An ideal linear amplifier relates the input power to the output power of a signal by a proportional gain factor.  Is this a valid statement? 

Anyways, since it is linear, linear scaling and the superposition principle hold. So I assume, when I set up the transfer characteristic of the amplifier, it can be applied to both single-tone and multi-tone signals, because of superposition. Is this the main advantage of a linear amplifier? There is also no saturation point, therefore linear amplifiers would probably have an increased efficiency compared to nonlinear amplifiers. Anything else?

The most important property of linear time invariant systems is that they obey              Y(w) = H(w) X(w).

In the presence of nonlinearities you get harmonic generation for single sinusoids, and mixture frequencies (sum and difference) for signals with >1 sinusoidal component.

This behavior is detrimental except when you specifically want it.

Y(J)S

So, I am currently working on a paper. I want to give a short introduction to ideal linear amplifiers. I was thinking about sth. like:

An ideal linear amplifier can be modelled and completely characterized by its impulse response h(t) in time domain or equivalently by its transfer function H(f) in frequency domain.
x(t) * h(t) = y(t)
X(f) H(f)  = Y(f)
Mathematically, linearity is defined by its scaling and superposition property:
L(\alpha x_1 + \beta x_2) = \alpha L(x_1) + \beta L(x_2)
An ideal linear amplifier relates the input and output voltage of a signal by a multiplicative gain:
gV_{in} = V_{out}
If a sum of multiple sinusoids is applied as input to the amplifier, the gain is the same for every sinusoid.

To me, unfortunately, this sounds like three different ways of starting the introduction. I want to link all three with a nice text. But I don't see yet, how the impulse response, for example, is linked with the multiplicative gain. And also, I only mention the linearity property, because this is why the gain is the same for every sinusoid of a multi-tone signal. 

Can you explain to me, how the impulse response and the gain are linked?

So, I'm going to rearrange things a bit:

An ideal amplifier just multiplies the input by some constant gain \(g\).  Part of that "ideal-ness" is that it's linear -- it's not ideal because it's linear, it's linear is because it's ideal.

You could describe such an amplifier's response with an impulse response or a transfer function in the frequency domain, but it would be boring (an impulse at \(t = 0\), or a frequency response of 1 across the board).

If you have a linear amplifier that is otherwise not ideal (presumably meaning that it's frequency response is not flat from DC to light), then by virtue of the fact that it is a linear system it can be completely characterized by an impulse response or, if it is also time-invariant, a frequency-domain transfer function.

I'm not sure how this all fits into your intended paper, but I think I'd make a point that an amplifier is in some sense just a system, and if it's linear it's linear like any other linear system.

Well, at the risk of confusing things... :-)

An ideal linear amplifier certainly relates the input signal to the output signal by a multiplicative gain. This signal is usually a voltage or a current.

The gain of input to output *power* will be affected by additional factors, such as amplifier input and output impedances, as well as load and source impedances.

Additionally, gain may be frequency dependent.

I suppose it comes down to "how ideal is ideal"?

Assume infinite input impedance, zero output impedance, frequency independent gain?

ok, thank you talldsp! 

So, my statement g P_in = P_out ist actually wrong. It has to be 

g V_in = V_out instead. 

Now, in this case I have another question: I consider the nonlinear characteristics of a realistic amplifier. In a book that I am using as reference, "Simulation of Communication Systems" by Jeruchim et. al., it says: 

"The construction of the nonlinear characteristics is made for different values of A under conditions where each value used can be considered fixed for a long time with respect to all time
constants involved." (this describes a continuous wave signal, right?).

"Thus, the average input power is P_in = A^2 / 2 and the average output power is P_out = g(A)^2 / 2 .The main point about interpretation of the AM/AM and AM/PM curves in simulation is that while they are obtained under static measurements, they are applied to dynamic situations. That is, we usually interpret g(·) and Phi as instantaneous transfer characteristics, whereas, as just discussed, the corresponding measurements imply an averaging. In fact, the notion of instantaneous, as applied to simulation, must be interpreted to
mean no faster than the simulation sampling rate. With this interpretation it can be shown that the power scale in the measured characteristics can be meaningfully interpreted as instantaneous power (a seeming oxymoron!), by which we mean the square of the instantaneous envelope." 

So, I really have troubles understanding this paragraph. A stands for the amplitude. So when my input signal is a voltage, can I somehow use the relationship P_in = A^2 / 2 to get from  g V_in = V_out to g P_in = P_out ? 

A better question is "when do we want an amplifier to be linear, and why?"

Yes, the ideal linear amplifier "relates the input power to the output power of a signal by a proportional gain factor".  This statement is accurate but not illuminating.

