Impulse response, transfer function and distortionless transmission

Impulse response and transfer function

• Impulse response $h(t)$: the output of an LTI system when the input is a unit impulse $\delta(t)$, with zero initial conditions. For any input, $y(t) = x(t) * h(t) = \int x(\lambda)h(t-\lambda)\,d\lambda$.
• Transfer function $H(f)$: the Fourier transform of the impulse response, $H(f) = \int h(t)e^{-j2\pi ft}\,dt = Y(f)/X(f)$. It gives the gain $|H(f)|$ and phase shift $\theta(f)$ the system applies to each frequency: $H(f) = |H(f)|e^{j\theta(f)}$.

Transfer function for distortionless transmission

Transmission is distortionless when the output is an exact replica of the input except for a constant gain $K$ and a constant delay $t_d$:

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

Take the Fourier transform of both sides, using the time-shift property $x(t - t_d) \leftrightarrow X(f)e^{-j2\pi f t_d}$:

$$Y(f) = K\,X(f)\,e^{-j2\pi f t_d}$$

Therefore

$$H(f) = \frac{Y(f)}{X(f)} = K\,e^{-j2\pi f t_d}$$

and the impulse response is $h(t) = K\,\delta(t - t_d)$.

Comparing with $H(f) = |H(f)|e^{j\theta(f)}$, the two conditions are:

1. Amplitude response constant: $|H(f)| = K$ for all frequencies in the signal band.
2. Phase response linear in frequency: $\theta(f) = -2\pi f t_d$, i.e. constant time delay $t_d = -\frac{1}{2\pi}\frac{d\theta}{df}$ for every component. (A phase of $-2\pi f t_d \pm n\pi$ is also acceptable; $\pm\pi$ only inverts the sign.)

Amplitude and phase response

 |H(f)|
    |
  K +-------------------------
    |
    +-------------------------> f
   -B            0           B

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

The amplitude response is a horizontal line at height $K$ and the phase response is a straight line through the origin with negative slope $-2\pi t_d$. In practice these need to hold only over the bandwidth $-B \le f \le B$ of the signal. If $|H(f)|$ varies, amplitude distortion results; if $\theta(f)$ is not linear, phase (delay) distortion results.