Channel Matrix Gain

Transmitted and Received Power in Communication Systems

Key Concepts

1. Transmitted Power (Ps):

   The average power of the transmitted signal is defined as the power that is radiated from the transmitter.

   For a BPSK signal, where the transmitted symbols are either +1 or -1, the average transmitted power is:

   Ps = 1

   Since the magnitude of both +1 and -1 is 1, the average power of the transmitted BPSK signal is 1.

2. Channel Coefficient (h):

   The channel coefficient h characterizes how the transmitted signal is affected as it propagates through the channel.

   It can be represented as a complex number:

   h = a + jb

   The magnitude squared of the channel coefficient |h|² gives us the channel gain, which describes how much the signal power is amplified or attenuated as it passes through the channel:

   |h|² = a² + b²

3. Received Power (Pr):

   The received power is calculated as the transmitted power multiplied by the channel gain:

   Pr = |h|² · Ps

   This reflects how the transmitted power is modified by the channel characteristics.

Example Calculation Breakdown

Let’s break down the calculations for the scenario where h = 4 + 3j:

1. Calculate Channel Gain

The channel coefficient:

h = 4 + 3j

The magnitude squared of the channel coefficient:

|h|² = 4² + 3² = 16 + 9 = 25

2. Transmitted Power

The average power of the transmitted signal:

Ps = 1

3. Received Power

Using the relationship:

Pr = |h|² · Ps

Substituting the values:

Pr = 25 · 1 = 25

Interpretation

• Transmitted Power (Ps = 1):

  This is the power of the signal being sent from the transmitter, regardless of the channel.

• Received Power (Pr = 25):

  This indicates the power of the signal at the receiver after being amplified by the channel. The channel gain, represented by the channel coefficient's magnitude squared (|h|² = 25), shows that the transmitted signal has been effectively increased in power by a factor of 25 due to the channel characteristics.

Conclusion

In summary, the reason the transmitted power is 1 and the received power is 25 is that the signal passing through the channel experiences an amplification (or gain) of 25 due to the characteristics of the channel represented by the channel coefficient. This is a fundamental aspect of how communication systems operate, illustrating the role of the channel in determining received signal strength.