Filters

The thermometer does not report continuously. Something asks it for a number, on a clock, at a steady rate, and each answer is a sample. In this course that clock ticks a thousand times a second, so a sample lasts a millisecond and a list of a thousand numbers is one second of measurement. Between two samples there is nothing recorded at all.

That is why charts here are sometimes drawn as separate sticks rather than as a line. A line says something about what happened between two samples, and nothing did. The course before this one, Sound and Signals, is where the sampling itself gets taken apart.

Because they behave differently across three neighbouring samples. The shape underneath is nearly the same in all three, so averaging three copies of nearly the same number gives you that number back. The wobble is a fresh accident at every sample, as likely to be above as below, so adding three of them together makes them partly cancel each other.

Partly, and not completely. Three numbers pointing in random directions do not add to zero, they add to something smaller than any one of them was likely to be. That is why the second lab reports a wobble that is smaller rather than gone, and why the number it reports is worth reading.

Something has to be assumed, and every choice is a little bit wrong. This course assumes that everything before the start of the list was zero, which is what a real filter sees when it is switched on with nothing behind it. The first few outputs are therefore too small, because they are averaging real readings with imaginary zeros.

The alternatives are to repeat the first sample backwards, to reflect the start of the list, or to refuse to produce an output until the window is full. Audio software argues about this and picture software argues about it more, because the edge of a photograph is where you can see the choice. It is worth knowing that the first few numbers out of any filter are not to be trusted.

Because it is an average of a stretch of time, and the honest label for an average is the middle of that stretch. A forty-one sample average taken at the newest sample is really a statement about twenty samples ago, with the newest twenty and the oldest twenty either side of it.

The lab measures the lag rather than working it out: it slides the output backwards over the input and reports the shift that lines the two up best. That is worth knowing as a technique. Sliding one list over another until it fits is how a radio finds the start of a packet and how a sonar finds an echo.

Because this signal was built rather than measured. The shape was written down first, the noise was added on top, and both are kept. So the chart can draw the answer faintly behind the working. The lab can also measure the damage exactly. It feeds the clean shape through the same filter on its own, and compares the two outputs.

A real measurement never offers that. It is why filters get argued about: without the answer to compare against, two people can disagree for ever about whether a bump was real. Anything you learn here about what a window costs is worth carrying to the case where you cannot check.

Only once somebody says what the signal is for. If the narrow peak in this lab is the event you are hunting, any window wide enough to flatten it has failed however clean the line looks. If the narrow peak is a fault in the sensor, the same window has succeeded.

What can be said without knowing the purpose is the shape of the trade. Averaging a window of forty-one takes the wobble down by roughly the square root of forty-one, which is a bit over six. The lag grows straight in proportion to the window. So doubling the window buys about forty per cent less noise and costs exactly twice the delay, and the returns get worse the further you push it.

From the hardware. The first filters of this kind were built as a chain of stores. Each store held the signal one sample later than the one before it, so a sample walked down the chain one step per tick. To get at the sample from three ticks ago you attached a wire at the third store along. Attaching a wire at a point in a line is called tapping it.

The chain is gone and the word stayed. A filter with forty-one taps is one with forty-one weights, needing forty-one multiplications for every output sample it produces, and Step 9 is about how much that costs.

It is not a convention somebody chose to be awkward. It falls out of what the operation actually is, and Step 4 shows that directly. Every input sample drops a scaled copy of the weight list into the output. Adding those copies up puts the weights on backwards, without anybody having decided anything.

For now the useful thing is to know that both orderings are used and have different names. Sliding the weights on forwards is called correlation, and it is what you want when you are hunting for a shape in a signal. Sliding them on backwards is convolution. It is what a filter does.

Then a steady input does not come out at the height it went in at. Weights adding to two double everything flat, weights adding to a half halve it, and weights adding to zero cancel a steady input completely. None of those is wrong, but only one of them is an average.

