How BER of Eve vs SNR(dB) is slightly change ?

Alice send the message to Bob. Bob receive the signal. Eve is passive eavesdropping attack. He is also received the signal whenever Alice send to Bob or Bob send to Alice.

Alice received signal: ra = h.v + nb

Bob received signal: rb = h.s +na

h = circularly symmetric gaussian distributed channel response with mean zero

na & nb are circularly symmetric gaussian distributed additive noise with mean zero Alice and Bob respectively

v = coded symbols transmitted by bob

s = coded symbols transmitted by alice

dot '.' = hadmard product

Now, Hadmard product between what we send and what we get that is:

wa = h.v.s + nb.s

wb = h.v.s + na.v

Now, quantization is applied at above equation and BER (Bit Error Rate) is calculated:

BER of Alice-Bob = [ | Summation of (Qa(i) - Qb(i)) | ] / N where i= 1,2,...N and N = total length of Qa

Qa and Qb is quantized sequence of Alice and Bob

Eve Side:

we = he.v.s + ne

he = circularly symmetric gaussian distributed channel response with mean zero

ne = circularly symmetric gaussian distributed additive noise with mean zero

Qe is quantized sequence of Eve

BER of Eve = [ | Summation of (Qa(i) - Qe(i)) | ] / N where i= 1,2,...N and N = total length of Qa

As SNR (signal to noise ratio) increases BER of Alice-Bob decreases. But BER of Eve should be slighlty change. For Example:

SNR (dB) BER of Alice-Bob BER of Eve

5 dB 0.3 0.45

10 dB 0.21 0.44

15 0.15 0.43

20 0.09 0.42

25 0.04 0.41

I get the approx same result of BER of Alice-Eve VS SNR (dB). But I didn't get the approx same result for BER of Eve vs SNR (dB).

My question is How BER of Eve is slightly change? What's the formula? Relation between BER and SNR?

Answer 1

[Rangesh]

Hi Himanshi,

I understand that you are interested in the variation in the Bit Error Rate (BER) of Alice-Eve link and its relationship with the signal-to-noise ratio (SNR).

Let's consider a model

where, y is the received signal, x is the transmitted signal and are Rayleigh fading channel response and noise, both of which are circularly symmetric Gaussian noise with zero mean respectively.

Please note that, we are considering Single-input-single-output system (SISO), which implies the dimension of the all the variable are one.

For this system, the BER can be expressed as:

(1)

(2)

From equation (1), we observe that the BER depends on the channel \distribution of the channel h . In our case, equation (2) considers . Therefore, the variation in the BER indicates how different the channel is between Alice-Bob and Alice-Eve for the same transmitted signal.

If Eve's channel experiences more multipath fading, shadowing, or other impairments, the effective SNR for Eve's received signal might not improve as much as for Alice-Bob link, resulting in a higher BER.

Conversely, if Eve has a better channel condition or is much closer to Alice than Bob, the received SNR would be higher, leading to a lower BER.

I hope this resolves your query.

[Himanshi]

I have clearly understood the relationship between BER and SNR. Thank you so much, Sir.