Analog domain and aliasing

[petert]

Hello all,

I have been having this question for a while, and haven't found a convincing explanation or proof yet.

When digitizing a signal, you should stay under the Nyquist frequency to avoid aliasing, that easily makes sense when looking at examples.
But is there really no aliasing or similar effect in analog components?

There are phase shifts and dampening of frequencies in filters, but why no aliasing?

Couldn't certain physical properties of materials cause similar effects? Rasterizing effect or "jumps" and similar patterns can occur in mechanical systems (think of a clock that translates continuous rotation or pendulum movement to step wise rotation of the clock's hands). Jumps/discrete steps also appear as discrete energy levels in Bohr's atom model (I know it has been superseded by Quantum mechanics, but it still describes the process quite accurately for hydrogen).

How do we know that analog components are really continuous and how much might it affect/add noise similar to aliasing to a signal as it gets processed by those components?
How can we know they really do not behave frequency dependent, beyond dampening and phase shifting?

[JohnnyMalaria]

Aliasing is the false assignment of a frequency due to sampling. There is no sampling in the analog domain. Analog signals are continuous as you say. The main issue with analog is random noise. In theory, if you could observe analog signals using a non-sampling technique (e.g., CRT scope) on extremely short timescales (say picoseconds) then you may start to observe quantum effects but in our worldly domain an analog signal is continuous.

[petert]

Thanks.

Aliasing is the false assignment of a frequency due to sampling.

What about the clock example which translates continuous to step-wise? Couldn't there be similar unintended effects in analog components that cause such interrupted/step-wise behavior and consequently aliasing?

on extremely short timescales (say picoseconds) then you may start to observe quantum effects but in our worldly domain an analog signal is continuous.

If I knew more, I would give other examples. But let's stay with quantum effects. Couldn't these aliasing effects propagate into the macroscopic world, since the alias frequency would be much lower?

[JohnnyMalaria]

With something like clock signal (e.g., a square wave), the major unwanted effects come from the finite time it takes to make the step change and also the bandwidth of the circuit plus other things. You may see distortions on the signal such as ringing or overshoot. But what in an analog clock is being sampled? Nothing. Aliasing is a phenomenon due to digital sampling of an analog signal (or synthesis of an analog signal from a digital signal).

EDIT: Re my comment about very short timescales etc, all kinds of weird stuff happens that inherently cannot be measured.

[vk6zgo]

Sampling is very similar to mixing, & yes, it is possible with analog sig "to end up with signals which are translated into the wrong part of a band of frequencies.
I started to write a long screed about this, but gave up.

Hint:- if in a mixer f1 is the higher frequency input, f2 the lower,
we would normally want to use f1- f2, or f1+f2, what would we get from f2-f1?

[tggzzz]

When digitizing a signal, you should stay under the Nyquist frequency to avoid aliasing, that easily makes sense when looking at examples.
But is there really no aliasing or similar effect in analog components?

Aliasing is sometimes inherent to the operation of a circuit. One easy example is "Tayloe mixer". I used the phenomena in 1979 to create an analogue bandpass filter with a Q of 4000 using 20% capacitors and 10% resistors. I "nicked" the concept from a Bell System Technical Journal article from April 1960.

One analogy for aliasing in the analogue domain is mixers.

[MattHollands]

Couldn't certain physical properties of materials cause similar effects? Rasterizing effect or "jumps" and similar patterns can occur in mechanical systems (think of a clock that translates continuous rotation or pendulum movement to step wise rotation of the clock's hands).

A mechanical clock is not analog. You’re taking an analog signal (the swinging pendulum), and creating a clock pulse and then counting the pulses. This is the electrical equivalent of passing a sine wave through a comparator and then passing those pulses to a counter chip. The system stops being analog once it enters the comparator.

Generally speaking, an important concept to understand is linear systems - most things in ordinary operation are linear systems. Among other things this means that a sine wave input will result in a sine wave output of the same frequency but potentially different phase and magnitude.

[JanJansen]

Try  drawing a saw wave in a limited ammount of blocks.
Now try that with a different frequency, you see it wont get exact to the top.
With square wave its different without aliasing, you can only get frequencys that fall exact in your blocks ( samplerate ).
The trick is to use sinewaves, they always go to top and bottom of your waveform.

[SiliconWizard]

Aliasing per se is a term usually associated with sampling.