This is a real trap in picture editing, where a blur whose weights add to 1.05 makes the whole photograph slowly brighter every time it is applied. It is why filter weights are usually normalised, meaning divided by their own total, before anybody uses them. The lab measures the total by running a flat input through rather than by adding the sliders up, so you can see the two agree.

Because you usually do not know it. A filter arrives as somebody else's code, or as a circuit, or as a room with a clap echoing round it. The impulse response is how you find out what it does without being shown the inside. This is a standard measurement: fire a starting pistol in a concert hall, record what comes back. You have the hall as a list of numbers.

Once you have that list you can apply the same room to a recording made in a dry studio, by convolving one with the other. That is what reverb plug-ins labelled convolution are doing, and the impulse responses they ship are real recordings of real halls.

Finite impulse response. Every filter in this course so far is one. The impulse response is the weight list, and the weight list has an end, so the response goes to exactly zero once the spike has walked off that end. Finite means it stops.

The word finite is only worth saying because the other kind exists. From Step 10 the output of the filter is fed back into it, and its impulse response never quite reaches zero. That one is called infinite impulse response, which is the reason anybody bothers to say finite about this one.

It does, provided the filter obeys two rules. Doubling the input must double the output, and putting in the sum of two signals must give the sum of what each would have given on its own. Together those are called linearity, and every filter in this course has them because all it ever does is multiply by fixed numbers and add.

Plenty of real things break those rules. A guitar distortion pedal is meant to: turn the input up and the output does not simply get louder, it changes shape. For anything like that the impulse response tells you about the one impulse you sent and nothing else, and a different approach is needed.

There is a hard ceiling and it is worth knowing where it is. The fastest thing a sampled list can hold is a wave that goes up, down, up, down, one sample each way. At a thousand samples a second that is five hundred complete cycles a second, and nothing faster can exist in the list at all. This is why every frequency axis in this course stops at five hundred.

Anything faster than that, arriving at the sensor, does not get recorded as itself. It comes back as a slower wave that was never there, which is called aliasing and is the main event of the course before this one. Here it is enough to know that five hundred is the top of the scale.

Wherever the slow part is the nuisance. A heart trace drifts up and down as the patient breathes and as the electrode gel dries out. That drift is far slower than any beat, so a high pass removes it and leaves the beats untouched. A microphone picks up rumble from traffic through the building, well below anything anyone is saying.

The same filter also shows up where nobody calls it a filter. Subtracting neighbouring pixels across a photograph leaves nothing on flat walls and something bright everywhere the brightness changes, which is edge detection. It is this step's arithmetic, applied sideways instead of forwards in time.

That is a band pass, and it is what a radio does to hear one station rather than the one next to it. There is also the opposite, a band stop, which is how a piece of equipment removes the fifty cycle hum of the mains and keeps everything either side of it.

Neither needs new machinery. A band pass is a low pass and a high pass applied one after the other, and that is still one list of weights. Running two weighted sums in a row can always be collapsed into a single longer weighted sum. Four names, one operation.

Because a sine is the one shape these filters cannot change. Put a sine in and what comes out is a sine at the same frequency, possibly shorter and possibly shifted along, but never a different shape. Nothing else has that property, so nothing else lets you describe the filter with two numbers per frequency.

Put a square wave in and the output is not a square wave, so there is no single height to write down. That does not make square waves useless: it makes them a mixture. Working out which sines a mixture is made of is the whole subject of the course after this one.

Not by looking for the tallest sample, which is the obvious method and is wrong. At four hundred cycles a second there are only two and a half samples per cycle, so the samples land wherever they land and mostly miss the actual peak. The tallest one you can see understates the wave, sometimes badly.

What the page does instead is ask which wave of that exact frequency best matches the whole output, using every sample rather than the luckiest one. It also throws away the first stretch of each run. A filter that has just been switched on is still filling up with the signal, and has not settled into its answer yet.

Because a five-sample average has frequencies it wipes out completely and frequencies in between where a little gets through. At two hundred cycles a second, exactly five samples span one whole cycle, so the window is averaging one complete cycle of the wave. One complete cycle of a sine adds to nothing. The output at that frequency is zero, and you can measure it.

