Signum signal; distortionless transmission

Signum signal

The signum (sign) function gives the sign of its argument:

$$\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}$$

It is an odd function, related to the unit step by $\text{sgn}(t) = 2u(t) - 1$, and its Fourier transform is $\frac{1}{j\pi f}$. It is used in defining the Hilbert transform: $\hat X(f) = -j\,\text{sgn}(f)X(f)$.

Distortionless transmission: significance and derivation

In a communication system the receiver must get the same waveform that was sent; otherwise pulses overlap (ISI), speech/music lose quality and video gets smeared. Transmission is distortionless when the output is only a scaled, delayed copy of the input:

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

Taking the Fourier transform (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 $|H(f)| = K$ and $\theta(f) = -2\pi ft_d$, and $h(t) = K\delta(t - t_d)$. The conditions are:

1. Constant amplitude response over the signal band (no amplitude distortion).
2. Linear phase response, i.e. constant delay $t_d$ for all components (no phase distortion).

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

Significance:

• Gives the requirement for designing channels, amplifiers and filters: flat gain and linear phase over the signal bandwidth.
• Explains amplitude distortion (non-flat gain) and phase/delay distortion (non-linear phase); the ear is not very sensitive to phase distortion, but video and digital data are.
• Shows why equalizers are used in telephone lines, modems and TV to correct the channel.
• Since an exact distortionless system is ideal, the conditions only need to hold over the bandwidth of the transmitted signal.