That said, there are many ways you can get frequency shifts in the analog domain. Amplitude or frequency modulation, for instance. Non-linear distortion (think of intermodulation distortion for instance).

[NorthGuy]

Sampling per-se doesn't create aliasing. Aliasing requires regular sampling. Cannot get it otherwise.

For example, there were no moire effects in photography before digital cameras. This is because the sensor's matrix is regular. Before digital, irregular sampling with grains on the film couldn't create it.

[petert]

Many good replies, thank you all. It is really appreciated.

For example, there were no moire effects in photography before digital cameras. This is because the sensor's matrix is regular. Before digital, irregular sampling with grains on the film couldn't create it.

That's a very good analogy, interesting.
Ofc I have to keep playing devil's advocate and keep pushing this idea. So assuming the granularity of the medium was limited, as it is also in a photographic film, and it was in a regular pattern, you would get aliasing? So in theory, similar effects could happen with "granular" material in capacitors, depending on how they were produced, I suppose.

Is there an inherent physical phenomenon that makes it unlikely a sufficiently regular "matrix" or pattern would happen in production processes of electrical components?

So in the end it is not only about under sampling (at some point the information is too few, and therefore ambiguous such that the real information cannot be restored), but it is also about randomness of sampling.

If it's random sampling you still might get errors as well, but since they are not regular, the faults do not repeat in regular patterns either. Therefore those random errors blend in more as evenly dispersed noise over your signal, instead of periodic one (=waves). So it would become grainy instead of leading to clearly identifiable Moiré patterns?

[T3sl4co1l]

To clarify:

1. Analog does it just fine; it's a mixing process.  Mixers, samplers, bucket-brigade devices, switched-capacitor filters, etc. are excellent practical examples.

2. It's a mixing process.  Namely, the product of the input signal with the sampling aperture.  The aperture has a short duration, approximated by a periodic impulse function.  The product of this signal, with an arbitrary input signal, is to copy the spectrum of the input signal around each harmonic of the impulse signal.

If the input signal has BW > Fs/2, then it will overlap itself (BW above a given harmonic, BW below the following harmonic, the harmonics being Fs apart), which is ambiguous.

Aliasing isn't a problem, in and of itself, but ambiguous signals are a problem.  The sampling theorem says that a signal can be reconstructed exactly when the signal is bandlimited to Fs/2.  If you don't need an exact reconstruction, you don't need this condition.

An interesting special case is equivalent time sampling: the input is not reconstructed exactly -- in fact (for an incremental-delay, triggered sample) it is divided.  The waveform is reconstructed, but at a rate below Fs/2 -- the bandwidth limitation is not violated, nor is the sampling theorem (you clearly have not reconstructed the original waveform one-to-one!).

3. Pedantic but important point -- sampling is analog or digital, it doesn't matter.  The domain of sampling is called discrete time.

(Incidentally, where continuous-time signals are typically transformed with the Laplace or Fourier transform, discrete-time signals are transformed with the Z transform (whereas s or j*w means frequency, Z simply means... delay one sample!).  Interestingly, there is a direct correspondence between these transforms, so that all our continuous-time tools still work, under a fairly simple mapping.)

Ofc I have to keep playing devil's advocate and keep pushing this idea. So assuming the granularity of the medium was limited, as it is also in a photographic film, and it was in a regular pattern, you would get aliasing? So in theory, similar effects could happen with "granular" material in capacitors, depending on how they were produced, I suppose.

You would get aliasing if the signal has more bandwidth than the film, yes.

This doesn't usually happen, because optics just aren't that great.  Even with very good optics, the scene's depth of field may frustrate that (even if stopped down very far).

In this case, we're talking spacial frequency -- sharpness, resolution.  It works exactly the same -- whereas an electronic signal is a function of time t, an image is a function of position (x, y).

An extreme example is photolithography (the process by which semiconductors are made), which is presently pushing ten nanometers.  Obviously, it helps that the exposed medium is a molecular resin; on this scale, a film emulsion looks like the Himalayas.

You can imagine, if a wafer were coated with a very regular colloid, so that a regular hexagonal monolayer sits on it, and that colloid were exposed to the patterns used to create CPUs, you would end up with a very strange image, as some particles are exposed while others are not; in fact, you would end up with Moire patterns (assuming a regular pattern of transistors) -- optical aliasing.

Is there an inherent physical phenomenon that makes it unlikely a sufficiently regular "matrix" or pattern would happen in production processes of electrical components?