Just above and below that, the window is averaging a bit more or a bit less than a whole cycle, so a small amount survives. Those small humps between the zeros are real, and they are called side lobes. They are the reason a plain moving average is a poor low pass however long you make it. Step 9 builds a better one.

Because a wall is heavy, and heavy things cannot shake quickly. A sound wave arriving at the wall is a push, then a pull, then a push. A slow wave pushes in the same direction for long enough to lean the whole wall a tiny distance. The moving wall then leans on the air in the next room. That is the bass getting through.

A fast wave reverses its push before the heavy wall has begun to move, so the wall simply never picks the wiggle up. The heavier the wall, the slower a wave has to be to move it, which is why a concrete wall has a lower knee than a plasterboard one. Same curve, different place.

Not to the eye. The curve bends gradually, so the eye picks out a region where the shelf is clearly over and the fall has clearly begun. Two people looking at the same curve would point a finger at two slightly different places, and neither would be wrong.

Engineers need to write one number in a datasheet, so they agreed on a convention: a particular height on the curve counts as the official knee. The next step says what the height is and why that height was the one chosen. The picture comes first, because the number only makes sense as a place on this picture.

Inside the equaliser, the signal is split in two. One copy goes straight through. The other goes through a filter that keeps only the top of the band, a high pass like Step 5's. The knob sets how much of that filtered copy is added back before the speaker. Full treble adds it all, and cutting the treble adds none of it, or even subtracts some.

Mixing a filtered copy with the original is itself just another filter. Two weighted sums added together can always be written as one longer weighted sum, so the knob is choosing between weight lists, exactly as the lab's handle does. The bass knob is the same trick built on a low pass.

Because the thing being halved is not the height. The power carried by a wave goes with the square of its height. A wave at 0.71 of its original height therefore carries 0.71 times 0.71 of its power, which is a half. The cutoff is the half-power point, and 0.71 is what half the power looks like when you are measuring heights.

Where 0.71 comes from is worth knowing: it is one divided by the square root of two, since squaring it has to give a half. You will meet the same number in the same role in electronics, acoustics and optics, always for this reason.

Plenty gets through. At the cutoff itself, by definition, 71 per cent of the height survives, which is most of it. The name misleads more than any other word in the subject, and the lab is worth using to break the habit. Shade in the passband, shade in the stopband, and look at how much of the axis is neither.

That middle stretch is the transition, and how wide it is turns out to be the expensive question. Step 9 is about the price.

They are a different way of writing the same gain, used because the interesting range is enormous. A filter's stopband can easily be a thousandth of its passband, and a chart from 1.0 down to 0.001 is unreadable with a straight scale. A decibel figure is twenty times the logarithm of the gain. So a gain of 1 is 0 decibels, a half is about minus 6, and a thousandth is minus 60. This course prints gains as plain numbers because they are easier to check against a chart.

Two pieces of the usual jargon fall straight out. The cutoff is called the minus three decibel point, because a gain of 0.71 works out to minus 3.01. Roll-off is usually quoted in decibels per octave, an octave being a doubling of frequency, so the lab's factor per doubling is the same statement in different clothes.

Because a short filter can only look at a short stretch of the signal, and telling two nearby frequencies apart takes time. Two waves 5 cycles a second apart stay in step with each other for a fifth of a second. Anything looking at less than that much signal cannot yet see any difference between them.

That is the whole reason for the trade, and it is not a limitation of any particular design. Taps are samples of the past, so more taps means a longer look back and a longer look back is the only thing that buys resolution in frequency.

Multiply it out and it usually is. Audio at 48,000 samples a second through a 101-tap filter is about five million multiplications a second, per channel. That is nothing for a laptop and a great deal for a hearing aid running off a battery the size of a lentil.

It is also why the shape of the weight list matters commercially. Suppose one design gets a given edge in 61 taps where another needs 101. That saves forty per cent of the work on every sample, for ever. A whole field of filter design exists to shave taps off.

