SNR, BER, and Bandwidth Study Notes for GATE EC 2022 Exam

In this article, you will find the Study Notes on SNR, BER, and Bandwidth which will cover the topics such as Noise in Digital Communication, Pe for Non-coherent ASK and Non-coherent FSK, Comparison of Digital Modulation Schemes.

Noise in Digital Communication

In digital communication for better SNR, matched filter is used whose impulse response h(t) is:

Consider a receiver model,

h(t) = Impulse response of LTI filter

x(t)=g(t) + ω(t)

T = Arbitrary observation interval

g(t) = Binary symbol 1 or 0

ω(t) = Sample function of white noise process of zero mean and power spectral density N₀/2

Function of receiver is to detect the pulse signal g(t) from received signal x(t). The design of the filter is to be optimized to minimize the effects of noise.

As the filter is linear

∴            x(t) = g₀(t) + n(t) 

where g_o(t) and n(t) are produced signal and noise components of input x(t) respectively.

For proper noise free detection, signal component g_o(t) > > noise component n(t). This can be done by making the instantaneous power in output signal g_o(t) measured at time t = T as large as possible compared with average power of output noise n(t) i.e. peak pulse signal to noise ratio should be maximized. 

Filter output due to noise ω(t) only is

For ⴄ to be maximum, we have to find H(f) for a given G(f).

By Schwarz inequality

Optimum of H(f) is given by

H_opt (f) = KG*(f)e^–j2πfT

Taking inverse fourier transform

 

Probability of Error for Coherent ASK, PSK, FSK

·           In ASK          

In digital communication system, reliability is commonly expressed in terms of probability of bit error or bit error rate (BER) measured at the receiver output. Clearly, the smaller the BER, the more reliable the communication system is. The average bit error probability is denoted by P_e.

The probability of bit error for coherent ASK system is given by

Where E_b is the bit energy, N₀ is the noise power density, and γ_b is the bit energy to noise density ratio.

·           In PSK         

If distance between two manage points is decreased, then probability of error will increase.

·           For FSK (coherent) probability of error is

For coherent binary FSK signal, we define the bit error probability as

Where E_b is the bit energy, N₀ is the noise power density, γ_b is the bit energy to noise density ratio.

Note:

For larger value of z, the Q(z) function can be approximated as

   

Pe for Non-coherent ASK and Non-coherent FSK

·           Probability of error for non-coherent ASK is

 if  

·           Probability of error for non-coherent FSK is

Probability of error for differential PSK is

 

• In case of FSK f₁ and f₂ are chosen such that f₁ = mf_s and f₂ = kf_s where m, k are integers.
• Bandwidth efficiency 

Comparison of Digital Modulation Schemes

 

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