It's always there, but whether you see a pattern, depends on whether that pattern is present in both signals (the image and the sensor, in this case).  If both are uncorrelated, then no correlation is observed at a large scale.  To observe a Moiré pattern of, say, 1mm pitch, one needs an image of, say, 0.01mm and a sensor pitch of 0.0101mm, and for that pattern to be regular (say, 1.0 +/- 0.1 mm), both pitches need to be accurate to as many decimal places -- that is, 0.01000 and 0.01010 respectively.

Moiré is a remarkable phenomenon for two reasons: one, order is not often seen in the world ("nature abhors a straight line"), let alone the long-distance regularity required to observe a large pattern; and two, when viewed from two locations (binocular vision!), a pattern formed from two screens (say), is very different from those two locations; they're very eye-catching.

Of course, the change in contrast over an aliased image, or the fluctuating intensity of an aliased signal, is nearly as jarring.

Tim

[dmills]

The aliasing you see at the movies is the wagon wheels seeming to rotate backwards in the old cowboy films.... The sampling is the 24FPS that the film is exposed at, and the signal is the angle of the spokes.

Aliasing is an artefact of sampling not quantisation (Two different processes).

Regards, Dan.

[SiliconWizard]

Yes. Sampling is discretization, not quantization, so it's not specifically digital. (This whole thread doesn't really belong here in fact.)

Aliasing *will* occur whenever the signal you're trying to represent has frequency components which frequency exceeds what the "canvas" you're using to represent it with can "store" (as an information) and from which the original signal can be reconstructed from. It can still happen with non-regular sampling.

Regular period sampling will just create regular patterns (which can be seen as moire in images, or typical frequency foldback sounds in audio), whereas with random or pseudo-random sampling, the patterns will be much less noticeable if at all, but there's still aliasing. It's just a lot less distracting for us humans because we can't notice the patterns, especially when dealing with images or audio.

Sampling is somewhat akin to amplitude modulation, it's the multiplication of a signal by a non-continuous function.

[NorthGuy]

Aliasing *will* occur whenever the signal you're trying to represent has frequency components which frequency exceeds what the "canvas" you're using to represent it with can "store" (as an information) and from which the original signal can be reconstructed from. It can still happen with non-regular sampling.

Regular period sampling will just create regular patterns (which can be seen as moire in images, or typical frequency foldback sounds in audio), whereas with random or pseudo-random sampling, the patterns will be much less noticeable if at all, but there's still aliasing. It's just a lot less distracting for us humans because we can't notice the patterns, especially when dealing with images or audio.

I would disagree with that. Aliasing requires regularity in both the signal and sampling. Say you sample a high-frequency sine wave using too low a frequency for your sampling, the frequencies get subtracted and you see a lower-frequency sine wave, which appears as a perfect sine wave, and there's no way for you to tell if this is the sine wave you see or it is an aliased sine wave of  higher frequency. Hence, you need a Nyquist filter to remove everything which can alias, thus giving you the certainty that you see the real thing.

If you sample white noise you will never get any aliasing no matter how low your sampling frequency is - you'll get the white noise anyway.

If you sample a perfect sine wave at random sampling intervals, you will see the sine wave if your sampling density is high enough. As your sampling density gets lower you will lose the ability to distinguish the sine wave and it'll eventually turn into white noise. However, there will be no aliasing - you will never see a sine wave of wrong frequency (except by pure coincidence as with monkey which can type encyclopedia by randomly banging a typewriter).

[RoGeorge]

How do we know that analog components are really continuous?

Many times, analog signals are not continuous, but jumps in the input signal doesn't matter.
It's the time jumps what creates the aliasing in digital domain.

Time is continuous in analog domain, and quantized in digital domain.

More intuitively said (for an infinitely long observation in time domain):

• If all you have are just snapshots of your world (like in digital domain), then you can never know for sure what happened between those snapshots. You simply don't have that information. For the particular case of a periodic signal faster than your snapshots, you will have aliasing.
• If you watch your world all the time (like in analog domain), then you know it all, there is no way to be tricked by faster signals, so aliasing won't happen.

Later edit:
-----------
Other more philosophical way to say that is:
What makes us unsure about the real frequency of a periodic signal (aliasing) is caused by the lack of complete information about that signal, and not by the shape of the input signal (like in your example with the energy levels jumps in the atom).