It is the impulse response a perfect brick-wall filter would have to have, and it is infinitely long. Cut it off at some number of taps and you have an approximation, which gets better the more taps you keep. The stick chart in the lab is that shape: a hump in the middle with smaller and smaller ripples to either side.

Notice the ripples go negative. A filter whose only job is to smooth ends up with weights that subtract in places, which looks wrong and is not. Those negative weights are what make the edge sharp, and in Step 12 they are also what makes it ring.

Because the memory is not stored in the coefficients, it is stored in the last output. That one number already contains a fraction of the output before it, which contains a fraction of the one before that, all the way back to the beginning. It is a running total that quietly forgets, and forgetting slowly is what makes it long.

A moving average has to keep every sample it is still using, so a window of fifty samples means fifty numbers in memory and fifty multiplications. The recursive version keeps one number and does two multiplications, for a similar amount of smoothing. What it gives up is any say over the shape of the forgetting, which is always the same decaying curve.

Almost certainly. It is the running average that games and dashboards use for a smoothed frame rate, and it is what a spreadsheet calls exponential smoothing. Somewhere in most pieces of software there is a line reading something like avg = 0.9 * avg + 0.1 * newValue. That line is this filter.

It is popular for the reason the lab measures: it is one line, it stores one number, and nobody has to keep a list. Whoever wrote it was usually not thinking about frequency responses, which does not stop it having one.

Three lines, and the middle one is the filter. The rest of this course could be built out of the same shape with more terms on the right.

let y = 0;                         // the last output, kept between samples
for (const x of input) {
  y = (1 - g) * x + g * y;         // the whole filter
  output.push(y);
}

Compare that with an FIR of the same smoothing power. The FIR needs an array of dozens of weights, a buffer of dozens of past samples, and a loop inside the loop. The saving is real, and it is why this shape turns up in so much shipped code.

Loud, briefly, and then not at all. In audio the numbers run past what the format can hold and the sound becomes a full-scale screech before the hardware protects itself or the output saturates flat. Anyone who has held a microphone too close to a speaker has heard the same effect built out of air rather than arithmetic. The howl is a loop handing back more than it received.

In a control system it is worse than a noise, because the numbers are steering something. A value doubling every few milliseconds reaches the machine's limit almost at once and stays there. That is a machine at full power in one direction, with nobody able to talk it down.

Because there is no route from the output back to the input. An FIR output is at most the biggest input multiplied by the sum of the sizes of the weights, and that is a fixed number however unpleasant it is. A filter with weights of a hundred each amplifies by a lot and then stops.

This is the reason FIR filters are chosen for anything where a failure is expensive, even when an IIR would do the same job with a tenth of the arithmetic. You cannot make one unstable by typing the wrong number, and that is worth a great many multiplications.

They work out where the loop exactly breaks even, for g or for whatever set of feedback numbers they have. Then they check that the actual values sit on the safe side. Those break-even values are called poles. The test is that all of them sit inside a circle of radius one. For the one-coefficient filter here, the circle is just the range from minus one to one, which is why 1.00 is the boundary you can watch the slider cross.

There is a second trap that the arithmetic on this page does not have. Real hardware stores its numbers with limited precision. A filter designed at 0.9998 may be stored as 1.0000, and become unstable on the device while being perfectly safe in the design. Splitting a long filter into a chain of short ones is the standard defence.

Because its centre of gravity is in the middle, and the middle is the same distance back whatever wave is passing. A nine-sample average weights the last nine samples equally, so the answer describes the moment four samples ago, for a slow wave and a fast one alike.

A feedback filter has no middle. Its impulse response starts tall and fades, so its centre of gravity sits near the front. How much a wave gets held back then depends on how much of that tail it is still in step with. That is why the second lab's table has one column standing still and one sliding.

They are the same shift measured against two different rulers. Delay measures it in samples, which is a fixed amount of time. Phase measures it as a fraction of one cycle of the wave being shifted. For a fast wave, the same shift is a much bigger fraction of a cycle than for a slow one.

