What is the definition of Amplifier?

What exactly does the term 'amplifier' come to our mind?
The general assumption, Amplifier is the device that increases the voltage/current amplitude of the signal without altering the nature of the signal.
From the above sentence, it states the observation of the amplifier. But, what exactly the amplifier does?
Before thinking about the exact definition of an amplifier, below shown are the two different networks.
 

                                                                                                                                                                    

The above two circuits which are different coincidentally perform the same operation i.e; increasing the strength or amplitude of a signal. But what differs them? 

Coming directly to the details, In transformers(step-up), we see that

                                          

                                           V2/V1=N and I1/I2=N

which in-turn P1=P2.

The power remains the same before and after enhancing the voltage levels.

But when it comes to an amplifier, the power value gets enhanced. 

So, Let me ask you again? What is the definition of an amplifier?

" The amplifier is an electronic device which enhances the amplitude of the power."

                                        

Does Frequency Modulation uses Envelope Detector?

The modulation involves the concept of variation of the amplitude, frequency or phase components of the carrier signal concerning the instantaneous amplitude of the message signal.
Generally, the carrier signal is a high-frequency component of the sinusoidal.
c(t) = Ac*cos(2πfct)
If the user understands the information without any distortion, then the communication is effective. 
Envelope Detection in AM:
Consider the modulated signal of AM, s(t)=(Ac+m(t))*cos(2πfct). 
Where m(t) is called the message signal.
Here the envelope is called a(t)=(Ac+m(t)).
The component a(t) contains the information of signal m(t) as the magnitude is continuously varying and hence can be detected.

Envelope Detection in FM:
In Frequency Modulation, the standard equation is:
s(t) = Accos(2πfct + 2πKf∫m(t)dt) 
Where Kf is called Frequency Sensitivity.
Considering Single tone modulation i.e., m(t)= Am*cos(2πfmt), Then
s(t) = Accos[2πfct+ (KfAm/fm)sin(2pifmt)] 
Let KfAm/fm = β, Then
s(t) = Accos[2πfct+ βsin(2pifmt)] 
s(t) = Accos(2πfct)*cos(β*sin(2πfmt)) - Acsin(2πfct)*sin(β*sin(2πfmt))

Coming to the detection of the envelope in s(t)   
The inphase component of s(t) is : Ac*cos(β*sin(2πfmt))
The quadrature component of s(t) is : Ac*sin(β*sin(2πfmt))

Then the magnitude of the envelope of s(t) is : 

which in turn results in |Ac|.
Generally, the detector captures the message signal if there is variation in the envelope. But in the case of Frequency modulated signal, the envelope remains constant i.e; |Ac|. So the information of message signal cannot be extracted and hence remains clueless.
There remains a path for demodulation. A discriminator is used in this case and hence the message signal can be detected.

Communication Systems: Bandwidth determination of a signal

In this blog, I am going to discuss a few basic things about the Bandwidth calculation of a signal.
A signal can be transmitted only when the bandwidth is finite i.e; band limited. How can we calculate the bandwidth of a signal? Consider a bandlimited signal in time-domain as well as in the frequency domain.

Here in the frequency domain, the range of frequencies lies in 
[-fm,fm]. As shown in the figure, the positive frequencies exist in the range [0,fm]. Hence the bandwidth is fm which is finite. 
Therefore we can design a channel whose bandwidth is fm, the minimum value, can be transmitted. 

The bandwidth of a few standard signals:

Can we transmit a delta signal?

In the frequency domain: M(f)=1 from [-inf,inf]. Clearly, the Bandwidth is infinite and practically a channel doesn't exist to transmit this signal. Hence the signal has to bandlimited for transmission. 

Also, can we transmit a rectangular pulse signal?

Here also the bandwidth is infinite, makes it impossible for transmission. But the energy of the signal at [-2π/T,2π/T] contains 99% of the total signal. Hence it is bandlimited to [-2π/T,2π/T] and is recovered. 
From these examples, we can conclude that we cannot judge whether the signal is bandlimited or band unlimited through time domain. It might have appeared that the signal is finite in the time but it does not mean it is suitable for transmission.