As an extension of the lack of complete information idea, if you can not see the whole spectrum, then you can have aliasing problems in analog domain, too. (i.e. if you have an analog modulator with an analog filter at output - so incomplete information about the spectrum - then you will not know if what you see at output is Fsignal+Fosc, or Fsignal-Fosc, so you can say it's a form of aliasing in analog domain, but it's the lack of information that makes us trouble, not the input signal).

[rhb]

Sampling is the multiplication of an analog signal by an analog  spike series.  In the frequency domain this is a convolution.  The Fourier transform of a regular spike series in time  is a regular spike series in the frequency domain.

Mixers are analog multipliers.

Aliasing occurs if the bandwidth of the analog signal is greater than the spacing of the spikes in the frequency domain.  This is why equivalent time sampling works.

If the the sampling spikes are random, then the Fourier transform is a single spike.  As a consequence, there is no aliasing.  This is a significant factor in why compressive sensing works.  TANSTAFL.  This requires that one compute the Fourier transform using least summed absolute error (L1) rather than least summed squared error (L2).  L2 has the property that it smears things in the process of finding the transform.  All conventional numerical Fourier transforms are L2.  L1 is much more compute intensive. So until recently it was not practical.

You can find a lot more on the subject in  the papers by Emmanuel Candes and David Donoho at statweb.stanford.edu. However, you'll need to understand Wiener, Shannon, Nyqist et al plus a good bit more to make sense of it.

The canonical work is, "Extrapolation, Interpolation and Smoothing of Stationary Time Series" by Norbert Wiener.  It appeared during WW II in a yellow cover indicating it was classified and was widely known as "the yellow peril" for the difficulty of the mathematics.

The best modern treatment is "Random Data" by Bendat and Piersol.

[T3sl4co1l]

Many times, analog signals are not continuous, but jumps in the input signal doesn't matter.
It's the time jumps what creates the aliasing in digital domain.

Time is continuous in analog domain, and quantized in digital domain.

No -- digital can be continuous-time as well.  Phase detectors are a simple example; they can't work otherwise!  There are unclocked CPU cores out there.  A lot of single-cycle instruction sets use the same techniques, then latch the result with a clock, probably mainly to get repeatable performance and to interface with clock-driven buses and peripherals.

The only reason clocked logic is so common, is because state machines are so easy to synthesize.

Tim

[JohnnyMalaria]

No -- digital can be continuous-time as well.  Phase detectors are a simple example; they can't work otherwise!
Tim

I'm very curious about this. An instrument I have built samples an analog photodetector signal at a fixed clock rate and uses one of two ways to determine phase (multiplication with a reference signal that has either been synthesized or from an analog reference digitized at the same rate as the signal using a SS&H ADC; or simply a digital equivalent of traditional phase sensitive detection).

[rhb]

Physical time is continuous.  Once you convert an analog signal to discrete samples, you have done the multiplication and you have the inevitable consequences of doing that.

An analog phase detector produces an analog output.  The phase difference between two sine waves is a  voltage which varies with the phase difference.  If the phase difference is constant, the voltage is DC.  This was how microwave measurements were done in the 30's and 40's and well into the 60's.   

There is no such thing as a continuous time digital signal. That is an oxymoron.

[JohnnyMalaria]

An analog phase detector produces an analog output.  The phase difference between two sine waves is a  voltage which varies with the phase difference.  If the phase difference is constant, the voltage is DC.  This was how microwave measurements were done in the 30's and 40's and well into the 60's.

Indeed.

Recently, I decided to do the blasphemous thing and choose analog detection over digital. I use two 4-quadrant multiplier/dividers to generate the I and Q components, filter the outputs and then digitize them. I'm much happier with this approach. The ADC demands are much less as are the subsequent numerical steps. When I did this originally 30 years ago, I used an EE&G-Princeton Applied Tech LIA. Nevertheless, the digital detection yields very good results.

Funny story (to me anyway). Someone patented a variant of my technique and said words to the effect, "Miller was an idiot for not using an all-digital design". Well, the instrument I built way back still beats the crapola out of existing commercial variants.

[T3sl4co1l]

There is no such thing as a continuous time digital signal. That is an oxymoron.

A continuous time digital signal is one where transitions happen at any function of real-valued time t.