So a delay of four samples is four samples at every frequency, but written as a phase it grows with frequency. That is where the word linear comes from: phase rising in a straight line means a constant delay. The lab reports samples, because samples are the thing a reader can count off the chart.

For listening, mostly nobody cares, and analogue equipment has bent phase for a century without anyone objecting. The ear is fairly forgiving about when the parts of a sound arrive relative to each other, within limits.

For measurement it matters a great deal. Say the job is to find the exact moment of a heartbeat, or the leading edge of a radar echo. A filter that moves the slow parts of that edge by a different amount from the fast parts has changed the answer. Anything doing timing uses a symmetric FIR for that reason, and pays the extra multiplications for a delay it can subtract back out.

Almost anywhere a knob changes tone. The treble control on a cheap guitar amplifier is close to being one resistor and one capacitor. The click removers on old record players, the smoothing after a power supply, and the touch sensitivity of a phone screen all lean on capacitors filling through resistors.

They are also hiding in places nobody calls a filter. A thermometer in a hot drink reads low for the first seconds because heat flows into it through a poor conductor and fills its small heat capacity. Same equation, same curve, with temperature standing in for charge.

Start the bucket empty and switch the input on. After one time constant the level has covered 63.2 per cent of the distance to its final value. After another time constant it covers 63.2 per cent of what remained, and so on. Each τ takes the same fraction of the leftover gap, which is why the curve never quite arrives. It is also why five time constants is the usual engineering answer for close enough, within about one per cent.

The number itself is 1 minus 1 over e, where e is the mathematical constant 2.718. It falls out of the differential equation, and you can check it against the lab rather than take it on trust.

The careful way to discretise the bucket keeps its behaviour exactly: the sampled decay matches the true curve at every sample, whatever the step size. The quick and obvious way, called an Euler step, just adds speed times step size on each tick. It is fine when the steps are small compared with τ, and it goes wrong when they are not.

Wrong here does not mean slightly off. A coarse Euler step can turn a decaying bucket into a growing oscillation, which is Step 11's instability arriving through the back door of a lazy approximation. The second lab lets you cross that line on purpose and watch it happen.

Every zero or pole placed off the horizontal axis in these labs brings a twin, mirrored below the axis. The reason is that the filter's own numbers, the ones the code multiplies by, have to be ordinary real numbers. A lone tilted point would demand weights that are not, and an input of real samples would come out imaginary, which no loudspeaker knows how to play.

Placed as a pair, the imaginary parts cancel and the weights come out real. The pair acts like one point as far as the response is concerned, so the labs let you steer one and carry its twin along.

Anywhere something prefers one frequency. A wine glass sings one note when flicked because the glass is a loop of stored motion with a pole near the circle at that note's angle. A car suspension bouncing after a bump, a bridge swaying in step with marching feet, and the whistle of a badly tuned microphone stage are the same picture.

Filter designers use resonance on purpose. A pole pair pulled close to the circle at one angle makes a filter that answers strongly to one band, which is how a radio's tuner leans towards one station. The radius sets how choosy it is, and how long it rings.

Continuous circuits like Step 13's RC pair have their own map, the s-plane, drawn with the Laplace transform. There the stable side is the left half of the plane rather than the inside of a circle. Sampling rolls that half-plane up into the unit circle, which is why the two stability rules are the same rule seen through different transforms.

Standard recipes such as the bilinear transform carry an analogue design across to the z-plane. The carrying bends the frequency axis a little, which is called warping, and a design moved across without accounting for it lands its cutoff in the wrong place.

Anyone whose filter has neighbours. A hearing aid team writes one so speech survives while the battery lasts the day. A phone team writes one so the radio does not leak into the channel next door, and the law in most countries makes some of those numbers compulsory. A heart monitor team writes one so the mains hum goes and the beat's timing stays.

The list is also how disagreements get settled. When the built filter disappoints, the question is not whether it seems fine but which promised number it misses. A specification is an argument that was had once, in advance, instead of forever afterwards.