Your second definition -- that superposition holds -- is probably more important.  If we have a signal that "works" because it can be separated into a bunch of components that are added together, then running it through a nonlinearity can scramble things up, essentially making those components interfere with each other.

Practical linear amplifiers do have saturation points, and must be operated so that those saturation points are not hit.  This is particularly irksome to operators that have to deal with "peaky" signals such as OFDM or a bunch of cell-phone calls all added together in a cell tower.

You might want to review your circuit theory: linear amplifiers are less efficient than some nonlinear amplifiers precisely because their gain elements do not saturate.  As Coop.aa1ww (Hi Coop -- ex KG7LI here) alluded, modern amplifiers try to take switching elements -- which are hideously nonlinear -- and build linear amps around them for exactly this reason.  Search on "class D" (baseband switching amps) and "class E" (RF switching amps) for more information.

There are cases where linear amps are unnecessary: if you have an RF signal whose information is entirely carried in the phase, or is switched on and off, then a class-C (horribly nonlinear) amplifier will amplify that signal splendidly, and more efficiently than a class B, AB, or A amplifier (all linear) ever could.

And, finally, there are cases where nonlinear amps are desirable -- guitar amps is the only case I can think of, but where would we be without Led Zepplin?

To elaborate, Class C "horribly nonlinear" amps have been used in FM broadcast transmitters, FM land mobile radios, GMSK cellphones, FSK and PSK data communications and good old CW (Morse Code) communications. All because of the increased efficiency and smaller component count which leads to reduced size and lower parts cost. For battery operated devices this also gives reduced weight and/or increased operating time.

Another case where a nonlinear amp is desirable includes frequency multiplication. Often a lower frequency signal is input to a non-linear device and a high frequency harmonic is selected by an output filter. This generates a stable, low phase noise signal at UHF and microwave frequencies. 

As John_G alluded, nonlinear (e.g., Class C) amps are often used in communications to increase transmit power efficiency.  This requires using constant-envelope waveforms (Like Tim Wescott mentioned), but is still often worth the effort.

Yes, I wasn't trying to denigrate class C amps (or any of the classes E through Z that make a constant-envelope wave more efficiently) -- I was just making the point that they are most definitely Not Linear.

No worries, it was really just addressing your comment about where do non-linear amps get employment if they're not in a rock or metal band.

Hey.  I'm blond.  Well, now I'm gray, but I used to be blond.

"There is also no saturation point, therefore linear amplifiers would probably have an increased efficiency compared to nonlinear amplifiers. "

Many linear amplifier architectures designed to provide significant bandwidth operate their active devices in a region that avoids "saturation" and cutoff in order to provide as proportional a relationship between input and output as possible.  In such cases the active devices are ever conducting current while maintaining varying voltage drops especially across their output terminals.  Such architectures are operating their active devices in Class-A or a combination of Class-A and Class-B.  The time spent in the conducting region with a non-zero or non-maximum voltage drop represents power dissipation in the devices themselves that is not part of the output signal.  Therefore it has always been a struggle to provide linear wideband gain with minimal device power consumption.

An expert in large signal analysis of various amplifying devices can comment in more detail but the whole shift toward switched devices in many applications is designed to take advantage of the fact that modern active can offer extremely low resistance in the maximally on state so that they dissipate very power in either the maximally on or maximally off state.  In this way switched systems (e.g. PWM) can offer significant variable drive without dissipating much power in comparison to the power delivered to the load.

Hello Lucky_12,

Adding to some of the answers regarding to your questions.

Linearity of any device, including amplifier, is relative parameter.  In other words, any device has linear and non linear behavior simultaneously.  As radio engineer, we quantify the non-linearity of any device via parameters like P1dB, IIP3, IIP2, etc.

The nonlinearity of the device distorts the signal via various mechanism.  In case of amplifier, the more nonlinear the amplifier is the more efficient in power consumption.

Therefore, there is tradeoff between linearity and efficiency.

To learn and quantify nonlinearity behavior, view https://ortenga.com/ortenga703/

Best regards,

Shahram Shafie

let's take audio decoder system with classD power amplifier as an example: 

non-linearity in classD amplifiers produces heavily saturated/clipped intermediate (PA) output. this saturation introduces harmonics in the PA output. 

from system view, the lowpass nature of the circuits which follows the PA controls the unwanted harmonic levels which gives same perceptual quality of the audio rendering out by the speaker. this is the reason why we hear speaker output through a class D amplifier sounds almost same as that from a (hypothetical) linear amplifier. 

so answer the above question is : the non-linearity can be taken cared in other signal processing stages so that overall system will still be linear - as far as the specific system characteristics. how to take care depends on the system and the characteristic which one want to keep linear. 

another potential examples are companding, predistortion, etc.