# Related to channel power gain

I am having a doubt that why channel power gain is denoted by $$|h|^2$$. For example, if the signal received is $$y(n) = hx(n)+w(n)$$

where $$x(n)$$ is transmitted signal with power $$P$$ , $$w(n)$$ is AWGN with power $$N$$ and $$h$$ rayleigh fading channel.

The SNR from above equation is given as

$$\mathrm{SNR} = \frac{|h|^2P}{N}\tag{1}$$

My query is why $$|h|^2$$ is written in equation $$(1)$$?

Any help in this regard is highly appreciated.

• SNR stands for signal to noise ratio, i.e., it is the ratio of the power of the input signal to the power of the noise. In this case, the signal power is given as $h \times x[n] \times x^*[n] \times h^*$, which is equal to $|h|^2 P$. The noise power is given as $N$. Take the ratio, and you get the SNR which includes the absolute value of $h$. Commented Sep 2, 2021 at 6:52
• Ok... Thank You so much...
– paru
Commented Sep 2, 2021 at 8:35

In general, the channel coefficient $$h$$ is a complex number so it has magnitude and phase. Physically, $$|h|^2$$, is the attenuation/power-loss due to the channel and that is why you include it in the signal power calculation. The signal power is:
$$\mathbb{E}\big[hx[n](hx[n])^* \big]=|h|^2\mathbb{E}\big[|x[n]|^2\big]=|h|^2P$$
• The $|h|^2$ is taken out of Expectation because it is not depending on $n$. Is my understanding correct?