Ripple is small wobble in the response where flatness was wanted. A passband that wavers between 0.99 and 1.01 instead of holding 1.00 has ripple, and so does a stopband floor that bumps along instead of lying still. It means some frequencies are treated a little differently from their neighbours.

It is accepted because it is bought at a discount. Allowing a per cent of wobble lets the same order fall much faster, so the edge sharpens without more taps. For music playback a per cent is inaudible and the sharper edge is not, so the trade is usually taken. For a measuring instrument it may be the other way round.

A stored recording offers a trick a live signal cannot. Run the filter forward, then run it again backwards over the result. The two passes shift every frequency by equal and opposite amounts, so the delays cancel to zero, and the filtering happens twice, which squares the response. It is called zero-phase filtering, and analysis software does it as a matter of course.

It needs future samples, so nothing live can use it. A controller or a radio receiving as the samples arrive cannot read tomorrow's input, and a method that is routine for stored measurements is simply impossible there. A specification should say which world it lives in.

As bad as digital audio gets. Saturation on a loud passage shaves the tops of the wave, which adds a hard buzz but keeps the shape. Wraparound replaces the top of the wave with its opposite extreme: a full-scale jump in the wrong direction on every loud sample. It plays as violent crackling far louder than the music.

Worse, inside a feedback filter that wrong-signed number goes back round the loop, so one overflow can leave the filter thrashing long after the loud moment has passed. That is why fixed-point designs reserve headroom, and why saturate is almost always the mode switched on.

Bigger processors do, which removes most of this step's problems in exchange for silicon and power. A hearing aid, a smoke alarm or a sensor on a coin cell cannot pay that. A fixed-point multiplier costs a fraction of the chip area and the battery of a floating-point one. A product made in millions multiplies those savings by millions.

Even where floating point is available, its rounding still exists, just further down. Long feedback filters can misbehave in float too, so the biquad habit and the test-what-you-ship rule survive at every budget.

A feedback filter carries its running memory between calls. Audio arrives in blocks of a few hundred samples, and code that resets the filter's memory at every block plays a small switch-on thump hundreds of times a second. Code that shares one memory between left and right channels quietly mixes them.

So a shipped filter defines who owns the state, what its start value is, and when it may be reset. The standard test is to stream the same recording in several different block sizes and compare against one uninterrupted run. Every difference is a state bug found before a customer finds it.

History and physics, in layers. Music settled on 44,100 samples a second because of how early digital audio was stored on video equipment, while film sound settled on 48,000. Telephone speech runs at 8,000, because long ago that was expensive enough. Radio hardware samples in the millions because its signals wiggle in the millions.

Any system touching two of those worlds converts between them, so rate changing is not an exotic corner. A phone call playing through a laptop speaker may cross three rates on its way to the air.

Because the ceiling was never about the signal, it was about the spacing of the samples. The fastest wave a list can record is up on one sample, down on the next. Keep every fourth sample and the gap between samples is four times longer. Up on one kept sample and down on the next is now a wave four times slower than before.

A 20,000 cycle wave in a list sampled at 48,000 is comfortably under the old ceiling of 24,000. Decimate by four and the new ceiling is 6,000. The wave cannot be recorded, so it folds back and lands at 4,000, indistinguishable from a real 4,000. The lab's safe pill is checking exactly this arithmetic.

Two devices both labelled 48 kHz run on two different crystals, and no two crystals agree exactly. One side of a cable produces samples a hair faster than the other consumes them. A buffer between them slowly fills, or slowly drains, and eventually something has to give. Dropping or repeating a whole sample gives an audible click.

The cure is asynchronous rate conversion: measure the true ratio between the two clocks, continuously, and resample by that ever-drifting factor so no sample is ever dropped whole. It is this step's machinery run with a ratio that is never a neat fraction and never quite constant.

On every phone call you make. The far end's voice comes out of your speaker, bounces round the room, and re-enters your microphone, and without cancellation the other person hears themselves half a second late. An adaptive filter learns the room's echo from the known speaker signal and subtracts it, relearning continuously as you move.

