Question

A common problem in over-the-air television signal transmission is multipath distortion of the received signal due to the transmitted signal bouncing off structures. Typically a strong main signal arrives at some time and a weaker ghost signal arrives later. So if the transmitted signal is x,() the recieved signal is where >> Kg and 1.x, . 1 What is the transfer function of this communication channel? 2. What would be the transfer function of an equalization system that would compensate for the effects of multipath?

Answer 1

Answer:

Part 1: The transfer function of the system is given as $K_me^{-j\omega t_m}+K_ge^{-j\omega t_g}$

Part 2: The transfer function of the equalization system is $\frac{1}{K_me^{-j\omega t_m}}$

Explanation:

Part 1

$TF_{sys}=\frac{FT[OP]}{FT[IP]}\\TF_{sys}=\frac{FT[K_mx_t(t-t_m)+K_gx_t(t-t_g]}{FT[x(t)]}\\TF_{sys}=\frac{K_mx_t(\omega)e^{-j\omega t_m}+K_gx_t(\omega)e^{-j\omega t_g}}{x_t(\omega)}\\TF_{sys}=\frac{x_t(\omega)[K_me^{-j\omega t_m}+K_ge^{-j\omega t_g}]}{x_t(\omega)}\\TF_{sys}=K_me^{-j\omega t_m}+K_ge^{-j\omega t_g}$

The transfer function of the system is  $K_me^{-j\omega t_m}+K_ge^{-j\omega t_g}$

Part 2

As the equalization system is a system which compensate the effects of the system thus if the transfer function of the equalization system is given as $TF_{eq}$

Than

$TF_{sys} \times TF_{eq}=1$

or

$TF_{eq}=\frac{1}{TF_{sys} }\\TF_{eq}=\frac{1}{K_me^{-j\omega t_m}+K_ge^{-j\omega t_g}}$

As it is given that Km>>Kg thus ignoring the values with Kg multiplier yields

$TF_{eq}=\frac{1}{K_me^{-j\omega t_m}}$

So the transfer function of the equalization system is $\frac{1}{K_me^{-j\omega t_m}}$