A discrete time digital signal is one where the instantaneous value of a discrete time digital signal is an array X[ i ], for time indices i.  Alternately, a function where transitions only happen at integer-valued times t * Tclk, and there is no change inbetween.

The former can only be Fourier transformed.  The latter can be Z- or Fourier- (using the stepwise format) transformed.

Note that a quantized digital signal is also an analog signal plus quantization noise; this remains true even if the signal is a single bit.  There is never a case where digital signals are not analog; we can take analytical shortcuts with them (event-driven logic analysis, Z transform, binary math, etc.), as long as the assumptions hold; but digital signals never cease being analog.  Digital is a subset.

Tim

[T3sl4co1l]

No -- digital can be continuous-time as well.  Phase detectors are a simple example; they can't work otherwise!
Tim

I'm very curious about this. An instrument I have built samples an analog photodetector signal at a fixed clock rate and uses one of two ways to determine phase (multiplication with a reference signal that has either been synthesized or from an analog reference digitized at the same rate as the signal using a SS&H ADC; or simply a digital equivalent of traditional phase sensitive detection).

Somewhat as a matter of definition, that is.  Suppose you have two arbitrary digital signals, coming into a Moore type state machine.  If the signals are clocked through latches to synchronize clock domains, heh, well... oops, right?

Which isn't to say it's not possible.  If you maintain independent clock domains (using a count from one to latch a count from the other), you get a frequency detector (very common in real systems).  The output is then a digital number, as continuous as you can get here; though the resolution-bandwidth is quite poor (to get N bits of resolution, you need 2^N counts).

Note that the analog PFD yields nearly Fclk bandwidth, full resolution, and some delay -- assuming a very good loop filter, of course (which is where most of the delay would be incurred, say, a few cycles worth of group delay for a sharp cutoff at Fclk/2).

Even with the latched inputs, you can still do something; that's essentially the huff-puff control loop.  If one input leads the other, the latched values no longer match, and the control slews away; if one input lags the other, the opposite happens.  As a result, the control huffs and puffs at the difference frequency.  This works even if the clock frequency itself is very low -- the, guess what -- aliasing(!) can be used, for example, to discipline a radio to discrete channel frequencies, while tuning continuously between them.

Tim

[ejeffrey]

I would disagree with that. Aliasing requires regularity in both the signal and sampling. Say you sample a high-frequency sine wave using too low a frequency for your sampling, the frequencies get subtracted and you see a lower-frequency sine wave, which appears as a perfect sine wave, and there's no way for you to tell if this is the sine wave you see or it is an aliased sine wave of  higher frequency. Hence, you need a Nyquist filter to remove everything which can alias, thus giving you the certainty that you see the real thing.

If you sample white noise you will never get any aliasing no matter how low your sampling frequency is - you'll get the white noise anyway.

That is absolutely not true.  I mean, it is true that you get white noise out (assuming perfect delta function sampling), but that noise is aliased down into the nyquist band and reduces your SNR.  After the aliasing happens, you can't fix it, as the higher frequency noise is indistinguishable from the lower frequency signal you care about.

If you sample a perfect sine wave at random sampling intervals, you will see the sine wave if your sampling density is high enough. As your sampling density gets lower you will lose the ability to distinguish the sine wave and it'll eventually turn into white noise. However, there will be no aliasing - you will never see a sine wave of wrong frequency (except by pure coincidence as with monkey which can type encyclopedia by randomly banging a typewriter).

This is more a matter of nomenclature.  Aliasing usually refers to the effect with periodic sampling, which is very simple where all signals are folded down into the first nyquist band.  Random sampling doesn't have a simple behavior in frequency space, but the same basic affect applies: for any given record of samples there are multiple analog waveforms that could have given rise to that, and you can't tell them apart.  They are just not related by a simple frequency shift.  If you have spectrally narrow high frequency signals, random sampling will spread them out which may be desirable.  However, the signal power is still there, just spread out differently. 

[ejeffrey]

There is no such thing as a continuous time digital signal. That is an oxymoron.

Except for comparators, most classic logic gates including AND, OR, NOT, NAND, XOR, and R-S latches, and PWM drives. PWM can actually be continuous or discrete or both (where the rising edge is triggered by a clock but the falling edge can happen any time).  You can even make a continuous time sigma-delta modulator if you want to.

Unless by digital you mean "stored on a computer" in which case of course you are right, but most people consider logic gates pretty much the definition of digital, and they don't require a clock in sight.