linearization of multiplier

Ask your digital TA for look-up tables.

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Lecture - 19 Analog Multipliers:

Look at:

This Article goes over Gilbert Mixer Basics and Linearization techniques such as emitter degradation, Feedback and other techniques.

By Stephen Yue fromthe University of Toronto ..."

CMOS Gilbert Cell Mixer:

Lecture 21: Balanced and Passive Mixers:

I always poke fun at the analog guy that without meaningful parameters (as in the OP's questions), anything is possible digital. The OP might not get a good grade, but a good laugh in class.

If it were linear, it wouldn't multiply. ;-)

Cheers! Rich

Linear in the second order.

See the datasheet for the LM13700, and you'll get the idea:

Cheers

Phil Hobbs

A good multiplier is *bilinear*, i.e. if you hold one input constant, the output is a linear function of the other input--it satisfies superposition and scale invariance. It isn't a nonlinear device unless you derive the X and Y inputs from a common source.

There are lots of bilinear things in electronics, e.g. the volume control on a stereo, DACs, and ADCs.

Cheers

Phil Hobbs

Or a (relatively) flat surface in 2D. The circuit is stable in concave and unstable in convex surface, if the third dimension is excitation.

You mean to say that rabbits are nonlinear??

Of course they're not linear. They're kinda spherical, like a cow.

OTOH, rabbets are very linear.

What do you specifically mean by linearizing the Gilbert multiplier cell?

The true gilbert cell multiplier can perform the following function:

Iout = IK.(IA-IB)/(IA+IB)

Which allows for IK.IX essentially as perfect multiplication, and does not require linearizing, usually. Are you maybe referring to a cell that is sometimes called a gilbert multiplier, but isnt really? The true gilbert cell has logging and anti-logging functions that gives "perfect" multiplication. Its why he got the patent.

Regarding another post in this thread, I did look at

Kevin Aylward B.Sc.

I'm going to introduce a semantic argument here, but:

I agree with your definition of bilinear up to the "isn't".

A system is linear, overall, if it obeys superposition -- overall. So if you have a system where y(x) = x1 * x2, then that system isn't linear because it doesn't obey superposition.

Viewed as a system, then, a multiplier isn't linear.

Yes, if you make a larger system with a multiplier as a component, and you apply some known and predefined signal to one of the inputs of the multiplier, then that larger system, as you've defined it, will be linear (albeit time varying, if your 'known and predefined signal' is time varying). But the multiplier itself is still a nonlinear system.

Just as an op-amp that's hooked up can usually be treated like a linear system, yet is made up of a whole bunch of transistors, which are fundamentally nonlinear in operation. You can coerce a collection of nonlinear components (or little systems, if you will) into acting like a linear system, you just can't do the reverse.

Granted, Rich was being a wise-ass: the OP wanted to know how to make a multiplier more ideal, and 'linearizing' is probably as good enough a term to use as anything ('idealizing' has the right denotation, but the connotation makes one think of putting the poor Gilbert cell on a pedestal with fresh flowers every day).

But I think he was correct from a systems analysis point of view, that the multiplier is -- by itself -- nonlinear.

Which doesn't really change much at all except that (and if there's any real point to this post at all, this is it) if you're doing systems analysis you'd darn well better know if you can treat your system as linear and time invariant, if you can treat it as linear but must take time variance or indeterminate parameters into account, or if it's just fundamentally nonlinear in operation, and must be treated as such.

Yes, it's just academic. The ideal multiplier is linear in the 2D sub- space in which it operates. In this sub-space, dY1(t) =3D dX1(t) * kX2(t) and dY2(t) =3D kX1(t) * dX2(t) and superposition of dY1 and dY2 works. The point is to make it flat and within the constraints of this subspace.

The "real" silicon multiplier is basically a log/anti-log machine utilizing the logarithmic characteristics of bipolar diodes. "Linearization" happens by pre-distorting input signals via compensating diodes.

CMOS "multipliers" aren't "linear", they're simply DPDT switches that implement RF mixing by "chopping". ...Jim Thompson

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That is exactly the definition of sub-space transformation. The domain is not linear, but the transformed (log) subspace is linear.

That's multiplexing, not multipling. Modulations with carrier frequencies are multiplers, even with switching frequencies, and they have to be linear for communications to work.

I'd actually call it mixing. :-) Or at least, a hard-switched mixer.

Apparently "binlinear" is the right term there?

...and with proper selection and care, can range 10-11 decades.