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# AWGN Channel - generating noise for required SNR

1. ## AWGN Channel

Hello,

In AWGN channel, how is white noise generated using MATLAB? I mean, I have a code at hand, but I did not understand it. For example, it says somewhere in the code:

y = s + 10^(-Eb_N0_dB(ii)/20)*n;

where s is the transmitted sequence, Eb_N0_dB is the SNR, and n is the additive white noise. Can anyone explain to me the noise term, please? Why do we use the expression 10^(-Eb_N0_dB(ii)/20)*n?

Regards

2. ## Re: AWGN Channel

Originally Posted by mathuranathan
The above code attempts to add noise (term - '10^(-Eb_N0_dB(ii)/20)*n') to a signal (term - 's').

the term y contains signal+noise.

The above code generates a noise term with required signal to noise ratio. Eb_N0_dB(ii) is a function that generates the scaling factor to generate required signal to noise ratio.

here ii is the input argument specifying the SNR in dB

For more details von how to generate noise for required SNR
http://gaussianwaves.blogspot.com/20...form-with.html

Regards
Mathuranathan

Actually, I still do not get why we use the expression 10^(-Eb_N0_dB(ii)/20)*n for the noise. Anyway, from the link you gave me, I found a built-in function in MATLAB which is AWGN() function that adds noise depending on the SNR. I tried it as in the following code:

Code:
```function BER
clear;
N=10^6;
SNRdB=[-3:10];
SNR=10.^(SNRdB/10);
b=rand(1,N)>0.5;
s=2*b-1;
for ii=1:length(SNRdB)
y=awgn(s,SNRdB(ii),'measured');
bHat=real(y)>0;
Diff=xor(bHat,b);
nErr(ii)=length(find(Diff));
end

simBER=nErr/N;
theoryBER=0.5*erfc(sqrt(SNR));
semilogy(SNRdB,simBER,'k*',SNRdB,theoryBER,'bx');
legend('Simualtion','Theory');```
but there is a shift between the theoritical and simulated results. Why is that? Is there anything wrong in the code?

3. ## Re: AWGN Channel

Originally Posted by mathuranathan
Hi
There is nothing wrong with the code.
The simulated value may be different from the theoretical value. The theoretical result uses a well defined equation whereas the simulated result is obtained from a set of generated inputs. Always the simulated result will show deviation from the theoretical result.

Regards
Mathuranathan
But when I replaced the addition of additive white Gaussian noise line y=awgn(s,SNRdB(ii),'measured'); by y = s + 10^(-Eb_N0_dB(ii)/20)*n; both results (the simulated and theoritical) collide. It does not make sense!!!

Regards

•

4. ## Re: AWGN Channel

Originally Posted by mathuranathan
Can u plot the graph in both cases to understand the situation ?

Regards
Mathuranathan

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5. ## Re: AWGN Channel

Originally Posted by mathuranathan
Hi saeddawoud
The term 10^(-Eb_N0_dB(ii)/20)*n is derived as follows

For AWGN channel with noise power spectral density N0 the standard deviation of noise is given by

σ = √(N0/2) --------> (1)

For BPSK the symbol energies are assumed to be normalized to 1 (since 0 represented by -1 and 1 represented by +1 )

Given Eb_N0_dB has to be converted to linear scale first
Eb/N0 = 10 ^(Eb_N0_dB/10); --->(2)

Now we need to find the sigma of noise according to the desired Eb/No ratio

if Eb=1 in (2)
then
1/N0 = 10 ^(Eb_N0_dB/10)
=> N0 = 1/(10 ^(Eb_N0_dB/10) ) ---> (3)

(3) in (1)
σ = √(1/(10 ^(Eb_N0_dB/10) )/2)
or
σ =1/√2 * 1/(10 ^(Eb_N0_dB/20)

this is how the term is derived

The problem here is not due to equation , it is due the way the awgn function is used.

The awgn measures the SNR (signal power/noise power) (note: everything in terms of power) but Eb/N0 is in terms of energy

For proper result if you want to use awgn function , translate the Eb/N0 to required SNR (in power) and then use the function

Regards
Mathuranathan
Ok, it is clear now. One last thing, why we multiply the variance by the noise in the term 10^(-Eb_N0_dB(ii)/20)*n?

6. ## Re: AWGN Channel

I have noticed that, when I set the range of errors from 10^-7 up to 0.5, the simulation curve stopped on 10^-6. Why is that?

7. ## Re: AWGN Channel - generating noise for required SNR

I think it is clear now the concept of adding AWGN to the transmitted signal over AWGN channel. Let me summarize:

For a specific SNR we need to add a noise term to the transmitted symbol such that the SNR is as specified, i.e.: if the SNR(dB)=15, we need to add a noise term such that the SNR(dB)=15 dB. To do this, we assume that the signal energy is unity (Why?), and then:

Code:
`y=s+10.^(-15/20)*n`
where y is the received signal, s is the transmitted signal, and n is the additive noise generated using randn function that generates normal RVs with zero mean and unity variance. the SNR (linear) of the above equation is:

$3\frac{1}{10^{-15/10}}=10^{15/10}=15\mbox{dB}$. But, is the assumption that the energy of the signal = 1 is valid?

Now, let us turn to the multipath fading channles. In this case the receiced signal will be:

Code:
`y=h*s+10.^(-15/20)*n`
the SNR in this case is:

$3\frac{|h|^2\,E_s}{\sigma_n^2}$

Shall we assume that the numerator is unity? I.e.: |h|^2*Es =1? Why?

8. ## Re: AWGN Channel

Originally Posted by mathuranathan
...
For proper result if you want to use awgn function , translate the Eb/N0 to required SNR (in power) and then use the function
...
How can I do that? I want to use the awgn function in my codes.

•

9. ## Re: AWGN Channel - generating noise for required SNR

Originally Posted by mathuranathan
S/N = Eb/N0 * R/B
where R is the data rate and B is the bandwidth of the channel.

here S/N is in terms of power (signal power / noise power).
Thank you very much for replying, but then, we need to specify the data rate and the bandwidth of the channel. How can we know the most appropriate values for these variables? and how will they affect the final result? Can you give me a simple example, please?

Regards

10. ## Re: AWGN Channel - generating noise for required SNR

Originally Posted by saeddawoud
Thank you very much for replying, but then, we need to specify the data rate and the bandwidth of the channel. How can we know the most appropriate values for these variables? and how will they affect the final result? Can you give me a simple example, please?

Regards
Is y=s-(20 log⁡(SNR)*n) the same with y = s + 10^(-Eb_N0_dB(ii)/20)*n?

---------- Post added at 05:28 ---------- Previous post was at 05:21 ----------

Originally Posted by saeddawoud
Hello,

In AWGN channel, how is white noise generated using MATLAB? I mean, I have a code at hand, but I did not understand it. For example, it says somewhere in the code:

y = s + 10^(-Eb_N0_dB(ii)/20)*n;

where s is the transmitted sequence, Eb_N0_dB is the SNR, and n is the additive white noise. Can anyone explain to me the noise term, please? Why do we use the expression 10^(-Eb_N0_dB(ii)/20)*n?

Regards
Is y=s-(20 log⁡(SNR)*n) the same with y = s + 10^(-Eb_N0_dB(ii)/20)*n?
please reply me as soon as possible. And then, i would like to know the value of n?

11. ## Re: AWGN Channel - generating noise for required SNR

Is SNR here in dB or linear?

12. ## Re: AWGN Channel - generating noise for required SNR

Originally Posted by David83
Is SNR here in dB or linear?
i think it is in dB!

13. ## Re: AWGN Channel - generating noise for required SNR

Originally Posted by rhi zan
i think it is in dB!
In the equation y = s + 10^(-Eb_N0_dB(ii)/20)*n, EB_N0_dB is the SNR is dB, and it is correct. Why do you need another formula? Your formula is not right. An alternative will be:

y=s+(1/sqrt(SNR))*n, where SNR here is in linear scale.

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14. ## Re: AWGN Channel - generating noise for required SNR

Thank u for ur reply!! but i need to change a little bit. how can i change the equation, y = s + 10^(-Eb_N0_dB(ii)/20)*n to another way? Please tell me! i am not so clear ! Thank u! Is this equation y=s+(1/sqrt(SNR))*n right and same with y = s + 10^(-Eb_N0_dB(ii)/20)*n?

15. ## Re: AWGN Channel - generating noise for required SNR

Originally Posted by rhi zan
Thank u for ur reply!! but i need to change a little bit. how can i change the equation, y = s + 10^(-Eb_N0_dB(ii)/20)*n to another way? Please tell me! i am not so clear ! Thank u! Is this equation y=s+(1/sqrt(SNR))*n right and same with y = s + 10^(-Eb_N0_dB(ii)/20)*n?
It is correct as long as SNR is in Linear scale, and s is BPSK. Do not forget to write SNR(ii) because SNR will be a vector.

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16. ## Re: AWGN Channel - generating noise for required SNR

How to include the path loss exponent and the distance between the source & destination in the channel H???

17. ## Re: AWGN Channel - generating noise for required SNR

i thinck it will be y=s+(1/sqrt(2*SNR))*n not ........ y=s+(1/sqrt(SNR))*n
because σ = √(N0/2)