Problem-1.Multipah channel models

A transmitter transmits a signal s(t) with carrier frequency f_c. The signal reaches the receiving antennas through three paths. The path gains are a₁, a₂, and a₃. The path delays are t₁, t₂, and t₃. The Doppler shifts at the carrier frequency are f₁, f₂, and f₃. What is the received signal? What is the transfer function characterizing the channel between the receiving and transmitting antenna?

Problem -2. Channel Parameters

Carrier frequency is 1 GHz. Moving speed is 2 m/sec. In the following, what are the reasonable numbers in indoors for the following parameters

a.) coherent bandwidth:

b.) coherent time:

c.) delay spread:

d.) Doppler spread:

Choose (just pick one value for each parameter) your answers from the following values:

1.) 1 sec; 2.)0.1 sec; 3.)0.01 sec; 4.)1 msec; 5.)0.1 msec; 6.)0.01 msec; 7.)1 msec; 8.)0.1 msec; 9.)0.01 msec; 10.)1 nsec; 11.)1 GHz; 12.)0.1 GHz; 13.)0.01 GHz; 14.)1 MHz; 15.)0.1 MHz; 16.)0.01 MHz; 17.)1 KHz; 18.)0.1 KHz; 19.)0.01 KHz; 20.) 1 Hz;

Problem 3.  System Design

The delay spread in a given environment T_D is 5 micro seconds.

The maximum Doppler spread F_D is 200Hz.

We would like to design a digital communication system in this environment such that

1. Distortion per data symbol is negligible

2. The time variation during a symbol period is negligible

What is the appropriate range of the symbol duration in this communication system?