page Guidelines. Facebook has cracked down on contests due to liability issues, so read through their strict rules ahead of time.)Below is an example from Canva's Facebook Page. Notice they pinned the post to the top of their profile.And here's another example, this time from Yoplait. They promoted their contest using a video to get more visibility on folks' News Feeds.10) Make a Facebook event page for your next webinar.While we’ve already covered sharing landing pages with dedicated content offerings such as ebooks or contests, webinars are another great format for capturing leads. While you can promote your webinar's sign-up form by posting them to your business' Timeline, another way to spread the word is by creating a Facebook Event with with a separate registration page on your website.Once you invite someone to a Facebook Event, you can encourage them to register on a separate landing page, where they'll become a lead. In terms of reaching new audiences, Facebook Events are also more visible than standard Facebook posts on the News Feed.Facebook also added new features that help businesses promote their events and see how they’re performing. For example, you can create ads for the desktop and mobile News Feed that boost awareness of events and drive responses.If you host events and webinars often, you can also use the Events tab on your Page to share with your followers in a single view. That way, people visiting your page can scroll through your upcoming events and webinars. (If you don’t see the tab on your Page, click "Manage Tabs" at the bottom of your tabs and reorder them so Events is one of the first to appear.) Read this blog post for more tips on hosting great webinars.11) Run targeted ads to extend your content's reach.One of the best things Facebook can do for your business is expand your reach to new audiences that are likely to be interested in your content -- and possibly become followers, leads, and even customers down the road. This is thanks to Facebook's very sophisticated targeting options, which let you target your ads to people based on things like location, age, gender, interests -- even the things they do off of Facebook.There are three, overarching formats for Facebook ads that I'll cover in brief here: boosted posts, right-hand column ads, and News Feed ads. The main distinction here is the placement of the ad, as well as the amount of writing and size of image that is allowed.Boosted Post: This is Facebook's way of letting marketers turn otherwise normal Facebook posts into ads by "boosting" them. The post will show organically to some users, but to get better reach, the admin will press "boost" on the post (shown only to admins, not to other users) so it shows to a larger number of fans and to targets you can select ahead of time.Right-Hand Column Ads: This is the most traditional on Facebook, it appears on the right side of a user’s Facebook News Feed. We often see less expensive clicks and conversions when using these ads, along with more advanced testing options.News Feed Ads: These appear directly in a user's News Feed and look more like native advertising, although you can also add a small CTA button. They're part of a tactic called "dark posts," which means using News Feed-style ads that don't actually get published to the News Feed of your Page. In our experience, these ads have a higher engagement rate than right-hand column ads (which makes your Page look super healthy), but they can also be more expensive.While we won't go too much more in depth on Facebook advertising (download our Facebook advertising ebook if you want to learn more), here are two examples of Facebook ads in users' News Feeds. This first one is a boosted post that targets people based on their Facebook connections:This second one is a News Feed ad, which lets you add a CTA button to the post -- in this case, "Sign Up." These CTAs are only available for News Feed ads.12) Run lead ads to simplify the mobile signup process.As if Facebook's addition of CTA buttons to its link ads wasn't exciting enough, Facebook added an entirely new feature called lead ads in 2015, which lets users sign up for lead-generating offers and content without leaving Facebook. It was created specifically to simplify the mobile sign-up processes by making it super easy for mobile users to fill out your forms.Why? Because the forms will autopopulate instead of mobile users having to pinch-and-zoom and type into tiny form fields. Basically, when you click on a lead ad, a form opens with your contact information automatically populated based on what you've shared with Facebook already, like name and email address. Talk about solving for form friction. Of course, you can edit your contact information before you click "Submit."Image Credit: FacebookWe won't go into too much detail about lead ads here, but creating them is easy: All you have to do is choose your ad creative, set your targeting and bidding type, and then customize your form fields. (Learn more from Facebook here.)How do you extract the leads you get from lead ads? If you're a HubSpot customer you can integrate Facebook Lead Ads directly with your HubSpot account. If you're not a HubSpot customer, you can export a CSV straight from your Facebook Page, download them from Ads Manager or Power Editor, or request it directly through the API.We hope you found these ideas for ways you can generate leads from Facebook helpful. Remember, though, that Facebook is constantly changing. While the ideas here are a strong start to success, nothing beats testing each strategy for your own audience.These are just a couple ways you can generate leads from Facebook, so we'd love to hear from you. How do you generate leads through Facebook? What works -- and doesn't work -- for your Facebook Fans?