# Spatial diversity: symbol error probability, upper bound

I am trying to understand the calculation upper bound that is given in book.

Can someone explain to me how to come from (5.5) to (5.7) ?

• Have you seen equation 3.44 as suggested? – MBaz Mar 4 at 15:35
• at the very least you could give us eq. 3.44 – Marcus Müller Mar 4 at 16:06
• @MarcusMüller I added – nani Mar 5 at 13:50
• @MBaz I added the equastion 3,44 – nani Mar 5 at 13:50

I think eq. 5.6 is unnecessary. To know where eq. 5.7 comes from, you need to realize that 5.7 is actually the average bit error rate. Which means that we find the expected value of eq 5.5 with respect to the effective channel. To do that, let

$$Y=\sum_{i=1}^M|h_i|^2$$

In case that $$h_i\sim\mathcal{CN}(0,1)$$ and i.i.d, $$Y$$ is a central Chi-square random variable with $$2M$$ degrees of freedom. The PDF of $$Y$$ is then given by

$$f_Y(y)=\frac{1}{(M-1)!}y^{M-1}e^{-y}$$

The ABER is then given by

$$\frac{1}{(M-1)!}\int_0^{\infty}\exp\left[-y\left(1+\frac{\rho\,d_{\text{min}}^2}{4M}\right)\right]y^{M-1}\,dy$$

From the table of integrals we have

$$\int_0^{\infty}x^ne^{-\mu\,x}\,dx = n!\mu^{-n-1}$$

So, the ABER can be evaluated to

$$\frac{1}{(M-1)!}(M-1)!\left[1+\frac{\rho\,d_{\text{min}}^2}{4M}\right]^{-M}$$

At high SNR $$\rho>>1$$, so, the ABER reduces to

$$\left(\frac{\rho\,d_{\text{min}}^2}{4M}\right)^{-M}$$

which means that the diversity order of this system is $$M$$. (You can add the scaler $$\bar{N_c}$$ to these equations, which has no effect on the calculations)

• So how i understood for the last equastion is E{hh^H} =I ( the identity matrix, size is the nu,ber of the transmit antennas) important: uncorrelated channel. But how does it change in case correlated channel. if E{hh^H} is not equal to the identity matrix? – nani Mar 5 at 13:54
• This is another question. I answered your question above. – BlackMath Mar 5 at 19:25
• After I answered your question, you ignored it and asked another question. You don't know what you want. – BlackMath Mar 5 at 22:28
• i just want to understand the concept of calculation using a different condition. I read a lot of information about it. – nani Mar 6 at 7:23
• M can have a different value, which depends on DO( diverssity order) – nani Mar 6 at 7:24