Analog communication components; distortionless transfer function

An analog communication system sends a continuous message (speech, music, video) from a source to a destination using continuous signals.

Main components

 [Info source]->[Input transducer]->[Transmitter]
                                         |
                    Noise, interference->[Channel]
                                         |
 [Destination]<-[Output transducer]<-[Receiver]

1. Information source: produces the message, e.g. a speaker's voice or a camera scene. The message may be non-electrical.
2. Input transducer: converts the message into an electrical signal (baseband) $m(t)$, e.g. microphone, camera.
3. Transmitter: prepares the signal for the channel. It amplifies, filters and modulates a high-frequency carrier (AM, FM), then power-amplifies it and couples it to the antenna or line. Modulation gives practical antenna size, multiplexing and channel matching.
4. Channel: the medium between transmitter and receiver, e.g. free space, coaxial cable, optical fibre. It attenuates and distorts the signal, and noise and interference are added here.
5. Receiver: extracts the message from the weak, corrupted received signal. It selects the wanted signal (tuning, filtering), amplifies it (RF/IF amplifiers), demodulates it to recover $m(t)$, and amplifies the baseband output.
6. Output transducer: converts the electrical signal back into the original form, e.g. loudspeaker, display.
7. Destination: the user who receives the message.

Transfer function for a distortionless system

A system is distortionless if the output has the same shape as the input; only a constant change in amplitude and a constant time delay are allowed:

$$y(t)=K\,x(t-t_d)$$

where $K$ is the gain (or attenuation) and $t_d$ is the delay. Taking the Fourier transform and using the time-shift property:

$$\begin{aligned} Y(f)&=K\,X(f)\,e^{-j2\pi ft_d}\\ H(f)&=\frac{Y(f)}{X(f)}=K\,e^{-j2\pi ft_d} \end{aligned}$$

So the conditions are:

• Amplitude response: $\lvert H(f)\rvert=K$, constant over the signal bandwidth (otherwise amplitude distortion).
• Phase response: $\theta(f)=-2\pi ft_d$, linear with frequency and passing through the origin (otherwise phase/delay distortion). Equivalently, the group delay $t_d=-\frac{1}{2\pi}\frac{d\theta}{df}$ is constant.

 |H(f)|                       theta(f)
   |                            |\
 K +-------------              | \  slope = -2*pi*td
   |                     ------+--\------> f
   +------------> f             |   \

In practice, it is enough for these conditions to hold only over the band occupied by the signal.