Noise cancelling headphones, hearing aids taming feedback whistle, and modems learning the shape of a telephone line are the same loop at different speeds. In each case the filter is cheap and ready-made, and the engineering is in choosing the reference and the step size.

Freshness against steadiness. A gain near one says believe the sensor, so the estimate follows every measurement including its noise: fast and jittery. A gain near zero says believe the model, so the estimate glides smoothly and reacts late when the world does something the model did not expect. It is the window length of Step 2 and the g of Step 10 wearing better mathematics.

The difference from those dials is that nobody turns this one by hand. The filter carries an estimate of its own uncertainty and computes the gain from it each tick, so the balance shifts on its own as conditions change.

When the state never leaves a trace in anything measured. A drone's height cannot be estimated from a compass, however clean the compass is, because height never affects it. Whether a state's effects eventually show up in the sensors is called observability, and it is a property of the physical arrangement, not of the filter's cleverness.

The trap in practice is buying a better version of the same blind sensor. Ten times less noise on the compass is still zero information about height. The fix is a different sensor, or a different motion that makes the hidden state leave fingerprints, and no tuning session can substitute for either.

Because an average lets every sample vote with its size, so one absurd sample drags the answer in proportion to how absurd it is. A three-sample average with a spike of 30 in it reports 10, which is the spike smeared into three samples of nonsense. The median only asks which value is in the middle. The spike sits at the extreme end of the sorted three, so it never touches the answer at all.

The trade is that on smooth wobbly noise, the kind Step 1 fought, the median does no better than the average and costs a sort per sample. The shape of the failure selects the tool.

Where the signal has structure a weighted sum cannot know. A photograph's noise sits next to textures, edges and repeated patterns. A model that has seen millions of photographs can tell grain from grass better than any fixed set of weights. Night-mode photos on a phone, denoised medical scans, and speech pulled out of a crowded room all lean on learned models today.

The same strength is the risk. A model that knows what photographs usually look like will helpfully invent usual-looking detail where the measurement was only noise. For a holiday photo that is fine, and for a scan a diagnosis rests on, it is not.

Whatever the filter is, ship its test as well as its code. Save the raw and filtered recordings, the exact coefficients or model version, the sample rate and the block size. Save the state reset policy and the timing figures too, and keep the labelled test set. After every change, replay the same corpus and compare against the promises of Step 15.

When a design tool or a model proposes a filter, treat the proposal as a hypothesis. Measure its response the way Step 6 did, its stability the way Step 11 did, its overflow the way Step 16 did, and its task accuracy against the labels. A suggestion is not a measurement.

The sticks at the top are the impulse response: one 1 was sent in and those are the numbers that came out. The curve in the middle is the frequency response, and every point on it is a sine that was built at that frequency, run through your filter, and measured. The line chart at the bottom is the same noisy test signal from Step 2, with your filter applied.

The pills report a flat input and the fastest possible wiggle, which are the two ends of the frequency curve. Those two numbers on their own tell you whether you have built a low pass or a high pass, before you look at anything in between.

Because the weights are no longer the only thing acting on the spike. The spike passes through the weights once, and then whatever came out is fed back round and added to the next output, and so on. The tail you see after the seventh stick is the loop talking to itself.

This is why the impulse response is the honest description and the weight list is not. Ask a filter what it does by sending a spike into it, and it answers for everything it contains, including the parts that are not written in the weight list.

Two things above all. The first is how anybody arrives at a set of weights on purpose rather than by dragging sliders, which is called filter design. The usual routes are two. Sample the shape you want and taper it, as Step 9 does, or let a program search for the weights that best match a target curve. The second gap is how to see a frequency response without running one sine at a time, and that is the whole subject of the next course.

Also missing: filters whose weights change while they run, which is how noise cancelling headphones work. Filters that change the sample rate as they go. And the arithmetic problems that come from storing coefficients in a fixed number of bits. Each of those starts from exactly what is in front of you here.